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

Optimal. Leaf size=313 \[ -\frac{3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{18 a^2 b^2 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (a f+5 b c)-\sqrt [3]{a} (a g+2 b d)\right )}{54 a^{8/3} b^{5/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+5 b c)-\sqrt [3]{a} (a g+2 b d)\right )}{27 a^{8/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} g+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right )}{9 \sqrt{3} a^{8/3} b^{5/3}}+\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.429386, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {1858, 1854, 1860, 31, 634, 617, 204, 628} \[ -\frac{3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{18 a^2 b^2 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (a f+5 b c)-\sqrt [3]{a} (a g+2 b d)\right )}{54 a^{8/3} b^{5/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+5 b c)-\sqrt [3]{a} (a g+2 b d)\right )}{27 a^{8/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} g+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right )}{9 \sqrt{3} a^{8/3} b^{5/3}}+\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1858

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*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]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1854

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

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]

Rubi steps

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

Mathematica [A]  time = 0.237521, size = 295, normalized size = 0.94 \[ \frac{\frac{9 a^{5/3} \left (a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right )}{\left (a+b x^3\right )^2}+\frac{3 a^{2/3} \left (-6 a^2 h+a b x (f+2 g x)+b^2 x (5 c+4 d x)\right )}{a+b x^3}+\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} g+2 \sqrt [3]{a} b d-a \sqrt [3]{b} f-5 b^{4/3} c\right )+2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} (-g)-2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right )-2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} g+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right )}{54 a^{8/3} b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 506, normalized size = 1.6 \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)/(b*x^3+a)^3,x)

[Out]

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

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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)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 20.8148, size = 15406, normalized size = 49.22 \begin{align*} \text{result too large to display} \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)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

-1/108*(36*a^2*b*h*x^3 - 12*(2*b^3*d + a*b^2*g)*x^5 - 6*(5*b^3*c + a*b^2*f)*x^4 + 18*a^2*b*e + 18*a^3*h - 6*(7
*a*b^2*d - a^2*b*g)*x^2 + 2*(a^2*b^4*x^6 + 2*a^3*b^3*x^3 + a^4*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b^4*c^3
+ 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(
a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)
*a*b^3)/(a^8*b^5))^(1/3) - 2*(1/2)^(2/3)*(10*b^2*c*d + a^2*f*g + (2*d*f + 5*c*g)*a*b)*(-I*sqrt(3) + 1)/(a^5*b^
3*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g
^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*
d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3)))*log(40*a*b^3*c*d^2 + 8*a^2*b^2*d^2*f + 1/4*(2*a^6*b^4*d + a^7*b^3*g)
*((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12
*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*
f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3) - 2*(1/2)^(2/3)*(10*b^2*c*d + a^2*f*g + (2
*d*f + 5*c*g)*a*b)*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 +
a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*
a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3)))^2 + 2*(5*a^3*b*c + a^4*f)
*g^2 - 1/2*(25*a^3*b^4*c^2 + 10*a^4*b^3*c*f + a^5*b^2*f^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b^4*c^3 + 8*a*b^
3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5)
+ (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(
a^8*b^5))^(1/3) - 2*(1/2)^(2/3)*(10*b^2*c*d + a^2*f*g + (2*d*f + 5*c*g)*a*b)*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b
^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*
g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*
c^2*f)*a*b^3)/(a^8*b^5))^(1/3))) + 8*(5*a^2*b^2*c*d + a^3*b*d*f)*g + (125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2
*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)*x) - 12*(4*a*b^2*c - a^2*b*f)*
x - ((a^2*b^4*x^6 + 2*a^3*b^3*x^3 + a^4*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b
^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3
- a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3
) - 2*(1/2)^(2/3)*(10*b^2*c*d + a^2*f*g + (2*d*f + 5*c*g)*a*b)*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b^4*c^3 + 8*a*b
^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5)
+ (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/
(a^8*b^5))^(1/3))) + 3*sqrt(1/3)*(a^2*b^4*x^6 + 2*a^3*b^3*x^3 + a^4*b^2)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*(
(125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2
+ a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3
- 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3) - 2*(1/2)^(2/3)*(10*b^2*c*d + a^2*f*g + (2*d*f + 5*c*g)*a*b)*(-I*sqrt(3)
+ 1)/(a^5*b^3*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g +
6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a
^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3)))^2*a^5*b^3 + 160*b^2*c*d + 32*a*b*d*f + 16*(5*a*b*c + a^2
*f)*g)/(a^5*b^3)))*log(-40*a*b^3*c*d^2 - 8*a^2*b^2*d^2*f - 1/4*(2*a^6*b^4*d + a^7*b^3*g)*((1/2)^(1/3)*(I*sqrt(
3) + 1)*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3
*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2
- (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3) - 2*(1/2)^(2/3)*(10*b^2*c*d + a^2*f*g + (2*d*f + 5*c*g)*a*b)*(-I
*sqrt(3) + 1)/(a^5*b^3*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^
2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4
*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3)))^2 - 2*(5*a^3*b*c + a^4*f)*g^2 + 1/2*(25*a^3*b^4
*c^2 + 10*a^4*b^3*c*f + a^5*b^2*f^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f
+ 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g
^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3) - 2*(1
/2)^(2/3)*(10*b^2*c*d + a^2*f*g + (2*d*f + 5*c*g)*a*b)*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b^4*c^3 + 8*a*b^3*d^3 +
75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*
b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5
))^(1/3))) - 8*(5*a^2*b^2*c*d + a^3*b*d*f)*g + 2*(125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^
2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)*x + 3/4*sqrt(1/3)*(50*a^3*b^4*c^2 + 20*a^4*b^3*c*f
+ 2*a^5*b^2*f^2 + (2*a^6*b^4*d + a^7*b^3*g)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b
^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3
- a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3
) - 2*(1/2)^(2/3)*(10*b^2*c*d + a^2*f*g + (2*d*f + 5*c*g)*a*b)*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b^4*c^3 + 8*a*b
^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5)
+ (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/
(a^8*b^5))^(1/3))))*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*
b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3
- 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3) - 2*(1/2)^(2/3)*
(10*b^2*c*d + a^2*f*g + (2*d*f + 5*c*g)*a*b)*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*
c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 -
a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3)))
^2*a^5*b^3 + 160*b^2*c*d + 32*a*b*d*f + 16*(5*a*b*c + a^2*f)*g)/(a^5*b^3))) - ((a^2*b^4*x^6 + 2*a^3*b^3*x^3 +
a^4*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*
f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b +
3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3) - 2*(1/2)^(2/3)*(10*b^2*c*d + a^2*
f*g + (2*d*f + 5*c*g)*a*b)*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2
*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6
*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3))) - 3*sqrt(1/3)*(a^
2*b^4*x^6 + 2*a^3*b^3*x^3 + a^4*b^2)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^
3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3
- a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3)
- 2*(1/2)^(2/3)*(10*b^2*c*d + a^2*f*g + (2*d*f + 5*c*g)*a*b)*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b^4*c^3 + 8*a*b^
3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5)
+ (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(
a^8*b^5))^(1/3)))^2*a^5*b^3 + 160*b^2*c*d + 32*a*b*d*f + 16*(5*a*b*c + a^2*f)*g)/(a^5*b^3)))*log(-40*a*b^3*c*d
^2 - 8*a^2*b^2*d^2*f - 1/4*(2*a^6*b^4*d + a^7*b^3*g)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b^4*c^3 + 8*a*b^3*d^3
+ 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125
*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^
5))^(1/3) - 2*(1/2)^(2/3)*(10*b^2*c*d + a^2*f*g + (2*d*f + 5*c*g)*a*b)*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b^4*c^3
+ 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(
a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)
*a*b^3)/(a^8*b^5))^(1/3)))^2 - 2*(5*a^3*b*c + a^4*f)*g^2 + 1/2*(25*a^3*b^4*c^2 + 10*a^4*b^3*c*f + a^5*b^2*f^2)
*((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12
*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*
f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3) - 2*(1/2)^(2/3)*(10*b^2*c*d + a^2*f*g + (2
*d*f + 5*c*g)*a*b)*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 +
a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*
a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3))) - 8*(5*a^2*b^2*c*d + a^3*
b*d*f)*g + 2*(125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6
*a^3*b*d*g^2 + a^4*g^3)*x - 3/4*sqrt(1/3)*(50*a^3*b^4*c^2 + 20*a^4*b^3*c*f + 2*a^5*b^2*f^2 + (2*a^6*b^4*d + a^
7*b^3*g)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*
f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b +
3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3) - 2*(1/2)^(2/3)*(10*b^2*c*d + a^2*
f*g + (2*d*f + 5*c*g)*a*b)*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2
*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6
*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3))))*sqrt(-(((1/2)^(1
/3)*(I*sqrt(3) + 1)*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3 + 12*a^2*b^2*d
^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*(5*c*f^2 - 4*d^
2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3) - 2*(1/2)^(2/3)*(10*b^2*c*d + a^2*f*g + (2*d*f + 5*c
*g)*a*b)*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b^4*c^3 + 8*a*b^3*d^3 + 75*a*b^3*c^2*f + 15*a^2*b^2*c*f^2 + a^3*b*f^3
+ 12*a^2*b^2*d^2*g + 6*a^3*b*d*g^2 + a^4*g^3)/(a^8*b^5) + (125*b^4*c^3 - a^4*g^3 + (f^3 - 6*d*g^2)*a^3*b + 3*
(5*c*f^2 - 4*d^2*g)*a^2*b^2 - (8*d^3 - 75*c^2*f)*a*b^3)/(a^8*b^5))^(1/3)))^2*a^5*b^3 + 160*b^2*c*d + 32*a*b*d*
f + 16*(5*a*b*c + a^2*f)*g)/(a^5*b^3))))/(a^2*b^4*x^6 + 2*a^3*b^3*x^3 + a^4*b^2)

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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)/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.08976, size = 467, normalized size = 1.49 \begin{align*} -\frac{{\left (2 \, b d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, b c + a f\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^{3} b} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c + \left (-a b^{2}\right )^{\frac{1}{3}} a b f - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d - \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 )}{27 \, a^{3} b^{3}} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c + \left (-a b^{2}\right )^{\frac{1}{3}} a b f + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d + \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 )}{54 \, a^{3} b^{3}} + \frac{4 \, b^{3} d x^{5} + 2 \, a b^{2} g x^{5} + 5 \, b^{3} c x^{4} + a b^{2} f x^{4} - 6 \, a^{2} b h x^{3} + 7 \, a b^{2} d x^{2} - a^{2} b g x^{2} + 8 \, a b^{2} c x - 2 \, a^{2} b f x - 3 \, a^{3} h - 3 \, a^{2} b e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2} b^{2}} \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)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

-1/27*(2*b*d*(-a/b)^(1/3) + a*g*(-a/b)^(1/3) + 5*b*c + a*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*b) +
1/27*sqrt(3)*(5*(-a*b^2)^(1/3)*b^2*c + (-a*b^2)^(1/3)*a*b*f - 2*(-a*b^2)^(2/3)*b*d - (-a*b^2)^(2/3)*a*g)*arcta
n(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*c + (-a*b^2)^(1/3)*a*b
*f + 2*(-a*b^2)^(2/3)*b*d + (-a*b^2)^(2/3)*a*g)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^3) + 1/18*(4*b
^3*d*x^5 + 2*a*b^2*g*x^5 + 5*b^3*c*x^4 + a*b^2*f*x^4 - 6*a^2*b*h*x^3 + 7*a*b^2*d*x^2 - a^2*b*g*x^2 + 8*a*b^2*c
*x - 2*a^2*b*f*x - 3*a^3*h - 3*a^2*b*e)/((b*x^3 + a)^2*a^2*b^2)