\(\int \frac {x (c+d x+e x^2)}{(a+b x^3)^{3/2}} \, dx\) [238]

Optimal result

Integrand size = 23, antiderivative size = 561 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^{3/2}} \, dx=-\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a b \sqrt {a+b x^3}}-\frac {2 d \sqrt {a+b x^3}}{3 a b}-\frac {2 c \sqrt {a+b x^3}}{3 a b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\sqrt {2-\sqrt {3}} c \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} a^{2/3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (b^{2/3} \left (c-\sqrt {3} c\right )+2 a^{2/3} e\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} a^{2/3} b^{4/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:

-2/3*x*(-b*d*x^2-b*c*x+a*e)/a/b/(b*x^3+a)^(1/2)-2/3*d*(b*x^3+a)^(1/2)/a/b- 
2/3*c*(b*x^3+a)^(1/2)/a/b^(2/3)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)+1/3*(1/2*6 
^(1/2)-1/2*2^(1/2))*c*(a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2 
/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)*EllipticE(((1-3^(1/2))*a 
^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x),I*3^(1/2)+2*I)*3^(1/4)/a 
^(2/3)/b^(2/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x 
)^2)^(1/2)/(b*x^3+a)^(1/2)+2/9*(1/2*6^(1/2)+1/2*2^(1/2))*(b^(2/3)*(c-3^(1/ 
2)*c)+2*a^(2/3)*e)*(a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3) 
*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)*EllipticF(((1-3^(1/2))*a^(1 
/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x),I*3^(1/2)+2*I)*3^(3/4)/a^(2 
/3)/b^(4/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2 
)^(1/2)/(b*x^3+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.19 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {-4 a (d+e x)+4 a e x \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a}\right )+3 b c x^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{2},\frac {5}{3},-\frac {b x^3}{a}\right )}{6 a b \sqrt {a+b x^3}} \] Input:

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

Output:

(-4*a*(d + e*x) + 4*a*e*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 
4/3, -((b*x^3)/a)] + 3*b*c*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 
3/2, 5/3, -((b*x^3)/a)])/(6*a*b*Sqrt[a + b*x^3])
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2367, 27, 2425, 793, 2417, 759, 2416}

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.

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

\(\Big \downarrow \) 2367

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2425

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

\(\Big \downarrow \) 793

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

\(\Big \downarrow \) 2417

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

\(\Big \downarrow \) 759

\(\displaystyle \frac {-b^{2/3} c \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx+\frac {2 \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (2 a^{2/3} e+\left (1-\sqrt {3}\right ) b^{2/3} c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-2 d \sqrt {a+b x^3}}{3 a b}-\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a b \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 2416

\(\displaystyle \frac {\frac {2 \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (2 a^{2/3} e+\left (1-\sqrt {3}\right ) b^{2/3} c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-b^{2/3} c \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )-2 d \sqrt {a+b x^3}}{3 a b}-\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a b \sqrt {a+b x^3}}\)

Input:

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

Output:

(-2*x*(a*e - b*c*x - b*d*x^2))/(3*a*b*Sqrt[a + b*x^3]) + (-2*d*Sqrt[a + b* 
x^3] - b^(2/3)*c*((2*Sqrt[a + b*x^3])/(b^(1/3)*((1 + Sqrt[3])*a^(1/3) + b^ 
(1/3)*x)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*(a^(1/3) + b^(1/3)*x)*Sqrt[ 
(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/ 
3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3 
])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(b^(1/3)*Sqrt[(a^(1/3)*(a^(1/3) 
 + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3])) + 
(2*Sqrt[2 + Sqrt[3]]*a^(1/3)*((1 - Sqrt[3])*b^(2/3)*c + 2*a^(2/3)*e)*(a^(1 
/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sq 
rt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b 
^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(3^(1/4)* 
b^(1/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1 
/3)*x)^2]*Sqrt[a + b*x^3]))/(3*a*b)
 

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 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

rule 793

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n) 
^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && 
 NeQ[p, -1]
 

rule 2367

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = 
 m + Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1) 
*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*x^ 
m*Pq, a + b*x^n, x]}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floo 
r[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))   I 
nt[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], 
 x], x], x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0 
] && LtQ[p, -1] && IGtQ[m, 0]
 

rule 2416

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S 
imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt 
[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) 
*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

rule 2417

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(c*r - (1 - Sqrt[3])*d*s)/r 
  Int[1/Sqrt[a + b*x^3], x], x] + Simp[d/r   Int[((1 - Sqrt[3])*s + r*x)/Sq 
rt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && NeQ[b*c^3 - 
2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

rule 2425

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Coeff[Pq, x, n - 
 1]   Int[x^(n - 1)*(a + b*x^n)^p, x], x] + Int[ExpandToSum[Pq - Coeff[Pq, 
x, n - 1]*x^(n - 1), x]*(a + b*x^n)^p, x] /; FreeQ[{a, b, p}, x] && PolyQ[P 
q, x] && IGtQ[n, 0] && Expon[Pq, x] == n - 1
 

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 765, normalized size of antiderivative = 1.36

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(765\)
default	\(\text {Expression too large to display}\)	\(782\)

Input:

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

Output:

-2*b*(-1/3*c/a/b*x^2+1/3*e*x/b^2+1/3*d/b^2)/((x^3+a/b)*b)^(1/2)-4/9*I*e/b^ 
2*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^ 
2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b* 
(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2 
)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b 
*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/ 
2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2) 
^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+2/9* 
I*c/a*3^(1/2)/b*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b* 
(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(- 
3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*( 
-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1 
/2)/(b*x^3+a)^(1/2)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3) 
)*EllipticE(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2 
)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2 
/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+1/b*(-a*b^2)^(1/ 
3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^ 
2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/ 
2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.20 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {2 \, {\left (2 \, {\left (a b e x^{3} + a^{2} e\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (b^{2} c x^{3} + a b c\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (b^{2} c x^{2} - a b e x - a b d\right )} \sqrt {b x^{3} + a}\right )}}{3 \, {\left (a b^{3} x^{3} + a^{2} b^{2}\right )}} \] Input:

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

Output:

2/3*(2*(a*b*e*x^3 + a^2*e)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) + (b^ 
2*c*x^3 + a*b*c)*sqrt(b)*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, 
 -4*a/b, x)) + (b^2*c*x^2 - a*b*e*x - a*b*d)*sqrt(b*x^3 + a))/(a*b^3*x^3 + 
 a^2*b^2)
 

Sympy [A] (verification not implemented)

Time = 3.40 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.19 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^{3/2}} \, dx=d \left (\begin {cases} - \frac {2}{3 b \sqrt {a + b x^{3}}} & \text {for}\: b \neq 0 \\\frac {x^{3}}{3 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {c x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {3}{2} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (\frac {5}{3}\right )} + \frac {e x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {3}{2} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (\frac {7}{3}\right )} \] Input:

integrate(x*(e*x**2+d*x+c)/(b*x**3+a)**(3/2),x)
 

Output:

d*Piecewise((-2/(3*b*sqrt(a + b*x**3)), Ne(b, 0)), (x**3/(3*a**(3/2)), Tru 
e)) + c*x**2*gamma(2/3)*hyper((2/3, 3/2), (5/3,), b*x**3*exp_polar(I*pi)/a 
)/(3*a**(3/2)*gamma(5/3)) + e*x**4*gamma(4/3)*hyper((4/3, 3/2), (7/3,), b* 
x**3*exp_polar(I*pi)/a)/(3*a**(3/2)*gamma(7/3))
 

Maxima [F]

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

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

Output:

integrate((e*x^2 + d*x + c)*x/(b*x^3 + a)^(3/2), x)
 

Giac [F]

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

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

Output:

integrate((e*x^2 + d*x + c)*x/(b*x^3 + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^{3/2}} \, dx=\int \frac {x\,\left (e\,x^2+d\,x+c\right )}{{\left (b\,x^3+a\right )}^{3/2}} \,d x \] Input:

int((x*(c + d*x + e*x^2))/(a + b*x^3)^(3/2),x)
 

Output:

int((x*(c + d*x + e*x^2))/(a + b*x^3)^(3/2), x)
 

Reduce [F]

\[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {-2 \sqrt {b \,x^{3}+a}\, d -6 \sqrt {b \,x^{3}+a}\, e x +6 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{2} e +6 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a b e \,x^{3}+3 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a b c +3 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) b^{2} c \,x^{3}}{3 b \left (b \,x^{3}+a \right )} \] Input:

int(x*(e*x^2+d*x+c)/(b*x^3+a)^(3/2),x)
 

Output:

( - 2*sqrt(a + b*x**3)*d - 6*sqrt(a + b*x**3)*e*x + 6*int(sqrt(a + b*x**3) 
/(a**2 + 2*a*b*x**3 + b**2*x**6),x)*a**2*e + 6*int(sqrt(a + b*x**3)/(a**2 
+ 2*a*b*x**3 + b**2*x**6),x)*a*b*e*x**3 + 3*int((sqrt(a + b*x**3)*x)/(a**2 
 + 2*a*b*x**3 + b**2*x**6),x)*a*b*c + 3*int((sqrt(a + b*x**3)*x)/(a**2 + 2 
*a*b*x**3 + b**2*x**6),x)*b**2*c*x**3)/(3*b*(a + b*x**3))