\(\int \frac {A+B x}{\sqrt {d+e x} (a-c x^2)^{5/2}} \, dx\) [287]

Optimal result

Integrand size = 27, antiderivative size = 519 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+e x} (a (B d-A e)+(A c d-a B e) x)}{3 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^{3/2}}-\frac {\sqrt {d+e x} \left (a e \left (A c d^2+4 a B d e-5 a A e^2\right )-\left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (c d^2+3 a e^2\right )\right ) x\right )}{6 a^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^2}}+\frac {\left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (c d^2+3 a e^2\right )\right ) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{6 a^{3/2} \sqrt {c} \left (c d^2-a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}-\frac {\left (4 A c d^2+a B d e-5 a A e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{6 a^{3/2} \sqrt {c} \left (c d^2-a e^2\right ) \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:

1/3*(e*x+d)^(1/2)*(a*(-A*e+B*d)+(A*c*d-B*a*e)*x)/a/(-a*e^2+c*d^2)/(-c*x^2+ 
a)^(3/2)-1/6*(e*x+d)^(1/2)*(a*e*(-5*A*a*e^2+A*c*d^2+4*B*a*d*e)-(4*A*c*d*(- 
2*a*e^2+c*d^2)+a*B*e*(3*a*e^2+c*d^2))*x)/a^2/(-a*e^2+c*d^2)^2/(-c*x^2+a)^( 
1/2)+1/6*(4*A*c*d*(-2*a*e^2+c*d^2)+a*B*e*(3*a*e^2+c*d^2))*(e*x+d)^(1/2)*(1 
-c*x^2/a)^(1/2)*EllipticE(1/2*(1-c^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)* 
(a^(1/2)*e/(c^(1/2)*d+a^(1/2)*e))^(1/2))/a^(3/2)/c^(1/2)/(-a*e^2+c*d^2)^2/ 
(c^(1/2)*(e*x+d)/(c^(1/2)*d+a^(1/2)*e))^(1/2)/(-c*x^2+a)^(1/2)-1/6*(-5*A*a 
*e^2+4*A*c*d^2+B*a*d*e)*(c^(1/2)*(e*x+d)/(c^(1/2)*d+a^(1/2)*e))^(1/2)*(1-c 
*x^2/a)^(1/2)*EllipticF(1/2*(1-c^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a 
^(1/2)*e/(c^(1/2)*d+a^(1/2)*e))^(1/2))/a^(3/2)/c^(1/2)/(-a*e^2+c*d^2)/(e*x 
+d)^(1/2)/(-c*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 26.77 (sec) , antiderivative size = 674, normalized size of antiderivative = 1.30 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^{5/2}} \, dx=\frac {\sqrt {a-c x^2} \left (\frac {(d+e x) \left (2 a \left (c d^2-a e^2\right ) (-a A e+A c d x+a B (d-e x))+\left (a-c x^2\right ) \left (4 A c^2 d^3 x+a^2 e^2 (-4 B d+5 A e+3 B e x)+a c d e (B d x-A (d+8 e x))\right )\right )}{\left (a-c x^2\right )^2}+\frac {e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (c d^2+3 a e^2\right )\right ) \left (a-c x^2\right )+i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (c d^2+3 a e^2\right )\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )+i \sqrt {a} \sqrt {c} e \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a B e \left (\sqrt {c} d-3 \sqrt {a} e\right )+A \left (4 c^{3/2} d^2+3 \sqrt {a} c d e-5 a \sqrt {c} e^2\right )\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{c e \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (a-c x^2\right )}\right )}{6 a^2 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}} \] Input:

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

Output:

(Sqrt[a - c*x^2]*(((d + e*x)*(2*a*(c*d^2 - a*e^2)*(-(a*A*e) + A*c*d*x + a* 
B*(d - e*x)) + (a - c*x^2)*(4*A*c^2*d^3*x + a^2*e^2*(-4*B*d + 5*A*e + 3*B* 
e*x) + a*c*d*e*(B*d*x - A*(d + 8*e*x)))))/(a - c*x^2)^2 + (e^2*Sqrt[-d + ( 
Sqrt[a]*e)/Sqrt[c]]*(4*A*c*d*(c*d^2 - 2*a*e^2) + a*B*e*(c*d^2 + 3*a*e^2))* 
(a - c*x^2) + I*Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e)*(4*A*c*d*(c*d^2 - 2*a*e^2) 
 + a*B*e*(c*d^2 + 3*a*e^2))*Sqrt[(e*(Sqrt[a]/Sqrt[c] + x))/(d + e*x)]*Sqrt 
[-(((Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*Arc 
Sinh[Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[a]*e 
)/(Sqrt[c]*d - Sqrt[a]*e)] + I*Sqrt[a]*Sqrt[c]*e*(Sqrt[c]*d - Sqrt[a]*e)*( 
a*B*e*(Sqrt[c]*d - 3*Sqrt[a]*e) + A*(4*c^(3/2)*d^2 + 3*Sqrt[a]*c*d*e - 5*a 
*Sqrt[c]*e^2))*Sqrt[(e*(Sqrt[a]/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((Sqrt[a]* 
e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d 
+ (Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[a]*e)/(Sqrt[c]*d 
- Sqrt[a]*e)])/(c*e*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(a - c*x^2))))/(6*a^2*( 
c*d^2 - a*e^2)^2*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {686, 27, 686, 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.

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

\(\Big \downarrow \) 686

\(\displaystyle \frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{3 a \left (a-c x^2\right )^{3/2} \left (c d^2-a e^2\right )}-\frac {\int -\frac {c \left (4 A c d^2+a B e d-5 a A e^2+3 e (A c d-a B e) x\right )}{2 \sqrt {d+e x} \left (a-c x^2\right )^{3/2}}dx}{3 a c \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {4 A c d^2+a B e d-5 a A e^2+3 e (A c d-a B e) x}{\sqrt {d+e x} \left (a-c x^2\right )^{3/2}}dx}{6 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{3 a \left (a-c x^2\right )^{3/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 686

\(\displaystyle \frac {-\frac {\int \frac {c e \left (a e \left (A c d^2+4 a B e d-5 a A e^2\right )+\left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (c d^2+3 a e^2\right )\right ) x\right )}{2 \sqrt {d+e x} \sqrt {a-c x^2}}dx}{a c \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (-5 a A e^2+4 a B d e+A c d^2\right )-x \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (3 a e^2+c d^2\right )\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{3 a \left (a-c x^2\right )^{3/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {e \int \frac {a e \left (A c d^2+4 a B e d-5 a A e^2\right )+\left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (c d^2+3 a e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (-5 a A e^2+4 a B d e+A c d^2\right )-x \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (3 a e^2+c d^2\right )\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{3 a \left (a-c x^2\right )^{3/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {-\frac {e \left (\frac {\left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (3 a e^2+c d^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}-\frac {\left (c d^2-a e^2\right ) \left (-5 a A e^2+a B d e+4 A c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (-5 a A e^2+4 a B d e+A c d^2\right )-x \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (3 a e^2+c d^2\right )\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{3 a \left (a-c x^2\right )^{3/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {-\frac {e \left (\frac {\sqrt {1-\frac {c x^2}{a}} \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (3 a e^2+c d^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {\left (c d^2-a e^2\right ) \left (-5 a A e^2+a B d e+4 A c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (-5 a A e^2+4 a B d e+A c d^2\right )-x \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (3 a e^2+c d^2\right )\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{3 a \left (a-c x^2\right )^{3/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {-\frac {e \left (-\frac {\left (c d^2-a e^2\right ) \left (-5 a A e^2+a B d e+4 A c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (3 a e^2+c d^2\right )\right ) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (-5 a A e^2+4 a B d e+A c d^2\right )-x \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (3 a e^2+c d^2\right )\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{3 a \left (a-c x^2\right )^{3/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {-\frac {e \left (-\frac {\left (c d^2-a e^2\right ) \left (-5 a A e^2+a B d e+4 A c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (3 a e^2+c d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (-5 a A e^2+4 a B d e+A c d^2\right )-x \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (3 a e^2+c d^2\right )\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{3 a \left (a-c x^2\right )^{3/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {-\frac {e \left (-\frac {\sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \left (-5 a A e^2+a B d e+4 A c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (3 a e^2+c d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (-5 a A e^2+4 a B d e+A c d^2\right )-x \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (3 a e^2+c d^2\right )\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{3 a \left (a-c x^2\right )^{3/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {-\frac {e \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (-5 a A e^2+a B d e+4 A c d^2\right ) \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (3 a e^2+c d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (-5 a A e^2+4 a B d e+A c d^2\right )-x \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (3 a e^2+c d^2\right )\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{3 a \left (a-c x^2\right )^{3/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {-\frac {e \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (-5 a A e^2+a B d e+4 A c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (3 a e^2+c d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (-5 a A e^2+4 a B d e+A c d^2\right )-x \left (4 A c d \left (c d^2-2 a e^2\right )+a B e \left (3 a e^2+c d^2\right )\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{3 a \left (a-c x^2\right )^{3/2} \left (c d^2-a e^2\right )}\)

Input:

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

Output:

(Sqrt[d + e*x]*(a*(B*d - A*e) + (A*c*d - a*B*e)*x))/(3*a*(c*d^2 - a*e^2)*( 
a - c*x^2)^(3/2)) + (-((Sqrt[d + e*x]*(a*e*(A*c*d^2 + 4*a*B*d*e - 5*a*A*e^ 
2) - (4*A*c*d*(c*d^2 - 2*a*e^2) + a*B*e*(c*d^2 + 3*a*e^2))*x))/(a*(c*d^2 - 
 a*e^2)*Sqrt[a - c*x^2])) - (e*((-2*Sqrt[a]*(4*A*c*d*(c*d^2 - 2*a*e^2) + a 
*B*e*(c*d^2 + 3*a*e^2))*Sqrt[d + e*x]*Sqrt[1 - (c*x^2)/a]*EllipticE[ArcSin 
[Sqrt[1 - (Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c]*d)/Sqrt[a] + e)]) 
/(Sqrt[c]*e*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[a - c*x 
^2]) + (2*Sqrt[a]*(c*d^2 - a*e^2)*(4*A*c*d^2 + a*B*d*e - 5*a*A*e^2)*Sqrt[( 
Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[1 - (c*x^2)/a]*EllipticF[ 
ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c]*d)/Sqrt[a] 
+ e)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[a - c*x^2])))/(2*a*(c*d^2 - a*e^2)))/ 
(6*a*(c*d^2 - a*e^2))
 

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 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])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(970\) vs. \(2(449)=898\).

Time = 8.79 (sec) , antiderivative size = 971, normalized size of antiderivative = 1.87

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (\frac {\left (-\frac {\left (A c d -B a e \right ) x}{3 a \left (a \,e^{2}-c \,d^{2}\right ) c^{2}}+\frac {A e -B d}{3 \left (a \,e^{2}-c \,d^{2}\right ) c^{2}}\right ) \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{\left (x^{2}-\frac {a}{c}\right )^{2}}-\frac {2 \left (-c e x -c d \right ) \left (-\frac {\left (8 A a c d \,e^{2}-4 A \,c^{2} d^{3}-3 B \,e^{3} a^{2}-B a c \,d^{2} e \right ) x}{12 a^{2} \left (a \,e^{2}-c \,d^{2}\right )^{2} c}+\frac {e \left (5 A a \,e^{2}-A c \,d^{2}-4 B a d e \right )}{12 a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\right )}{\sqrt {\left (x^{2}-\frac {a}{c}\right ) \left (-c e x -c d \right )}}+\frac {2 \left (\frac {5 A a \,e^{2}-4 A c \,d^{2}-B a d e}{6 \left (a \,e^{2}-c \,d^{2}\right ) a^{2}}-\frac {e^{2} \left (5 A a \,e^{2}-A c \,d^{2}-4 B a d e \right )}{12 a \left (a \,e^{2}-c \,d^{2}\right )^{2}}+\frac {d \left (8 A a c d \,e^{2}-4 A \,c^{2} d^{3}-3 B \,e^{3} a^{2}-B a c \,d^{2} e \right )}{6 a^{2} \left (a \,e^{2}-c \,d^{2}\right )^{2}}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {e \left (8 A a c d \,e^{2}-4 A \,c^{2} d^{3}-3 B \,e^{3} a^{2}-B a c \,d^{2} e \right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )+\frac {\sqrt {a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{c}\right )}{6 a^{2} \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\)	\(971\)
default	\(\text {Expression too large to display}\)	\(4765\)

Input:

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

Output:

((e*x+d)*(-c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)*((-1/3*(A*c*d-B* 
a*e)/a/(a*e^2-c*d^2)/c^2*x+1/3*(A*e-B*d)/(a*e^2-c*d^2)/c^2)*(-c*e*x^3-c*d* 
x^2+a*e*x+a*d)^(1/2)/(x^2-a/c)^2-2*(-c*e*x-c*d)*(-1/12*(8*A*a*c*d*e^2-4*A* 
c^2*d^3-3*B*a^2*e^3-B*a*c*d^2*e)/a^2/(a*e^2-c*d^2)^2/c*x+1/12*e*(5*A*a*e^2 
-A*c*d^2-4*B*a*d*e)/a/c/(a*e^2-c*d^2)^2)/((x^2-a/c)*(-c*e*x-c*d))^(1/2)+2* 
(1/6/(a*e^2-c*d^2)*(5*A*a*e^2-4*A*c*d^2-B*a*d*e)/a^2-1/12*e^2*(5*A*a*e^2-A 
*c*d^2-4*B*a*d*e)/a/(a*e^2-c*d^2)^2+1/6*d*(8*A*a*c*d*e^2-4*A*c^2*d^3-3*B*a 
^2*e^3-B*a*c*d^2*e)/a^2/(a*e^2-c*d^2)^2)*(d/e-1/c*(a*c)^(1/2))*((x+d/e)/(d 
/e-1/c*(a*c)^(1/2)))^(1/2)*((x-1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1 
/2)*((x+1/c*(a*c)^(1/2))/(-d/e+1/c*(a*c)^(1/2)))^(1/2)/(-c*e*x^3-c*d*x^2+a 
*e*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2),((-d/e+1/c 
*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2))+1/6*e*(8*A*a*c*d*e^2-4*A*c^2* 
d^3-3*B*a^2*e^3-B*a*c*d^2*e)/a^2/(a*e^2-c*d^2)^2*(d/e-1/c*(a*c)^(1/2))*((x 
+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2)*((x-1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1 
/2)))^(1/2)*((x+1/c*(a*c)^(1/2))/(-d/e+1/c*(a*c)^(1/2)))^(1/2)/(-c*e*x^3-c 
*d*x^2+a*e*x+a*d)^(1/2)*((-d/e-1/c*(a*c)^(1/2))*EllipticE(((x+d/e)/(d/e-1/ 
c*(a*c)^(1/2)))^(1/2),((-d/e+1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2 
))+1/c*(a*c)^(1/2)*EllipticF(((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2),((-d/e+ 
1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 828, normalized size of antiderivative = 1.60 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/18*((4*A*a^2*c^2*d^4 + B*a^3*c*d^3*e - 11*A*a^3*c*d^2*e^2 - 9*B*a^4*d*e 
^3 + 15*A*a^4*e^4 + (4*A*c^4*d^4 + B*a*c^3*d^3*e - 11*A*a*c^3*d^2*e^2 - 9* 
B*a^2*c^2*d*e^3 + 15*A*a^2*c^2*e^4)*x^4 - 2*(4*A*a*c^3*d^4 + B*a^2*c^2*d^3 
*e - 11*A*a^2*c^2*d^2*e^2 - 9*B*a^3*c*d*e^3 + 15*A*a^3*c*e^4)*x^2)*sqrt(-c 
*e)*weierstrassPInverse(4/3*(c*d^2 + 3*a*e^2)/(c*e^2), -8/27*(c*d^3 - 9*a* 
d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e) + 3*(4*A*a^2*c^2*d^3*e + B*a^3*c*d^2*e^ 
2 - 8*A*a^3*c*d*e^3 + 3*B*a^4*e^4 + (4*A*c^4*d^3*e + B*a*c^3*d^2*e^2 - 8*A 
*a*c^3*d*e^3 + 3*B*a^2*c^2*e^4)*x^4 - 2*(4*A*a*c^3*d^3*e + B*a^2*c^2*d^2*e 
^2 - 8*A*a^2*c^2*d*e^3 + 3*B*a^3*c*e^4)*x^2)*sqrt(-c*e)*weierstrassZeta(4/ 
3*(c*d^2 + 3*a*e^2)/(c*e^2), -8/27*(c*d^3 - 9*a*d*e^2)/(c*e^3), weierstras 
sPInverse(4/3*(c*d^2 + 3*a*e^2)/(c*e^2), -8/27*(c*d^3 - 9*a*d*e^2)/(c*e^3) 
, 1/3*(3*e*x + d)/e)) - 3*(2*B*a^2*c^2*d^3*e - 3*A*a^2*c^2*d^2*e^2 - 6*B*a 
^3*c*d*e^3 + 7*A*a^3*c*e^4 - (4*A*c^4*d^3*e + B*a*c^3*d^2*e^2 - 8*A*a*c^3* 
d*e^3 + 3*B*a^2*c^2*e^4)*x^3 + (A*a*c^3*d^2*e^2 + 4*B*a^2*c^2*d*e^3 - 5*A* 
a^2*c^2*e^4)*x^2 + (6*A*a*c^3*d^3*e - B*a^2*c^2*d^2*e^2 - 10*A*a^2*c^2*d*e 
^3 + 5*B*a^3*c*e^4)*x)*sqrt(-c*x^2 + a)*sqrt(e*x + d))/(a^4*c^3*d^4*e - 2* 
a^5*c^2*d^2*e^3 + a^6*c*e^5 + (a^2*c^5*d^4*e - 2*a^3*c^4*d^2*e^3 + a^4*c^3 
*e^5)*x^4 - 2*(a^3*c^4*d^4*e - 2*a^4*c^3*d^2*e^3 + a^5*c^2*e^5)*x^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^{5/2}} \, dx=\left (\int \frac {\sqrt {e x +d}}{\sqrt {-c \,x^{2}+a}\, a^{2} d^{2}-\sqrt {-c \,x^{2}+a}\, a^{2} e^{2} x^{2}-2 \sqrt {-c \,x^{2}+a}\, a c \,d^{2} x^{2}+2 \sqrt {-c \,x^{2}+a}\, a c \,e^{2} x^{4}+\sqrt {-c \,x^{2}+a}\, c^{2} d^{2} x^{4}-\sqrt {-c \,x^{2}+a}\, c^{2} e^{2} x^{6}}d x \right ) a d +\left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, x}{-c^{3} e \,x^{7}-c^{3} d \,x^{6}+3 a \,c^{2} e \,x^{5}+3 a \,c^{2} d \,x^{4}-3 a^{2} c e \,x^{3}-3 a^{2} c d \,x^{2}+a^{3} e x +a^{3} d}d x \right ) b -\left (\int \frac {\sqrt {e x +d}\, x}{\sqrt {-c \,x^{2}+a}\, a^{2} d^{2}-\sqrt {-c \,x^{2}+a}\, a^{2} e^{2} x^{2}-2 \sqrt {-c \,x^{2}+a}\, a c \,d^{2} x^{2}+2 \sqrt {-c \,x^{2}+a}\, a c \,e^{2} x^{4}+\sqrt {-c \,x^{2}+a}\, c^{2} d^{2} x^{4}-\sqrt {-c \,x^{2}+a}\, c^{2} e^{2} x^{6}}d x \right ) a e \] Input:

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

Output:

int(sqrt(d + e*x)/(sqrt(a - c*x**2)*a**2*d**2 - sqrt(a - c*x**2)*a**2*e**2 
*x**2 - 2*sqrt(a - c*x**2)*a*c*d**2*x**2 + 2*sqrt(a - c*x**2)*a*c*e**2*x** 
4 + sqrt(a - c*x**2)*c**2*d**2*x**4 - sqrt(a - c*x**2)*c**2*e**2*x**6),x)* 
a*d + int((sqrt(d + e*x)*sqrt(a - c*x**2)*x)/(a**3*d + a**3*e*x - 3*a**2*c 
*d*x**2 - 3*a**2*c*e*x**3 + 3*a*c**2*d*x**4 + 3*a*c**2*e*x**5 - c**3*d*x** 
6 - c**3*e*x**7),x)*b - int((sqrt(d + e*x)*x)/(sqrt(a - c*x**2)*a**2*d**2 
- sqrt(a - c*x**2)*a**2*e**2*x**2 - 2*sqrt(a - c*x**2)*a*c*d**2*x**2 + 2*s 
qrt(a - c*x**2)*a*c*e**2*x**4 + sqrt(a - c*x**2)*c**2*d**2*x**4 - sqrt(a - 
 c*x**2)*c**2*e**2*x**6),x)*a*e