\(\int \frac {(A+B x) (a-c x^2)^{3/2}}{(d+e x)^{5/2}} \, dx\) [270]

Optimal result

Integrand size = 27, antiderivative size = 432 \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=-\frac {4 \left (9 a B e^2-4 c d (8 B d-5 A e)-c e (8 B d-5 A e) x\right ) \sqrt {a-c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (8 B d-5 A e+3 B e x) \left (a-c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}+\frac {8 \sqrt {a} \sqrt {c} \left (9 a B e^2-4 c d (8 B d-5 A e)\right ) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{15 e^5 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}+\frac {8 \sqrt {a} \sqrt {c} \left (32 B c d^3-20 A c d^2 e-17 a B d e^2+5 a A e^3\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{15 e^5 \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:

-4/15*(9*B*a*e^2-4*c*d*(-5*A*e+8*B*d)-c*e*(-5*A*e+8*B*d)*x)*(-c*x^2+a)^(1/ 
2)/e^4/(e*x+d)^(1/2)+2/15*(3*B*e*x-5*A*e+8*B*d)*(-c*x^2+a)^(3/2)/e^2/(e*x+ 
d)^(3/2)+8/15*a^(1/2)*c^(1/2)*(9*B*a*e^2-4*c*d*(-5*A*e+8*B*d))*(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))/e^5/(c^(1/2)*(e*x+d)/(c^(1/2 
)*d+a^(1/2)*e))^(1/2)/(-c*x^2+a)^(1/2)+8/15*a^(1/2)*c^(1/2)*(5*A*a*e^3-20* 
A*c*d^2*e-17*B*a*d*e^2+32*B*c*d^3)*(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))/e^5/(e*x+d)^(1/2)/(-c*x^ 
2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 25.82 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.37 \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {\sqrt {a-c x^2} \left (-\frac {2 \left (5 a A e^3+5 a B e^2 (2 d+3 e x)+5 A c e \left (8 d^2+10 d e x+e^2 x^2\right )-B c \left (64 d^3+80 d^2 e x+8 d e^2 x^2-3 e^3 x^3\right )\right )}{e^4 (d+e x)}-\frac {8 \left (e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (32 B c d^2-20 A c d e-9 a B e^2\right ) \left (-a+c x^2\right )+i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (-32 B c d^2+20 A c d e+9 a B e^2\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )+i \sqrt {a} \sqrt {c} e \left (-32 B c d^2+8 \sqrt {a} B \sqrt {c} d e+20 A c d e+9 a B e^2-5 \sqrt {a} A \sqrt {c} e^2\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )\right )}{e^6 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}\right )}{15 \sqrt {d+e x}} \] Input:

Integrate[((A + B*x)*(a - c*x^2)^(3/2))/(d + e*x)^(5/2),x]
 

Output:

(Sqrt[a - c*x^2]*((-2*(5*a*A*e^3 + 5*a*B*e^2*(2*d + 3*e*x) + 5*A*c*e*(8*d^ 
2 + 10*d*e*x + e^2*x^2) - B*c*(64*d^3 + 80*d^2*e*x + 8*d*e^2*x^2 - 3*e^3*x 
^3)))/(e^4*(d + e*x)) - (8*(e^2*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(32*B*c*d^2 
 - 20*A*c*d*e - 9*a*B*e^2)*(-a + c*x^2) + I*Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e 
)*(-32*B*c*d^2 + 20*A*c*d*e + 9*a*B*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)*El 
lipticE[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*Sqrt[a]*Sqrt[c]*e*(-32*B*c*d^2 
 + 8*Sqrt[a]*B*Sqrt[c]*d*e + 20*A*c*d*e + 9*a*B*e^2 - 5*Sqrt[a]*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)]))/(e^6*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c*x^2))))/(15*Sqrt[d + e*x 
])
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {681, 25, 681, 25, 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.

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

\(\Big \downarrow \) 681

\(\displaystyle \frac {2 \left (a-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}-\frac {2 \int -\frac {(3 a B e+c (8 B d-5 A e) x) \sqrt {a-c x^2}}{(d+e x)^{3/2}}dx}{5 e^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \int \frac {(3 a B e+c (8 B d-5 A e) x) \sqrt {a-c x^2}}{(d+e x)^{3/2}}dx}{5 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\)

\(\Big \downarrow \) 681

\(\displaystyle \frac {2 \left (-\frac {2 \int -\frac {c \left (a e (8 B d-5 A e)-\left (9 a B e^2-4 c d (8 B d-5 A e)\right ) x\right )}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{3 e^2}-\frac {2 \sqrt {a-c x^2} \left (9 a B e^2-c e x (8 B d-5 A e)-4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (\frac {2 \int \frac {c \left (a e (8 B d-5 A e)-\left (9 a B e^2-4 c d (8 B d-5 A e)\right ) x\right )}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{3 e^2}-\frac {2 \sqrt {a-c x^2} \left (9 a B e^2-c e x (8 B d-5 A e)-4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {2 c \int \frac {a e (8 B d-5 A e)-\left (9 a B e^2-4 c d (8 B d-5 A e)\right ) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{3 e^2}-\frac {2 \sqrt {a-c x^2} \left (9 a B e^2-c e x (8 B d-5 A e)-4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {2 \left (\frac {2 c \left (-\frac {\left (5 a A e^3-17 a B d e^2-20 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {\left (9 a B e^2-4 c d (8 B d-5 A e)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}\right )}{3 e^2}-\frac {2 \sqrt {a-c x^2} \left (9 a B e^2-c e x (8 B d-5 A e)-4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {2 \left (\frac {2 c \left (-\frac {\left (5 a A e^3-17 a B d e^2-20 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {\sqrt {1-\frac {c x^2}{a}} \left (9 a B e^2-4 c d (8 B d-5 A e)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}\right )}{3 e^2}-\frac {2 \sqrt {a-c x^2} \left (9 a B e^2-c e x (8 B d-5 A e)-4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {2 \left (\frac {2 c \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (9 a B e^2-4 c d (8 B d-5 A e)\right ) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {\left (5 a A e^3-17 a B d e^2-20 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{3 e^2}-\frac {2 \sqrt {a-c x^2} \left (9 a B e^2-c e x (8 B d-5 A e)-4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 \left (\frac {2 c \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (9 a B e^2-4 c d (8 B d-5 A e)\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {\left (5 a A e^3-17 a B d e^2-20 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{3 e^2}-\frac {2 \sqrt {a-c x^2} \left (9 a B e^2-c e x (8 B d-5 A e)-4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {2 \left (\frac {2 c \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (9 a B e^2-4 c d (8 B d-5 A e)\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {\sqrt {1-\frac {c x^2}{a}} \left (5 a A e^3-17 a B d e^2-20 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}\right )}{3 e^2}-\frac {2 \sqrt {a-c x^2} \left (9 a B e^2-c e x (8 B d-5 A e)-4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {2 \left (\frac {2 c \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (5 a A e^3-17 a B d e^2-20 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (9 a B e^2-4 c d (8 B d-5 A e)\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{3 e^2}-\frac {2 \sqrt {a-c x^2} \left (9 a B e^2-c e x (8 B d-5 A e)-4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 \left (\frac {2 c \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (5 a A e^3-17 a B d e^2-20 A c d^2 e+32 B c d^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (9 a B e^2-4 c d (8 B d-5 A e)\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{3 e^2}-\frac {2 \sqrt {a-c x^2} \left (9 a B e^2-c e x (8 B d-5 A e)-4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\)

Input:

Int[((A + B*x)*(a - c*x^2)^(3/2))/(d + e*x)^(5/2),x]
 

Output:

(2*(8*B*d - 5*A*e + 3*B*e*x)*(a - c*x^2)^(3/2))/(15*e^2*(d + e*x)^(3/2)) + 
 (2*((-2*(9*a*B*e^2 - 4*c*d*(8*B*d - 5*A*e) - c*e*(8*B*d - 5*A*e)*x)*Sqrt[ 
a - c*x^2])/(3*e^2*Sqrt[d + e*x]) + (2*c*((2*Sqrt[a]*(9*a*B*e^2 - 4*c*d*(8 
*B*d - 5*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)])/(Sqrt[c] 
*e*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[a - c*x^2]) + (2 
*Sqrt[a]*(32*B*c*d^3 - 20*A*c*d^2*e - 17*a*B*d*e^2 + 5*a*A*e^3)*Sqrt[(Sqrt 
[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[1 - (c*x^2)/a]*EllipticF[ArcS 
in[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])))/(3*e^2)))/(5*e^2)
 

Defintions of rubi rules used

rule 25

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

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 681

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) 
 + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ 
(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim 
p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] 
, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || 
EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ[m + 2 
*p + 1, 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. \(988\) vs. \(2(362)=724\).

Time = 15.36 (sec) , antiderivative size = 989, normalized size of antiderivative = 2.29

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (-\frac {2 \left (A a \,e^{3}-A c \,d^{2} e -B a d \,e^{2}+B c \,d^{3}\right ) \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{3 e^{6} \left (x +\frac {d}{e}\right )^{2}}-\frac {2 \left (-c e \,x^{2}+a e \right ) \left (8 A c d e +3 B a \,e^{2}-11 B c \,d^{2}\right )}{3 e^{5} \sqrt {\left (x +\frac {d}{e}\right ) \left (-c e \,x^{2}+a e \right )}}-\frac {2 B c x \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{5 e^{3}}-\frac {2 \left (\frac {c^{2} \left (A e -2 B d \right )}{e^{3}}-\frac {4 B \,c^{2} d}{5 e^{3}}\right ) \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{3 c e}+\frac {2 \left (-\frac {c \left (2 A a \,e^{3}-3 A c \,d^{2} e -4 B a d \,e^{2}+4 B c \,d^{3}\right )}{e^{5}}+\frac {\left (A a \,e^{3}-A c \,d^{2} e -B a d \,e^{2}+B c \,d^{3}\right ) c}{3 e^{5}}-\frac {\left (8 A c d e +3 B a \,e^{2}-11 B c \,d^{2}\right ) c d}{3 e^{5}}+\frac {2 B c a d}{5 e^{3}}+\frac {\left (\frac {c^{2} \left (A e -2 B d \right )}{e^{3}}-\frac {4 B \,c^{2} d}{5 e^{3}}\right ) a}{3 c}\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 {2 \left (-\frac {c \left (2 A c d e +2 B a \,e^{2}-3 B c \,d^{2}\right )}{e^{4}}-\frac {\left (8 A c d e +3 B a \,e^{2}-11 B c \,d^{2}\right ) c}{3 e^{4}}+\frac {3 B c a}{5 e^{2}}-\frac {2 \left (\frac {c^{2} \left (A e -2 B d \right )}{e^{3}}-\frac {4 B \,c^{2} d}{5 e^{3}}\right ) d}{3 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 )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\)	\(989\)
risch	\(\text {Expression too large to display}\)	\(2172\)
default	\(\text {Expression too large to display}\)	\(3669\)

Input:

int((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(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)*(-2/3*(A*a*e^3-A 
*c*d^2*e-B*a*d*e^2+B*c*d^3)/e^6*(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e) 
^2-2/3*(-c*e*x^2+a*e)*(8*A*c*d*e+3*B*a*e^2-11*B*c*d^2)/e^5/((x+d/e)*(-c*e* 
x^2+a*e))^(1/2)-2/5*B*c/e^3*x*(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)-2/3*(c^2/ 
e^3*(A*e-2*B*d)-4/5*B*c^2/e^3*d)/c/e*(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)+2* 
(-c*(2*A*a*e^3-3*A*c*d^2*e-4*B*a*d*e^2+4*B*c*d^3)/e^5+1/3*(A*a*e^3-A*c*d^2 
*e-B*a*d*e^2+B*c*d^3)*c/e^5-1/3*(8*A*c*d*e+3*B*a*e^2-11*B*c*d^2)*c/e^5*d+2 
/5*B*c/e^3*a*d+1/3*(c^2/e^3*(A*e-2*B*d)-4/5*B*c^2/e^3*d)/c*a)*(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))+2*(-c/e^ 
4*(2*A*c*d*e+2*B*a*e^2-3*B*c*d^2)-1/3*(8*A*c*d*e+3*B*a*e^2-11*B*c*d^2)*c/e 
^4+3/5*B*c/e^2*a-2/3*(c^2/e^3*(A*e-2*B*d)-4/5*B*c^2/e^3*d)/e*d)*(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 = 513, normalized size of antiderivative = 1.19 \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (4 \, {\left (32 \, B c d^{5} - 20 \, A c d^{4} e - 33 \, B a d^{3} e^{2} + 15 \, A a d^{2} e^{3} + {\left (32 \, B c d^{3} e^{2} - 20 \, A c d^{2} e^{3} - 33 \, B a d e^{4} + 15 \, A a e^{5}\right )} x^{2} + 2 \, {\left (32 \, B c d^{4} e - 20 \, A c d^{3} e^{2} - 33 \, B a d^{2} e^{3} + 15 \, A a d e^{4}\right )} x\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 12 \, {\left (32 \, B c d^{4} e - 20 \, A c d^{3} e^{2} - 9 \, B a d^{2} e^{3} + {\left (32 \, B c d^{2} e^{3} - 20 \, A c d e^{4} - 9 \, B a e^{5}\right )} x^{2} + 2 \, {\left (32 \, B c d^{3} e^{2} - 20 \, A c d^{2} e^{3} - 9 \, B a d e^{4}\right )} x\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (3 \, B c e^{5} x^{3} - 64 \, B c d^{3} e^{2} + 40 \, A c d^{2} e^{3} + 10 \, B a d e^{4} + 5 \, A a e^{5} - {\left (8 \, B c d e^{4} - 5 \, A c e^{5}\right )} x^{2} - 5 \, {\left (16 \, B c d^{2} e^{3} - 10 \, A c d e^{4} - 3 \, B a e^{5}\right )} x\right )} \sqrt {-c x^{2} + a} \sqrt {e x + d}\right )}}{45 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \] Input:

integrate((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(5/2),x, algorithm="fricas")
 

Output:

2/45*(4*(32*B*c*d^5 - 20*A*c*d^4*e - 33*B*a*d^3*e^2 + 15*A*a*d^2*e^3 + (32 
*B*c*d^3*e^2 - 20*A*c*d^2*e^3 - 33*B*a*d*e^4 + 15*A*a*e^5)*x^2 + 2*(32*B*c 
*d^4*e - 20*A*c*d^3*e^2 - 33*B*a*d^2*e^3 + 15*A*a*d*e^4)*x)*sqrt(-c*e)*wei 
erstrassPInverse(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) + 12*(32*B*c*d^4*e - 20*A*c*d^3*e^2 - 9*B*a*d^ 
2*e^3 + (32*B*c*d^2*e^3 - 20*A*c*d*e^4 - 9*B*a*e^5)*x^2 + 2*(32*B*c*d^3*e^ 
2 - 20*A*c*d^2*e^3 - 9*B*a*d*e^4)*x)*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), weierstrassPInvers 
e(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*(3*B*c*e^5*x^3 - 64*B*c*d^3*e^2 + 40*A*c*d^2*e^3 + 10*B* 
a*d*e^4 + 5*A*a*e^5 - (8*B*c*d*e^4 - 5*A*c*e^5)*x^2 - 5*(16*B*c*d^2*e^3 - 
10*A*c*d*e^4 - 3*B*a*e^5)*x)*sqrt(-c*x^2 + a)*sqrt(e*x + d))/(e^8*x^2 + 2* 
d*e^7*x + d^2*e^6)
 

Sympy [F]

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

integrate((B*x+A)*(-c*x**2+a)**(3/2)/(e*x+d)**(5/2),x)
 

Output:

Integral((A + B*x)*(a - c*x**2)**(3/2)/(d + e*x)**(5/2), x)
 

Maxima [F]

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

integrate((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(5/2),x, algorithm="maxima")
 

Output:

integrate((-c*x^2 + a)^(3/2)*(B*x + A)/(e*x + d)^(5/2), x)
 

Giac [F]

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

integrate((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(5/2),x, algorithm="giac")
 

Output:

integrate((-c*x^2 + a)^(3/2)*(B*x + A)/(e*x + d)^(5/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int \frac {{\left (a-c\,x^2\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \] Input:

int(((a - c*x^2)^(3/2)*(A + B*x))/(d + e*x)^(5/2),x)
 

Output:

int(((a - c*x^2)^(3/2)*(A + B*x))/(d + e*x)^(5/2), x)
 

Reduce [F]

\[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\text {too large to display} \] Input:

int((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(5/2),x)
 

Output:

(2*( - 18*sqrt(d + e*x)*sqrt(a - c*x**2)*a**2*b*e**3 - 25*sqrt(d + e*x)*sq 
rt(a - c*x**2)*a**2*c*d*e**2 + 40*sqrt(d + e*x)*sqrt(a - c*x**2)*a*b*c*d** 
2*e + 21*sqrt(d + e*x)*sqrt(a - c*x**2)*a*b*c*d*e**2*x + 30*sqrt(d + e*x)* 
sqrt(a - c*x**2)*a*c**2*d**2*e*x - 5*sqrt(d + e*x)*sqrt(a - c*x**2)*a*c**2 
*d*e**2*x**2 - 48*sqrt(d + e*x)*sqrt(a - c*x**2)*b*c**2*d**3*x + 8*sqrt(d 
+ e*x)*sqrt(a - c*x**2)*b*c**2*d**2*e*x**2 - 3*sqrt(d + e*x)*sqrt(a - c*x* 
*2)*b*c**2*d*e**2*x**3 + 9*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d* 
*3 + 3*a*d**2*e*x + 3*a*d*e**2*x**2 + a*e**3*x**3 - c*d**3*x**2 - 3*c*d**2 
*e*x**3 - 3*c*d*e**2*x**4 - c*e**3*x**5),x)*a**2*b*c*d**2*e**4 + 18*int((s 
qrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d**3 + 3*a*d**2*e*x + 3*a*d*e**2*x* 
*2 + a*e**3*x**3 - c*d**3*x**2 - 3*c*d**2*e*x**3 - 3*c*d*e**2*x**4 - c*e** 
3*x**5),x)*a**2*b*c*d*e**5*x + 9*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2) 
/(a*d**3 + 3*a*d**2*e*x + 3*a*d*e**2*x**2 + a*e**3*x**3 - c*d**3*x**2 - 3* 
c*d**2*e*x**3 - 3*c*d*e**2*x**4 - c*e**3*x**5),x)*a**2*b*c*e**6*x**2 + 27* 
int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d**3 + 3*a*d**2*e*x + 3*a*d*e 
**2*x**2 + a*e**3*x**3 - c*d**3*x**2 - 3*c*d**2*e*x**3 - 3*c*d*e**2*x**4 - 
 c*e**3*x**5),x)*a*b*c**2*d**4*e**2 + 54*int((sqrt(d + e*x)*sqrt(a - c*x** 
2)*x**2)/(a*d**3 + 3*a*d**2*e*x + 3*a*d*e**2*x**2 + a*e**3*x**3 - c*d**3*x 
**2 - 3*c*d**2*e*x**3 - 3*c*d*e**2*x**4 - c*e**3*x**5),x)*a*b*c**2*d**3*e* 
*3*x + 27*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d**3 + 3*a*d**2*...