\(\int \frac {(A+B x) (a-c x^2)^{3/2}}{(d+e x)^{3/2}} \, dx\) [269]

Optimal result

Integrand size = 27, antiderivative size = 443 \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {4 \sqrt {d+e x} \left (5 a B e^2-4 c d (8 B d-7 A e)+3 c e (8 B d-7 A e) x\right ) \sqrt {a-c x^2}}{35 e^4}+\frac {2 (8 B d-7 A e+B e x) \left (a-c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {8 \sqrt {a} \sqrt {c} \left (32 B c d^3-28 A c d^2 e-29 a B d e^2+21 a A e^3\right ) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{35 e^5 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}-\frac {8 \sqrt {a} \left (c d^2-a e^2\right ) \left (32 B c d^2-28 A c d e-5 a B e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{35 \sqrt {c} e^5 \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:

4/35*(e*x+d)^(1/2)*(5*B*a*e^2-4*c*d*(-7*A*e+8*B*d)+3*c*e*(-7*A*e+8*B*d)*x) 
*(-c*x^2+a)^(1/2)/e^4+2/7*(B*e*x-7*A*e+8*B*d)*(-c*x^2+a)^(3/2)/e^2/(e*x+d) 
^(1/2)+8/35*a^(1/2)*c^(1/2)*(21*A*a*e^3-28*A*c*d^2*e-29*B*a*d*e^2+32*B*c*d 
^3)*(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)-8/35*a^(1/2)*(-a*e^2 
+c*d^2)*(-28*A*c*d*e-5*B*a*e^2+32*B*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))/c^(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 = 26.23 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.39 \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \sqrt {a-c x^2} \left (-35 (B d-A e) \left (c d^2-a e^2\right )-\left (29 B c d^2-21 A c d e-15 a B e^2\right ) (d+e x)+c e (13 B d-7 A e) x (d+e x)-5 B c e^2 x^2 (d+e x)+\frac {4 \left (e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (32 B c d^3-28 A c d^2 e-29 a B d e^2+21 a A e^3\right ) \left (a-c x^2\right )+i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (32 B c d^3-28 A c d^2 e-29 a B d e^2+21 a A e^3\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )+i \sqrt {a} e \left (-\sqrt {c} d+\sqrt {a} e\right ) \left (B \left (-32 c d^2-24 \sqrt {a} \sqrt {c} d e+5 a e^2\right )+7 A \left (4 c d e+3 \sqrt {a} \sqrt {c} e^2\right )\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )\right )}{e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (a-c x^2\right )}\right )}{35 e^4 \sqrt {d+e x}} \] Input:

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

Output:

(2*Sqrt[a - c*x^2]*(-35*(B*d - A*e)*(c*d^2 - a*e^2) - (29*B*c*d^2 - 21*A*c 
*d*e - 15*a*B*e^2)*(d + e*x) + c*e*(13*B*d - 7*A*e)*x*(d + e*x) - 5*B*c*e^ 
2*x^2*(d + e*x) + (4*(e^2*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(32*B*c*d^3 - 28* 
A*c*d^2*e - 29*a*B*d*e^2 + 21*a*A*e^3)*(a - c*x^2) + I*Sqrt[c]*(Sqrt[c]*d 
- Sqrt[a]*e)*(32*B*c*d^3 - 28*A*c*d^2*e - 29*a*B*d*e^2 + 21*a*A*e^3)*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*Sq 
rt[a]*e*(-(Sqrt[c]*d) + Sqrt[a]*e)*(B*(-32*c*d^2 - 24*Sqrt[a]*Sqrt[c]*d*e 
+ 5*a*e^2) + 7*A*(4*c*d*e + 3*Sqrt[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^2*Sqrt[-d + (Sqrt[ 
a]*e)/Sqrt[c]]*(a - c*x^2))))/(35*e^4*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {681, 25, 682, 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.

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

\(\Big \downarrow \) 681

\(\displaystyle \frac {2 \left (a-c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}-\frac {6 \int -\frac {(a B e+c (8 B d-7 A e) x) \sqrt {a-c x^2}}{\sqrt {d+e x}}dx}{7 e^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {6 \int \frac {(a B e+c (8 B d-7 A e) x) \sqrt {a-c x^2}}{\sqrt {d+e x}}dx}{7 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {6 \left (\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (5 a B e^2+3 c e x (8 B d-7 A e)-4 c d (8 B d-7 A e)\right )}{15 e^2}-\frac {4 \int \frac {c \left (a e \left (8 B c d^2-7 A c e d-5 a B e^2\right )+c \left (32 B c d^3-28 A c e d^2-29 a B e^2 d+21 a A e^3\right ) x\right )}{2 \sqrt {d+e x} \sqrt {a-c x^2}}dx}{15 c e^2}\right )}{7 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6 \left (\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (5 a B e^2+3 c e x (8 B d-7 A e)-4 c d (8 B d-7 A e)\right )}{15 e^2}-\frac {2 \int \frac {a e \left (8 B c d^2-7 A c e d-5 a B e^2\right )+c \left (32 B c d^3-28 A c e d^2-29 a B e^2 d+21 a A e^3\right ) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{15 e^2}\right )}{7 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {6 \left (\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (5 a B e^2+3 c e x (8 B d-7 A e)-4 c d (8 B d-7 A e)\right )}{15 e^2}-\frac {2 \left (\frac {c \left (21 a A e^3-29 a B d e^2-28 A c d^2 e+32 B c d^3\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}-\frac {\left (c d^2-a e^2\right ) \left (-5 a B e^2-28 A c d e+32 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{15 e^2}\right )}{7 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {6 \left (\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (5 a B e^2+3 c e x (8 B d-7 A e)-4 c d (8 B d-7 A e)\right )}{15 e^2}-\frac {2 \left (\frac {c \sqrt {1-\frac {c x^2}{a}} \left (21 a A e^3-29 a B d e^2-28 A c d^2 e+32 B c d^3\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {\left (c d^2-a e^2\right ) \left (-5 a B e^2-28 A c d e+32 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{15 e^2}\right )}{7 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {6 \left (\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (5 a B e^2+3 c e x (8 B d-7 A e)-4 c d (8 B d-7 A e)\right )}{15 e^2}-\frac {2 \left (-\frac {\left (c d^2-a e^2\right ) \left (-5 a B e^2-28 A c d e+32 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (21 a A e^3-29 a B d e^2-28 A c d^2 e+32 B c d^3\right ) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{15 e^2}\right )}{7 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {6 \left (\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (5 a B e^2+3 c e x (8 B d-7 A e)-4 c d (8 B d-7 A e)\right )}{15 e^2}-\frac {2 \left (-\frac {\left (c d^2-a e^2\right ) \left (-5 a B e^2-28 A c d e+32 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (21 a A e^3-29 a B d e^2-28 A c d^2 e+32 B c d^3\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{15 e^2}\right )}{7 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {6 \left (\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (5 a B e^2+3 c e x (8 B d-7 A e)-4 c d (8 B d-7 A e)\right )}{15 e^2}-\frac {2 \left (-\frac {\sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \left (-5 a B e^2-28 A c d e+32 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (21 a A e^3-29 a B d e^2-28 A c d^2 e+32 B c d^3\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{15 e^2}\right )}{7 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {6 \left (\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (5 a B e^2+3 c e x (8 B d-7 A e)-4 c d (8 B d-7 A e)\right )}{15 e^2}-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (-5 a B e^2-28 A c d e+32 B c d^2\right ) \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (21 a A e^3-29 a B d e^2-28 A c d^2 e+32 B c d^3\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{15 e^2}\right )}{7 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {6 \left (\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (5 a B e^2+3 c e x (8 B d-7 A e)-4 c d (8 B d-7 A e)\right )}{15 e^2}-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (-5 a B e^2-28 A c d e+32 B c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (21 a A e^3-29 a B d e^2-28 A c d^2 e+32 B c d^3\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{15 e^2}\right )}{7 e^2}+\frac {2 \left (a-c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\)

Input:

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

Output:

(2*(8*B*d - 7*A*e + B*e*x)*(a - c*x^2)^(3/2))/(7*e^2*Sqrt[d + e*x]) + (6*( 
(2*Sqrt[d + e*x]*(5*a*B*e^2 - 4*c*d*(8*B*d - 7*A*e) + 3*c*e*(8*B*d - 7*A*e 
)*x)*Sqrt[a - c*x^2])/(15*e^2) - (2*((-2*Sqrt[a]*Sqrt[c]*(32*B*c*d^3 - 28* 
A*c*d^2*e - 29*a*B*d*e^2 + 21*a*A*e^3)*Sqrt[d + e*x]*Sqrt[1 - (c*x^2)/a]*E 
llipticE[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]*(c*d^2 - a*e^2)*(32*B*c*d^2 - 28*A*c*d*e - 5*a*B* 
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])))/(15*e^2)))/( 
7*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])
 

rule 682

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1136\) vs. \(2(373)=746\).

Time = 8.69 (sec) , antiderivative size = 1137, normalized size of antiderivative = 2.57

method	result	size
risch	\(\text {Expression too large to display}\)	\(1137\)
elliptic	\(\text {Expression too large to display}\)	\(1178\)
default	\(\text {Expression too large to display}\)	\(2439\)

Input:

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

Output:

2/35*(-5*B*c*e^2*x^2-7*A*c*e^2*x+13*B*c*d*e*x+21*A*c*d*e+15*B*a*e^2-29*B*c 
*d^2)*(e*x+d)^(1/2)/e^4*(-c*x^2+a)^(1/2)-1/35/e^4*((49*A*a*e^3-77*A*c*d^2* 
e-81*B*a*d*e^2+93*B*c*d^3)*(a*c)^(1/2)*2^(1/2)*((x+1/c*(a*c)^(1/2))*c/(a*c 
)^(1/2))^(1/2)*((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2)*(-2*(x-1/c*(a*c)^(1/2 
))*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(1/2*2^(1/2)*((x+1/c*(a*c)^(1/2))*c/(a*c)^(1/2))^(1/2),(- 
2/c*(a*c)^(1/2)/(d/e-1/c*(a*c)^(1/2)))^(1/2))-d/e*EllipticF(1/2*2^(1/2)*(( 
x+1/c*(a*c)^(1/2))*c/(a*c)^(1/2))^(1/2),(-2/c*(a*c)^(1/2)/(d/e-1/c*(a*c)^( 
1/2)))^(1/2)))-35*(A*a^2*e^5-2*A*a*c*d^2*e^3+A*c^2*d^4*e-B*a^2*d*e^4+2*B*a 
*c*d^3*e^2-B*c^2*d^5)/e*(-2*(-c*e*x^2+a*e)/(a*e^2-c*d^2)/((x+d/e)*(-c*e*x^ 
2+a*e))^(1/2)-d/(a*e^2-c*d^2)*(a*c)^(1/2)*2^(1/2)*((x+1/c*(a*c)^(1/2))*c/( 
a*c)^(1/2))^(1/2)*((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2)*(-2*(x-1/c*(a*c)^( 
1/2))*c/(a*c)^(1/2))^(1/2)/(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)*EllipticF(1/ 
2*2^(1/2)*((x+1/c*(a*c)^(1/2))*c/(a*c)^(1/2))^(1/2),(-2/c*(a*c)^(1/2)/(d/e 
-1/c*(a*c)^(1/2)))^(1/2))-e/(a*e^2-c*d^2)*(a*c)^(1/2)*2^(1/2)*((x+1/c*(a*c 
)^(1/2))*c/(a*c)^(1/2))^(1/2)*((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2)*(-2*(x 
-1/c*(a*c)^(1/2))*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(1/2*2^(1/2)*((x+1/c*(a*c)^(1/2))*c/(a*c)^ 
(1/2))^(1/2),(-2/c*(a*c)^(1/2)/(d/e-1/c*(a*c)^(1/2)))^(1/2))-d/e*EllipticF 
(1/2*2^(1/2)*((x+1/c*(a*c)^(1/2))*c/(a*c)^(1/2))^(1/2),(-2/c*(a*c)^(1/2...
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.14 \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (4 \, {\left (32 \, B c^{2} d^{5} - 28 \, A c^{2} d^{4} e - 53 \, B a c d^{3} e^{2} + 42 \, A a c d^{2} e^{3} + 15 \, B a^{2} d e^{4} + {\left (32 \, B c^{2} d^{4} e - 28 \, A c^{2} d^{3} e^{2} - 53 \, B a c d^{2} e^{3} + 42 \, A a c d e^{4} + 15 \, B a^{2} e^{5}\right )} x\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 12 \, {\left (32 \, B c^{2} d^{4} e - 28 \, A c^{2} d^{3} e^{2} - 29 \, B a c d^{2} e^{3} + 21 \, A a c d e^{4} + {\left (32 \, B c^{2} d^{3} e^{2} - 28 \, A c^{2} d^{2} e^{3} - 29 \, B a c d e^{4} + 21 \, A a c e^{5}\right )} x\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (5 \, B c^{2} e^{5} x^{3} + 64 \, B c^{2} d^{3} e^{2} - 56 \, A c^{2} d^{2} e^{3} - 50 \, B a c d e^{4} + 35 \, A a c e^{5} - {\left (8 \, B c^{2} d e^{4} - 7 \, A c^{2} e^{5}\right )} x^{2} + {\left (16 \, B c^{2} d^{2} e^{3} - 14 \, A c^{2} d e^{4} - 15 \, B a c e^{5}\right )} x\right )} \sqrt {-c x^{2} + a} \sqrt {e x + d}\right )}}{105 \, {\left (c e^{7} x + c d e^{6}\right )}} \] Input:

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

Output:

-2/105*(4*(32*B*c^2*d^5 - 28*A*c^2*d^4*e - 53*B*a*c*d^3*e^2 + 42*A*a*c*d^2 
*e^3 + 15*B*a^2*d*e^4 + (32*B*c^2*d^4*e - 28*A*c^2*d^3*e^2 - 53*B*a*c*d^2* 
e^3 + 42*A*a*c*d*e^4 + 15*B*a^2*e^5)*x)*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) + 12*(32*B*c^2*d^4*e - 28*A*c^2*d^3*e^2 - 29*B*a*c*d^2*e^3 + 21*A* 
a*c*d*e^4 + (32*B*c^2*d^3*e^2 - 28*A*c^2*d^2*e^3 - 29*B*a*c*d*e^4 + 21*A*a 
*c*e^5)*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), 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*(5*B*c^2 
*e^5*x^3 + 64*B*c^2*d^3*e^2 - 56*A*c^2*d^2*e^3 - 50*B*a*c*d*e^4 + 35*A*a*c 
*e^5 - (8*B*c^2*d*e^4 - 7*A*c^2*e^5)*x^2 + (16*B*c^2*d^2*e^3 - 14*A*c^2*d* 
e^4 - 15*B*a*c*e^5)*x)*sqrt(-c*x^2 + a)*sqrt(e*x + d))/(c*e^7*x + c*d*e^6)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*( - 10*sqrt(d + e*x)*sqrt(a - c*x**2)*a**2*b*e**3 - 7*sqrt(d + e*x)*sqr 
t(a - c*x**2)*a**2*c*d*e**2 + 8*sqrt(d + e*x)*sqrt(a - c*x**2)*a*b*c*d**2* 
e + 15*sqrt(d + e*x)*sqrt(a - c*x**2)*a*b*c*d*e**2*x + 14*sqrt(d + e*x)*sq 
rt(a - c*x**2)*a*c**2*d**2*e*x - 7*sqrt(d + e*x)*sqrt(a - c*x**2)*a*c**2*d 
*e**2*x**2 - 16*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 - 5*sqrt(d + e*x)*sqrt(a - c*x**2 
)*b*c**2*d*e**2*x**3 - 5*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d**2 
 + 2*a*d*e*x + a*e**2*x**2 - c*d**2*x**2 - 2*c*d*e*x**3 - c*e**2*x**4),x)* 
a**2*b*c*d*e**4 - 5*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d**2 + 2* 
a*d*e*x + a*e**2*x**2 - c*d**2*x**2 - 2*c*d*e*x**3 - c*e**2*x**4),x)*a**2* 
b*c*e**5*x - 28*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d**2 + 2*a*d* 
e*x + a*e**2*x**2 - c*d**2*x**2 - 2*c*d*e*x**3 - c*e**2*x**4),x)*a**2*c**2 
*d**2*e**3 - 28*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d**2 + 2*a*d* 
e*x + a*e**2*x**2 - c*d**2*x**2 - 2*c*d*e*x**3 - c*e**2*x**4),x)*a**2*c**2 
*d*e**4*x + 37*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d**2 + 2*a*d*e 
*x + a*e**2*x**2 - c*d**2*x**2 - 2*c*d*e*x**3 - c*e**2*x**4),x)*a*b*c**2*d 
**3*e**2 + 37*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d**2 + 2*a*d*e* 
x + a*e**2*x**2 - c*d**2*x**2 - 2*c*d*e*x**3 - c*e**2*x**4),x)*a*b*c**2*d* 
*2*e**3*x + 28*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d**2 + 2*a*d*e 
*x + a*e**2*x**2 - c*d**2*x**2 - 2*c*d*e*x**3 - c*e**2*x**4),x)*a*c**3*...