3.413 \(\int \frac{x^3 (c+d x+e x^2+f x^3+g x^4+h x^5)}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=311 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right )}{18 a^{2/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right )}{9 a^{2/3} b^{8/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^{4/3} g+2 \sqrt [3]{a} b d-4 a \sqrt [3]{b} f+b^{4/3} c\right )}{3 \sqrt{3} a^{2/3} b^{8/3}}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 b^2 \left (a+b x^3\right )}+\frac{(b e-2 a h) \log \left (a+b x^3\right )}{3 b^3}+\frac{f x}{b^2}+\frac{g x^2}{2 b^2}+\frac{h x^3}{3 b^2} \]

[Out]

(f*x)/b^2 + (g*x^2)/(2*b^2) + (h*x^3)/(3*b^2) - (x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(3*b^2*(a +
b*x^3)) - ((b^(4/3)*c + 2*a^(1/3)*b*d - 4*a*b^(1/3)*f - 5*a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a
^(1/3))])/(3*Sqrt[3]*a^(2/3)*b^(8/3)) + ((b^(1/3)*(b*c - 4*a*f) - a^(1/3)*(2*b*d - 5*a*g))*Log[a^(1/3) + b^(1/
3)*x])/(9*a^(2/3)*b^(8/3)) - ((b^(1/3)*(b*c - 4*a*f) - a^(1/3)*(2*b*d - 5*a*g))*Log[a^(2/3) - a^(1/3)*b^(1/3)*
x + b^(2/3)*x^2])/(18*a^(2/3)*b^(8/3)) + ((b*e - 2*a*h)*Log[a + b*x^3])/(3*b^3)

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Rubi [A]  time = 0.639639, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {1828, 1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right )}{18 a^{2/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right )}{9 a^{2/3} b^{8/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^{4/3} g+2 \sqrt [3]{a} b d-4 a \sqrt [3]{b} f+b^{4/3} c\right )}{3 \sqrt{3} a^{2/3} b^{8/3}}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 b^2 \left (a+b x^3\right )}+\frac{(b e-2 a h) \log \left (a+b x^3\right )}{3 b^3}+\frac{f x}{b^2}+\frac{g x^2}{2 b^2}+\frac{h x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^2,x]

[Out]

(f*x)/b^2 + (g*x^2)/(2*b^2) + (h*x^3)/(3*b^2) - (x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(3*b^2*(a +
b*x^3)) - ((b^(4/3)*c + 2*a^(1/3)*b*d - 4*a*b^(1/3)*f - 5*a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a
^(1/3))])/(3*Sqrt[3]*a^(2/3)*b^(8/3)) + ((b^(1/3)*(b*c - 4*a*f) - a^(1/3)*(2*b*d - 5*a*g))*Log[a^(1/3) + b^(1/
3)*x])/(9*a^(2/3)*b^(8/3)) - ((b^(1/3)*(b*c - 4*a*f) - a^(1/3)*(2*b*d - 5*a*g))*Log[a^(2/3) - a^(1/3)*b^(1/3)*
x + b^(2/3)*x^2])/(18*a^(2/3)*b^(8/3)) + ((b*e - 2*a*h)*Log[a + b*x^3])/(3*b^3)

Rule 1828

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

Rule 1887

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rule 1871

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 1860

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 260

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

Rubi steps

\begin{align*} \int \frac{x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^2} \, dx &=-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 b^2 \left (a+b x^3\right )}-\frac{\int \frac{-a b (b c-a f)-2 a b (b d-a g) x-3 a b (b e-a h) x^2-3 a b^2 f x^3-3 a b^2 g x^4-3 a b^2 h x^5}{a+b x^3} \, dx}{3 a b^3}\\ &=-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 b^2 \left (a+b x^3\right )}-\frac{\int \left (-3 a b f-3 a b g x-3 a b h x^2-\frac{a b (b c-4 a f)+a b (2 b d-5 a g) x+3 a b (b e-2 a h) x^2}{a+b x^3}\right ) \, dx}{3 a b^3}\\ &=\frac{f x}{b^2}+\frac{g x^2}{2 b^2}+\frac{h x^3}{3 b^2}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 b^2 \left (a+b x^3\right )}+\frac{\int \frac{a b (b c-4 a f)+a b (2 b d-5 a g) x+3 a b (b e-2 a h) x^2}{a+b x^3} \, dx}{3 a b^3}\\ &=\frac{f x}{b^2}+\frac{g x^2}{2 b^2}+\frac{h x^3}{3 b^2}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 b^2 \left (a+b x^3\right )}+\frac{\int \frac{a b (b c-4 a f)+a b (2 b d-5 a g) x}{a+b x^3} \, dx}{3 a b^3}+\frac{(b e-2 a h) \int \frac{x^2}{a+b x^3} \, dx}{b^2}\\ &=\frac{f x}{b^2}+\frac{g x^2}{2 b^2}+\frac{h x^3}{3 b^2}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 b^2 \left (a+b x^3\right )}+\frac{(b e-2 a h) \log \left (a+b x^3\right )}{3 b^3}+\frac{\int \frac{\sqrt [3]{a} \left (2 a b^{4/3} (b c-4 a f)+a^{4/3} b (2 b d-5 a g)\right )+\sqrt [3]{b} \left (-a b^{4/3} (b c-4 a f)+a^{4/3} b (2 b d-5 a g)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{5/3} b^{10/3}}+\frac{\left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{2/3} b^{7/3}}\\ &=\frac{f x}{b^2}+\frac{g x^2}{2 b^2}+\frac{h x^3}{3 b^2}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 b^2 \left (a+b x^3\right )}+\frac{\left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{8/3}}+\frac{(b e-2 a h) \log \left (a+b x^3\right )}{3 b^3}+\frac{\left (b^{4/3} c+2 \sqrt [3]{a} b d-4 a \sqrt [3]{b} f-5 a^{4/3} g\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 \sqrt [3]{a} b^{7/3}}-\frac{\left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{2/3} b^{8/3}}\\ &=\frac{f x}{b^2}+\frac{g x^2}{2 b^2}+\frac{h x^3}{3 b^2}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 b^2 \left (a+b x^3\right )}+\frac{\left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{8/3}}-\frac{\left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{8/3}}+\frac{(b e-2 a h) \log \left (a+b x^3\right )}{3 b^3}+\frac{\left (b^{4/3} c+2 \sqrt [3]{a} b d-4 a \sqrt [3]{b} f-5 a^{4/3} g\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{2/3} b^{8/3}}\\ &=\frac{f x}{b^2}+\frac{g x^2}{2 b^2}+\frac{h x^3}{3 b^2}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 b^2 \left (a+b x^3\right )}-\frac{\left (b^{4/3} c+2 \sqrt [3]{a} b d-4 a \sqrt [3]{b} f-5 a^{4/3} g\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{8/3}}+\frac{\left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{8/3}}-\frac{\left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{8/3}}+\frac{(b e-2 a h) \log \left (a+b x^3\right )}{3 b^3}\\ \end{align*}

Mathematica [A]  time = 0.212374, size = 294, normalized size = 0.95 \[ \frac{-\frac{6 \left (a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right )}{a+b x^3}-\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^{4/3} g-2 \sqrt [3]{a} b d-4 a \sqrt [3]{b} f+b^{4/3} c\right )}{a^{2/3}}+\frac{2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^{4/3} g-2 \sqrt [3]{a} b d-4 a \sqrt [3]{b} f+b^{4/3} c\right )}{a^{2/3}}+\frac{2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (5 a^{4/3} g-2 \sqrt [3]{a} b d+4 a \sqrt [3]{b} f-b^{4/3} c\right )}{a^{2/3}}+6 (b e-2 a h) \log \left (a+b x^3\right )+18 b f x+9 b g x^2+6 b h x^3}{18 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^2,x]

[Out]

(18*b*f*x + 9*b*g*x^2 + 6*b*h*x^3 - (6*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(a + b*x^3) + (2*Sqr
t[3]*b^(1/3)*(-(b^(4/3)*c) - 2*a^(1/3)*b*d + 4*a*b^(1/3)*f + 5*a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/S
qrt[3]])/a^(2/3) + (2*b^(1/3)*(b^(4/3)*c - 2*a^(1/3)*b*d - 4*a*b^(1/3)*f + 5*a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*
x])/a^(2/3) - (b^(1/3)*(b^(4/3)*c - 2*a^(1/3)*b*d - 4*a*b^(1/3)*f + 5*a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)
*x + b^(2/3)*x^2])/a^(2/3) + 6*(b*e - 2*a*h)*Log[a + b*x^3])/(18*b^3)

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Maple [B]  time = 0.011, size = 533, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^2,x)

[Out]

1/3*h*x^3/b^2+1/2*g*x^2/b^2+f*x/b^2+1/3/b^2/(b*x^3+a)*x^2*a*g-1/3*x^2*d/(b*x^3+a)/b+1/3/b^2*a*x/(b*x^3+a)*f-1/
3/b*x/(b*x^3+a)*c-1/3/b^3/(b*x^3+a)*a^2*h+1/3/b^2*a/(b*x^3+a)*e-4/9/b^3*a*f/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))+
2/9/b^3*a*f/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-4/9/b^3*a*f/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3
^(1/2)*(2/(1/b*a)^(1/3)*x-1))+1/9/b^2*c/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-1/18/b^2*c/(1/b*a)^(2/3)*ln(x^2-(1/b
*a)^(1/3)*x+(1/b*a)^(2/3))+1/9/b^2*c/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+5/9/b^3*a
*g/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-5/18/b^3*a*g/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-5/9/b^3*
a*g*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-2/9/b^2/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*
d+1/9/b^2/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*d+2/9/b^2*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/
2)*(2/(1/b*a)^(1/3)*x-1))*d-2/3/b^3*ln(b*x^3+a)*a*h+1/3/b^2*ln(b*x^3+a)*e

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.08641, size = 475, normalized size = 1.53 \begin{align*} -\frac{{\left (2 \, a h - b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a b^{4}} - \frac{a^{2} h +{\left (b^{2} d - a b g\right )} x^{2} - a b e +{\left (b^{2} c - a b f\right )} x}{3 \,{\left (b x^{3} + a\right )} b^{3}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a b^{4}} - \frac{{\left (2 \, b^{4} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, a b^{3} g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + b^{4} c - 4 \, a b^{3} f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{5}} + \frac{2 \, b^{4} h x^{3} + 3 \, b^{4} g x^{2} + 6 \, b^{4} f x}{6 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^2,x, algorithm="giac")

[Out]

-1/3*(2*a*h - b*e)*log(abs(b*x^3 + a))/b^3 + 1/9*sqrt(3)*((-a*b^2)^(1/3)*b^2*c - 4*(-a*b^2)^(1/3)*a*b*f - 2*(-
a*b^2)^(2/3)*b*d + 5*(-a*b^2)^(2/3)*a*g)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^4) - 1/3*(
a^2*h + (b^2*d - a*b*g)*x^2 - a*b*e + (b^2*c - a*b*f)*x)/((b*x^3 + a)*b^3) + 1/18*((-a*b^2)^(1/3)*b^2*c - 4*(-
a*b^2)^(1/3)*a*b*f + 2*(-a*b^2)^(2/3)*b*d - 5*(-a*b^2)^(2/3)*a*g)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*
b^4) - 1/9*(2*b^4*d*(-a/b)^(1/3) - 5*a*b^3*g*(-a/b)^(1/3) + b^4*c - 4*a*b^3*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)
^(1/3)))/(a*b^5) + 1/6*(2*b^4*h*x^3 + 3*b^4*g*x^2 + 6*b^4*f*x)/b^6