\(\int \frac {(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx\) [124]

Optimal result

Integrand size = 25, antiderivative size = 202 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx=-\frac {2 (B d+A e) \sqrt {d+e x}}{c}-\frac {2 B (d+e x)^{3/2}}{3 c}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}} \] Output:

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

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.28 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx=-\frac {2 c \sqrt {d+e x} (4 B d+3 A e+B e x)+\frac {3 \left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {-c d-\sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a}}+\frac {3 \left (-\sqrt {a} B+A \sqrt {c}\right ) \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )^2 \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{3 c^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {653, 25, 653, 25, 27, 654, 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.

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

\(\Big \downarrow \) 653

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 653

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 654

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(-2*B*(d + e*x)^(3/2))/(3*c) + (-2*(B*d + A*e)*Sqrt[d + e*x] + 2*(((Sqrt[a 
]*B - A*Sqrt[c])*(Sqrt[c]*d - Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e 
*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(2*Sqrt[a]*c^(3/4)) + ((Sqrt[a]*B + A*S 
qrt[c])*(Sqrt[c]*d + Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt 
[Sqrt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c^(3/4))))/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 653

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), 
 x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c   Int[(d + e*x)^(m 
- 1)*(Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x]/(a + c*x^2)), x], x] /; Fr 
eeQ[{a, c, d, e, f, g}, x] && FractionQ[m] && GtQ[m, 0]
 

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 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 = 271, normalized size of antiderivative = 1.34

method	result	size
pseudoelliptic	\(\frac {-\frac {2 \left (B e x +3 A e +4 B d \right ) \sqrt {e x +d}}{3}-\frac {\left (-A a c \,e^{3}-A \,c^{2} d^{2} e -2 B a d \,e^{2} c +2 A \sqrt {a c \,e^{2}}\, c d e +B \sqrt {a c \,e^{2}}\, a \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (A a c \,e^{3}+A \,c^{2} d^{2} e +2 B a d \,e^{2} c +2 A \sqrt {a c \,e^{2}}\, c d e +B \sqrt {a c \,e^{2}}\, a \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{c}\)	\(271\)
risch	\(-\frac {2 \left (B e x +3 A e +4 B d \right ) \sqrt {e x +d}}{3 c}-\frac {\left (-A a c \,e^{3}-A \,c^{2} d^{2} e -2 B a d \,e^{2} c +2 A \sqrt {a c \,e^{2}}\, c d e +B \sqrt {a c \,e^{2}}\, a \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (A a c \,e^{3}+A \,c^{2} d^{2} e +2 B a d \,e^{2} c +2 A \sqrt {a c \,e^{2}}\, c d e +B \sqrt {a c \,e^{2}}\, a \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\)	\(276\)
derivativedivides	\(-\frac {2 \left (\frac {B \left (e x +d \right )^{\frac {3}{2}}}{3}+A e \sqrt {e x +d}+B d \sqrt {e x +d}\right )}{c}+\frac {\left (A a c \,e^{3}+A \,c^{2} d^{2} e +2 B a d \,e^{2} c +2 A \sqrt {a c \,e^{2}}\, c d e +B \sqrt {a c \,e^{2}}\, a \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (-A a c \,e^{3}-A \,c^{2} d^{2} e -2 B a d \,e^{2} c +2 A \sqrt {a c \,e^{2}}\, c d e +B \sqrt {a c \,e^{2}}\, a \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\)	\(287\)
default	\(-\frac {2 \left (\frac {B \left (e x +d \right )^{\frac {3}{2}}}{3}+A e \sqrt {e x +d}+B d \sqrt {e x +d}\right )}{c}-\frac {\left (-A a c \,e^{3}-A \,c^{2} d^{2} e -2 B a d \,e^{2} c -2 A \sqrt {a c \,e^{2}}\, c d e -B \sqrt {a c \,e^{2}}\, a \,e^{2}-B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (A a c \,e^{3}+A \,c^{2} d^{2} e +2 B a d \,e^{2} c -2 A \sqrt {a c \,e^{2}}\, c d e -B \sqrt {a c \,e^{2}}\, a \,e^{2}-B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\)	\(291\)

Input:

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

Output:

1/c*(-2/3*(B*e*x+3*A*e+4*B*d)*(e*x+d)^(1/2)-(-A*a*c*e^3-A*c^2*d^2*e-2*B*a* 
d*e^2*c+2*A*(a*c*e^2)^(1/2)*c*d*e+B*(a*c*e^2)^(1/2)*a*e^2+B*(a*c*e^2)^(1/2 
)*c*d^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))+(A*a*c*e^3+A*c^2*d^2*e+2*B*a*d*e^ 
2*c+2*A*(a*c*e^2)^(1/2)*c*d*e+B*(a*c*e^2)^(1/2)*a*e^2+B*(a*c*e^2)^(1/2)*c* 
d^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. 4480 vs. \(2 (148) = 296\).

Time = 1.97 (sec) , antiderivative size = 4480, normalized size of antiderivative = 22.18 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (148) = 296\).

Time = 0.19 (sec) , antiderivative size = 550, normalized size of antiderivative = 2.72 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx=-\frac {{\left (2 \, \sqrt {a c} B a c^{3} d^{2} e^{2} - 2 \, \sqrt {a c} A a c^{3} d e^{3} - {\left (\sqrt {a c} a c d^{2} + \sqrt {a c} a^{2} e^{2}\right )} B c^{2} e^{2} + {\left (a c^{3} d^{2} e - a^{2} c^{2} e^{3}\right )} A {\left | c \right |} {\left | e \right |} + {\left (a c^{3} d^{3} - a^{2} c^{2} d e^{2}\right )} B {\left | c \right |} {\left | e \right |} + {\left (\sqrt {a c} c^{4} d^{3} e + \sqrt {a c} a c^{3} d e^{3}\right )} A\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{4} d + \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} - a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{4} d - \sqrt {a c} a c^{3} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | e \right |}} + \frac {{\left (2 \, \sqrt {a c} B a c^{3} d^{2} e^{2} - 2 \, \sqrt {a c} A a c^{3} d e^{3} - {\left (\sqrt {a c} a c d^{2} + \sqrt {a c} a^{2} e^{2}\right )} B c^{2} e^{2} - {\left (a c^{3} d^{2} e - a^{2} c^{2} e^{3}\right )} A {\left | c \right |} {\left | e \right |} - {\left (a c^{3} d^{3} - a^{2} c^{2} d e^{2}\right )} B {\left | c \right |} {\left | e \right |} + {\left (\sqrt {a c} c^{4} d^{3} e + \sqrt {a c} a c^{3} d e^{3}\right )} A\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{4} d - \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} - a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{4} d + \sqrt {a c} a c^{3} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | e \right |}} - \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B c^{2} + 3 \, \sqrt {e x + d} B c^{2} d + 3 \, \sqrt {e x + d} A c^{2} e\right )}}{3 \, c^{3}} \] Input:

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

Output:

-(2*sqrt(a*c)*B*a*c^3*d^2*e^2 - 2*sqrt(a*c)*A*a*c^3*d*e^3 - (sqrt(a*c)*a*c 
*d^2 + sqrt(a*c)*a^2*e^2)*B*c^2*e^2 + (a*c^3*d^2*e - a^2*c^2*e^3)*A*abs(c) 
*abs(e) + (a*c^3*d^3 - a^2*c^2*d*e^2)*B*abs(c)*abs(e) + (sqrt(a*c)*c^4*d^3 
*e + sqrt(a*c)*a*c^3*d*e^3)*A)*arctan(sqrt(e*x + d)/sqrt(-(c^4*d + sqrt(c^ 
8*d^2 - (c^4*d^2 - a*c^3*e^2)*c^4))/c^4))/((a*c^4*d - sqrt(a*c)*a*c^3*e)*s 
qrt(-c^2*d - sqrt(a*c)*c*e)*abs(e)) + (2*sqrt(a*c)*B*a*c^3*d^2*e^2 - 2*sqr 
t(a*c)*A*a*c^3*d*e^3 - (sqrt(a*c)*a*c*d^2 + sqrt(a*c)*a^2*e^2)*B*c^2*e^2 - 
 (a*c^3*d^2*e - a^2*c^2*e^3)*A*abs(c)*abs(e) - (a*c^3*d^3 - a^2*c^2*d*e^2) 
*B*abs(c)*abs(e) + (sqrt(a*c)*c^4*d^3*e + sqrt(a*c)*a*c^3*d*e^3)*A)*arctan 
(sqrt(e*x + d)/sqrt(-(c^4*d - sqrt(c^8*d^2 - (c^4*d^2 - a*c^3*e^2)*c^4))/c 
^4))/((a*c^4*d + sqrt(a*c)*a*c^3*e)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(e)) - 
 2/3*((e*x + d)^(3/2)*B*c^2 + 3*sqrt(e*x + d)*B*c^2*d + 3*sqrt(e*x + d)*A* 
c^2*e)/c^3
 

Mupad [B] (verification not implemented)

Time = 6.54 (sec) , antiderivative size = 7560, normalized size of antiderivative = 37.43 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

- ((2*A*e - 2*B*d)/c + (4*B*d)/c)*(d + e*x)^(1/2) - atan(((((8*(4*A*a^2*c^ 
4*e^5 - 4*A*a*c^5*d^2*e^3 - 4*B*a*c^5*d^3*e^2 + 4*B*a^2*c^4*d*e^4))/c^2 - 
64*a*c^4*d*e^2*(d + e*x)^(1/2)*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)^( 
1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*c^4*e^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) 
 + 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e 
^2 + A^2*a*c*e^3*(a^3*c^7)^(1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B 
*a^2*c^5*d^2*e + 6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2))*((B^ 
2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*c^ 
4*e^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 
3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e^2 + A^2*a*c*e^3*(a^3*c^7)^(1/2) + 
3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e + 6*A*B*a*c*d*e^2*(a 
^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2) + (d + e*x)^(1/2)*(16*B^2*a^3*e^6 + 16*A 
^2*c^3*d^4*e^2 + 16*A^2*a^2*c*e^6 + 96*A^2*a*c^2*d^2*e^4 + 16*B^2*a*c^2*d^ 
4*e^2 + 96*B^2*a^2*c*d^2*e^4 + 128*A*B*a^2*c*d*e^5 + 128*A*B*a*c^2*d^3*e^3 
))*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B 
*a^3*c^4*e^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^( 
1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e^2 + A^2*a*c*e^3*(a^3*c^7)^( 
1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e + 6*A*B*a*c*d 
*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2)*1i - (((8*(4*A*a^2*c^4*e^5 - 4*A* 
a*c^5*d^2*e^3 - 4*B*a*c^5*d^3*e^2 + 4*B*a^2*c^4*d*e^4))/c^2 + 64*a*c^4*...
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 540, normalized size of antiderivative = 2.67 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx=\frac {-6 \sqrt {a}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}}\right ) b e -6 \sqrt {a}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}}\right ) c d +6 \sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}}\right ) a e +6 \sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}}\right ) b d -3 \sqrt {a}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (-\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) b e -3 \sqrt {a}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (-\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) c d +3 \sqrt {a}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) b e +3 \sqrt {a}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) c d -3 \sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (-\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) a e -3 \sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (-\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) b d +3 \sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) a e +3 \sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) b d -12 \sqrt {e x +d}\, a c e -16 \sqrt {e x +d}\, b c d -4 \sqrt {e x +d}\, b c e x}{6 c^{2}} \] Input:

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

Output:

( - 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*e - 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)))*c*d 
+ 6*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*e + 6*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)))*b*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*e - 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))*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*e + 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))*c*d - 3*sqrt(c)*sqrt(sqr 
t(c)*sqrt(a)*e + c*d)*log( - sqrt(sqrt(c)*sqrt(a)*e + c*d) + sqrt(c)*sqrt( 
d + e*x))*a*e - 3*sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e + c*d)*log( - sqrt(sqrt(c 
)*sqrt(a)*e + c*d) + sqrt(c)*sqrt(d + e*x))*b*d + 3*sqrt(c)*sqrt(sqrt(c)*s 
qrt(a)*e + c*d)*log(sqrt(sqrt(c)*sqrt(a)*e + c*d) + sqrt(c)*sqrt(d + e*x)) 
*a*e + 3*sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e + c*d)*log(sqrt(sqrt(c)*sqrt(a)*e 
+ c*d) + sqrt(c)*sqrt(d + e*x))*b*d - 12*sqrt(d + e*x)*a*c*e - 16*sqrt(d + 
 e*x)*b*c*d - 4*sqrt(d + e*x)*b*c*e*x)/(6*c**2)