\(\int \frac {x}{(d+e x) (a+b x+c x^2)^{5/2}} \, dx\) [48]

Optimal result

Integrand size = 23, antiderivative size = 306 \[ \int \frac {x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 (a (2 c d-b e)+c (b d-2 a e) x)}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (4 a c e (b d-2 a e) (2 c d-b e)-d \left (b c d-b^2 e+2 a c e\right ) \left (4 b c d-3 b^2 e+4 a c e\right )+c \left (4 c e (b d-2 a e)^2-d (2 c d-b e) \left (4 b c d-3 b^2 e+4 a c e\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}-\frac {d e^3 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 10.50 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.99 \[ \int \frac {x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {-2 b c d x+a (-4 c d+2 b e+4 c e x)}{3 \left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))^{3/2}}-\frac {2 \left (3 b^4 d e^2+b^3 c d e (-7 d+3 e x)+2 b^2 c d \left (-3 a e^2+c d (2 d-7 e x)\right )+8 a c^2 e \left (c d^2 x+a e (3 d-2 e x)\right )+4 b c \left (-2 a^2 e^3+2 c^2 d^3 x+a c d e (d+3 e x)\right )\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2+e (-b d+a e)\right )^2 \sqrt {a+x (b+c x)}}+\frac {d e^3 \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{\left (c d^2+e (-b d+a e)\right )^{5/2}} \] Input:

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

Output:

(-2*b*c*d*x + a*(-4*c*d + 2*b*e + 4*c*e*x))/(3*(b^2 - 4*a*c)*(-(c*d^2) + e 
*(b*d - a*e))*(a + x*(b + c*x))^(3/2)) - (2*(3*b^4*d*e^2 + b^3*c*d*e*(-7*d 
 + 3*e*x) + 2*b^2*c*d*(-3*a*e^2 + c*d*(2*d - 7*e*x)) + 8*a*c^2*e*(c*d^2*x 
+ a*e*(3*d - 2*e*x)) + 4*b*c*(-2*a^2*e^3 + 2*c^2*d^3*x + a*c*d*e*(d + 3*e* 
x))))/(3*(b^2 - 4*a*c)^2*(c*d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + x*(b + c*x) 
]) + (d*e^3*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*( 
-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(5/2)
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1235, 27, 1235, 27, 1154, 219}

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.

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

\(\Big \downarrow \) 1235

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1235

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(2*(a*(2*c*d - b*e) + c*(b*d - 2*a*e)*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e 
+ a*e^2)*(a + b*x + c*x^2)^(3/2)) + ((2*(4*a*c*e*(b*d - 2*a*e)*(2*c*d - b* 
e) - d*(b*c*d - b^2*e + 2*a*c*e)*(4*b*c*d - 3*b^2*e + 4*a*c*e) + c*(4*c*e* 
(b*d - 2*a*e)^2 - d*(2*c*d - b*e)*(4*b*c*d - 3*b^2*e + 4*a*c*e))*x))/((b^2 
 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x + c*x^2]) - (3*(b^2 - 4*a*c 
)*d*e^3*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a* 
e^2]*Sqrt[a + b*x + c*x^2])])/(c*d^2 - b*d*e + a*e^2)^(3/2))/(3*(b^2 - 4*a 
*c)*(c*d^2 - b*d*e + a*e^2))
 

Defintions of rubi rules used

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(838\) vs. \(2(290)=580\).

Time = 1.38 (sec) , antiderivative size = 839, normalized size of antiderivative = 2.74

method	result	size
default	\(\frac {\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}}{e}-\frac {d \left (\frac {e^{2}}{3 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}-\frac {\left (b e -2 c d \right ) e \left (\frac {\frac {4 c \left (x +\frac {d}{e}\right )}{3}+\frac {2 \left (b e -2 c d \right )}{3 e}}{\left (\frac {4 c \left (a \,e^{2}-b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}+\frac {16 c \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{3 {\left (\frac {4 c \left (a \,e^{2}-b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right )}^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )}+\frac {e^{2} \left (\frac {e^{2}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {\left (b e -2 c d \right ) e \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (\frac {4 c \left (a \,e^{2}-b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{a \,e^{2}-b d e +c \,d^{2}}\right )}{e^{2}}\)	\(839\)

Input:

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

Output:

1/e*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2 
*c*x+b)/(c*x^2+b*x+a)^(1/2))-d/e^2*(1/3/(a*e^2-b*d*e+c*d^2)*e^2/(c*(x+d/e) 
^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-1/2*(b*e-2*c*d)*e/ 
(a*e^2-b*d*e+c*d^2)*(2/3*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d 
^2)/e^2-(b*e-2*c*d)^2/e^2)/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e 
+c*d^2)/e^2)^(3/2)+16/3*c/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)^ 
2*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b* 
d*e+c*d^2)/e^2)^(1/2))+1/(a*e^2-b*d*e+c*d^2)*e^2*(1/(a*e^2-b*d*e+c*d^2)*e^ 
2/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-(b*e-2 
*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+ 
c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b* 
d*e+c*d^2)/e^2)^(1/2)-1/(a*e^2-b*d*e+c*d^2)*e^2/((a*e^2-b*d*e+c*d^2)/e^2)^ 
(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+ 
c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e 
^2)^(1/2))/(x+d/e))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2277 vs. \(2 (290) = 580\).

Time = 1.40 (sec) , antiderivative size = 4596, normalized size of antiderivative = 15.02 \[ \int \frac {x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

integrate(x/(e*x+d)/(c*x**2+b*x+a)**(5/2),x)
 

Output:

Integral(x/((d + e*x)*(a + b*x + c*x**2)**(5/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume((b/e-(2*c*d)/e^2)^2>0)', see `as 
sume?` for
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8378 vs. \(2 (290) = 580\).

Time = 0.47 (sec) , antiderivative size = 8378, normalized size of antiderivative = 27.38 \[ \int \frac {x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-2*d*e^3*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt( 
-c*d^2 + b*d*e - a*e^2))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2 
*e^2 - 2*a*b*d*e^3 + a^2*e^4)*sqrt(-c*d^2 + b*d*e - a*e^2)) - 2/3*((((8*b* 
c^10*d^15 - 62*b^2*c^9*d^14*e + 8*a*c^10*d^14*e + 207*b^3*c^8*d^13*e^2 + 1 
2*a*b*c^9*d^13*e^2 - 388*b^4*c^7*d^12*e^3 - 276*a*b^2*c^8*d^12*e^3 + 32*a^ 
2*c^9*d^12*e^3 + 445*b^5*c^6*d^11*e^4 + 938*a*b^3*c^7*d^11*e^4 + 48*a^2*b* 
c^8*d^11*e^4 - 318*b^6*c^5*d^10*e^5 - 1530*a*b^4*c^6*d^10*e^5 - 810*a^2*b^ 
2*c^7*d^10*e^5 + 24*a^3*c^8*d^10*e^5 + 137*b^7*c^4*d^9*e^6 + 1392*a*b^5*c^ 
5*d^9*e^6 + 2165*a^2*b^3*c^6*d^9*e^6 + 340*a^3*b*c^7*d^9*e^6 - 32*b^8*c^3* 
d^8*e^7 - 712*a*b^6*c^4*d^8*e^7 - 2640*a^2*b^4*c^5*d^8*e^7 - 1720*a^3*b^2* 
c^6*d^8*e^7 - 80*a^4*c^7*d^8*e^7 + 3*b^9*c^2*d^7*e^8 + 186*a*b^7*c^3*d^7*e 
^8 + 1638*a^2*b^5*c^4*d^7*e^8 + 2940*a^3*b^3*c^5*d^7*e^8 + 840*a^4*b*c^6*d 
^7*e^8 - 18*a*b^8*c^2*d^6*e^9 - 478*a^2*b^6*c^3*d^6*e^9 - 2260*a^3*b^4*c^4 
*d^6*e^9 - 2130*a^4*b^2*c^5*d^6*e^9 - 200*a^5*c^6*d^6*e^9 + 45*a^2*b^7*c^2 
*d^5*e^10 + 736*a^3*b^5*c^3*d^5*e^10 + 2105*a^4*b^3*c^4*d^5*e^10 + 948*a^5 
*b*c^5*d^5*e^10 - 60*a^3*b^6*c^2*d^4*e^11 - 780*a^4*b^4*c^3*d^4*e^11 - 133 
2*a^5*b^2*c^4*d^4*e^11 - 192*a^6*c^5*d^4*e^11 + 45*a^4*b^5*c^2*d^3*e^12 + 
602*a^5*b^3*c^3*d^3*e^12 + 512*a^6*b*c^4*d^3*e^12 - 18*a^5*b^4*c^2*d^2*e^1 
3 - 326*a^6*b^2*c^3*d^2*e^13 - 88*a^7*c^4*d^2*e^13 + 3*a^6*b^3*c^2*d*e^14 
+ 108*a^7*b*c^3*d*e^14 - 16*a^8*c^3*e^15)*x/(b^4*c^8*d^16 - 8*a*b^2*c^9...
 

Mupad [F(-1)]

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

int(x/((d + e*x)*(a + b*x + c*x^2)^(5/2)),x)
 

Output:

int(x/((d + e*x)*(a + b*x + c*x^2)^(5/2)), x)
 

Reduce [F]

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

int(x/(e*x+d)/(c*x^2+b*x+a)^(5/2),x)
 

Output:

int(x/(e*x+d)/(c*x^2+b*x+a)^(5/2),x)