\(\int \frac {\sqrt {a-b x^2} (A+B x+C x^2)}{x^5 \sqrt {c+d x}} \, dx\) [175]

Optimal result

Integrand size = 35, antiderivative size = 738 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx=-\frac {A \sqrt {c+d x} \sqrt {a-b x^2}}{4 c x^4}-\frac {(8 B c-7 A d) \sqrt {c+d x} \sqrt {a-b x^2}}{24 c^2 x^3}-\frac {\left (8 a c (6 c C-5 B d)-A \left (12 b c^2-35 a d^2\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{96 a c^3 x^2}+\frac {\left (4 b c^2 (16 B c-11 A d)+3 a d \left (48 c^2 C-40 B c d+35 A d^2\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{192 a c^4 x}-\frac {\sqrt {b} \left (4 b c^2 (16 B c-11 A d)+3 a d \left (48 c^2 C-40 B c d+35 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{192 \sqrt {a} c^4 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {\sqrt {b} \left (4 b c^2 (16 B c-5 A d)+a d \left (48 c^2 C-40 B c d+35 A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{192 \sqrt {a} c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\left (A \left (16 b^2 c^4+24 a b c^2 d^2-35 a^2 d^4\right )-8 a c \left (a d^2 (6 c C-5 B d)-4 b c^2 (2 c C-B d)\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{64 a c^4 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-1/4*A*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/c/x^4-1/24*(-7*A*d+8*B*c)*(d*x+c)^(1 
/2)*(-b*x^2+a)^(1/2)/c^2/x^3-1/96*(8*a*c*(-5*B*d+6*C*c)-A*(-35*a*d^2+12*b* 
c^2))*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a/c^3/x^2+1/192*(4*b*c^2*(-11*A*d+16* 
B*c)+3*a*d*(35*A*d^2-40*B*c*d+48*C*c^2))*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a/ 
c^4/x-1/192*b^(1/2)*(4*b*c^2*(-11*A*d+16*B*c)+3*a*d*(35*A*d^2-40*B*c*d+48* 
C*c^2))*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1 
/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(1/2 
)/c^4/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)+1/192*b^(1/2) 
*(4*b*c^2*(-5*A*d+16*B*c)+a*d*(35*A*d^2-40*B*c*d+48*C*c^2))*((d*x+c)/(c+a^ 
(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticF(1/2*(1-b^(1/2)*x/a^ 
(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(1 
/2)/c^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)+1/64*(A*(-35*a^2*d^4+24*a*b*c^2*d^2 
+16*b^2*c^4)-8*a*c*(a*d^2*(-5*B*d+6*C*c)-4*b*c^2*(-B*d+2*C*c)))*((d*x+c)/( 
c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticPi(1/2*(1-b^(1/2) 
*x/a^(1/2))^(1/2)*2^(1/2),2,2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2 
))/a/c^4/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 32.74 (sec) , antiderivative size = 2679, normalized size of antiderivative = 3.63 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx=\text {Result too large to show} \] Input:

Integrate[(Sqrt[a - b*x^2]*(A + B*x + C*x^2))/(x^5*Sqrt[c + d*x]),x]
 

Output:

(-1/4*A/(c*x^4) + (-8*B*c + 7*A*d)/(24*c^2*x^3) + (12*A*b*c^2 - 48*a*c^2*C 
 + 40*a*B*c*d - 35*a*A*d^2)/(96*a*c^3*x^2) + (64*b*B*c^3 - 44*A*b*c^2*d + 
144*a*c^2*C*d - 120*a*B*c*d^2 + 105*a*A*d^3)/(192*a*c^4*x))*Sqrt[c + d*x]* 
Sqrt[a - b*x^2] + (d*Sqrt[a - (b*(c + d*x)^2*(-1 + c/(c + d*x))^2)/d^2]*(- 
64*b^2*B*c^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 44*A*b^2*c^3*d*Sqrt[-c + (Sq 
rt[a]*d)/Sqrt[b]] - 144*a*b*c^3*C*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 120*a 
*b*B*c^2*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 105*a*A*b*c*d^3*Sqrt[-c + (S 
qrt[a]*d)/Sqrt[b]] - (64*b^2*B*c^6*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d* 
x)^2 + (44*A*b^2*c^5*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 - (144* 
a*b*c^5*C*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 + (184*a*b*B*c^4*d 
^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 - (149*a*A*b*c^3*d^3*Sqrt[- 
c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 + (144*a^2*c^3*C*d^3*Sqrt[-c + (Sqrt 
[a]*d)/Sqrt[b]])/(c + d*x)^2 - (120*a^2*B*c^2*d^4*Sqrt[-c + (Sqrt[a]*d)/Sq 
rt[b]])/(c + d*x)^2 + (105*a^2*A*c*d^5*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c 
+ d*x)^2 + (128*b^2*B*c^5*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x) - (88* 
A*b^2*c^4*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x) + (288*a*b*c^4*C*d*S 
qrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x) - (240*a*b*B*c^3*d^2*Sqrt[-c + (S 
qrt[a]*d)/Sqrt[b]])/(c + d*x) + (210*a*A*b*c^2*d^3*Sqrt[-c + (Sqrt[a]*d)/S 
qrt[b]])/(c + d*x) + (I*Sqrt[b]*c*(Sqrt[b]*c - Sqrt[a]*d)*(4*b*c^2*(16*B*c 
 - 11*A*d) + 3*a*d*(48*c^2*C - 40*B*c*d + 35*A*d^2))*Sqrt[1 - c/(c + d*...
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1907\) vs. \(2(738)=1476\).

Time = 14.33 (sec) , antiderivative size = 1907, normalized size of antiderivative = 2.58, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.629, Rules used = {2355, 628, 25, 2352, 25, 2352, 2352, 2351, 600, 509, 508, 327, 512, 511, 321, 633, 632, 186, 413, 412, 7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 2355

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {\sqrt {a-b x^2}}{x^5 \sqrt {c+d x}}dx+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 628

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \int -\frac {-\frac {5 b d x^2}{c}+2 b x+\frac {7 a d}{c}}{x^4 \sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {1}{8} \int \frac {-\frac {5 b d x^2}{c}+2 b x+\frac {7 a d}{c}}{x^4 \sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 2352

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {\int -\frac {\frac {21 a b d^2 x^2}{c}-2 a b d x+a \left (12 b c-\frac {35 a d^2}{c}\right )}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{6 a c}+\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {\int \frac {\frac {21 a b d^2 x^2}{c}-2 a b d x+a \left (12 b c-\frac {35 a d^2}{c}\right )}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 2352

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {-\frac {\int \frac {d \left (44 b c-\frac {105 a d^2}{c}\right ) a^2-b d \left (12 b c-\frac {35 a d^2}{c}\right ) x^2 a-2 b \left (12 b c^2+7 a d^2\right ) x a}{x^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 2352

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {-\frac {-\frac {\int \frac {b d^2 \left (44 b c-\frac {105 a d^2}{c}\right ) x^2 a^2+3 \left (-\frac {35 a^2 d^4}{c}+24 a b c d^2+16 b^2 c^3\right ) a^2+2 b d \left (12 b c^2-35 a d^2\right ) x a^2}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (44 b c^2-105 a d^2\right )}{c^2 x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 2351

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {-\frac {-\frac {3 a^2 \left (-\frac {35 a^2 d^4}{c}+24 a b c d^2+16 b^2 c^3\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\int \frac {-70 a^3 b d^3+a^2 b \left (44 b c-\frac {105 a d^2}{c}\right ) x d^2+24 a^2 b^2 c^2 d}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (44 b c^2-105 a d^2\right )}{c^2 x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 600

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {-\frac {-\frac {3 a^2 \left (-\frac {35 a^2 d^4}{c}+24 a b c d^2+16 b^2 c^3\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx-5 a^2 b d \left (4 b c^2-7 a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+a^2 b d \left (44 b c-\frac {105 a d^2}{c}\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (44 b c^2-105 a d^2\right )}{c^2 x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 509

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {-\frac {-\frac {3 a^2 \left (-\frac {35 a^2 d^4}{c}+24 a b c d^2+16 b^2 c^3\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx-5 a^2 b d \left (4 b c^2-7 a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {a^2 b d \sqrt {1-\frac {b x^2}{a}} \left (44 b c-\frac {105 a d^2}{c}\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (44 b c^2-105 a d^2\right )}{c^2 x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 508

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {-\frac {-\frac {-\frac {2 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (44 b c-\frac {105 a d^2}{c}\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+3 a^2 \left (-\frac {35 a^2 d^4}{c}+24 a b c d^2+16 b^2 c^3\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx-5 a^2 b d \left (4 b c^2-7 a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (44 b c^2-105 a d^2\right )}{c^2 x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 327

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {-\frac {-\frac {3 a^2 \left (-\frac {35 a^2 d^4}{c}+24 a b c d^2+16 b^2 c^3\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx-5 a^2 b d \left (4 b c^2-7 a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (44 b c-\frac {105 a d^2}{c}\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (44 b c^2-105 a d^2\right )}{c^2 x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 512

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {-\frac {-\frac {3 a^2 \left (-\frac {35 a^2 d^4}{c}+24 a b c d^2+16 b^2 c^3\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {5 a^2 b d \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-7 a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-\frac {2 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (44 b c-\frac {105 a d^2}{c}\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (44 b c^2-105 a d^2\right )}{c^2 x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 511

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {-\frac {-\frac {\frac {10 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-7 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {a-b x^2} \sqrt {c+d x}}+3 a^2 \left (-\frac {35 a^2 d^4}{c}+24 a b c d^2+16 b^2 c^3\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (44 b c-\frac {105 a d^2}{c}\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (44 b c^2-105 a d^2\right )}{c^2 x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 321

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {-\frac {-\frac {3 a^2 \left (-\frac {35 a^2 d^4}{c}+24 a b c d^2+16 b^2 c^3\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {10 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-7 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (44 b c-\frac {105 a d^2}{c}\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (44 b c^2-105 a d^2\right )}{c^2 x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 633

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {-\frac {-\frac {\frac {3 a^2 \sqrt {1-\frac {b x^2}{a}} \left (-\frac {35 a^2 d^4}{c}+24 a b c d^2+16 b^2 c^3\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}+\frac {10 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-7 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (44 b c-\frac {105 a d^2}{c}\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (44 b c^2-105 a d^2\right )}{c^2 x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 632

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {-\frac {-\frac {\frac {3 a^2 \sqrt {1-\frac {b x^2}{a}} \left (-\frac {35 a^2 d^4}{c}+24 a b c d^2+16 b^2 c^3\right ) \int \frac {1}{x \sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}} \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+d x}}dx}{\sqrt {a-b x^2}}+\frac {10 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-7 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (44 b c-\frac {105 a d^2}{c}\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (44 b c^2-105 a d^2\right )}{c^2 x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 186

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {-\frac {-\frac {-\frac {6 a^2 \sqrt {1-\frac {b x^2}{a}} \left (-\frac {35 a^2 d^4}{c}+24 a b c d^2+16 b^2 c^3\right ) \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2}}+\frac {10 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-7 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (44 b c-\frac {105 a d^2}{c}\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (44 b c^2-105 a d^2\right )}{c^2 x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 413

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {-\frac {-\frac {-\frac {6 a^2 \sqrt {1-\frac {b x^2}{a}} \left (-\frac {35 a^2 d^4}{c}+24 a b c d^2+16 b^2 c^3\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}+\frac {10 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-7 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (44 b c-\frac {105 a d^2}{c}\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (44 b c^2-105 a d^2\right )}{c^2 x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx\)

\(\Big \downarrow \) 412

\(\displaystyle \int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^5}dx+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {-\frac {-\frac {\frac {10 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-7 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (44 b c-\frac {105 a d^2}{c}\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {6 a^2 \sqrt {1-\frac {b x^2}{a}} \left (-\frac {35 a^2 d^4}{c}+24 a b c d^2+16 b^2 c^3\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (44 b c^2-105 a d^2\right )}{c^2 x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\sqrt {c+d x} \sqrt {a-b x^2} C}{d x^4}+\frac {(B d-c C) \sqrt {c+d x} \sqrt {a-b x^2}}{d^2 x^5}\right )dx+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 x^3}-\frac {-\frac {-\frac {\frac {10 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-7 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 a^{5/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (44 b c-\frac {105 a d^2}{c}\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {6 a^2 \sqrt {1-\frac {b x^2}{a}} \left (-\frac {35 a^2 d^4}{c}+24 a b c d^2+16 b^2 c^3\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (44 b c^2-105 a d^2\right )}{c^2 x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}}{6 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{4 c x^4}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {b} \left (8 b c^2+3 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) C}{24 \sqrt {a} c^2 d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {\sqrt {b} \left (8 b c^2+a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) C}{24 \sqrt {a} c d \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\left (4 b-\frac {a d^2}{c^2}\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) C}{8 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\left (\frac {3 d^2}{c^2}+\frac {8 b}{a}\right ) \sqrt {c+d x} \sqrt {a-b x^2} C}{24 d x}-\frac {\sqrt {c+d x} \sqrt {a-b x^2} C}{12 c x^2}-\frac {\sqrt {c+d x} \sqrt {a-b x^2} C}{3 d x^3}+\frac {5 \sqrt {b} (c C-B d) \left (4 b c^2-3 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{192 \sqrt {a} c^3 d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} (c C-B d) \left (44 b c^2-5 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{192 \sqrt {a} c^2 d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {(c C-B d) \left (16 b^2 c^4-8 a b d^2 c^2+5 a^2 d^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{64 a c^3 d^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{8} \left (\frac {7 d \sqrt {c+d x} \sqrt {a-b x^2}}{3 c^2 x^3}-\frac {-\frac {\sqrt {c+d x} \sqrt {a-b x^2} \left (12 b-\frac {35 a d^2}{c^2}\right )}{2 x^2}-\frac {-\frac {a d \sqrt {c+d x} \sqrt {a-b x^2} \left (44 b c^2-105 a d^2\right )}{c^2 x}-\frac {-\frac {2 \sqrt {b} d \left (44 b c-\frac {105 a d^2}{c}\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) a^{5/2}}{\sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {10 \sqrt {b} d \left (4 b c^2-7 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) a^{5/2}}{\sqrt {c+d x} \sqrt {a-b x^2}}-\frac {6 \left (-\frac {35 a^2 d^4}{c}+24 a b c d^2+16 b^2 c^3\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) a^2}{\sqrt {a-b x^2} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}}{2 a c}}{4 a c}}{6 a c}\right )-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{4 c x^4}\right )-\frac {5 (c C-B d) \left (4 b c^2-3 a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{192 a c^3 d x}-\frac {(c C-B d) \left (\frac {5 d^2}{c^2}+\frac {12 b}{a}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{96 d^2 x^2}+\frac {(c C-B d) \sqrt {c+d x} \sqrt {a-b x^2}}{24 c d x^3}+\frac {(c C-B d) \sqrt {c+d x} \sqrt {a-b x^2}}{4 d^2 x^4}\)

Input:

Int[(Sqrt[a - b*x^2]*(A + B*x + C*x^2))/(x^5*Sqrt[c + d*x]),x]
 

Output:

((c*C - B*d)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(4*d^2*x^4) - (C*Sqrt[c + d*x] 
*Sqrt[a - b*x^2])/(3*d*x^3) + ((c*C - B*d)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/ 
(24*c*d*x^3) - (C*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(12*c*x^2) - ((c*C - B*d) 
*((12*b)/a + (5*d^2)/c^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(96*d^2*x^2) - (5 
*(c*C - B*d)*(4*b*c^2 - 3*a*d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(192*a*c^3 
*d*x) + (C*((8*b)/a + (3*d^2)/c^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(24*d*x) 
 + (5*Sqrt[b]*(c*C - B*d)*(4*b*c^2 - 3*a*d^2)*Sqrt[c + d*x]*Sqrt[1 - (b*x^ 
2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqr 
t[b]*c)/Sqrt[a] + d)])/(192*Sqrt[a]*c^3*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b 
]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (Sqrt[b]*C*(8*b*c^2 + 3*a*d^2)*Sqrt[c 
 + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]] 
/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(24*Sqrt[a]*c^2*d*Sqrt[(Sqrt[ 
b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (Sqrt[b]*(c*C - 
B*d)*(44*b*c^2 - 5*a*d^2)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d) 
]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[ 
2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(192*Sqrt[a]*c^2*d*Sqrt[c + d*x]*Sq 
rt[a - b*x^2]) + (Sqrt[b]*C*(8*b*c^2 + a*d^2)*Sqrt[(Sqrt[b]*(c + d*x))/(Sq 
rt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[ 
b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(24*Sqrt[a]*c*d 
*Sqrt[c + d*x]*Sqrt[a - b*x^2]) + (C*(4*b - (a*d^2)/c^2)*Sqrt[(Sqrt[b]*...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 186

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ 
)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2   Subst[Int[1/(Simp[b*c - a*d 
- b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ 
d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, 
g, h}, x] && GtQ[(d*e - c*f)/d, 0]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 509

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq 
rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], 
x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 512

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 
2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 628

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*Sqrt[(a_) + (b_.)*(x_)^2], x 
_Symbol] :> Simp[c^(n - 1/2)*(e*x)^(m + 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2]/( 
e*(m + 1))), x] - Simp[1/(2*e*(m + 1))   Int[((e*x)^(m + 1)/(Sqrt[c + d*x]* 
Sqrt[a + b*x^2]))*ExpandToSum[(2*a*c^(n + 1/2)*(m + 1) + a*c^(n - 1/2)*d*(2 
*m + 3)*x + 2*b*c^(n + 1/2)*(m + 2)*x^2 + b*c^(n - 1/2)*d*(2*m + 5)*x^3 - 2 
*a*(m + 1)*(c + d*x)^(n + 1/2) - 2*b*(m + 1)*x^2*(c + d*x)^(n + 1/2))/x, x] 
, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n + 3/2, 0] && LtQ[m, -1] && 
IntegerQ[2*m]
 

rule 632

Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : 
> With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a]   Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - 
q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 633

Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : 
> Simp[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(x*Sqrt[c + d*x]*Sqrt[1 
+ b*(x^2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2351

Int[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S 
ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + 
Simp[PolynomialRemainder[Px, x, x]   Int[(c + d*x)^n*((a + b*x^2)^p/x), x], 
 x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
 

rule 2352

Int[((Px_)*((e_.)*(x_))^(m_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x 
_)^2]), x_Symbol] :> With[{Px0 = Coefficient[Px, x, 0]}, Simp[Px0*(e*x)^(m 
+ 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2]/(a*c*e*(m + 1))), x] + Simp[1/(2*a*c*e* 
(m + 1))   Int[((e*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[ 
2*a*c*(m + 1)*((Px - Px0)/x) - Px0*(a*d*(2*m + 3) + 2*b*c*(m + 2)*x + b*d*( 
2*m + 5)*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, 
x] && LtQ[m, -1]
 

rule 2355

Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) 
^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(e*x)^m*(c + d* 
x)^(n + 1)*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c + d*x, x]   I 
nt[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p} 
, x] && PolynomialQ[Px, x] && LtQ[n, 0]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 7.96 (sec) , antiderivative size = 1132, normalized size of antiderivative = 1.53

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1132\)
risch	\(\text {Expression too large to display}\)	\(1362\)
default	\(\text {Expression too large to display}\)	\(5800\)

Input:

int((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^5/(d*x+c)^(1/2),x,method=_RETURNVERBO 
SE)
                                                                                    
                                                                                    
 

Output:

((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*(-1/4*A/c/x^4*(- 
b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+1/24*(7*A*d-8*B*c)/c^2*(-b*d*x^3-b*c*x^2+ 
a*d*x+a*c)^(1/2)/x^3-1/96/a/c^3*(35*A*a*d^2-12*A*b*c^2-40*B*a*c*d+48*C*a*c 
^2)*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/x^2+1/192*(105*A*a*d^3-44*A*b*c^2*d 
-120*B*a*c*d^2+64*B*b*c^3+144*C*a*c^2*d)/a/c^4*(-b*d*x^3-b*c*x^2+a*d*x+a*c 
)^(1/2)/x+1/96*b*d*(35*A*a*d^2-12*A*b*c^2-40*B*a*c*d+48*C*a*c^2)/a/c^3*(c/ 
d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/ 
2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/ 
2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b* 
(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)) 
+1/192*(105*A*a*d^3-44*A*b*c^2*d-120*B*a*c*d^2+64*B*b*c^3+144*C*a*c^2*d)*d 
/a*b/c^4*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1 
/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1 
/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((-c/d-1/b*(a*b) 
^(1/2))*EllipticE(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^( 
1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+1/b*(a*b)^(1/2)*EllipticF(((x+c/d)/(c 
/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2))) 
^(1/2)))-1/64*(35*A*a^2*d^4-24*A*a*b*c^2*d^2-16*A*b^2*c^4-40*B*a^2*c*d^3+3 
2*B*a*b*c^3*d+48*C*a^2*c^2*d^2-64*C*a*b*c^4)/c^5/a*(c/d-1/b*(a*b)^(1/2))*( 
(x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*...
 

Fricas [F]

\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a}}{\sqrt {d x + c} x^{5}} \,d x } \] Input:

integrate((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^5/(d*x+c)^(1/2),x, algorithm="f 
ricas")
 

Output:

integral((C*x^2 + B*x + A)*sqrt(-b*x^2 + a)*sqrt(d*x + c)/(d*x^6 + c*x^5), 
 x)
 

Sympy [F]

\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx=\int \frac {\sqrt {a - b x^{2}} \left (A + B x + C x^{2}\right )}{x^{5} \sqrt {c + d x}}\, dx \] Input:

integrate((-b*x**2+a)**(1/2)*(C*x**2+B*x+A)/x**5/(d*x+c)**(1/2),x)
 

Output:

Integral(sqrt(a - b*x**2)*(A + B*x + C*x**2)/(x**5*sqrt(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a}}{\sqrt {d x + c} x^{5}} \,d x } \] Input:

integrate((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^5/(d*x+c)^(1/2),x, algorithm="m 
axima")
 

Output:

integrate((C*x^2 + B*x + A)*sqrt(-b*x^2 + a)/(sqrt(d*x + c)*x^5), x)
 

Giac [F]

\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a}}{\sqrt {d x + c} x^{5}} \,d x } \] Input:

integrate((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^5/(d*x+c)^(1/2),x, algorithm="g 
iac")
 

Output:

integrate((C*x^2 + B*x + A)*sqrt(-b*x^2 + a)/(sqrt(d*x + c)*x^5), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx=\int \frac {\sqrt {a-b\,x^2}\,\left (C\,x^2+B\,x+A\right )}{x^5\,\sqrt {c+d\,x}} \,d x \] Input:

int(((a - b*x^2)^(1/2)*(A + B*x + C*x^2))/(x^5*(c + d*x)^(1/2)),x)
 

Output:

int(((a - b*x^2)^(1/2)*(A + B*x + C*x^2))/(x^5*(c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx=\int \frac {\sqrt {-b \,x^{2}+a}\, \left (C \,x^{2}+B x +A \right )}{x^{5} \sqrt {d x +c}}d x \] Input:

int((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^5/(d*x+c)^(1/2),x)
 

Output:

int((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^5/(d*x+c)^(1/2),x)