\(\int \frac {(A+B x) (d+e x)^{5/2}}{(a-c x^2)^2} \, dx\) [129]

Optimal result

Integrand size = 25, antiderivative size = 280 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\frac {2 B e^2 \sqrt {d+e x}}{c^2}+\frac {\sqrt {d+e x} \left (a \left (B c d^2+2 A c d e+a B e^2\right )+c \left (A c d^2+2 a B d e+a A e^2\right ) x\right )}{2 a c^2 \left (a-c x^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (2 A c d-5 a B e+3 \sqrt {a} A \sqrt {c} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \left (2 A c d-5 a B e-3 \sqrt {a} A \sqrt {c} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}} \] Output:

2*B*e^2*(e*x+d)^(1/2)/c^2+1/2*(e*x+d)^(1/2)*(a*(2*A*c*d*e+B*a*e^2+B*c*d^2) 
+c*(A*a*e^2+A*c*d^2+2*B*a*d*e)*x)/a/c^2/(-c*x^2+a)-1/4*(c^(1/2)*d-a^(1/2)* 
e)^(3/2)*(2*A*c*d-5*B*a*e+3*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^(9/4)+1/4*(c^(1/2)*d+a^(1/2)*e 
)^(3/2)*(2*A*c*d-5*B*a*e-3*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^(9/4)
 

Mathematica [A] (verified)

Time = 1.95 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.15 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {d+e x} \left (5 a^2 B e^2+A c^2 d^2 x+a c \left (A e (2 d+e x)+B \left (d^2+2 d e x-4 e^2 x^2\right )\right )\right )}{-a+c x^2}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^2 \left (2 A c d-5 a B e-3 \sqrt {a} A \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (2 A c d-5 a B e+3 \sqrt {a} A \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{4 a^{3/2} c^2} \] Input:

Integrate[((A + B*x)*(d + e*x)^(5/2))/(a - c*x^2)^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {684, 27, 653, 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.

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

\(\Big \downarrow \) 684

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 653

\(\displaystyle \frac {\frac {2 e \sqrt {d+e x} (5 a B e+A c d)}{c}-\frac {\int -\frac {2 A c d \left (c d^2-2 a e^2\right )-5 a B e \left (c d^2+a e^2\right )+c e \left (A c d^2-10 a B e d-3 a A e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}}{4 a c}+\frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{2 a c \left (a-c x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {2 A c d \left (c d^2-2 a e^2\right )-5 a B e \left (c d^2+a e^2\right )+c e \left (A c d^2-10 a B e d-3 a A e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}+\frac {2 e \sqrt {d+e x} (5 a B e+A c d)}{c}}{4 a c}+\frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{2 a c \left (a-c x^2\right )}\)

\(\Big \downarrow \) 654

\(\displaystyle \frac {\frac {2 \int -\frac {e \left ((A c d+5 a B e) \left (c d^2-a e^2\right )+c \left (A c d^2-10 a B e d-3 a A e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c}+\frac {2 e \sqrt {d+e x} (5 a B e+A c d)}{c}}{4 a c}+\frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{2 a c \left (a-c x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {2 e \sqrt {d+e x} (5 a B e+A c d)}{c}-\frac {2 \int \frac {e \left ((A c d+5 a B e) \left (c d^2-a e^2\right )+c \left (A c d^2-10 a B e d-3 a A e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c}}{4 a c}+\frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{2 a c \left (a-c x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 e \sqrt {d+e x} (5 a B e+A c d)}{c}-\frac {2 e \int \frac {(A c d+5 a B e) \left (c d^2-a e^2\right )+c \left (A c d^2-10 a B e d-3 a A e^2\right ) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c}}{4 a c}+\frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{2 a c \left (a-c x^2\right )}\)

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

Int[((A + B*x)*(d + e*x)^(5/2))/(a - c*x^2)^2,x]
 

Output:

((d + e*x)^(3/2)*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(2*a*c*(a - c*x^2)) 
+ ((2*e*(A*c*d + 5*a*B*e)*Sqrt[d + e*x])/c - (2*e*(((Sqrt[c]*d - Sqrt[a]*e 
)^(3/2)*(2*A*c*d - 5*a*B*e + 3*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[c]* 
d + Sqrt[a]*e)^(3/2)*(2*A*c*d - 5*a*B*e - 3*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) 
))/c)/(4*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 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 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 = 1.92 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.36

method	result	size
pseudoelliptic	\(\frac {-c e \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (\frac {\left (A c \,d^{2}-3 \left (A e +\frac {10 B d}{3}\right ) e a \right ) \sqrt {a c \,e^{2}}}{4}-\frac {A \,c^{2} d^{3}}{2}+a d e \left (A e +\frac {5 B d}{4}\right ) c +\frac {5 B \,e^{3} a^{2}}{4}\right ) \left (-c \,x^{2}+a \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-c e \left (\frac {\left (-A c \,d^{2}+3 \left (A e +\frac {10 B d}{3}\right ) e a \right ) \sqrt {a c \,e^{2}}}{4}-\frac {A \,c^{2} d^{3}}{2}+a d e \left (A e +\frac {5 B d}{4}\right ) c +\frac {5 B \,e^{3} a^{2}}{4}\right ) \left (-c \,x^{2}+a \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {e x +d}\, \sqrt {a c \,e^{2}}\, \left (\frac {A \,c^{2} d^{2} x}{2}+a \left (\left (\frac {1}{2} A x -2 B \,x^{2}\right ) e^{2}+e \left (B x +A \right ) d +\frac {B \,d^{2}}{2}\right ) c +\frac {5 B \,e^{2} a^{2}}{2}\right )\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, c^{2} \left (-c \,x^{2}+a \right ) a}\)	\(380\)
derivativedivides	\(2 e^{2} \left (\frac {B \sqrt {e x +d}}{c^{2}}+\frac {\frac {\frac {c \left (A a \,e^{2}+A c \,d^{2}+2 B a d e \right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a e}+\frac {\left (A a c d \,e^{2}-A \,c^{2} d^{3}+B \,e^{3} a^{2}-B a c \,d^{2} e \right ) \sqrt {e x +d}}{4 a e}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}}+\frac {c \left (-\frac {\left (4 A a c d \,e^{2}-2 A \,c^{2} d^{3}+5 B \,e^{3} a^{2}+5 B a c \,d^{2} e +3 A \sqrt {a c \,e^{2}}\, a \,e^{2}-A \sqrt {a c \,e^{2}}\, c \,d^{2}+10 B \sqrt {a c \,e^{2}}\, a d e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-4 A a c d \,e^{2}+2 A \,c^{2} d^{3}-5 B \,e^{3} a^{2}-5 B a c \,d^{2} e +3 A \sqrt {a c \,e^{2}}\, a \,e^{2}-A \sqrt {a c \,e^{2}}\, c \,d^{2}+10 B \sqrt {a c \,e^{2}}\, a d e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 a e}}{c^{2}}\right )\)	\(422\)
default	\(2 e^{2} \left (\frac {B \sqrt {e x +d}}{c^{2}}+\frac {\frac {\frac {c \left (A a \,e^{2}+A c \,d^{2}+2 B a d e \right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a e}+\frac {\left (A a c d \,e^{2}-A \,c^{2} d^{3}+B \,e^{3} a^{2}-B a c \,d^{2} e \right ) \sqrt {e x +d}}{4 a e}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}}+\frac {c \left (-\frac {\left (4 A a c d \,e^{2}-2 A \,c^{2} d^{3}+5 B \,e^{3} a^{2}+5 B a c \,d^{2} e +3 A \sqrt {a c \,e^{2}}\, a \,e^{2}-A \sqrt {a c \,e^{2}}\, c \,d^{2}+10 B \sqrt {a c \,e^{2}}\, a d e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-4 A a c d \,e^{2}+2 A \,c^{2} d^{3}-5 B \,e^{3} a^{2}-5 B a c \,d^{2} e +3 A \sqrt {a c \,e^{2}}\, a \,e^{2}-A \sqrt {a c \,e^{2}}\, c \,d^{2}+10 B \sqrt {a c \,e^{2}}\, a d e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 a e}}{c^{2}}\right )\)	\(422\)
risch	\(\frac {2 B \,e^{2} \sqrt {e x +d}}{c^{2}}+\frac {2 e^{2} \left (\frac {-\frac {c \left (A a \,e^{2}+A c \,d^{2}+2 B a d e \right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a e}-\frac {\left (A a c d \,e^{2}-A \,c^{2} d^{3}+B \,e^{3} a^{2}-B a c \,d^{2} e \right ) \sqrt {e x +d}}{4 a e}}{c \left (e x +d \right )^{2}-2 c d \left (e x +d \right )-a \,e^{2}+c \,d^{2}}+\frac {c \left (-\frac {\left (4 A a c d \,e^{2}-2 A \,c^{2} d^{3}+5 B \,e^{3} a^{2}+5 B a c \,d^{2} e +3 A \sqrt {a c \,e^{2}}\, a \,e^{2}-A \sqrt {a c \,e^{2}}\, c \,d^{2}+10 B \sqrt {a c \,e^{2}}\, a d e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-4 A a c d \,e^{2}+2 A \,c^{2} d^{3}-5 B \,e^{3} a^{2}-5 B a c \,d^{2} e +3 A \sqrt {a c \,e^{2}}\, a \,e^{2}-A \sqrt {a c \,e^{2}}\, c \,d^{2}+10 B \sqrt {a c \,e^{2}}\, a d e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 a e}\right )}{c^{2}}\)	\(424\)

Input:

int((B*x+A)*(e*x+d)^(5/2)/(-c*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/(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)*(-c*e*((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*(1/4*(A*c*d^2-3*(A*e+10/3*B 
*d)*e*a)*(a*c*e^2)^(1/2)-1/2*A*c^2*d^3+a*d*e*(A*e+5/4*B*d)*c+5/4*B*e^3*a^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)*(-c*e*(1/4*(-A*c*d^2+3*(A*e+10/3*B*d)*e*a)*(a 
*c*e^2)^(1/2)-1/2*A*c^2*d^3+a*d*e*(A*e+5/4*B*d)*c+5/4*B*e^3*a^2)*(-c*x^2+a 
)*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)*(1/2*A*c^2*d^2*x+a*((1/2*A* 
x-2*B*x^2)*e^2+e*(B*x+A)*d+1/2*B*d^2)*c+5/2*B*e^2*a^2)))/c^2/(-c*x^2+a)/a
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5611 vs. \(2 (223) = 446\).

Time = 10.98 (sec) , antiderivative size = 5611, normalized size of antiderivative = 20.04 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^(5/2)/(-c*x^2+a)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(e*x+d)**(5/2)/(-c*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} - a\right )}^{2}} \,d x } \] Input:

integrate((B*x+A)*(e*x+d)^(5/2)/(-c*x^2+a)^2,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 755 vs. \(2 (223) = 446\).

Time = 0.30 (sec) , antiderivative size = 755, normalized size of antiderivative = 2.70 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^(5/2)/(-c*x^2+a)^2,x, algorithm="giac")
 

Output:

2*sqrt(e*x + d)*B*e^2/c^2 - 1/4*(10*sqrt(a*c)*B*a^3*d*e^4*abs(c) - (sqrt(a 
*c)*c*d^2*e - 3*sqrt(a*c)*a*e^3)*A*a^2*e^2*abs(c) - (a*c^2*d^3*e - a^2*c*d 
*e^3)*A*abs(a)*abs(c)*abs(e) - 5*(a^2*c*d^2*e^2 - a^3*e^4)*B*abs(a)*abs(c) 
*abs(e) + 2*(sqrt(a*c)*a*c^2*d^4*e - 2*sqrt(a*c)*a^2*c*d^2*e^3)*A*abs(c) - 
 5*(sqrt(a*c)*a^2*c*d^3*e^2 + sqrt(a*c)*a^3*d*e^4)*B*abs(c))*arctan(sqrt(e 
*x + d)/sqrt(-(a*c^3*d + sqrt(a^2*c^6*d^2 - (a*c^3*d^2 - a^2*c^2*e^2)*a*c^ 
3))/(a*c^3)))/((a^2*c^3*d - sqrt(a*c)*a^2*c^2*e)*sqrt(-c^2*d - sqrt(a*c)*c 
*e)*abs(a)*abs(e)) + 1/4*(10*B*a^3*c*d*e^4*abs(c) - (c^2*d^2*e - 3*a*c*e^3 
)*A*a^2*e^2*abs(c) + (sqrt(a*c)*c^2*d^3*e - sqrt(a*c)*a*c*d*e^3)*A*abs(a)* 
abs(c)*abs(e) + 5*(sqrt(a*c)*a*c*d^2*e^2 - sqrt(a*c)*a^2*e^4)*B*abs(a)*abs 
(c)*abs(e) + 2*(a*c^3*d^4*e - 2*a^2*c^2*d^2*e^3)*A*abs(c) - 5*(a^2*c^2*d^3 
*e^2 + a^3*c*d*e^4)*B*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a*c^3*d - sqrt(a 
^2*c^6*d^2 - (a*c^3*d^2 - a^2*c^2*e^2)*a*c^3))/(a*c^3)))/((a^2*c^3*e + sqr 
t(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^2*d^2*e - sqrt(e*x + d)*A*c^2*d^3*e + 2*(e*x + d)^(3/2)*B*a*c 
*d*e^2 - sqrt(e*x + d)*B*a*c*d^2*e^2 + (e*x + d)^(3/2)*A*a*c*e^3 + sqrt(e* 
x + d)*A*a*c*d*e^3 + sqrt(e*x + d)*B*a^2*e^4)/(((e*x + d)^2*c - 2*(e*x + d 
)*c*d + c*d^2 - a*e^2)*a*c^2)
 

Mupad [B] (verification not implemented)

Time = 6.61 (sec) , antiderivative size = 9253, normalized size of antiderivative = 33.05 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((320*B*a^5*c^4*e^6 + 64*A*a^4*c^5*d*e^5 - 64*A*a^3*c^6*d^3*e^3 - 3 
20*B*a^4*c^5*d^2*e^4)/(8*a^3*c^3) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2 
*a^3*c^8*d^5 - 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 2 
5*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^9)^( 
1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^ 
9)^(1/2) + 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e - 75* 
B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 - 70*A*B*a*c*d*e^ 
4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2))*((4*A^2*a^3*c^8*d^5 - 25*B^2*a^2*e 
^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30* 
A*B*a^6*c^5*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 
 + 75*B^2*a^6*c^5*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) + 30*A*B*c^2*d^3*e 
^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e - 75*B^2*a*c*d^2*e^3*(a^9*c^9)^( 
1/2) + 30*A*B*a^5*c^6*d^2*e^3 - 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6* 
c^9))^(1/2) + ((d + e*x)^(1/2)*(25*B^2*a^4*e^8 + 4*A^2*c^4*d^6*e^2 + 9*A^2 
*a^3*c*e^8 + 10*A^2*a^2*c^2*d^2*e^6 + 25*B^2*a^2*c^2*d^4*e^4 - 15*A^2*a*c^ 
3*d^4*e^4 + 150*B^2*a^3*c*d^2*e^6 + 100*A*B*a^3*c*d*e^7 - 20*A*B*a*c^3*d^5 
*e^3))/(a^2*c))*((4*A^2*a^3*c^8*d^5 - 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15* 
A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 + 5*A^2* 
c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 
- 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) + 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - ...
 

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 1558, normalized size of antiderivative = 5.56 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(e*x+d)^(5/2)/(-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**2*c*e**2 + 10*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*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 - 6*sqrt(a)*sqrt(s 
qrt(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*e**2*x**2 - 10*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)/(sq 
rt(c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)))*c**3*d**2*x**2 - 10*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 + 10*sq 
rt(c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(s 
qrt(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 + 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**2*c*e**2 + 5*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*b*c*d*e - 2*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e + c*d)*log( ...