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

Optimal. Leaf size=337 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (4 b e-7 a h)+b^{2/3} (2 b c-5 a f)\right )}{18 \sqrt [3]{a} b^{10/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (4 b e-7 a h)+b^{2/3} (2 b c-5 a f)\right )}{9 \sqrt [3]{a} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-4 a^{2/3} b e+7 a^{5/3} h-5 a b^{2/3} f+2 b^{5/3} c\right )}{3 \sqrt{3} \sqrt [3]{a} b^{10/3}}+\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 b^3 \left (a+b x^3\right )}+\frac{(b d-2 a g) \log \left (a+b x^3\right )}{3 b^3}+\frac{x (b e-2 a h)}{b^3}+\frac{f x^2}{2 b^2}+\frac{g x^3}{3 b^2}+\frac{h x^4}{4 b^2} \]

[Out]

((b*e - 2*a*h)*x)/b^3 + (f*x^2)/(2*b^2) + (g*x^3)/(3*b^2) + (h*x^4)/(4*b^2) + (x*(a*(b*e - a*h) - b*(b*c - a*f
)*x - b*(b*d - a*g)*x^2))/(3*b^3*(a + b*x^3)) - ((2*b^(5/3)*c - 4*a^(2/3)*b*e - 5*a*b^(2/3)*f + 7*a^(5/3)*h)*A
rcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(1/3)*b^(10/3)) - ((b^(2/3)*(2*b*c - 5*a*f) + a
^(2/3)*(4*b*e - 7*a*h))*Log[a^(1/3) + b^(1/3)*x])/(9*a^(1/3)*b^(10/3)) + ((b^(2/3)*(2*b*c - 5*a*f) + a^(2/3)*(
4*b*e - 7*a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(1/3)*b^(10/3)) + ((b*d - 2*a*g)*Log[a +
b*x^3])/(3*b^3)

________________________________________________________________________________________

Rubi [A]  time = 0.717088, antiderivative size = 337, 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} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (4 b e-7 a h)+b^{2/3} (2 b c-5 a f)\right )}{18 \sqrt [3]{a} b^{10/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (4 b e-7 a h)+b^{2/3} (2 b c-5 a f)\right )}{9 \sqrt [3]{a} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-4 a^{2/3} b e+7 a^{5/3} h-5 a b^{2/3} f+2 b^{5/3} c\right )}{3 \sqrt{3} \sqrt [3]{a} b^{10/3}}+\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 b^3 \left (a+b x^3\right )}+\frac{(b d-2 a g) \log \left (a+b x^3\right )}{3 b^3}+\frac{x (b e-2 a h)}{b^3}+\frac{f x^2}{2 b^2}+\frac{g x^3}{3 b^2}+\frac{h x^4}{4 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.397708, size = 334, normalized size = 0.99 \[ \frac{-\frac{12 b^{2/3} \left (a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right )}{a+b x^3}+\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (4 a^{2/3} b^{4/3} e-7 a^{5/3} \sqrt [3]{b} h-5 a b f+2 b^2 c\right )}{\sqrt [3]{a}}+\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-4 a^{2/3} b^{4/3} e+7 a^{5/3} \sqrt [3]{b} h+5 a b f-2 b^2 c\right )}{\sqrt [3]{a}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-4 a^{2/3} b^{4/3} e+7 a^{5/3} \sqrt [3]{b} h-5 a b f+2 b^2 c\right )}{\sqrt [3]{a}}+12 b^{2/3} (b d-2 a g) \log \left (a+b x^3\right )+36 b^{2/3} x (b e-2 a h)+18 b^{5/3} f x^2+12 b^{5/3} g x^3+9 b^{5/3} h x^4}{36 b^{11/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(36*b^(2/3)*(b*e - 2*a*h)*x + 18*b^(5/3)*f*x^2 + 12*b^(5/3)*g*x^3 + 9*b^(5/3)*h*x^4 - (12*b^(2/3)*(b^2*c*x^2 +
a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(a + b*x^3) - (4*Sqrt[3]*(2*b^2*c - 4*a^(2/3)*b^(4/3)*e - 5*a*b*f + 7
*a^(5/3)*b^(1/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (4*(-2*b^2*c - 4*a^(2/3)*b^(4/3)*e
+ 5*a*b*f + 7*a^(5/3)*b^(1/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + (2*(2*b^2*c + 4*a^(2/3)*b^(4/3)*e - 5*a*b
*f - 7*a^(5/3)*b^(1/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3) + 12*b^(2/3)*(b*d - 2*a*g)*L
og[a + b*x^3])/(36*b^(11/3))

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Maple [B]  time = 0.011, size = 562, 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^4*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^2,x)

[Out]

-7/18/b^4*a^2*h/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+1/3*g*x^3/b^2+1/4*h*x^4/b^2+1/2*f*x^2/b^2-
2/b^3*a*h*x-1/3/b*x^2/(b*x^3+a)*c-2/9/b^2*c/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+1/9/b^2*c/(1/b*a)^(1/3)*ln(x^2-(
1/b*a)^(1/3)*x+(1/b*a)^(2/3))+2/9/b^3*a/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*e-1/3/b^3/(b*x^3+a
)*a^2*g+1/3/b^2/(b*x^3+a)*d*a-2/3/b^3*ln(b*x^3+a)*a*g+5/9/b^3*a*f/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-5/18/b^3*a
*f/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+1/3/b^2*a*x^2/(b*x^3+a)*f+1/3/b^2*x*a/(b*x^3+a)*e-4/9/b
^3*a/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*e-4/9/b^3*a/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1
/3))*e+1/b^2*x*e+1/3/b^2*ln(b*x^3+a)*d+2/9/b^2*c*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1
))+7/9/b^4*a^2*h/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-5/9/b^3*a*f*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/
b*a)^(1/3)*x-1))-1/3/b^3/(b*x^3+a)*a^2*h*x+7/9/b^4*a^2*h/(1/b*a)^(2/3)*3^(1/2)*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(x^4*(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^4*(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**4*(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 = 1.09132, size = 512, normalized size = 1.52 \begin{align*} \frac{{\left (b d - 2 \, a g\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} + \frac{\sqrt{3}{\left (7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b e - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b c + 5 \, \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 b^{4}} + \frac{a b d - a^{2} g -{\left (b^{2} c - a b f\right )} x^{2} -{\left (a^{2} h - a b e\right )} x}{3 \,{\left (b x^{3} + a\right )} b^{3}} + \frac{{\left (7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b e + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b c - 5 \, \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 b^{4}} - \frac{{\left (2 \, b^{6} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, a b^{5} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 7 \, a^{2} b^{4} h - 4 \, a b^{5} 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 b^{7}} + \frac{3 \, b^{6} h x^{4} + 4 \, b^{6} g x^{3} + 6 \, b^{6} f x^{2} - 24 \, a b^{5} h x + 12 \, b^{6} x e}{12 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(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*(b*d - 2*a*g)*log(abs(b*x^3 + a))/b^3 + 1/9*sqrt(3)*(7*(-a*b^2)^(1/3)*a^2*h - 4*(-a*b^2)^(1/3)*a*b*e - 2*(
-a*b^2)^(2/3)*b*c + 5*(-a*b^2)^(2/3)*a*f)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^4) + 1/3*
(a*b*d - a^2*g - (b^2*c - a*b*f)*x^2 - (a^2*h - a*b*e)*x)/((b*x^3 + a)*b^3) + 1/18*(7*(-a*b^2)^(1/3)*a^2*h - 4
*(-a*b^2)^(1/3)*a*b*e + 2*(-a*b^2)^(2/3)*b*c - 5*(-a*b^2)^(2/3)*a*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/
(a*b^4) - 1/9*(2*b^6*c*(-a/b)^(1/3) - 5*a*b^5*f*(-a/b)^(1/3) + 7*a^2*b^4*h - 4*a*b^5*e)*(-a/b)^(1/3)*log(abs(x
- (-a/b)^(1/3)))/(a*b^7) + 1/12*(3*b^6*h*x^4 + 4*b^6*g*x^3 + 6*b^6*f*x^2 - 24*a*b^5*h*x + 12*b^6*x*e)/b^8