\(\int \frac {(A+B x) (d+e x)^{5/2}}{(a-c x^2)^{5/2}} \, dx\) [284]

Optimal result

Integrand size = 27, antiderivative size = 434 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^{5/2}} \, dx=\frac {(d+e x)^{3/2} (a (B d+A e)+(A c d+a B e) x)}{3 a c \left (a-c x^2\right )^{3/2}}+\frac {\sqrt {d+e x} \left (a e (A c d-5 a B e)+c \left (4 A c d^2-5 a B d e-3 a A e^2\right ) x\right )}{6 a^2 c^2 \sqrt {a-c x^2}}+\frac {\left (4 A c d^2-5 a B d e-3 a A e^2\right ) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{6 a^{3/2} c^{3/2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}-\frac {(4 A c d-5 a B e) \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{6 a^{3/2} c^{5/2} \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 25.93 (sec) , antiderivative size = 606, normalized size of antiderivative = 1.40 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^{5/2}} \, dx=\frac {\sqrt {a-c x^2} \left (-\frac {(d+e x) \left (5 a^3 B e^2+4 A c^3 d^2 x^3+a^2 c \left (A e (-3 d+e x)+B \left (-2 d^2+d e x-7 e^2 x^2\right )\right )-a c^2 x \left (5 B d e x^2+A \left (6 d^2+d e x+3 e^2 x^2\right )\right )\right )}{a^2 c^2 \left (a-c x^2\right )^2}+\frac {e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (4 A c d^2-5 a B d e-3 a A e^2\right ) \left (-a+c x^2\right )-i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (4 A c d^2-5 a B d e-3 a A e^2\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )+i \sqrt {a} e \left (-\sqrt {c} d+\sqrt {a} e\right ) \left (4 A c d-5 a B e+3 \sqrt {a} A \sqrt {c} e\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{a^2 c^2 e \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}\right )}{6 \sqrt {d+e x}} \] Input:

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

Output:

(Sqrt[a - c*x^2]*(-(((d + e*x)*(5*a^3*B*e^2 + 4*A*c^3*d^2*x^3 + a^2*c*(A*e 
*(-3*d + e*x) + B*(-2*d^2 + d*e*x - 7*e^2*x^2)) - a*c^2*x*(5*B*d*e*x^2 + A 
*(6*d^2 + d*e*x + 3*e^2*x^2))))/(a^2*c^2*(a - c*x^2)^2)) + (e^2*Sqrt[-d + 
(Sqrt[a]*e)/Sqrt[c]]*(4*A*c*d^2 - 5*a*B*d*e - 3*a*A*e^2)*(-a + c*x^2) - I* 
Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e)*(4*A*c*d^2 - 5*a*B*d*e - 3*a*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*ArcSinh[Sqrt[-d + (Sqrt[a]*e)/Sqrt[c] 
]/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[a]*e)/(Sqrt[c]*d - Sqrt[a]*e)] + I*Sqr 
t[a]*e*(-(Sqrt[c]*d) + Sqrt[a]*e)*(4*A*c*d - 5*a*B*e + 3*Sqrt[a]*A*Sqrt[c] 
*e)*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 
)])/(a^2*c^2*e*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c*x^2))))/(6*Sqrt[d + 
e*x])
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {684, 27, 685, 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.

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

\(\Big \downarrow \) 684

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 685

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

Input:

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

Output:

((d + e*x)^(3/2)*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(3*a*c*(a - c*x^2)^( 
3/2)) + ((Sqrt[d + e*x]*(a*e*(A*c*d - 5*a*B*e) + c*(4*A*c*d^2 - a*e*(5*B*d 
 + 3*A*e))*x))/(a*c*Sqrt[a - c*x^2]) - (e*((-2*Sqrt[a]*Sqrt[c]*(4*A*c*d^2 
- a*e*(5*B*d + 3*A*e))*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)])/ 
(e*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[a - c*x^2]) + (2 
*Sqrt[a]*(4*A*c*d - 5*a*B*e)*(c*d^2 - a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqr 
t[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))/(6*a*c)
 

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 684

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*((a*(e*f + d*g 
) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Simp[1/(2*a*c*(p + 1))   Int[ 
(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^ 
2*f*(2*p + 3) + e*(a*e*g*m - c*d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a 
, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] 
 && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])
 

rule 685

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(d + e*x)^m*(a + c*x^2)^(p + 1)*((a*g - c*f*x)/(2*a*c 
*(p + 1))), x] - Simp[1/(2*a*c*(p + 1))   Int[(d + e*x)^(m - 1)*(a + c*x^2) 
^(p + 1)*Simp[a*e*g*m - c*d*f*(2*p + 3) - c*e*f*(m + 2*p + 3)*x, x], x], x] 
 /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 0] && (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. \(844\) vs. \(2(364)=728\).

Time = 6.30 (sec) , antiderivative size = 845, normalized size of antiderivative = 1.95

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (\frac {\left (\frac {\left (A a \,e^{2}+A c \,d^{2}+2 B a d e \right ) x}{3 c^{3} a}+\frac {2 A c d e +B a \,e^{2}+B c \,d^{2}}{3 c^{4}}\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 (3 A a \,e^{2}-4 A c \,d^{2}+5 B a d e \right ) x}{12 c^{2} a^{2}}-\frac {\left (A c d +7 B a e \right ) e}{12 c^{3} a}\right )}{\sqrt {\left (x^{2}-\frac {a}{c}\right ) \left (-c e x -c d \right )}}+\frac {2 \left (\frac {B \,e^{3}}{c^{2}}-\frac {4 A a c d \,e^{2}-4 A \,c^{2} d^{3}+7 B \,e^{3} a^{2}+5 B a c \,d^{2} e}{6 c^{2} a^{2}}+\frac {e^{2} \left (A c d +7 B a e \right )}{12 c^{2} a}+\frac {d \left (3 A a \,e^{2}-4 A c \,d^{2}+5 B a d e \right )}{6 c \,a^{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 {\left (3 A a \,e^{2}-4 A c \,d^{2}+5 B a d e \right ) e \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} c \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\)	\(845\)
default	\(\text {Expression too large to display}\)	\(3500\)

Input:

int((B*x+A)*(e*x+d)^(5/2)/(-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*a*e^2+A 
*c*d^2+2*B*a*d*e)/c^3/a*x+1/3*(2*A*c*d*e+B*a*e^2+B*c*d^2)/c^4)*(-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*(3*A*a*e^2-4*A*c 
*d^2+5*B*a*d*e)/c^2/a^2*x-1/12*(A*c*d+7*B*a*e)*e/c^3/a)/((x^2-a/c)*(-c*e*x 
-c*d))^(1/2)+2*(B*e^3/c^2-1/6/c^2*(4*A*a*c*d*e^2-4*A*c^2*d^3+7*B*a^2*e^3+5 
*B*a*c*d^2*e)/a^2+1/12/c^2*e^2*(A*c*d+7*B*a*e)/a+1/6/c*d*(3*A*a*e^2-4*A*c* 
d^2+5*B*a*d*e)/a^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)*Ellip 
ticF(((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*(3*A*a*e^2-4*A*c*d^2+5*B*a*d*e)*e/a^2/c*(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))*Ellipti 
cE(((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 = 0.10 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.42 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^{5/2}} \, dx=-\frac {{\left (4 \, A a^{2} c^{2} d^{3} - 5 \, B a^{3} c d^{2} e - 6 \, A a^{3} c d e^{2} + 15 \, B a^{4} e^{3} + {\left (4 \, A c^{4} d^{3} - 5 \, B a c^{3} d^{2} e - 6 \, A a c^{3} d e^{2} + 15 \, B a^{2} c^{2} e^{3}\right )} x^{4} - 2 \, {\left (4 \, A a c^{3} d^{3} - 5 \, B a^{2} c^{2} d^{2} e - 6 \, A a^{2} c^{2} d e^{2} + 15 \, B a^{3} c e^{3}\right )} x^{2}\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (4 \, A a^{2} c^{2} d^{2} e - 5 \, B a^{3} c d e^{2} - 3 \, A a^{3} c e^{3} + {\left (4 \, A c^{4} d^{2} e - 5 \, B a c^{3} d e^{2} - 3 \, A a c^{3} e^{3}\right )} x^{4} - 2 \, {\left (4 \, A a c^{3} d^{2} e - 5 \, B a^{2} c^{2} d e^{2} - 3 \, A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (2 \, B a^{2} c^{2} d^{2} e + 3 \, A a^{2} c^{2} d e^{2} - 5 \, B a^{3} c e^{3} - {\left (4 \, A c^{4} d^{2} e - 5 \, B a c^{3} d e^{2} - 3 \, A a c^{3} e^{3}\right )} x^{3} + {\left (A a c^{3} d e^{2} + 7 \, B a^{2} c^{2} e^{3}\right )} x^{2} + {\left (6 \, A a c^{3} d^{2} e - B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3}\right )} x\right )} \sqrt {-c x^{2} + a} \sqrt {e x + d}}{18 \, {\left (a^{2} c^{5} e x^{4} - 2 \, a^{3} c^{4} e x^{2} + a^{4} c^{3} e\right )}} \] Input:

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

Output:

-1/18*((4*A*a^2*c^2*d^3 - 5*B*a^3*c*d^2*e - 6*A*a^3*c*d*e^2 + 15*B*a^4*e^3 
 + (4*A*c^4*d^3 - 5*B*a*c^3*d^2*e - 6*A*a*c^3*d*e^2 + 15*B*a^2*c^2*e^3)*x^ 
4 - 2*(4*A*a*c^3*d^3 - 5*B*a^2*c^2*d^2*e - 6*A*a^2*c^2*d*e^2 + 15*B*a^3*c* 
e^3)*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^2*e 
 - 5*B*a^3*c*d*e^2 - 3*A*a^3*c*e^3 + (4*A*c^4*d^2*e - 5*B*a*c^3*d*e^2 - 3* 
A*a*c^3*e^3)*x^4 - 2*(4*A*a*c^3*d^2*e - 5*B*a^2*c^2*d*e^2 - 3*A*a^2*c^2*e^ 
3)*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), 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*(2*B*a^2*c^ 
2*d^2*e + 3*A*a^2*c^2*d*e^2 - 5*B*a^3*c*e^3 - (4*A*c^4*d^2*e - 5*B*a*c^3*d 
*e^2 - 3*A*a*c^3*e^3)*x^3 + (A*a*c^3*d*e^2 + 7*B*a^2*c^2*e^3)*x^2 + (6*A*a 
*c^3*d^2*e - B*a^2*c^2*d*e^2 - A*a^2*c^2*e^3)*x)*sqrt(-c*x^2 + a)*sqrt(e*x 
 + d))/(a^2*c^5*e*x^4 - 2*a^3*c^4*e*x^2 + a^4*c^3*e)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(a - c*x**2)*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**5*d*e**4 + 5*sqrt(a - c*x**2)*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**4*b*d**2*e**3 - 7*sq 
rt(a - c*x**2)*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**4*c*d**3*e**2 - 2*sqrt(a - c*x**2)*int(sqrt(d + e*x)/(sq 
rt(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**4*c*d*e**4*x**2 - 
5*sqrt(a - c*x**2)*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**3*b*c*d**4*e - 10*sqrt(a - c*x**2)*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...