\(\int \frac {A+B x+C x^2}{x (c+d x)^{3/2} (a-b x^2)^{3/2}} \, dx\) [220]

Optimal result

Integrand size = 35, antiderivative size = 611 \[ \int \frac {A+B x+C x^2}{x (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {A b c+a c C-a B d+(b B c-A b d-a C d) x}{a \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {d \left (b c^2 (B c-2 A d)-a d \left (4 c^2 C-3 B c d+2 A d^2\right )\right ) \sqrt {a-b x^2}}{a c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {\sqrt {b} \left (b c^2 (B c-2 A d)-a d \left (4 c^2 C-3 B c d+2 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} c \left (b c^2-a d^2\right )^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {(b B c-A b d-a C d) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} \sqrt {b} \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 A \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a c \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

(A*b*c+C*a*c-B*a*d+(-A*b*d+B*b*c-C*a*d)*x)/a/(-a*d^2+b*c^2)/(d*x+c)^(1/2)/ 
(-b*x^2+a)^(1/2)-d*(b*c^2*(-2*A*d+B*c)-a*d*(2*A*d^2-3*B*c*d+4*C*c^2))*(-b* 
x^2+a)^(1/2)/a/c/(-a*d^2+b*c^2)^2/(d*x+c)^(1/2)+b^(1/2)*(b*c^2*(-2*A*d+B*c 
)-a*d*(2*A*d^2-3*B*c*d+4*C*c^2))*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*Ellipt 
icE(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/(-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)-(-A*b*d+B*b*c-C*a*d)*((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)/b^(1/2)/ 
(-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))/a/c/(d*x+c) 
^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 30.15 (sec) , antiderivative size = 1424, normalized size of antiderivative = 2.33 \[ \int \frac {A+B x+C x^2}{x (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

Integrate[(A + B*x + C*x^2)/(x*(c + d*x)^(3/2)*(a - b*x^2)^(3/2)),x]
 

Output:

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

Rubi [F]

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.

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

\(\Big \downarrow \) 2351

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 688

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

Input:

Int[(A + B*x + C*x^2)/(x*(c + d*x)^(3/2)*(a - b*x^2)^(3/2)),x]
 

Output:

$Aborted
 

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 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 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 637

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[(a + b*x^2)^p/Sqrt[c + d*x], x^m*(c + d*x)^(n + 1 
/2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p + 1/2] && IntegerQ[n 
 + 1/2] && IntegerQ[m]
 

rule 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1280\) vs. \(2(532)=1064\).

Time = 6.27 (sec) , antiderivative size = 1281, normalized size of antiderivative = 2.10

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1281\)
default	\(\text {Expression too large to display}\)	\(4185\)

Input:

int((C*x^2+B*x+A)/x/(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),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*b*d*(-1/2/a*( 
2*A*a*d^3+2*A*b*c^2*d-3*B*a*c*d^2-B*b*c^3+4*C*a*c^2*d)/c/(a*d^2-b*c^2)^2*x 
^2+1/2*(A*b*d-B*b*c+C*a*d)/a/(a*d^2-b*c^2)/b/d*x+1/2*(2*A*a^2*d^4+A*a*b*c^ 
2*d^2+A*b^2*c^4-2*B*a^2*c*d^3-2*B*a*b*c^3*d+3*C*a^2*c^2*d^2+C*a*b*c^4)/a/( 
a^2*d^4-2*a*b*c^2*d^2+b^2*c^4)/b/d/c)/(-(x^3+c/d*x^2-a*x/b-a*c/b/d)*b*d)^( 
1/2)+2*(1/2*(5*A*a*b*d^3-A*b^2*c^2*d-6*B*a*b*c*d^2+2*B*b^2*c^3+3*C*a^2*d^3 
+C*a*b*c^2*d)/a/(a^2*d^4-2*a*b*c^2*d^2+b^2*c^4)-(A*b*d-B*b*c+C*a*d)/a/(a*d 
^2-b*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)*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*(-3/2*b*d/a*(2*A*a*d^3+2*A*b*c^2*d-3*B*a*c*d^2-B*b*c^3+4* 
C*a*c^2*d)/(a^2*d^4-2*a*b*c^2*d^2+b^2*c^4)/c+2*b*d/a*(2*A*a*d^3+2*A*b*c^2* 
d-3*B*a*c*d^2-B*b*c^3+4*C*a*c^2*d)/c/(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)*((-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^2*A*(c/d-1/...
 

Fricas [F]

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

integrate((C*x^2+B*x+A)/x/(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x, algorithm="fri 
cas")
 

Output:

integral((C*x^2 + B*x + A)*sqrt(-b*x^2 + a)*sqrt(d*x + c)/(b^2*d^2*x^7 + 2 
*b^2*c*d*x^6 - 4*a*b*c*d*x^4 + 2*a^2*c*d*x^2 + (b^2*c^2 - 2*a*b*d^2)*x^5 + 
 a^2*c^2*x - (2*a*b*c^2 - a^2*d^2)*x^3), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{x (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((C*x**2+B*x+A)/x/(d*x+c)**(3/2)/(-b*x**2+a)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((C*x^2+B*x+A)/x/(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x, algorithm="max 
ima")
 

Output:

integrate((C*x^2 + B*x + A)/((-b*x^2 + a)^(3/2)*(d*x + c)^(3/2)*x), x)
 

Giac [F]

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

integrate((C*x^2+B*x+A)/x/(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x, algorithm="gia 
c")
 

Output:

integrate((C*x^2 + B*x + A)/((-b*x^2 + a)^(3/2)*(d*x + c)^(3/2)*x), x)
 

Mupad [F(-1)]

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

int((A + B*x + C*x^2)/(x*(a - b*x^2)^(3/2)*(c + d*x)^(3/2)),x)
 

Output:

int((A + B*x + C*x^2)/(x*(a - b*x^2)^(3/2)*(c + d*x)^(3/2)), x)
 

Reduce [F]

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

int((C*x^2+B*x+A)/x/(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x)
 

Output:

int((C*x^2+B*x+A)/x/(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x)