\(\int \frac {A+B x}{(d+e x)^{3/2} (a-c x^2)^2} \, dx\) [133]

Optimal result

Integrand size = 25, antiderivative size = 303 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^2} \, dx=-\frac {e \left (A c d^2-6 a B d e+5 a A e^2\right )}{2 a \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}+\frac {a (B d-A e)+(A c d-a B e) x}{2 a \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a-c x^2\right )}-\frac {\left (2 A c d+3 a B e-5 \sqrt {a} A \sqrt {c} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {\left (2 A c d+3 a B e+5 \sqrt {a} A \sqrt {c} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 3.31 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.20 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {a} \left (A c^2 d^2 x (d+e x)+a^2 e^2 (5 B d-4 A e+B e x)+a c \left (B d \left (d^2-d e x-6 e^2 x^2\right )+A e \left (-2 d^2-d e x+5 e^2 x^2\right )\right )\right )}{\left (c d^2-a e^2\right )^2 \sqrt {d+e x} \left (-a+c x^2\right )}+\frac {\left (2 A c d+3 a B e+5 \sqrt {a} A \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\left (\sqrt {c} d+\sqrt {a} e\right )^2 \sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {\left (2 A c d+3 a B e-5 \sqrt {a} A \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\left (\sqrt {c} d-\sqrt {a} e\right )^2 \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{4 a^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.25, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {686, 27, 655, 25, 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.

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

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 655

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 654

\(\displaystyle \frac {\frac {2 \int -\frac {e \left (A c d \left (c d^2-13 a e^2\right )+3 a B e \left (3 c d^2+a e^2\right )+c \left (A c d^2-6 a B e d+5 a A e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c d^2-a e^2}-\frac {2 e \left (5 a A e^2-6 a B d e+A c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{4 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{2 a \left (a-c x^2\right ) \sqrt {d+e x} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {2 \int \frac {e \left (A c d \left (c d^2-13 a e^2\right )+3 a B e \left (3 c d^2+a e^2\right )+c \left (A c d^2-6 a B e d+5 a A e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c d^2-a e^2}-\frac {2 e \left (5 a A e^2-6 a B d e+A c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{4 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{2 a \left (a-c x^2\right ) \sqrt {d+e x} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {2 e \int \frac {A c d \left (c d^2-13 a e^2\right )+3 a B e \left (3 c d^2+a e^2\right )+c \left (A c d^2-6 a B e d+5 a A e^2\right ) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c d^2-a e^2}-\frac {2 e \left (5 a A e^2-6 a B d e+A c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{4 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{2 a \left (a-c x^2\right ) \sqrt {d+e x} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(a*(B*d - A*e) + (A*c*d - a*B*e)*x)/(2*a*(c*d^2 - a*e^2)*Sqrt[d + e*x]*(a 
- c*x^2)) + ((-2*e*(A*c*d^2 - 6*a*B*d*e + 5*a*A*e^2))/((c*d^2 - a*e^2)*Sqr 
t[d + e*x]) - (2*e*(((Sqrt[c]*d + Sqrt[a]*e)^2*(2*A*c*d + 3*a*B*e - 5*Sqrt 
[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]* 
e]])/(2*Sqrt[a]*c^(1/4)*e*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) - ((Sqrt[c]*d - Sqr 
t[a]*e)^2*(2*A*c*d + 3*a*B*e + 5*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqr 
t[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c^(1/4)*e*Sqrt[Sqrt[c 
]*d + Sqrt[a]*e])))/(c*d^2 - a*e^2))/(4*a*(c*d^2 - a*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 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 655

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

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 = 1.92 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.42

method	result	size
pseudoelliptic	\(-\frac {2 \left (c e \left (\frac {\left (A c \,d^{2}+5 e \left (A e -\frac {6 B d}{5}\right ) a \right ) \sqrt {a c \,e^{2}}}{8}-\frac {A \,c^{2} d^{3}}{4}+a d e \left (A e -\frac {3 B d}{8}\right ) c -\frac {3 B \,e^{3} a^{2}}{8}\right ) \sqrt {e x +d}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-c \,x^{2}+a \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\left (c e \left (\frac {\left (-A c \,d^{2}-5 e \left (A e -\frac {6 B d}{5}\right ) a \right ) \sqrt {a c \,e^{2}}}{8}-\frac {A \,c^{2} d^{3}}{4}+a d e \left (A e -\frac {3 B d}{8}\right ) c -\frac {3 B \,e^{3} a^{2}}{8}\right ) \sqrt {e x +d}\, \left (-c \,x^{2}+a \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\left (-\frac {A \,d^{2} x \left (e x +d \right ) c^{2}}{4}+\frac {a \left (-\frac {B \,d^{3}}{2}+e \left (\frac {B x}{2}+A \right ) d^{2}+\frac {e^{2} x \left (6 B x +A \right ) d}{2}-\frac {5 A \,e^{3} x^{2}}{2}\right ) c}{2}+\left (-\frac {5 B d}{4}+e \left (-\frac {B x}{4}+A \right )\right ) e^{2} a^{2}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\right )}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {e x +d}\, \left (-c \,x^{2}+a \right ) \left (a \,e^{2}-c \,d^{2}\right )^{2} a}\)	\(431\)
derivativedivides	\(2 e^{2} \left (-\frac {A e -B d}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {e x +d}}-\frac {\frac {-\frac {c \left (A a \,e^{2}+A c \,d^{2}-2 B a d e \right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a e}+\frac {\left (3 A a c d \,e^{2}+A \,c^{2} d^{3}-B \,e^{3} a^{2}-3 B a c \,d^{2} e \right ) \sqrt {e x +d}}{4 a e}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}}+\frac {c \left (-\frac {\left (-8 A a c d \,e^{2}+2 A \,c^{2} d^{3}+3 B \,e^{3} a^{2}+3 B a c \,d^{2} e +5 A \sqrt {a c \,e^{2}}\, a \,e^{2}+A \sqrt {a c \,e^{2}}\, c \,d^{2}-6 B \sqrt {a c \,e^{2}}\, a d e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (8 A a c d \,e^{2}-2 A \,c^{2} d^{3}-3 B \,e^{3} a^{2}-3 B a c \,d^{2} e +5 A \sqrt {a c \,e^{2}}\, a \,e^{2}+A \sqrt {a c \,e^{2}}\, c \,d^{2}-6 B \sqrt {a c \,e^{2}}\, a d e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 a e}}{\left (a \,e^{2}-c \,d^{2}\right )^{2}}\right )\)	\(452\)
default	\(2 e^{2} \left (-\frac {A e -B d}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {e x +d}}-\frac {\frac {-\frac {c \left (A a \,e^{2}+A c \,d^{2}-2 B a d e \right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a e}+\frac {\left (3 A a c d \,e^{2}+A \,c^{2} d^{3}-B \,e^{3} a^{2}-3 B a c \,d^{2} e \right ) \sqrt {e x +d}}{4 a e}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}}+\frac {c \left (-\frac {\left (-8 A a c d \,e^{2}+2 A \,c^{2} d^{3}+3 B \,e^{3} a^{2}+3 B a c \,d^{2} e +5 A \sqrt {a c \,e^{2}}\, a \,e^{2}+A \sqrt {a c \,e^{2}}\, c \,d^{2}-6 B \sqrt {a c \,e^{2}}\, a d e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (8 A a c d \,e^{2}-2 A \,c^{2} d^{3}-3 B \,e^{3} a^{2}-3 B a c \,d^{2} e +5 A \sqrt {a c \,e^{2}}\, a \,e^{2}+A \sqrt {a c \,e^{2}}\, c \,d^{2}-6 B \sqrt {a c \,e^{2}}\, a d e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 a e}}{\left (a \,e^{2}-c \,d^{2}\right )^{2}}\right )\)	\(452\)

Input:

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

Output:

-2*(c*e*(1/8*(A*c*d^2+5*e*(A*e-6/5*B*d)*a)*(a*c*e^2)^(1/2)-1/4*A*c^2*d^3+a 
*d*e*(A*e-3/8*B*d)*c-3/8*B*e^3*a^2)*(e*x+d)^(1/2)*((c*d+(a*c*e^2)^(1/2))*c 
)^(1/2)*(-c*x^2+a)*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2) 
)+(c*e*(1/8*(-A*c*d^2-5*e*(A*e-6/5*B*d)*a)*(a*c*e^2)^(1/2)-1/4*A*c^2*d^3+a 
*d*e*(A*e-3/8*B*d)*c-3/8*B*e^3*a^2)*(e*x+d)^(1/2)*(-c*x^2+a)*arctanh(c*(e* 
x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))+(-1/4*A*d^2*x*(e*x+d)*c^2+1/2* 
a*(-1/2*B*d^3+e*(1/2*B*x+A)*d^2+1/2*e^2*x*(6*B*x+A)*d-5/2*A*e^3*x^2)*c+(-5 
/4*B*d+e*(-1/4*B*x+A))*e^2*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))/((-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)/(e*x+d)^(1/2)/(-c*x^2+a)/( 
a*e^2-c*d^2)^2/a
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1970 vs. \(2 (244) = 488\).

Time = 0.50 (sec) , antiderivative size = 1970, normalized size of antiderivative = 6.50 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(6*(a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5)^2*B*a*c*d*e^2*abs(c) - (a 
*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5)^2*(c^2*d^2*e + 5*a*c*e^3)*A*abs(c) 
 - (sqrt(a*c)*c^4*d^7*e - 15*sqrt(a*c)*a*c^3*d^5*e^3 + 27*sqrt(a*c)*a^2*c^ 
2*d^3*e^5 - 13*sqrt(a*c)*a^3*c*d*e^7)*A*abs(a*c^2*d^4*e - 2*a^2*c*d^2*e^3 
+ a^3*e^5)*abs(c) - 3*(3*sqrt(a*c)*a*c^3*d^6*e^2 - 5*sqrt(a*c)*a^2*c^2*d^4 
*e^4 + sqrt(a*c)*a^3*c*d^2*e^6 + sqrt(a*c)*a^4*e^8)*B*abs(a*c^2*d^4*e - 2* 
a^2*c*d^2*e^3 + a^3*e^5)*abs(c) + 2*(a*c^7*d^12*e - 8*a^2*c^6*d^10*e^3 + 2 
2*a^3*c^5*d^8*e^5 - 28*a^4*c^4*d^6*e^7 + 17*a^5*c^3*d^4*e^9 - 4*a^6*c^2*d^ 
2*e^11)*A*abs(c) + 3*(a^2*c^6*d^11*e^2 - 3*a^3*c^5*d^9*e^4 + 2*a^4*c^4*d^7 
*e^6 + 2*a^5*c^3*d^5*e^8 - 3*a^6*c^2*d^3*e^10 + a^7*c*d*e^12)*B*abs(c))*ar 
ctan(sqrt(e*x + d)/sqrt(-(a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4 + sq 
rt((a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)^2 - (a*c^3*d^6 - 3*a^2*c^ 
2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*(a*c^3*d^4 - 2*a^2*c^2*d^2*e^2 + a^ 
3*c*e^4)))/(a*c^3*d^4 - 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/((a^2*c^5*d^8*e - 
 4*a^3*c^4*d^6*e^3 + 6*a^4*c^3*d^4*e^5 - 4*a^5*c^2*d^2*e^7 + a^6*c*e^9 - s 
qrt(a*c)*a*c^5*d^9 + 4*sqrt(a*c)*a^2*c^4*d^7*e^2 - 6*sqrt(a*c)*a^3*c^3*d^5 
*e^4 + 4*sqrt(a*c)*a^4*c^2*d^3*e^6 - sqrt(a*c)*a^5*c*d*e^8)*sqrt(-c^2*d - 
sqrt(a*c)*c*e)*abs(a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5)) + 1/4*(6*(a*c 
^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5)^2*sqrt(a*c)*B*a*d*e^2*abs(c) - (a*c^ 
2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5)^2*(sqrt(a*c)*c*d^2*e + 5*sqrt(a*c)...
 

Mupad [B] (verification not implemented)

Time = 9.54 (sec) , antiderivative size = 19787, normalized size of antiderivative = 65.30 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(((d + e*x)*(B*a^2*e^4 - A*c^2*d^3*e - 11*A*a*c*d*e^3 + 11*B*a*c*d^2*e^2)) 
/(2*a*(a*e^2 - c*d^2)^2) - (2*(A*e^3 - B*d*e^2))/(a*e^2 - c*d^2) + (c*(d + 
 e*x)^2*(5*A*a*e^3 - 6*B*a*d*e^2 + A*c*d^2*e))/(2*a*(a*e^2 - c*d^2)^2))/(( 
a*e^2 - c*d^2)*(d + e*x)^(1/2) - c*(d + e*x)^(5/2) + 2*c*d*(d + e*x)^(3/2) 
) - atan((((d + e*x)^(1/2)*(800*A^2*a^12*c^4*e^20 + 288*B^2*a^13*c^3*e^20 
+ 128*A^2*a^3*c^13*d^18*e^2 - 1760*A^2*a^4*c^12*d^16*e^4 + 10240*A^2*a^5*c 
^11*d^14*e^6 - 30848*A^2*a^6*c^10*d^12*e^8 + 52480*A^2*a^7*c^9*d^10*e^10 - 
 51008*A^2*a^8*c^8*d^8*e^12 + 25600*A^2*a^9*c^7*d^6*e^14 - 3200*A^2*a^10*c 
^6*d^4*e^16 - 2432*A^2*a^11*c^5*d^2*e^18 + 288*B^2*a^5*c^11*d^16*e^4 - 576 
0*B^2*a^7*c^9*d^12*e^8 + 18432*B^2*a^8*c^8*d^10*e^10 - 25920*B^2*a^9*c^7*d 
^8*e^12 + 18432*B^2*a^10*c^6*d^6*e^14 - 5760*B^2*a^11*c^5*d^4*e^16 - 3456* 
A*B*a^12*c^4*d*e^19 + 384*A*B*a^4*c^12*d^17*e^3 - 3840*A*B*a^5*c^11*d^15*e 
^5 + 11520*A*B*a^6*c^10*d^13*e^7 - 9984*A*B*a^7*c^9*d^11*e^9 - 15360*A*B*a 
^8*c^8*d^9*e^11 + 43776*A*B*a^9*c^7*d^7*e^13 - 42240*A*B*a^10*c^6*d^5*e^15 
 + 19200*A*B*a^11*c^5*d^3*e^17) + (-(4*A^2*a^3*c^5*d^7 + 9*B^2*a^3*e^7*(a^ 
9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c 
^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 - 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 4 
5*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 + 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) 
 - 30*A*B*a^7*c*e^7 + 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) + 154*A^2*a*c^2*d^2 
*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e + 45*B^2*a*c^2*d^4*e^3*(a^9*c...
 

Reduce [B] (verification not implemented)

Time = 1.22 (sec) , antiderivative size = 3362, normalized size of antiderivative = 11.10 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 10*sqrt(a)*sqrt(d + e*x)*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*e**4 + 18*sqrt(a)* 
sqrt(d + e*x)*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*e**3 - 18*sqrt(a)*sqrt(d + e* 
x)*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**2*e**2 + 10*sqrt(a)*sqrt(d + e*x)*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)))*a**2*c**2*e**4*x**2 + 6*sqrt(a)*sqrt(d + e*x)*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*d**3*e - 18*sqrt(a)*sqrt(d + e*x)*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*d*e**3*x**2 + 4*sqrt(a)*sqrt(d + e*x)*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*c**3*d** 
4 + 18*sqrt(a)*sqrt(d + e*x)*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*c**3*d**2*e**2*x**2 - 6 
*sqrt(a)*sqrt(d + e*x)*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + e*x)*c 
)/(sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)))*b*c**3*d**3*e*x**2 - 4*sqrt(a)* 
sqrt(d + e*x)*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**4*d**4*x**2 + 6*sqrt(c)*sqrt(d + e*x) 
*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(sqr...