\(\int \frac {(A+B x) (d+e x)^{3/2}}{(a-c x^2)^3} \, dx\) [136]

Optimal result

Integrand size = 25, antiderivative size = 350 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx=\frac {\sqrt {d+e x} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} (a A e-3 (2 A c d-a B e) x)}{16 a^2 c \left (a-c x^2\right )}+\frac {3 \left (a B e \left (2 \sqrt {c} d-\sqrt {a} e\right )-A \left (4 c^{3/2} d^2-2 \sqrt {a} c d e-a \sqrt {c} e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{7/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}-\frac {3 \left (a B e \left (2 \sqrt {c} d+\sqrt {a} e\right )-A \left (4 c^{3/2} d^2+2 \sqrt {a} c d e-a \sqrt {c} e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{7/4} \sqrt {\sqrt {c} d+\sqrt {a} e}} \] Output:

1/4*(e*x+d)^(1/2)*(a*(A*e+B*d)+(A*c*d+B*a*e)*x)/a/c/(-c*x^2+a)^2-1/16*(e*x 
+d)^(1/2)*(a*A*e-3*(2*A*c*d-B*a*e)*x)/a^2/c/(-c*x^2+a)+3/32*(a*B*e*(2*c^(1 
/2)*d-a^(1/2)*e)-A*(4*c^(3/2)*d^2-2*a^(1/2)*c*d*e-a*c^(1/2)*e^2))*arctanh( 
c^(1/4)*(e*x+d)^(1/2)/(c^(1/2)*d-a^(1/2)*e)^(1/2))/a^(5/2)/c^(7/4)/(c^(1/2 
)*d-a^(1/2)*e)^(1/2)-3/32*(a*B*e*(2*c^(1/2)*d+a^(1/2)*e)-A*(4*c^(3/2)*d^2+ 
2*a^(1/2)*c*d*e-a*c^(1/2)*e^2))*arctanh(c^(1/4)*(e*x+d)^(1/2)/(c^(1/2)*d+a 
^(1/2)*e)^(1/2))/a^(5/2)/c^(7/4)/(c^(1/2)*d+a^(1/2)*e)^(1/2)
 

Mathematica [A] (verified)

Time = 3.60 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.01 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {c} \sqrt {d+e x} \left (6 A c^2 d x^3-a^2 (4 B d+3 A e+B e x)-a c x \left (10 A d+A e x+3 B e x^2\right )\right )}{\left (a-c x^2\right )^2}-\frac {3 \left (a B e \left (2 \sqrt {c} d+\sqrt {a} e\right )+A \left (-4 c^{3/2} d^2-2 \sqrt {a} c d e+a \sqrt {c} e^2\right )\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {3 \left (a B e \left (-2 \sqrt {c} d+\sqrt {a} e\right )+A \left (4 c^{3/2} d^2-2 \sqrt {a} c d e-a \sqrt {c} e^2\right )\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{32 a^{5/2} c^{3/2}} \] Input:

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

Output:

((-2*Sqrt[a]*Sqrt[c]*Sqrt[d + e*x]*(6*A*c^2*d*x^3 - a^2*(4*B*d + 3*A*e + B 
*e*x) - a*c*x*(10*A*d + A*e*x + 3*B*e*x^2)))/(a - c*x^2)^2 - (3*(a*B*e*(2* 
Sqrt[c]*d + Sqrt[a]*e) + A*(-4*c^(3/2)*d^2 - 2*Sqrt[a]*c*d*e + a*Sqrt[c]*e 
^2))*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + 
Sqrt[a]*e)])/Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e] - (3*(a*B*e*(-2*Sqrt[c]*d + 
Sqrt[a]*e) + A*(4*c^(3/2)*d^2 - 2*Sqrt[a]*c*d*e - a*Sqrt[c]*e^2))*ArcTan[( 
Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/ 
Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e])/(32*a^(5/2)*c^(3/2))
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.17, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {684, 27, 686, 27, 654, 25, 27, 1480, 221}

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.

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

\(\Big \downarrow \) 684

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 654

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

\(\displaystyle \frac {-\frac {3 e \left (\frac {\left (a B e \left (2 \sqrt {c} d-\sqrt {a} e\right )-A \left (-2 \sqrt {a} c d e-a \sqrt {c} e^2+4 c^{3/2} d^2\right )\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}-\frac {\left (a B e \left (\sqrt {a} e+2 \sqrt {c} d\right )-A \left (2 \sqrt {a} c d e-a \sqrt {c} e^2+4 c^{3/2} d^2\right )\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}\right )}{2 a}-\frac {\sqrt {d+e x} \left (a A e \left (c d^2-a e^2\right )-3 x \left (c d^2-a e^2\right ) (2 A c d-a B e)\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a c}+\frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {3 e \left (\frac {\left (a B e \left (\sqrt {a} e+2 \sqrt {c} d\right )-A \left (2 \sqrt {a} c d e-a \sqrt {c} e^2+4 c^{3/2} d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} c^{3/4} e \sqrt {\sqrt {a} e+\sqrt {c} d}}-\frac {\left (a B e \left (2 \sqrt {c} d-\sqrt {a} e\right )-A \left (-2 \sqrt {a} c d e-a \sqrt {c} e^2+4 c^{3/2} d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} c^{3/4} e \sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 a}-\frac {\sqrt {d+e x} \left (a A e \left (c d^2-a e^2\right )-3 x \left (c d^2-a e^2\right ) (2 A c d-a B e)\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a c}+\frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\)

Input:

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

Output:

(Sqrt[d + e*x]*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(4*a*c*(a - c*x^2)^2) 
+ (-1/2*(Sqrt[d + e*x]*(a*A*e*(c*d^2 - a*e^2) - 3*(2*A*c*d - a*B*e)*(c*d^2 
 - a*e^2)*x))/(a*(c*d^2 - a*e^2)*(a - c*x^2)) - (3*e*(-1/2*((a*B*e*(2*Sqrt 
[c]*d - Sqrt[a]*e) - A*(4*c^(3/2)*d^2 - 2*Sqrt[a]*c*d*e - a*Sqrt[c]*e^2))* 
ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c^( 
3/4)*e*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + ((a*B*e*(2*Sqrt[c]*d + Sqrt[a]*e) - 
A*(4*c^(3/2)*d^2 + 2*Sqrt[a]*c*d*e - a*Sqrt[c]*e^2))*ArcTanh[(c^(1/4)*Sqrt 
[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c^(3/4)*e*Sqrt[Sqrt[c] 
*d + Sqrt[a]*e])))/(2*a))/(8*a*c)
 

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 654

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), 
x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* 
x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
 

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 686

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

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 1.80 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.96

method	result	size
pseudoelliptic	\(\frac {-\frac {3 e \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (\left (2 A c d -B a e \right ) \sqrt {a c \,e^{2}}+c \left (-4 A c \,d^{2}+a e \left (A e +2 B d \right )\right )\right ) \left (-c \,x^{2}+a \right )^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32}+\frac {3 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-\frac {e \left (\left (-2 A c d +B a e \right ) \sqrt {a c \,e^{2}}+c \left (-4 A c \,d^{2}+a e \left (A e +2 B d \right )\right )\right ) \left (-c \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {e x +d}\, \sqrt {a c \,e^{2}}\, \left (-2 A \,c^{2} d \,x^{3}+\frac {10 x \left (A d +\frac {e x \left (3 B x +A \right )}{10}\right ) a c}{3}+\left (\frac {4 B d}{3}+e \left (\frac {B x}{3}+A \right )\right ) a^{2}\right )\right )}{16}}{a^{2} c \left (-c \,x^{2}+a \right )^{2} \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\)	\(337\)
default	\(2 e^{4} \left (\frac {-\frac {3 \left (2 A c d -B a e \right ) \left (e x +d \right )^{\frac {7}{2}}}{32 a^{2} e^{3}}+\frac {\left (A a \,e^{2}+18 A c \,d^{2}-9 B a d e \right ) \left (e x +d \right )^{\frac {5}{2}}}{32 a^{2} e^{3}}+\frac {\left (8 A a c d \,e^{2}-18 A \,c^{2} d^{3}+B \,e^{3} a^{2}+9 B a c \,d^{2} e \right ) \left (e x +d \right )^{\frac {3}{2}}}{32 a^{2} e^{3} c}+\frac {3 \left (a \,e^{2}-c \,d^{2}\right ) \left (A a \,e^{2}-2 A c \,d^{2}+B a d e \right ) \sqrt {e x +d}}{32 a^{2} e^{3} c}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}\right )^{2}}+\frac {\frac {3 \left (-A a c \,e^{2}+4 A \,c^{2} d^{2}-2 B a c d e -2 A \sqrt {a c \,e^{2}}\, c d +B \sqrt {a c \,e^{2}}\, a e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{64 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {3 \left (A a c \,e^{2}-4 A \,c^{2} d^{2}+2 B a c d e -2 A \sqrt {a c \,e^{2}}\, c d +B \sqrt {a c \,e^{2}}\, a e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{64 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{a^{2} e^{3}}\right )\)	\(426\)
derivativedivides	\(-2 e^{4} \left (-\frac {-\frac {3 \left (2 A c d -B a e \right ) \left (e x +d \right )^{\frac {7}{2}}}{32 a^{2} e^{3}}+\frac {\left (A a \,e^{2}+18 A c \,d^{2}-9 B a d e \right ) \left (e x +d \right )^{\frac {5}{2}}}{32 a^{2} e^{3}}+\frac {\left (8 A a c d \,e^{2}-18 A \,c^{2} d^{3}+B \,e^{3} a^{2}+9 B a c \,d^{2} e \right ) \left (e x +d \right )^{\frac {3}{2}}}{32 a^{2} e^{3} c}+\frac {3 \left (a \,e^{2}-c \,d^{2}\right ) \left (A a \,e^{2}-2 A c \,d^{2}+B a d e \right ) \sqrt {e x +d}}{32 a^{2} e^{3} c}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}\right )^{2}}-\frac {3 \left (\frac {\left (-A a c \,e^{2}+4 A \,c^{2} d^{2}-2 B a c d e -2 A \sqrt {a c \,e^{2}}\, c d +B \sqrt {a c \,e^{2}}\, a e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (A a c \,e^{2}-4 A \,c^{2} d^{2}+2 B a c d e -2 A \sqrt {a c \,e^{2}}\, c d +B \sqrt {a c \,e^{2}}\, a e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32 a^{2} e^{3}}\right )\)	\(427\)

Input:

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

Output:

3/16*(-1/2*e*((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*((2*A*c*d-B*a*e)*(a*c*e^2)^(1 
/2)+c*(-4*A*c*d^2+a*e*(A*e+2*B*d)))*(-c*x^2+a)^2*arctan(c*(e*x+d)^(1/2)/(( 
-c*d+(a*c*e^2)^(1/2))*c)^(1/2))+((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*(-1/2*e*( 
(-2*A*c*d+B*a*e)*(a*c*e^2)^(1/2)+c*(-4*A*c*d^2+a*e*(A*e+2*B*d)))*(-c*x^2+a 
)^2*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))+((c*d+(a*c*e^ 
2)^(1/2))*c)^(1/2)*(e*x+d)^(1/2)*(a*c*e^2)^(1/2)*(-2*A*c^2*d*x^3+10/3*x*(A 
*d+1/10*e*x*(3*B*x+A))*a*c+(4/3*B*d+e*(1/3*B*x+A))*a^2)))/((c*d+(a*c*e^2)^ 
(1/2))*c)^(1/2)/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)/a^2/c/(-c 
*x^2+a)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4176 vs. \(2 (280) = 560\).

Time = 2.71 (sec) , antiderivative size = 4176, normalized size of antiderivative = 11.93 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 731 vs. \(2 (280) = 560\).

Time = 0.27 (sec) , antiderivative size = 731, normalized size of antiderivative = 2.09 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-3/32*(2*B*a*c^3*d^2*e^2 + 2*A*a*c^3*d*e^3 - B*a^2*c^2*e^4 - sqrt(a*c)*B*a 
*c*d*e^2*abs(c)*abs(e) + (2*sqrt(a*c)*c^2*d^2*e - sqrt(a*c)*a*c*e^3)*A*abs 
(c)*abs(e) - (4*c^4*d^3*e - a*c^3*d*e^3)*A)*arctan(sqrt(e*x + d)/sqrt(-(a^ 
2*c^2*d + sqrt(a^4*c^4*d^2 - (a^2*c^2*d^2 - a^3*c*e^2)*a^2*c^2))/(a^2*c^2) 
))/((a^3*c^3*e - sqrt(a*c)*a^2*c^3*d)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(e)) 
 - 3/32*(2*B*a*c^3*d^2*e^2 + 2*A*a*c^3*d*e^3 - B*a^2*c^2*e^4 + sqrt(a*c)*B 
*a*c*d*e^2*abs(c)*abs(e) - (2*sqrt(a*c)*c^2*d^2*e - sqrt(a*c)*a*c*e^3)*A*a 
bs(c)*abs(e) - (4*c^4*d^3*e - a*c^3*d*e^3)*A)*arctan(sqrt(e*x + d)/sqrt(-( 
a^2*c^2*d - sqrt(a^4*c^4*d^2 - (a^2*c^2*d^2 - a^3*c*e^2)*a^2*c^2))/(a^2*c^ 
2)))/((a^3*c^3*e + sqrt(a*c)*a^2*c^3*d)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(e 
)) - 1/16*(6*(e*x + d)^(7/2)*A*c^2*d*e - 18*(e*x + d)^(5/2)*A*c^2*d^2*e + 
18*(e*x + d)^(3/2)*A*c^2*d^3*e - 6*sqrt(e*x + d)*A*c^2*d^4*e - 3*(e*x + d) 
^(7/2)*B*a*c*e^2 + 9*(e*x + d)^(5/2)*B*a*c*d*e^2 - 9*(e*x + d)^(3/2)*B*a*c 
*d^2*e^2 + 3*sqrt(e*x + d)*B*a*c*d^3*e^2 - (e*x + d)^(5/2)*A*a*c*e^3 - 8*( 
e*x + d)^(3/2)*A*a*c*d*e^3 + 9*sqrt(e*x + d)*A*a*c*d^2*e^3 - (e*x + d)^(3/ 
2)*B*a^2*e^4 - 3*sqrt(e*x + d)*B*a^2*d*e^4 - 3*sqrt(e*x + d)*A*a^2*e^5)/(( 
(e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 - a*e^2)^2*a^2*c)
 

Mupad [B] (verification not implemented)

Time = 9.46 (sec) , antiderivative size = 7239, normalized size of antiderivative = 20.68 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((3*(B*a*e^2 - 2*A*c*d*e)*(d + e*x)^(7/2))/(16*a^2) + ((d + e*x)^(5/2)*(A* 
a*e^3 - 9*B*a*d*e^2 + 18*A*c*d^2*e))/(16*a^2) + (3*(d + e*x)^(1/2)*(A*a^2* 
e^5 + B*a^2*d*e^4 + 2*A*c^2*d^4*e - 3*A*a*c*d^2*e^3 - B*a*c*d^3*e^2))/(16* 
a^2*c) + ((d + e*x)^(3/2)*(B*a^2*e^4 - 18*A*c^2*d^3*e + 8*A*a*c*d*e^3 + 9* 
B*a*c*d^2*e^2))/(16*a^2*c))/(c^2*(d + e*x)^4 + a^2*e^4 + c^2*d^4 + (6*c^2* 
d^2 - 2*a*c*e^2)*(d + e*x)^2 - (4*c^2*d^3 - 4*a*c*d*e^2)*(d + e*x) - 4*c^2 
*d*(d + e*x)^3 - 2*a*c*d^2*e^2) + atan(((((3*(4096*A*a^6*c^4*e^5 + 4096*B* 
a^6*c^4*d*e^4 - 8192*A*a^5*c^5*d^2*e^3))/(4096*a^6*c^2) - 64*a*c^4*d*e^2*( 
d + e*x)^(1/2)*(-(9*(B^2*a*e^5*(a^15*c^7)^(1/2) - 16*A^2*a^5*c^7*d^5 + A^2 
*c*e^5*(a^15*c^7)^(1/2) + 20*A^2*a^6*c^6*d^3*e^2 - 4*B^2*a^7*c^5*d^3*e^2 + 
 2*A*B*a^8*c^4*e^5 - 5*A^2*a^7*c^5*d*e^4 + 3*B^2*a^8*c^4*d*e^4 + 16*A*B*a^ 
6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) - 16*A*B*a^7*c^5*d^2*e^3))/(4 
096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2))*(-(9*(B^2*a*e^5*(a^15*c^7)^(1/2 
) - 16*A^2*a^5*c^7*d^5 + A^2*c*e^5*(a^15*c^7)^(1/2) + 20*A^2*a^6*c^6*d^3*e 
^2 - 4*B^2*a^7*c^5*d^3*e^2 + 2*A*B*a^8*c^4*e^5 - 5*A^2*a^7*c^5*d*e^4 + 3*B 
^2*a^8*c^4*d*e^4 + 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) - 
 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2) + (( 
d + e*x)^(1/2)*(9*B^2*a^3*e^6 + 144*A^2*c^3*d^4*e^2 + 9*A^2*a^2*c*e^6 - 36 
*A^2*a*c^2*d^2*e^4 + 36*B^2*a^2*c*d^2*e^4 - 144*A*B*a*c^2*d^3*e^3))/(64*a^ 
4))*(-(9*(B^2*a*e^5*(a^15*c^7)^(1/2) - 16*A^2*a^5*c^7*d^5 + A^2*c*e^5*(...
 

Reduce [B] (verification not implemented)

Time = 7.21 (sec) , antiderivative size = 3475, normalized size of antiderivative = 9.93 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(6*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*s 
qrt(sqrt(c)*sqrt(a)*e - c*d)))*a**3*b*e**3 - 18*sqrt(a)*sqrt(sqrt(c)*sqrt( 
a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)) 
)*a**3*c*d*e**2 - 12*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + 
e*x)*c)/(sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)))*a**2*b*c*d**2*e - 12*sqrt 
(a)*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(sqr 
t(c)*sqrt(a)*e - c*d)))*a**2*b*c*e**3*x**2 + 24*sqrt(a)*sqrt(sqrt(c)*sqrt( 
a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)) 
)*a**2*c**2*d**3 + 36*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + 
 e*x)*c)/(sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)))*a**2*c**2*d*e**2*x**2 + 
24*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*s 
qrt(sqrt(c)*sqrt(a)*e - c*d)))*a*b*c**2*d**2*e*x**2 + 6*sqrt(a)*sqrt(sqrt( 
c)*sqrt(a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e 
 - c*d)))*a*b*c**2*e**3*x**4 - 48*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e - c*d)*at 
an((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)))*a*c**3*d**3* 
x**2 - 18*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sq 
rt(c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)))*a*c**3*d*e**2*x**4 - 12*sqrt(a)*sqrt 
(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(sqrt(c)*sqr 
t(a)*e - c*d)))*b*c**3*d**2*e*x**4 + 24*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e - c 
*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)))*c**...