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

Optimal. Leaf size=347 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (a g+5 b d)-\sqrt [3]{a} (a h+2 b e)\right )}{54 a^{8/3} b^{5/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a g+5 b d)-\sqrt [3]{a} (a h+2 b e)\right )}{27 a^{8/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{9 \sqrt{3} a^{8/3} b^{5/3}}+\frac{x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (-3 b x^2 (3 b c-a f)+a (a g+5 b d)+2 a x (a h+2 b e)\right )}{18 a^3 b \left (a+b x^3\right )}-\frac{c \log \left (a+b x^3\right )}{3 a^3}+\frac{c \log (x)}{a^3} \]

[Out]

(x*(a*(b*d - a*g) + a*(b*e - a*h)*x - b*(b*c - a*f)*x^2))/(6*a^2*b*(a + b*x^3)^2) + (x*(a*(5*b*d + a*g) + 2*a*
(2*b*e + a*h)*x - 3*b*(3*b*c - a*f)*x^2))/(18*a^3*b*(a + b*x^3)) - ((5*b^(4/3)*d + 2*a^(1/3)*b*e + a*b^(1/3)*g
+ a^(4/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(5/3)) + (c*Log[x])/a^3
+ ((b^(1/3)*(5*b*d + a*g) - a^(1/3)*(2*b*e + a*h))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(8/3)*b^(5/3)) - ((b^(1/3)*
(5*b*d + a*g) - a^(1/3)*(2*b*e + a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(5/3)) -
(c*Log[a + b*x^3])/(3*a^3)

________________________________________________________________________________________

Rubi [A]  time = 0.723069, antiderivative size = 345, normalized size of antiderivative = 0.99, number of steps used = 12, 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} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (a h+2 b e)}{\sqrt [3]{b}}+a g+5 b d\right )}{54 a^{8/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a g+5 b d)-\sqrt [3]{a} (a h+2 b e)\right )}{27 a^{8/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{9 \sqrt{3} a^{8/3} b^{5/3}}+\frac{x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (-3 b x^2 (3 b c-a f)+a (a g+5 b d)+2 a x (a h+2 b e)\right )}{18 a^3 b \left (a+b x^3\right )}-\frac{c \log \left (a+b x^3\right )}{3 a^3}+\frac{c \log (x)}{a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a*(b*d - a*g) + a*(b*e - a*h)*x - b*(b*c - a*f)*x^2))/(6*a^2*b*(a + b*x^3)^2) + (x*(a*(5*b*d + a*g) + 2*a*
(2*b*e + a*h)*x - 3*b*(3*b*c - a*f)*x^2))/(18*a^3*b*(a + b*x^3)) - ((5*b^(4/3)*d + 2*a^(1/3)*b*e + a*b^(1/3)*g
+ a^(4/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(5/3)) + (c*Log[x])/a^3
+ ((b^(1/3)*(5*b*d + a*g) - a^(1/3)*(2*b*e + a*h))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(8/3)*b^(5/3)) - ((5*b*d +
a*g - (a^(1/3)*(2*b*e + a*h))/b^(1/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(4/3)) -
(c*Log[a + b*x^3])/(3*a^3)

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 \left (a+b x^3\right )^3} \, dx &=\frac{x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}-\frac{\int \frac{-6 b^2 c-b (5 b d+a g) x-2 b (2 b e+a h) x^2+3 b^2 \left (\frac{b c}{a}-f\right ) x^3}{x \left (a+b x^3\right )^2} \, dx}{6 a b^2}\\ &=\frac{x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{\int \frac{18 b^3 c+2 b^2 (5 b d+a g) x+2 b^2 (2 b e+a h) x^2}{x \left (a+b x^3\right )} \, dx}{18 a^2 b^3}\\ &=\frac{x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{\int \left (\frac{18 b^3 c}{a x}+\frac{2 b^2 \left (a (5 b d+a g)+a (2 b e+a h) x-9 b^2 c x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^3}\\ &=\frac{x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{c \log (x)}{a^3}+\frac{\int \frac{a (5 b d+a g)+a (2 b e+a h) x-9 b^2 c x^2}{a+b x^3} \, dx}{9 a^3 b}\\ &=\frac{x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{c \log (x)}{a^3}+\frac{\int \frac{a (5 b d+a g)+a (2 b e+a h) x}{a+b x^3} \, dx}{9 a^3 b}-\frac{(b c) \int \frac{x^2}{a+b x^3} \, dx}{a^3}\\ &=\frac{x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{c \log (x)}{a^3}-\frac{c \log \left (a+b x^3\right )}{3 a^3}+\frac{\int \frac{\sqrt [3]{a} \left (2 a \sqrt [3]{b} (5 b d+a g)+a^{4/3} (2 b e+a h)\right )+\sqrt [3]{b} \left (-a \sqrt [3]{b} (5 b d+a g)+a^{4/3} (2 b e+a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{11/3} b^{4/3}}+\frac{\left (5 b d+a g-\frac{\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{8/3} b}\\ &=\frac{x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{c \log (x)}{a^3}+\frac{\left (5 b d+a g-\frac{\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{4/3}}-\frac{c \log \left (a+b x^3\right )}{3 a^3}+\frac{\left (5 b^{4/3} d+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+a^{4/3} h\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{7/3} b^{4/3}}-\frac{\left (5 b d+a g-\frac{\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\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}{54 a^{8/3} b^{4/3}}\\ &=\frac{x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{c \log (x)}{a^3}+\frac{\left (5 b d+a g-\frac{\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{4/3}}-\frac{\left (5 b d+a g-\frac{\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{4/3}}-\frac{c \log \left (a+b x^3\right )}{3 a^3}+\frac{\left (5 b^{4/3} d+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+a^{4/3} h\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{8/3} b^{5/3}}\\ &=\frac{x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}-\frac{\left (5 b^{4/3} d+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+a^{4/3} h\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} b^{5/3}}+\frac{c \log (x)}{a^3}+\frac{\left (5 b d+a g-\frac{\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{4/3}}-\frac{\left (5 b d+a g-\frac{\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{4/3}}-\frac{c \log \left (a+b x^3\right )}{3 a^3}\\ \end{align*}

Mathematica [A]  time = 0.274023, size = 311, normalized size = 0.9 \[ \frac{\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e-a \sqrt [3]{b} g-5 b^{4/3} d\right )}{b^{5/3}}+\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} (-h)-2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{5/3}}-\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{5/3}}-\frac{9 a^2 (a (f+x (g+h x))-b (c+x (d+e x)))}{b \left (a+b x^3\right )^2}+\frac{3 a (a x (g+2 h x)+6 b c+b x (5 d+4 e x))}{b \left (a+b x^3\right )}-18 c \log \left (a+b x^3\right )+54 c \log (x)}{54 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*a*(6*b*c + b*x*(5*d + 4*e*x) + a*x*(g + 2*h*x)))/(b*(a + b*x^3)) - (9*a^2*(-(b*(c + x*(d + e*x))) + a*(f +
x*(g + h*x))))/(b*(a + b*x^3)^2) - (2*Sqrt[3]*a^(1/3)*(5*b^(4/3)*d + 2*a^(1/3)*b*e + a*b^(1/3)*g + a^(4/3)*h)
*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(5/3) + 54*c*Log[x] + (2*a^(1/3)*(5*b^(4/3)*d - 2*a^(1/3)*b*e
+ a*b^(1/3)*g - a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x])/b^(5/3) + (a^(1/3)*(-5*b^(4/3)*d + 2*a^(1/3)*b*e - a*b^(1
/3)*g + a^(4/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(5/3) - 18*c*Log[a + b*x^3])/(54*a^3)

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Maple [B]  time = 0.017, size = 618, normalized size = 1.8 \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/(b*x^3+a)^3,x)

[Out]

1/27/a/b^2/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*g-1/3*c*ln(b*x^3+a)/a^3+1/2/a/(b*x^
3+a)^2*c+2/9/a^2/(b*x^3+a)^2*x^5*b*e+7/18/a/(b*x^3+a)^2*x^2*e+1/9/a/(b*x^3+a)^2*x^5*h+1/18/a/(b*x^3+a)^2*x^4*g
-1/18/(b*x^3+a)^2/b*x^2*h-1/9/(b*x^3+a)^2/b*x*g+5/18/a^2/(b*x^3+a)^2*b*x^4*d+1/27/a/b^2/(1/b*a)^(2/3)*ln(x+(1/
b*a)^(1/3))*g-1/54/a/b^2/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*g-1/27/a/b^2/(1/b*a)^(1/3)*ln(x+(
1/b*a)^(1/3))*h+1/54/a/b^2/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*h+1/27/a/b^2*3^(1/2)/(1/b*a)^(1
/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*h-1/6/b/(b*x^3+a)^2*f-5/54/a^2/b/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1
/3)*x+(1/b*a)^(2/3))*d+5/27/a^2/b/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*d+4/9/a/(b*x^3+a)^2*x*d-2/27/a^2/b/(1/b*a)
^(1/3)*ln(x+(1/b*a)^(1/3))*e+2/27/a^2/b*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*e+1/3/
a^2/(b*x^3+a)^2*b*c*x^3+c*ln(x)/a^3+5/27/a^2/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))
*d+1/27/a^2/b/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*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((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x/(b*x^3+a)^3,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/(b*x^3+a)^3,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/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.09791, size = 529, normalized size = 1.52 \begin{align*} -\frac{c \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac{c \log \left ({\left | x \right |}\right )}{a^{3}} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d + \left (-a b^{2}\right )^{\frac{1}{3}} a b g - \left (-a b^{2}\right )^{\frac{2}{3}} a h - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\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 )}{27 \, a^{3} b^{3}} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d + \left (-a b^{2}\right )^{\frac{1}{3}} a b g + \left (-a b^{2}\right )^{\frac{2}{3}} a h + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b^{3}} + \frac{6 \, a b^{2} c x^{3} + 2 \,{\left (a^{2} b h + 2 \, a b^{2} e\right )} x^{5} +{\left (5 \, a b^{2} d + a^{2} b g\right )} x^{4} + 9 \, a^{2} b c - 3 \, a^{3} f -{\left (a^{3} h - 7 \, a^{2} b e\right )} x^{2} + 2 \,{\left (4 \, a^{2} b d - a^{3} g\right )} x}{18 \,{\left (b x^{3} + a\right )}^{2} a^{3} b} - \frac{{\left (a^{5} b^{2} h \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, a^{4} b^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 5 \, a^{4} b^{3} d + a^{5} b^{2} g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{7} 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/(b*x^3+a)^3,x, algorithm="giac")

[Out]

-1/3*c*log(abs(b*x^3 + a))/a^3 + c*log(abs(x))/a^3 + 1/27*sqrt(3)*(5*(-a*b^2)^(1/3)*b^2*d + (-a*b^2)^(1/3)*a*b
*g - (-a*b^2)^(2/3)*a*h - 2*(-a*b^2)^(2/3)*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b^3
) + 1/54*(5*(-a*b^2)^(1/3)*b^2*d + (-a*b^2)^(1/3)*a*b*g + (-a*b^2)^(2/3)*a*h + 2*(-a*b^2)^(2/3)*b*e)*log(x^2 +
x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^3) + 1/18*(6*a*b^2*c*x^3 + 2*(a^2*b*h + 2*a*b^2*e)*x^5 + (5*a*b^2*d + a
^2*b*g)*x^4 + 9*a^2*b*c - 3*a^3*f - (a^3*h - 7*a^2*b*e)*x^2 + 2*(4*a^2*b*d - a^3*g)*x)/((b*x^3 + a)^2*a^3*b) -
1/27*(a^5*b^2*h*(-a/b)^(1/3) + 2*a^4*b^3*(-a/b)^(1/3)*e + 5*a^4*b^3*d + a^5*b^2*g)*(-a/b)^(1/3)*log(abs(x - (
-a/b)^(1/3)))/(a^7*b^3)