\(\int \frac {x^2}{(d+e x) (a+b x+c x^2)^{5/2}} \, dx\) [47]

Optimal result

Integrand size = 25, antiderivative size = 349 \[ \int \frac {x^2}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (a (b d-2 a e)+\left (b^2 d-2 a c d-a b e\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (b^4 d^2 e-24 a^2 c^2 d^2 e-4 a b c d \left (c d^2-2 a e^2\right )+2 a b^2 e \left (7 c d^2+2 a e^2\right )-b^3 \left (c d^3+8 a d e^2\right )+c \left (5 b^3 d^2 e-8 a c d \left (c d^2-2 a e^2\right )+4 a b e \left (c d^2+2 a e^2\right )-2 b^2 \left (c d^3+8 a d e^2\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}+\frac {d^2 e^2 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{5/2}} \] Output:

1/3*(-2*a*(-2*a*e+b*d)-2*(-a*b*e-2*a*c*d+b^2*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*(b^4*d^2*e-24*a^2*c^2*d^2*e-4*a*b*c*d*(- 
2*a*e^2+c*d^2)+2*a*b^2*e*(2*a*e^2+7*c*d^2)-b^3*(8*a*d*e^2+c*d^3)+c*(5*b^3* 
d^2*e-8*a*c*d*(-2*a*e^2+c*d^2)+4*a*b*e*(2*a*e^2+c*d^2)-2*b^2*(8*a*d*e^2+c* 
d^3))*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^2*e^2* 
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 = 12.04 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.02 \[ \int \frac {x^2}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (-2 a^2 e+b^2 d x-2 a c d x+a b (d-e x)\right )}{3 \left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))^{3/2}}-\frac {2 \left (b^4 d^2 e+2 b^2 \left (2 a^2 e^3-c^2 d^3 x+a c d e (7 d-8 e x)\right )-b^3 d \left (8 a e^2+c d (d-5 e x)\right )-8 a c^2 d \left (c d^2 x+a e (3 d-2 e x)\right )+4 a b c \left (c d^2 (-d+e x)+2 a e^2 (d+e x)\right )\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2+e (-b d+a e)\right )^2 \sqrt {a+x (b+c x)}}+\frac {d^2 e^2 \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^{5/2}}-\frac {d^2 e^2 \log \left (-b d+2 a e-2 c d x+b e x+2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}\right )}{\left (c d^2+e (-b d+a e)\right )^{5/2}} \] Input:

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

Output:

(2*(-2*a^2*e + b^2*d*x - 2*a*c*d*x + a*b*(d - e*x)))/(3*(b^2 - 4*a*c)*(-(c 
*d^2) + e*(b*d - a*e))*(a + x*(b + c*x))^(3/2)) - (2*(b^4*d^2*e + 2*b^2*(2 
*a^2*e^3 - c^2*d^3*x + a*c*d*e*(7*d - 8*e*x)) - b^3*d*(8*a*e^2 + c*d*(d - 
5*e*x)) - 8*a*c^2*d*(c*d^2*x + a*e*(3*d - 2*e*x)) + 4*a*b*c*(c*d^2*(-d + e 
*x) + 2*a*e^2*(d + 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^2*e^2*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e) 
)^(5/2) - (d^2*e^2*Log[-(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 = 0.61 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1264, 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.

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

\(\Big \downarrow \) 1264

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1235

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

rule 1264

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d + e*x) 
^m*(f + g*x)^n, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[(d + e*x 
)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[ 
(d + e*x)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 1]}, Simp[(b*R - 2*a*S + ( 
2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + S 
imp[1/((p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*E 
xpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*R - b*S) 
)/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 1] 
 && LtQ[p, -1] && ILtQ[m, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(935\) vs. \(2(335)=670\).

Time = 1.49 (sec) , antiderivative size = 936, normalized size of antiderivative = 2.68

method	result	size
default	\(\frac {-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\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}}\right )}{2 c}}{e}+\frac {d^{2} \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^{3}}-\frac {d \left (\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}}\right )}{e^{2}}\)	\(936\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2297 vs. \(2 (333) = 666\).

Time = 1.25 (sec) , antiderivative size = 4636, normalized size of antiderivative = 13.28 \[ \int \frac {x^2}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^2/(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. 8210 vs. \(2 (333) = 666\).

Time = 0.49 (sec) , antiderivative size = 8210, normalized size of antiderivative = 23.52 \[ \int \frac {x^2}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

2*d^2*e^2*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*((((2*b 
^2*c^9*d^15 + 8*a*c^10*d^15 - 17*b^3*c^8*d^14*e - 52*a*b*c^9*d^14*e + 60*b 
^4*c^7*d^13*e^2 + 172*a*b^2*c^8*d^13*e^2 + 32*a^2*c^9*d^13*e^2 - 115*b^5*c 
^6*d^12*e^3 - 406*a*b^3*c^7*d^12*e^3 - 176*a^2*b*c^8*d^12*e^3 + 130*b^6*c^ 
5*d^11*e^4 + 710*a*b^4*c^6*d^11*e^4 + 534*a^2*b^2*c^7*d^11*e^4 + 24*a^3*c^ 
8*d^11*e^4 - 87*b^7*c^4*d^10*e^5 - 848*a*b^5*c^5*d^10*e^5 - 1195*a^2*b^3*c 
^6*d^10*e^5 - 108*a^3*b*c^7*d^10*e^5 + 32*b^8*c^3*d^9*e^6 + 632*a*b^6*c^4* 
d^9*e^6 + 1840*a^2*b^4*c^5*d^9*e^6 + 520*a^3*b^2*c^6*d^9*e^6 - 80*a^4*c^7* 
d^9*e^6 - 5*b^9*c^2*d^8*e^7 - 262*a*b^7*c^3*d^8*e^7 - 1722*a^2*b^5*c^4*d^8 
*e^7 - 1540*a^3*b^3*c^5*d^8*e^7 + 280*a^4*b*c^6*d^8*e^7 + 46*a*b^8*c^2*d^7 
*e^8 + 866*a^2*b^6*c^3*d^7*e^8 + 2220*a^3*b^4*c^4*d^7*e^8 + 110*a^4*b^2*c^ 
5*d^7*e^8 - 200*a^5*c^6*d^7*e^8 - 179*a^2*b^7*c^2*d^6*e^9 - 1504*a^3*b^5*c 
^3*d^6*e^9 - 1255*a^4*b^3*c^4*d^6*e^9 + 500*a^5*b*c^5*d^6*e^9 + 388*a^3*b^ 
6*c^2*d^5*e^10 + 1460*a^4*b^4*c^3*d^5*e^10 + 12*a^5*b^2*c^4*d^5*e^10 - 192 
*a^6*c^5*d^5*e^10 - 515*a^4*b^5*c^2*d^4*e^11 - 742*a^5*b^3*c^3*d^4*e^11 + 
288*a^6*b*c^4*d^4*e^11 + 430*a^5*b^4*c^2*d^3*e^12 + 122*a^6*b^2*c^3*d^3*e^ 
12 - 88*a^7*c^4*d^3*e^12 - 221*a^6*b^3*c^2*d^2*e^13 + 44*a^7*b*c^3*d^2*e^1 
3 + 64*a^7*b^2*c^2*d*e^14 - 16*a^8*c^3*d*e^14 - 8*a^8*b*c^2*e^15)*x/(b^...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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