\(\int \frac {(A+B x) (d+e x)^{3/2}}{(a-c x^2)^2} \, dx\) [130]

Optimal result

Integrand size = 25, antiderivative size = 238 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^2} \, dx=\frac {\sqrt {d+e x} (a (B d+A e)+(A c d+a B e) x)}{2 a c \left (a-c x^2\right )}-\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \left (2 A c d-3 a B e+\sqrt {a} A \sqrt {c} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{7/4}}+\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} \left (2 A c d-3 a B e-\sqrt {a} A \sqrt {c} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{7/4}} \] Output:

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

Mathematica [A] (verified)

Time = 1.84 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.07 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {a} c \sqrt {d+e x} (a A e+A c d x+a B (d+e x))}{-a+c x^2}-\sqrt {-c d-\sqrt {a} \sqrt {c} e} \left (2 A c d-3 a B e-\sqrt {a} A \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )+\sqrt {-c d+\sqrt {a} \sqrt {c} e} \left (2 A c d-3 a B e+\sqrt {a} A \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{4 a^{3/2} c^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {684, 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.

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

\(\Big \downarrow \) 684

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 654

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(Sqrt[d + e*x]*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(2*a*c*(a - c*x^2)) - 
(e*((Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(2*A*c*d - 3*a*B*e + 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^(3/4)*e) - (Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(2*A*c*d - 3*a*B*e - Sqrt[a]*A*S 
qrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(2 
*Sqrt[a]*c^(3/4)*e)))/(2*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 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 = 1.66 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.28

method	result	size
pseudoelliptic	\(\frac {-\frac {\left (\left (A c d -3 B a e \right ) \sqrt {a c \,e^{2}}+c \left (-2 A c \,d^{2}+a e \left (A e +3 B d \right )\right )\right ) e \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-c \,x^{2}+a \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-\frac {\left (\left (-A c d +3 B a e \right ) \sqrt {a c \,e^{2}}+c \left (-2 A c \,d^{2}+a e \left (A e +3 B d \right )\right )\right ) e \left (-c \,x^{2}+a \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\left (c x A d +a \left (B d +e \left (B x +A \right )\right )\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {e x +d}\, \sqrt {a c \,e^{2}}\right )}{2 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a c \left (-c \,x^{2}+a \right )}\)	\(305\)
derivativedivides	\(2 e^{2} \left (\frac {\frac {\left (A c d +B a e \right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a c e}+\frac {A \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {e x +d}}{4 a c e}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}}+\frac {-\frac {\left (A a c \,e^{2}-2 A \,c^{2} d^{2}+3 B a c d e -A \sqrt {a c \,e^{2}}\, c d +3 B \sqrt {a c \,e^{2}}\, a e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-A a c \,e^{2}+2 A \,c^{2} d^{2}-3 B a c d e -A \sqrt {a c \,e^{2}}\, c d +3 B \sqrt {a c \,e^{2}}\, a e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 a e}\right )\)	\(322\)
default	\(2 e^{2} \left (\frac {\frac {\left (A c d +B a e \right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a c e}+\frac {A \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {e x +d}}{4 a c e}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}}+\frac {-\frac {\left (A a c \,e^{2}-2 A \,c^{2} d^{2}+3 B a c d e -A \sqrt {a c \,e^{2}}\, c d +3 B \sqrt {a c \,e^{2}}\, a e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-A a c \,e^{2}+2 A \,c^{2} d^{2}-3 B a c d e -A \sqrt {a c \,e^{2}}\, c d +3 B \sqrt {a c \,e^{2}}\, a e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 a e}\right )\)	\(322\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2327 vs. \(2 (183) = 366\).

Time = 0.33 (sec) , antiderivative size = 2327, normalized size of antiderivative = 9.78 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*((a*c^2*x^2 - a^2*c)*sqrt((4*A^2*c^2*d^3 - 12*A*B*a*c*d^2*e + 6*A*B*a^ 
2*e^3 + a^3*c^3*sqrt((36*A^2*B^2*c^2*d^2*e^4 - 12*(9*A*B^3*a*c + A^3*B*c^2 
)*d*e^5 + (81*B^4*a^2 + 18*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7)) + 3*(3*B 
^2*a^2 - A^2*a*c)*d*e^2)/(a^3*c^3))*log((24*A^3*B*c^3*d^3*e^2 - 4*(27*A^2* 
B^2*a*c^2 + A^4*c^3)*d^2*e^3 + 6*(27*A*B^3*a^2*c + A^3*B*a*c^2)*d*e^4 - (8 
1*B^4*a^3 - A^4*a*c^2)*e^5)*sqrt(e*x + d) + (6*A^2*B*a^2*c^3*d*e^3 - (9*A* 
B^2*a^3*c^2 + A^3*a^2*c^3)*e^4 - (2*A*a^3*c^6*d - 3*B*a^4*c^5*e)*sqrt((36* 
A^2*B^2*c^2*d^2*e^4 - 12*(9*A*B^3*a*c + A^3*B*c^2)*d*e^5 + (81*B^4*a^2 + 1 
8*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7)))*sqrt((4*A^2*c^2*d^3 - 12*A*B*a*c 
*d^2*e + 6*A*B*a^2*e^3 + a^3*c^3*sqrt((36*A^2*B^2*c^2*d^2*e^4 - 12*(9*A*B^ 
3*a*c + A^3*B*c^2)*d*e^5 + (81*B^4*a^2 + 18*A^2*B^2*a*c + A^4*c^2)*e^6)/(a 
^3*c^7)) + 3*(3*B^2*a^2 - A^2*a*c)*d*e^2)/(a^3*c^3))) - (a*c^2*x^2 - a^2*c 
)*sqrt((4*A^2*c^2*d^3 - 12*A*B*a*c*d^2*e + 6*A*B*a^2*e^3 + a^3*c^3*sqrt((3 
6*A^2*B^2*c^2*d^2*e^4 - 12*(9*A*B^3*a*c + A^3*B*c^2)*d*e^5 + (81*B^4*a^2 + 
 18*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7)) + 3*(3*B^2*a^2 - A^2*a*c)*d*e^2 
)/(a^3*c^3))*log((24*A^3*B*c^3*d^3*e^2 - 4*(27*A^2*B^2*a*c^2 + A^4*c^3)*d^ 
2*e^3 + 6*(27*A*B^3*a^2*c + A^3*B*a*c^2)*d*e^4 - (81*B^4*a^3 - A^4*a*c^2)* 
e^5)*sqrt(e*x + d) - (6*A^2*B*a^2*c^3*d*e^3 - (9*A*B^2*a^3*c^2 + A^3*a^2*c 
^3)*e^4 - (2*A*a^3*c^6*d - 3*B*a^4*c^5*e)*sqrt((36*A^2*B^2*c^2*d^2*e^4 - 1 
2*(9*A*B^3*a*c + A^3*B*c^2)*d*e^5 + (81*B^4*a^2 + 18*A^2*B^2*a*c + A^4*...
 

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]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (183) = 366\).

Time = 0.27 (sec) , antiderivative size = 546, normalized size of antiderivative = 2.29 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^2} \, dx=\frac {{\left (3 \, \sqrt {a c} B a^{2} c^{3} d^{2} e^{2} + \sqrt {a c} A a^{2} c^{3} d e^{3} - 3 \, \sqrt {a c} B a^{3} c^{2} e^{4} + {\left (a c^{3} d^{2} e - a^{2} c^{2} e^{3}\right )} A {\left | a \right |} {\left | c \right |} {\left | e \right |} - {\left (2 \, \sqrt {a c} a c^{4} d^{3} e - \sqrt {a c} a^{2} c^{3} d e^{3}\right )} A\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{2} d + \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{4} d - \sqrt {a c} a^{2} c^{3} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a \right |} {\left | e \right |}} - \frac {{\left (3 \, B a^{2} c^{3} d^{2} e^{2} + A a^{2} c^{3} d e^{3} - 3 \, B a^{3} c^{2} e^{4} - {\left (\sqrt {a c} c^{2} d^{2} e - \sqrt {a c} a c e^{3}\right )} A {\left | a \right |} {\left | c \right |} {\left | e \right |} - {\left (2 \, a c^{4} d^{3} e - a^{2} c^{3} d e^{3}\right )} A\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{2} d - \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{3} e + \sqrt {a c} a c^{3} d\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a \right |} {\left | e \right |}} - \frac {{\left (e x + d\right )}^{\frac {3}{2}} A c d e - \sqrt {e x + d} A c d^{2} e + {\left (e x + d\right )}^{\frac {3}{2}} B a e^{2} + \sqrt {e x + d} A a e^{3}}{2 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} - a e^{2}\right )} a c} \] Input:

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

Output:

1/4*(3*sqrt(a*c)*B*a^2*c^3*d^2*e^2 + sqrt(a*c)*A*a^2*c^3*d*e^3 - 3*sqrt(a* 
c)*B*a^3*c^2*e^4 + (a*c^3*d^2*e - a^2*c^2*e^3)*A*abs(a)*abs(c)*abs(e) - (2 
*sqrt(a*c)*a*c^4*d^3*e - sqrt(a*c)*a^2*c^3*d*e^3)*A)*arctan(sqrt(e*x + d)/ 
sqrt(-(a*c^2*d + sqrt(a^2*c^4*d^2 - (a*c^2*d^2 - a^2*c*e^2)*a*c^2))/(a*c^2 
)))/((a^2*c^4*d - sqrt(a*c)*a^2*c^3*e)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(a) 
*abs(e)) - 1/4*(3*B*a^2*c^3*d^2*e^2 + A*a^2*c^3*d*e^3 - 3*B*a^3*c^2*e^4 - 
(sqrt(a*c)*c^2*d^2*e - sqrt(a*c)*a*c*e^3)*A*abs(a)*abs(c)*abs(e) - (2*a*c^ 
4*d^3*e - a^2*c^3*d*e^3)*A)*arctan(sqrt(e*x + d)/sqrt(-(a*c^2*d - sqrt(a^2 
*c^4*d^2 - (a*c^2*d^2 - a^2*c*e^2)*a*c^2))/(a*c^2)))/((a^2*c^3*e + sqrt(a* 
c)*a*c^3*d)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(a)*abs(e)) - 1/2*((e*x + d)^( 
3/2)*A*c*d*e - sqrt(e*x + d)*A*c*d^2*e + (e*x + d)^(3/2)*B*a*e^2 + sqrt(e* 
x + d)*A*a*e^3)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 - a*e^2)*a*c)
 

Mupad [B] (verification not implemented)

Time = 6.49 (sec) , antiderivative size = 5212, normalized size of antiderivative = 21.90 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((64*A*a^4*c^4*e^5 - 64*A*a^3*c^5*d^2*e^3)/(8*a^3*c^2) - 64*a*c^4*d 
*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^6*d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2) + A 
^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2 
*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e - 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64 
*a^6*c^7))^(1/2))*((4*A^2*a^3*c^6*d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2) + A^2* 
c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^ 
5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e - 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^ 
6*c^7))^(1/2) + ((d + e*x)^(1/2)*(9*B^2*a^3*e^6 + 4*A^2*c^3*d^4*e^2 + A^2* 
a^2*c*e^6 - 3*A^2*a*c^2*d^2*e^4 + 9*B^2*a^2*c*d^2*e^4 - 12*A*B*a*c^2*d^3*e 
^3))/a^2)*((4*A^2*a^3*c^6*d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2) + A^2*c*e^3*(a 
^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d* 
e^2 - 12*A*B*a^4*c^5*d^2*e - 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c^7))^ 
(1/2)*1i - (((64*A*a^4*c^4*e^5 - 64*A*a^3*c^5*d^2*e^3)/(8*a^3*c^2) + 64*a* 
c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^6*d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2 
) + A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 
9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e - 6*A*B*c*d*e^2*(a^9*c^7)^(1/2) 
)/(64*a^6*c^7))^(1/2))*((4*A^2*a^3*c^6*d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2) + 
 A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B 
^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e - 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/( 
64*a^6*c^7))^(1/2) - ((d + e*x)^(1/2)*(9*B^2*a^3*e^6 + 4*A^2*c^3*d^4*e^...
 

Reduce [B] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 870, normalized size of antiderivative = 3.66 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(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*b*e - 4*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*c*d 
 - 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)))*b*c*e*x**2 + 4*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**2*d*x**2 - 2*sqrt(c)*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*e + 2*sqrt(c)*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*e*x**2 + 3*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e + c*d)*log( - sqr 
t(sqrt(c)*sqrt(a)*e + c*d) + sqrt(c)*sqrt(d + e*x))*a*b*e - 2*sqrt(a)*sqrt 
(sqrt(c)*sqrt(a)*e + c*d)*log( - sqrt(sqrt(c)*sqrt(a)*e + c*d) + sqrt(c)*s 
qrt(d + e*x))*a*c*d - 3*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e + c*d)*log( - sqrt( 
sqrt(c)*sqrt(a)*e + c*d) + sqrt(c)*sqrt(d + e*x))*b*c*e*x**2 + 2*sqrt(a)*s 
qrt(sqrt(c)*sqrt(a)*e + c*d)*log( - sqrt(sqrt(c)*sqrt(a)*e + c*d) + sqrt(c 
)*sqrt(d + e*x))*c**2*d*x**2 - 3*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e + c*d)*log 
(sqrt(sqrt(c)*sqrt(a)*e + c*d) + sqrt(c)*sqrt(d + e*x))*a*b*e + 2*sqrt(a)* 
sqrt(sqrt(c)*sqrt(a)*e + c*d)*log(sqrt(sqrt(c)*sqrt(a)*e + c*d) + sqrt(c)* 
sqrt(d + e*x))*a*c*d + 3*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e + c*d)*log(sqrt(sq 
rt(c)*sqrt(a)*e + c*d) + sqrt(c)*sqrt(d + e*x))*b*c*e*x**2 - 2*sqrt(a)*...