\(\int \frac {(A+B x) \sqrt {d+e x}}{(a-c x^2)^2} \, dx\) [131]

Optimal result

Integrand size = 25, antiderivative size = 225 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^2} \, dx=\frac {(a B+A c x) \sqrt {d+e x}}{2 a c \left (a-c x^2\right )}-\frac {\left (2 A c d-a B e-\sqrt {a} A \sqrt {c} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{5/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (2 A c d-a B e+\sqrt {a} A \sqrt {c} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{5/4} \sqrt {\sqrt {c} d+\sqrt {a} e}} \] Output:

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

Mathematica [A] (verified)

Time = 2.18 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.27 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {c} (a B+A c x) \sqrt {d+e x}}{-a+c x^2}-\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \left (2 A c d-a B e+\sqrt {a} A \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {c} d+\sqrt {a} e}+\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \left (2 A c d-a B e-\sqrt {a} A \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {c} d-\sqrt {a} e}}{4 a^{3/2} c^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {685, 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.

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

\(\Big \downarrow \) 685

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 654

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

((a*B + A*c*x)*Sqrt[d + e*x])/(2*a*c*(a - c*x^2)) - (e*(-1/2*((A*Sqrt[c] - 
 (2*A*c*d - a*B*e)/(Sqrt[a]*e))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[ 
c]*d - Sqrt[a]*e]])/(c^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) - ((A*Sqrt[c] + 
(2*A*c*d - a*B*e)/(Sqrt[a]*e))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c 
]*d + Sqrt[a]*e]])/(2*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[a]*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 685

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(d + e*x)^m*(a + c*x^2)^(p + 1)*((a*g - c*f*x)/(2*a*c 
*(p + 1))), x] - Simp[1/(2*a*c*(p + 1))   Int[(d + e*x)^(m - 1)*(a + c*x^2) 
^(p + 1)*Simp[a*e*g*m - c*d*f*(2*p + 3) - c*e*f*(m + 2*p + 3)*x, x], x], x] 
 /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 0] && (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 = 1.51 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(2 e^{2} \left (\frac {\frac {A \left (e x +d \right )^{\frac {3}{2}}}{4 a e}-\frac {\left (A c d -B a e \right ) \sqrt {e x +d}}{4 a e c}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}}+\frac {\frac {\left (2 A c d -B a e -A \sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (-2 A c d +B a e -A \sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 a e}\right )\)	\(244\)
default	\(2 e^{2} \left (\frac {\frac {A \left (e x +d \right )^{\frac {3}{2}}}{4 a e}-\frac {\left (A c d -B a e \right ) \sqrt {e x +d}}{4 a e c}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}}+\frac {\frac {\left (2 A c d -B a e -A \sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (-2 A c d +B a e -A \sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 a e}\right )\)	\(244\)
pseudoelliptic	\(\frac {c e \left (A c d -\frac {B a e}{2}-\frac {A \sqrt {a c \,e^{2}}}{2}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-c \,x^{2}+a \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (c \left (A c d -\frac {B a e}{2}+\frac {A \sqrt {a c \,e^{2}}}{2}\right ) e \left (-c \,x^{2}+a \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\left (A c x +B a \right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {e x +d}\, \sqrt {a c \,e^{2}}\right )}{2 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a c \left (-c \,x^{2}+a \right )}\)	\(253\)

Input:

int((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3195 vs. \(2 (170) = 340\).

Time = 3.68 (sec) , antiderivative size = 3195, normalized size of antiderivative = 14.20 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

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

integrate((B*x+A)*(e*x+d)**(1/2)/(-c*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^2,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 449 vs. \(2 (170) = 340\).

Time = 0.20 (sec) , antiderivative size = 449, normalized size of antiderivative = 2.00 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^2} \, dx=\frac {{\left (2 \, A a c^{3} d^{2} e - B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3} - \sqrt {a c} A c d e {\left | a \right |} {\left | c \right |} {\left | e \right |} + \sqrt {a c} B a e^{2} {\left | a \right |} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{2} d + \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{2} e - \sqrt {a c} a c^{2} d\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a \right |} {\left | e \right |}} + \frac {{\left (2 \, A a c^{3} d^{2} e - B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3} + \sqrt {a c} A c d e {\left | a \right |} {\left | c \right |} {\left | e \right |} - \sqrt {a c} B a e^{2} {\left | a \right |} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{2} d - \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{2} e + \sqrt {a c} a c^{2} d\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a \right |} {\left | e \right |}} - \frac {{\left (e x + d\right )}^{\frac {3}{2}} A c e - \sqrt {e x + d} A c d e + \sqrt {e x + d} B a e^{2}}{2 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} - a e^{2}\right )} a c} \] Input:

integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^2,x, algorithm="giac")
 

Output:

1/4*(2*A*a*c^3*d^2*e - B*a^2*c^2*d*e^2 - A*a^2*c^2*e^3 - sqrt(a*c)*A*c*d*e 
*abs(a)*abs(c)*abs(e) + sqrt(a*c)*B*a*e^2*abs(a)*abs(c)*abs(e))*arctan(sqr 
t(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^2*e - sqrt(a*c)*a*c^2*d)*sqrt(-c^2*d - sqrt(a*c)*c* 
e)*abs(a)*abs(e)) + 1/4*(2*A*a*c^3*d^2*e - B*a^2*c^2*d*e^2 - A*a^2*c^2*e^3 
 + sqrt(a*c)*A*c*d*e*abs(a)*abs(c)*abs(e) - sqrt(a*c)*B*a*e^2*abs(a)*abs(c 
)*abs(e))*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^2*e + sqrt(a*c)*a*c^2*d)*sqrt(- 
c^2*d + sqrt(a*c)*c*e)*abs(a)*abs(e)) - 1/2*((e*x + d)^(3/2)*A*c*e - sqrt( 
e*x + d)*A*c*d*e + sqrt(e*x + d)*B*a*e^2)/(((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 = 7.14 (sec) , antiderivative size = 5062, normalized size of antiderivative = 22.50 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 1528, normalized size of antiderivative = 6.79 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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