\(\int \frac {A+B x+C x^2}{x (c+d x)^{5/2} \sqrt {a-b x^2}} \, dx\) [203]

Optimal result

Integrand size = 35, antiderivative size = 619 \[ \int \frac {A+B x+C x^2}{x (c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=-\frac {2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a-b x^2}}{3 c \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {2 \left (3 a d^2 \left (c^2 C-A d^2\right )+b c^2 \left (c^2 C-4 B c d+7 A d^2\right )\right ) \sqrt {a-b x^2}}{3 c^2 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {b} \left (3 a d^2 \left (c^2 C-A d^2\right )+b c^2 \left (c^2 C-4 B c d+7 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 c^2 d \left (b c^2-a d^2\right )^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \left (c^2 C-B c d+A d^2\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 c d \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 A \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-2/3*(A*d^2-B*c*d+C*c^2)*(-b*x^2+a)^(1/2)/c/(-a*d^2+b*c^2)/(d*x+c)^(3/2)-2 
/3*(3*a*d^2*(-A*d^2+C*c^2)+b*c^2*(7*A*d^2-4*B*c*d+C*c^2))*(-b*x^2+a)^(1/2) 
/c^2/(-a*d^2+b*c^2)^2/(d*x+c)^(1/2)+2/3*a^(1/2)*b^(1/2)*(3*a*d^2*(-A*d^2+C 
*c^2)+b*c^2*(7*A*d^2-4*B*c*d+C*c^2))*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*El 
lipticE(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))/c^2/d/(-a*d^2+b*c^2)^2/((d*x+c)/(c+a^(1/2)*d/b^(1/2 
)))^(1/2)/(-b*x^2+a)^(1/2)-2/3*a^(1/2)*b^(1/2)*(A*d^2-B*c*d+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 
))/c/d/(-a*d^2+b*c^2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-2*A*((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))/c^2/ 
(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 31.24 (sec) , antiderivative size = 1728, normalized size of antiderivative = 2.79 \[ \int \frac {A+B x+C x^2}{x (c+d x)^{5/2} \sqrt {a-b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*Sqrt[a - b*x^2]*(-(c*(c*(b*c^2 - a*d^2)*(c^2*C - B*c*d + A*d^2) + (3*a* 
d^2*(c^2*C - A*d^2) + b*c^2*(c^2*C - 4*B*c*d + 7*A*d^2))*(c + d*x))) - ((c 
 + d*x)*(b^2*c^7*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 4*b^2*B*c^6*d*Sqrt[-c 
+ (Sqrt[a]*d)/Sqrt[b]] + 7*A*b^2*c^5*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 
2*a*b*c^5*C*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 4*a*b*B*c^4*d^3*Sqrt[-c + 
 (Sqrt[a]*d)/Sqrt[b]] - 10*a*A*b*c^3*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 
3*a^2*c^3*C*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 3*a^2*A*c*d^6*Sqrt[-c + ( 
Sqrt[a]*d)/Sqrt[b]] - 2*b^2*c^6*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) 
 + 8*b^2*B*c^5*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) - 14*A*b^2*c^4*d 
^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) - 6*a*b*c^4*C*d^2*Sqrt[-c + (S 
qrt[a]*d)/Sqrt[b]]*(c + d*x) + 6*a*A*b*c^2*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[ 
b]]*(c + d*x) + b^2*c^5*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - 4*b 
^2*B*c^4*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 + 7*A*b^2*c^3*d^2*Sq 
rt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 + 3*a*b*c^3*C*d^2*Sqrt[-c + (Sqrt 
[a]*d)/Sqrt[b]]*(c + d*x)^2 - 3*a*A*b*c*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] 
*(c + d*x)^2 - I*Sqrt[b]*c*(Sqrt[b]*c - Sqrt[a]*d)*(3*a*d^2*(c^2*C - A*d^2 
) + b*c^2*(c^2*C - 4*B*c*d + 7*A*d^2))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + 
 d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*Elli 
pticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c 
+ Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - I*d*(Sqrt[b]*c - Sqrt[a]*d)*(3*...
 

Rubi [A] (verified)

Time = 3.10 (sec) , antiderivative size = 1037, normalized size of antiderivative = 1.68, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.514, Rules used = {2351, 635, 633, 632, 186, 413, 412, 688, 27, 688, 27, 600, 509, 508, 327, 512, 511, 321}

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.

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

\(\Big \downarrow \) 2351

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

\(\Big \downarrow \) 635

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

\(\Big \downarrow \) 633

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

\(\Big \downarrow \) 632

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

\(\Big \downarrow \) 186

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 412

\(\displaystyle A \left (\int \frac {-\frac {x d^2}{c^2}-\frac {2 d}{c}}{(c+d x)^{5/2} \sqrt {a-b x^2}}dx-\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}\right )+\int \frac {B+C x}{(c+d x)^{5/2} \sqrt {a-b x^2}}dx\)

\(\Big \downarrow \) 688

\(\displaystyle A \left (\frac {2 \int -\frac {d \left (3 c \left (2 b-\frac {a d^2}{c^2}\right )-b d x\right )}{2 c (c+d x)^{3/2} \sqrt {a-b x^2}}dx}{3 \left (b c^2-a d^2\right )}-\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 d^2 \sqrt {a-b x^2}}{3 c (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )+\frac {2 \int \frac {3 (b B c-a C d)+b (c C-B d) x}{2 (c+d x)^{3/2} \sqrt {a-b x^2}}dx}{3 \left (b c^2-a d^2\right )}-\frac {2 \sqrt {a-b x^2} (c C-B d)}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle A \left (-\frac {d \int \frac {3 \left (2 b c-\frac {a d^2}{c}\right )-b d x}{(c+d x)^{3/2} \sqrt {a-b x^2}}dx}{3 c \left (b c^2-a d^2\right )}-\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 d^2 \sqrt {a-b x^2}}{3 c (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )+\frac {\int \frac {3 (b B c-a C d)+b (c C-B d) x}{(c+d x)^{3/2} \sqrt {a-b x^2}}dx}{3 \left (b c^2-a d^2\right )}-\frac {2 \sqrt {a-b x^2} (c C-B d)}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 688

\(\displaystyle A \left (-\frac {d \left (\frac {2 \int \frac {b \left (2 c \left (3 b c^2-a d^2\right )+d \left (7 b c^2-3 a d^2\right ) x\right )}{2 c \sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {2 d \sqrt {a-b x^2} \left (7 b c^2-3 a d^2\right )}{c \sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 c \left (b c^2-a d^2\right )}-\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 d^2 \sqrt {a-b x^2}}{3 c (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )+\frac {\frac {2 \int \frac {b \left (3 b B c^2-a d (4 c C-B d)-\left (3 a C d^2+b c (c C-4 B d)\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}-\frac {2 \sqrt {a-b x^2} \left (3 a C d^2+b c (c C-4 B d)\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}-\frac {2 \sqrt {a-b x^2} (c C-B d)}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle A \left (-\frac {d \left (\frac {b \int \frac {2 c \left (3 b c^2-a d^2\right )+d \left (7 b c^2-3 a d^2\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{c \left (b c^2-a d^2\right )}+\frac {2 d \sqrt {a-b x^2} \left (7 b c^2-3 a d^2\right )}{c \sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 c \left (b c^2-a d^2\right )}-\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 d^2 \sqrt {a-b x^2}}{3 c (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )+\frac {\frac {b \int \frac {3 b B c^2-a d (4 c C-B d)-\left (3 a C d^2+b c (c C-4 B d)\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}-\frac {2 \sqrt {a-b x^2} \left (3 a C d^2+b c (c C-4 B d)\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}-\frac {2 \sqrt {a-b x^2} (c C-B d)}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle A \left (-\frac {d \left (\frac {b \left (\left (7 b c^2-3 a d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx-c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\right )}{c \left (b c^2-a d^2\right )}+\frac {2 d \sqrt {a-b x^2} \left (7 b c^2-3 a d^2\right )}{c \sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 c \left (b c^2-a d^2\right )}-\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 d^2 \sqrt {a-b x^2}}{3 c (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )+\frac {\frac {b \left (\frac {\left (b c^2-a d^2\right ) (c C-B d) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {\left (3 a C d^2+b c (c C-4 B d)\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )}{b c^2-a d^2}-\frac {2 \sqrt {a-b x^2} \left (3 a C d^2+b c (c C-4 B d)\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}-\frac {2 \sqrt {a-b x^2} (c C-B d)}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle A \left (-\frac {d \left (\frac {b \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (7 b c^2-3 a d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\right )}{c \left (b c^2-a d^2\right )}+\frac {2 d \sqrt {a-b x^2} \left (7 b c^2-3 a d^2\right )}{c \sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 c \left (b c^2-a d^2\right )}-\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 d^2 \sqrt {a-b x^2}}{3 c (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )+\frac {\frac {b \left (\frac {\left (b c^2-a d^2\right ) (c C-B d) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {\sqrt {1-\frac {b x^2}{a}} \left (3 a C d^2+b c (c C-4 B d)\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{b c^2-a d^2}-\frac {2 \sqrt {a-b x^2} \left (3 a C d^2+b c (c C-4 B d)\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}-\frac {2 \sqrt {a-b x^2} (c C-B d)}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

\(\displaystyle -\frac {2 \sqrt {a-b x^2} (c C-B d)}{3 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {\frac {b \left (\frac {2 \sqrt {a} \left (3 a C d^2+b c (c C-4 B d)\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {(c C-B d) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{b c^2-a d^2}-\frac {2 \left (3 a C d^2+b c (c C-4 B d)\right ) \sqrt {a-b x^2}}{\left (b c^2-a d^2\right ) \sqrt {c+d x}}}{3 \left (b c^2-a d^2\right )}+A \left (-\frac {2 \sqrt {a-b x^2} d^2}{3 c \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {\left (\frac {2 d \sqrt {a-b x^2} \left (7 b c^2-3 a d^2\right )}{c \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {b \left (-\frac {2 \sqrt {a} \left (7 b c^2-3 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\right )}{c \left (b c^2-a d^2\right )}\right ) d}{3 c \left (b c^2-a d^2\right )}-\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}\right )\)

\(\Big \downarrow \) 512

\(\displaystyle -\frac {2 \sqrt {a-b x^2} (c C-B d)}{3 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {\frac {b \left (\frac {2 \sqrt {a} \left (3 a C d^2+b c (c C-4 B d)\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {(c C-B d) \left (b c^2-a d^2\right ) \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{b c^2-a d^2}-\frac {2 \left (3 a C d^2+b c (c C-4 B d)\right ) \sqrt {a-b x^2}}{\left (b c^2-a d^2\right ) \sqrt {c+d x}}}{3 \left (b c^2-a d^2\right )}+A \left (-\frac {2 \sqrt {a-b x^2} d^2}{3 c \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {\left (\frac {2 d \sqrt {a-b x^2} \left (7 b c^2-3 a d^2\right )}{c \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {b \left (-\frac {2 \sqrt {a} \left (7 b c^2-3 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {c \left (b c^2-a d^2\right ) \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}\right )}{c \left (b c^2-a d^2\right )}\right ) d}{3 c \left (b c^2-a d^2\right )}-\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}\right )\)

\(\Big \downarrow \) 511

\(\displaystyle -\frac {2 \sqrt {a-b x^2} (c C-B d)}{3 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {\frac {b \left (\frac {2 \sqrt {a} \left (3 a C d^2+b c (c C-4 B d)\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 )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} (c C-B d) \left (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}} \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 {b} d \sqrt {c+d x} \sqrt {a-b x^2}}\right )}{b c^2-a d^2}-\frac {2 \left (3 a C d^2+b c (c C-4 B d)\right ) \sqrt {a-b x^2}}{\left (b c^2-a d^2\right ) \sqrt {c+d x}}}{3 \left (b c^2-a d^2\right )}+A \left (-\frac {2 \sqrt {a-b x^2} d^2}{3 c \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {\left (\frac {2 d \sqrt {a-b x^2} \left (7 b c^2-3 a d^2\right )}{c \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {b \left (\frac {2 \sqrt {a} c \left (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}} \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 {b} \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (7 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 )}{\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )}{c \left (b c^2-a d^2\right )}\right ) d}{3 c \left (b c^2-a d^2\right )}-\frac {2 \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 )}{c^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}}}}\right )\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {2 \sqrt {a-b x^2} (c C-B d)}{3 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {\frac {b \left (\frac {2 \sqrt {a} \left (3 a C d^2+b c (c C-4 B d)\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 )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} (c C-B d) \left (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 )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}\right )}{b c^2-a d^2}-\frac {2 \left (3 a C d^2+b c (c C-4 B d)\right ) \sqrt {a-b x^2}}{\left (b c^2-a d^2\right ) \sqrt {c+d x}}}{3 \left (b c^2-a d^2\right )}+A \left (-\frac {2 \sqrt {a-b x^2} d^2}{3 c \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {\left (\frac {2 d \sqrt {a-b x^2} \left (7 b c^2-3 a d^2\right )}{c \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {b \left (\frac {2 \sqrt {a} c \left (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 )}{\sqrt {b} \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (7 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 )}{\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )}{c \left (b c^2-a d^2\right )}\right ) d}{3 c \left (b c^2-a d^2\right )}-\frac {2 \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 )}{c^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}}}}\right )\)

Input:

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

Output:

(-2*(c*C - B*d)*Sqrt[a - b*x^2])/(3*(b*c^2 - a*d^2)*(c + d*x)^(3/2)) + ((- 
2*(3*a*C*d^2 + b*c*(c*C - 4*B*d))*Sqrt[a - b*x^2])/((b*c^2 - a*d^2)*Sqrt[c 
 + d*x]) + (b*((2*Sqrt[a]*(3*a*C*d^2 + b*c*(c*C - 4*B*d))*Sqrt[c + d*x]*Sq 
rt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], 
 (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sq 
rt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (2*Sqrt[a]*(c*C - B*d)*(b*c^2 - 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)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(b*c^2 - a* 
d^2))/(3*(b*c^2 - a*d^2)) + A*((-2*d^2*Sqrt[a - b*x^2])/(3*c*(b*c^2 - a*d^ 
2)*(c + d*x)^(3/2)) - (d*((2*d*(7*b*c^2 - 3*a*d^2)*Sqrt[a - b*x^2])/(c*(b* 
c^2 - a*d^2)*Sqrt[c + d*x]) + (b*((-2*Sqrt[a]*(7*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]]/S 
qrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*Sqrt[(Sqrt[b]*(c + d*x 
))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*c*(b*c^2 - a*d^2 
)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*El 
lipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/ 
Sqrt[a] + d)])/(Sqrt[b]*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(c*(b*c^2 - a*d^2 
))))/(3*c*(b*c^2 - a*d^2)) - (2*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (Sqrt[a]*d*(1 
 - (Sqrt[b]*x)/Sqrt[a]))/(Sqrt[b]*c + Sqrt[a]*d)]*EllipticPi[2, ArcSin[...
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 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 635

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

rule 688

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( 
(m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2))   Int[(d + 
 e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m 
 + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] 
&& (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1101\) vs. \(2(532)=1064\).

Time = 5.64 (sec) , antiderivative size = 1102, normalized size of antiderivative = 1.78

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1102\)
default	\(\text {Expression too large to display}\)	\(8465\)

Input:

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

Output:

((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*(2/3/(a*d^2-b*c^ 
2)/d^2*(A*d^2-B*c*d+C*c^2)/c*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^2+ 
2/3*(-b*d*x^2+a*d)/(a*d^2-b*c^2)^2/d*(3*A*a*d^4-7*A*b*c^2*d^2+4*B*b*c^3*d- 
3*C*a*c^2*d^2-C*b*c^4)/c^2/((x+c/d)*(-b*d*x^2+a*d))^(1/2)+2*(-1/3*b/d*(A*d 
^2-B*c*d+C*c^2)/(a*d^2-b*c^2)/c+1/3*b/c/d*(3*A*a*d^4-7*A*b*c^2*d^2+4*B*b*c 
^3*d-3*C*a*c^2*d^2-C*b*c^4)/(a*d^2-b*c^2)^2)*(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))+2/3*b*(3*A*a*d^4-7*A*b*c^ 
2*d^2+4*B*b*c^3*d-3*C*a*c^2*d^2-C*b*c^4)/(a*d^2-b*c^2)^2/c^2*(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)))-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)*d*EllipticPi(((x+c/d)/(c/d-...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x+C x^2}{x (c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x}{-b \,d^{3} x^{5}-3 b c \,d^{2} x^{4}+a \,d^{3} x^{3}-3 b \,c^{2} d \,x^{3}+3 a c \,d^{2} x^{2}-b \,c^{3} x^{2}+3 a \,c^{2} d x +a \,c^{3}}d x \right ) c +\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{3} x^{6}-3 b c \,d^{2} x^{5}+a \,d^{3} x^{4}-3 b \,c^{2} d \,x^{4}+3 a c \,d^{2} x^{3}-b \,c^{3} x^{3}+3 a \,c^{2} d \,x^{2}+a \,c^{3} x}d x \right ) a +\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{3} x^{5}-3 b c \,d^{2} x^{4}+a \,d^{3} x^{3}-3 b \,c^{2} d \,x^{3}+3 a c \,d^{2} x^{2}-b \,c^{3} x^{2}+3 a \,c^{2} d x +a \,c^{3}}d x \right ) b \] Input:

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

Output:

int((sqrt(c + d*x)*sqrt(a - b*x**2)*x)/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2 
*x**2 + a*d**3*x**3 - b*c**3*x**2 - 3*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - b* 
d**3*x**5),x)*c + int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c**3*x + 3*a*c** 
2*d*x**2 + 3*a*c*d**2*x**3 + a*d**3*x**4 - b*c**3*x**3 - 3*b*c**2*d*x**4 - 
 3*b*c*d**2*x**5 - b*d**3*x**6),x)*a + int((sqrt(c + d*x)*sqrt(a - b*x**2) 
)/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 - b*c**3*x**2 - 3 
*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - b*d**3*x**5),x)*b