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

Optimal. Leaf size=301 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (a h+2 b e)+b^{2/3} (4 b c-a f)\right )}{18 a^{7/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (a h+2 b e)+b^{2/3} (4 b c-a f)\right )}{9 a^{7/3} b^{4/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+4 b^{5/3} c\right )}{3 \sqrt{3} a^{7/3} b^{4/3}}+\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 a^2 b \left (a+b x^3\right )}-\frac{d \log \left (a+b x^3\right )}{3 a^2}-\frac{c}{a^2 x}+\frac{d \log (x)}{a^2} \]

[Out]

-(c/(a^2*x)) + (x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(3*a^2*b*(a + b*x^3)) + ((4*b^(5/3)*c
- 2*a^(2/3)*b*e - a*b^(2/3)*f - a^(5/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7
/3)*b^(4/3)) + (d*Log[x])/a^2 + ((b^(2/3)*(4*b*c - a*f) + a^(2/3)*(2*b*e + a*h))*Log[a^(1/3) + b^(1/3)*x])/(9*
a^(7/3)*b^(4/3)) - ((b^(2/3)*(4*b*c - a*f) + a^(2/3)*(2*b*e + a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*
x^2])/(18*a^(7/3)*b^(4/3)) - (d*Log[a + b*x^3])/(3*a^2)

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Rubi [A]  time = 0.593431, antiderivative size = 301, 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 = {1829, 1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (a h+2 b e)+b^{2/3} (4 b c-a f)\right )}{18 a^{7/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (a h+2 b e)+b^{2/3} (4 b c-a f)\right )}{9 a^{7/3} b^{4/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+4 b^{5/3} c\right )}{3 \sqrt{3} a^{7/3} b^{4/3}}+\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 a^2 b \left (a+b x^3\right )}-\frac{d \log \left (a+b x^3\right )}{3 a^2}-\frac{c}{a^2 x}+\frac{d \log (x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c/(a^2*x)) + (x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(3*a^2*b*(a + b*x^3)) + ((4*b^(5/3)*c
- 2*a^(2/3)*b*e - a*b^(2/3)*f - a^(5/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7
/3)*b^(4/3)) + (d*Log[x])/a^2 + ((b^(2/3)*(4*b*c - a*f) + a^(2/3)*(2*b*e + a*h))*Log[a^(1/3) + b^(1/3)*x])/(9*
a^(7/3)*b^(4/3)) - ((b^(2/3)*(4*b*c - a*f) + a^(2/3)*(2*b*e + a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*
x^2])/(18*a^(7/3)*b^(4/3)) - (d*Log[a + b*x^3])/(3*a^2)

Rule 1829

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = Polynomi
alQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1
)*x^m*Pq, a + b*x^n, x], i}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[x^m*(a + b*x^n)^(p + 1)*Expand
ToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)*Coeff[R, x, i]*x^(i - m))/a, {i, 0, n - 1}], x], x], x] - S
imp[(x*R*(a + b*x^n)^(p + 1))/(a^2*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

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

Mathematica [A]  time = 0.298581, size = 285, normalized size = 0.95 \[ -\frac{\frac{6 a \left (a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right )}{b \left (a+b x^3\right )}+\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^{2/3} b e+a^{5/3} h-a b^{2/3} f+4 b^{5/3} c\right )}{b^{4/3}}-\frac{2 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^{2/3} b e+a^{5/3} h-a b^{2/3} f+4 b^{5/3} c\right )}{b^{4/3}}+\frac{2 \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (2 a^{2/3} b e+a^{5/3} h+a b^{2/3} f-4 b^{5/3} c\right )}{b^{4/3}}+6 a d \log \left (a+b x^3\right )+\frac{18 a c}{x}-18 a d \log (x)}{18 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.014, size = 517, 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((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^2/(b*x^3+a)^2,x)

[Out]

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

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

[Out]

Timed out

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Giac [A]  time = 1.10244, size = 473, normalized size = 1.57 \begin{align*} -\frac{d \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac{d \log \left ({\left | x \right |}\right )}{a^{2}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b e + 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} b c - \left (-a b^{2}\right )^{\frac{2}{3}} a f\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^{3} b^{2}} - \frac{4 \, b^{2} c x^{3} - a b f x^{3} + a^{2} h x^{2} - a b x^{2} e - a b d x + a^{2} g x + 3 \, a b c}{3 \,{\left (b x^{4} + a x\right )} a^{2} b} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b e - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} b c + \left (-a b^{2}\right )^{\frac{2}{3}} a f\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^{3} b^{2}} + \frac{{\left (4 \, a^{2} b^{4} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} b^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{4} b^{2} h - 2 \, a^{3} b^{3} e\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^{5} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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