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

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

[Out]

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

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Rubi [A]  time = 0.988245, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {1836, 1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )}{6 b^{8/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{\sqrt{3} b^{8/3}}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )}{3 b^{8/3}}+\frac{x (b c-a f)}{b^2}+\frac{x^2 (b d-a g)}{2 b^2}+\frac{x^3 (b e-a h)}{3 b^2}-\frac{a (b e-a h) \log \left (a+b x^3\right )}{3 b^3}+\frac{f x^4}{4 b}+\frac{g x^5}{5 b}+\frac{h x^6}{6 b} \]

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),x]

[Out]

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

Rule 1836

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq =
Coeff[Pq, x, q]}, Dist[1/(b*(m + q + n*p + 1)), Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(Pq - Pqq*x^q) - a
*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x] + Simp[(Pqq*(c*x)^(m + q - n + 1)*(a + b*x^n)^(p + 1)
)/(b*c^(q - n + 1)*(m + q + n*p + 1)), x]] /; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || Integ
erQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 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 )}{a+b x^3} \, dx &=\frac{h x^6}{6 b}+\frac{\int \frac{x^3 \left (6 b c+6 b d x+6 (b e-a h) x^2+6 b f x^3+6 b g x^4\right )}{a+b x^3} \, dx}{6 b}\\ &=\frac{g x^5}{5 b}+\frac{h x^6}{6 b}+\frac{\int \frac{x^3 \left (30 b^2 c+30 b (b d-a g) x+30 b (b e-a h) x^2+30 b^2 f x^3\right )}{a+b x^3} \, dx}{30 b^2}\\ &=\frac{f x^4}{4 b}+\frac{g x^5}{5 b}+\frac{h x^6}{6 b}+\frac{\int \frac{x^3 \left (120 b^2 (b c-a f)+120 b^2 (b d-a g) x+120 b^2 (b e-a h) x^2\right )}{a+b x^3} \, dx}{120 b^3}\\ &=\frac{f x^4}{4 b}+\frac{g x^5}{5 b}+\frac{h x^6}{6 b}+\frac{\int \left (120 b (b c-a f)+120 b (b d-a g) x+120 b (b e-a h) x^2-\frac{120 \left (a b (b c-a f)+a b (b d-a g) x+a b (b e-a h) x^2\right )}{a+b x^3}\right ) \, dx}{120 b^3}\\ &=\frac{(b c-a f) x}{b^2}+\frac{(b d-a g) x^2}{2 b^2}+\frac{(b e-a h) x^3}{3 b^2}+\frac{f x^4}{4 b}+\frac{g x^5}{5 b}+\frac{h x^6}{6 b}-\frac{\int \frac{a b (b c-a f)+a b (b d-a g) x+a b (b e-a h) x^2}{a+b x^3} \, dx}{b^3}\\ &=\frac{(b c-a f) x}{b^2}+\frac{(b d-a g) x^2}{2 b^2}+\frac{(b e-a h) x^3}{3 b^2}+\frac{f x^4}{4 b}+\frac{g x^5}{5 b}+\frac{h x^6}{6 b}-\frac{\int \frac{a b (b c-a f)+a b (b d-a g) x}{a+b x^3} \, dx}{b^3}-\frac{(a (b e-a h)) \int \frac{x^2}{a+b x^3} \, dx}{b^2}\\ &=\frac{(b c-a f) x}{b^2}+\frac{(b d-a g) x^2}{2 b^2}+\frac{(b e-a h) x^3}{3 b^2}+\frac{f x^4}{4 b}+\frac{g x^5}{5 b}+\frac{h x^6}{6 b}-\frac{a (b e-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-a f)+a^{4/3} b (b d-a g)\right )+\sqrt [3]{b} \left (-a b^{4/3} (b c-a f)+a^{4/3} b (b d-a g)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{2/3} b^{10/3}}-\frac{\left (\sqrt [3]{a} \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{7/3}}\\ &=\frac{(b c-a f) x}{b^2}+\frac{(b d-a g) x^2}{2 b^2}+\frac{(b e-a h) x^3}{3 b^2}+\frac{f x^4}{4 b}+\frac{g x^5}{5 b}+\frac{h x^6}{6 b}-\frac{\sqrt [3]{a} \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}-\frac{a (b e-a h) \log \left (a+b x^3\right )}{3 b^3}-\frac{\left (a^{2/3} \left (b^{4/3} c+\sqrt [3]{a} b d-a \sqrt [3]{b} f-a^{4/3} g\right )\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b^{7/3}}+\frac{\left (\sqrt [3]{a} \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )\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}{6 b^{8/3}}\\ &=\frac{(b c-a f) x}{b^2}+\frac{(b d-a g) x^2}{2 b^2}+\frac{(b e-a h) x^3}{3 b^2}+\frac{f x^4}{4 b}+\frac{g x^5}{5 b}+\frac{h x^6}{6 b}-\frac{\sqrt [3]{a} \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}+\frac{\sqrt [3]{a} \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{8/3}}-\frac{a (b e-a h) \log \left (a+b x^3\right )}{3 b^3}-\frac{\left (\sqrt [3]{a} \left (b^{4/3} c+\sqrt [3]{a} b d-a \sqrt [3]{b} f-a^{4/3} g\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{b^{8/3}}\\ &=\frac{(b c-a f) x}{b^2}+\frac{(b d-a g) x^2}{2 b^2}+\frac{(b e-a h) x^3}{3 b^2}+\frac{f x^4}{4 b}+\frac{g x^5}{5 b}+\frac{h x^6}{6 b}+\frac{\sqrt [3]{a} \left (b^{4/3} c+\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 )}{\sqrt{3} b^{8/3}}-\frac{\sqrt [3]{a} \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}+\frac{\sqrt [3]{a} \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{8/3}}-\frac{a (b e-a h) \log \left (a+b x^3\right )}{3 b^3}\\ \end{align*}

Mathematica [A]  time = 0.238224, size = 299, normalized size = 0.96 \[ \frac{10 \sqrt [3]{a} \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} g-\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )-20 \sqrt [3]{a} \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} g-\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )-20 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} g-\sqrt [3]{a} b d+a \sqrt [3]{b} f-b^{4/3} c\right )+60 b x (b c-a f)+30 b x^2 (b d-a g)+20 b x^3 (b e-a h)+20 a (a h-b e) \log \left (a+b x^3\right )+15 b^2 f x^4+12 b^2 g x^5+10 b^2 h x^6}{60 b^3} \]

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),x]

[Out]

(60*b*(b*c - a*f)*x + 30*b*(b*d - a*g)*x^2 + 20*b*(b*e - a*h)*x^3 + 15*b^2*f*x^4 + 12*b^2*g*x^5 + 10*b^2*h*x^6
- 20*Sqrt[3]*a^(1/3)*b^(1/3)*(-(b^(4/3)*c) - 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]] - 20*a^(1/3)*b^(1/3)*(b^(4/3)*c - a^(1/3)*b*d - a*b^(1/3)*f + a^(4/3)*g)*Log[a^(1/3) + b^(1
/3)*x] + 10*a^(1/3)*b^(1/3)*(b^(4/3)*c - 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] + 20*a*(-(b*e) + a*h)*Log[a + b*x^3])/(60*b^3)

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Maple [B]  time = 0.005, size = 505, 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(x^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x)

[Out]

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

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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),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),x, algorithm="fricas")

[Out]

Timed out

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Sympy [B]  time = 29.5259, size = 842, normalized size = 2.69 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} b^{9} + t^{2} \left (- 27 a^{2} b^{6} h + 27 a b^{7} e\right ) + t \left (9 a^{4} b^{3} h^{2} - 18 a^{3} b^{4} e h + 9 a^{3} b^{4} f g - 9 a^{2} b^{5} c g - 9 a^{2} b^{5} d f + 9 a^{2} b^{5} e^{2} + 9 a b^{6} c d\right ) - a^{6} h^{3} + 3 a^{5} b e h^{2} - 3 a^{5} b f g h + a^{5} b g^{3} + 3 a^{4} b^{2} c g h + 3 a^{4} b^{2} d f h - 3 a^{4} b^{2} d g^{2} - 3 a^{4} b^{2} e^{2} h + 3 a^{4} b^{2} e f g - a^{4} b^{2} f^{3} - 3 a^{3} b^{3} c d h - 3 a^{3} b^{3} c e g + 3 a^{3} b^{3} c f^{2} + 3 a^{3} b^{3} d^{2} g - 3 a^{3} b^{3} d e f + a^{3} b^{3} e^{3} - 3 a^{2} b^{4} c^{2} f + 3 a^{2} b^{4} c d e - a^{2} b^{4} d^{3} + a b^{5} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} a b^{6} g - 9 t^{2} b^{7} d - 6 t a^{3} b^{3} g h + 6 t a^{2} b^{4} d h + 6 t a^{2} b^{4} e g + 3 t a^{2} b^{4} f^{2} - 6 t a b^{5} c f - 6 t a b^{5} d e + 3 t b^{6} c^{2} + a^{5} g h^{2} - a^{4} b d h^{2} - 2 a^{4} b e g h - a^{4} b f^{2} h + 2 a^{4} b f g^{2} + 2 a^{3} b^{2} c f h - 2 a^{3} b^{2} c g^{2} + 2 a^{3} b^{2} d e h - 4 a^{3} b^{2} d f g + a^{3} b^{2} e^{2} g + a^{3} b^{2} e f^{2} - a^{2} b^{3} c^{2} h + 4 a^{2} b^{3} c d g - 2 a^{2} b^{3} c e f + 2 a^{2} b^{3} d^{2} f - a^{2} b^{3} d e^{2} + a b^{4} c^{2} e - 2 a b^{4} c d^{2}}{a^{4} b g^{3} - 3 a^{3} b^{2} d g^{2} + a^{3} b^{2} f^{3} - 3 a^{2} b^{3} c f^{2} + 3 a^{2} b^{3} d^{2} g + 3 a b^{4} c^{2} f - a b^{4} d^{3} - b^{5} c^{3}} \right )} \right )\right )} + \frac{f x^{4}}{4 b} + \frac{g x^{5}}{5 b} + \frac{h x^{6}}{6 b} - \frac{x^{3} \left (a h - b e\right )}{3 b^{2}} - \frac{x^{2} \left (a g - b d\right )}{2 b^{2}} - \frac{x \left (a f - b c\right )}{b^{2}} \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),x)

[Out]

RootSum(27*_t**3*b**9 + _t**2*(-27*a**2*b**6*h + 27*a*b**7*e) + _t*(9*a**4*b**3*h**2 - 18*a**3*b**4*e*h + 9*a*
*3*b**4*f*g - 9*a**2*b**5*c*g - 9*a**2*b**5*d*f + 9*a**2*b**5*e**2 + 9*a*b**6*c*d) - a**6*h**3 + 3*a**5*b*e*h*
*2 - 3*a**5*b*f*g*h + a**5*b*g**3 + 3*a**4*b**2*c*g*h + 3*a**4*b**2*d*f*h - 3*a**4*b**2*d*g**2 - 3*a**4*b**2*e
**2*h + 3*a**4*b**2*e*f*g - a**4*b**2*f**3 - 3*a**3*b**3*c*d*h - 3*a**3*b**3*c*e*g + 3*a**3*b**3*c*f**2 + 3*a*
*3*b**3*d**2*g - 3*a**3*b**3*d*e*f + a**3*b**3*e**3 - 3*a**2*b**4*c**2*f + 3*a**2*b**4*c*d*e - a**2*b**4*d**3
+ a*b**5*c**3, Lambda(_t, _t*log(x + (9*_t**2*a*b**6*g - 9*_t**2*b**7*d - 6*_t*a**3*b**3*g*h + 6*_t*a**2*b**4*
d*h + 6*_t*a**2*b**4*e*g + 3*_t*a**2*b**4*f**2 - 6*_t*a*b**5*c*f - 6*_t*a*b**5*d*e + 3*_t*b**6*c**2 + a**5*g*h
**2 - a**4*b*d*h**2 - 2*a**4*b*e*g*h - a**4*b*f**2*h + 2*a**4*b*f*g**2 + 2*a**3*b**2*c*f*h - 2*a**3*b**2*c*g**
2 + 2*a**3*b**2*d*e*h - 4*a**3*b**2*d*f*g + a**3*b**2*e**2*g + a**3*b**2*e*f**2 - a**2*b**3*c**2*h + 4*a**2*b*
*3*c*d*g - 2*a**2*b**3*c*e*f + 2*a**2*b**3*d**2*f - a**2*b**3*d*e**2 + a*b**4*c**2*e - 2*a*b**4*c*d**2)/(a**4*
b*g**3 - 3*a**3*b**2*d*g**2 + a**3*b**2*f**3 - 3*a**2*b**3*c*f**2 + 3*a**2*b**3*d**2*g + 3*a*b**4*c**2*f - a*b
**4*d**3 - b**5*c**3)))) + f*x**4/(4*b) + g*x**5/(5*b) + h*x**6/(6*b) - x**3*(a*h - b*e)/(3*b**2) - x**2*(a*g
- b*d)/(2*b**2) - x*(a*f - b*c)/b**2

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Giac [A]  time = 1.09472, size = 477, normalized size = 1.52 \begin{align*} \frac{{\left (a^{2} h - a 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 - \left (-a b^{2}\right )^{\frac{1}{3}} a b f - \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 )}{3 \, b^{4}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b f + \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 )}{6 \, b^{4}} + \frac{10 \, b^{5} h x^{6} + 12 \, b^{5} g x^{5} + 15 \, b^{5} f x^{4} - 20 \, a b^{4} h x^{3} + 20 \, b^{5} x^{3} e + 30 \, b^{5} d x^{2} - 30 \, a b^{4} g x^{2} + 60 \, b^{5} c x - 60 \, a b^{4} f x}{60 \, b^{6}} + \frac{{\left (a b^{12} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{2} b^{11} g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a b^{12} c - a^{2} b^{11} f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{13}} \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),x, algorithm="giac")

[Out]

1/3*(a^2*h - a*b*e)*log(abs(b*x^3 + a))/b^3 - 1/3*sqrt(3)*((-a*b^2)^(1/3)*b^2*c - (-a*b^2)^(1/3)*a*b*f - (-a*b
^2)^(2/3)*b*d + (-a*b^2)^(2/3)*a*g)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^4 - 1/6*((-a*b^2)^
(1/3)*b^2*c - (-a*b^2)^(1/3)*a*b*f + (-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))/b^4 + 1/60*(10*b^5*h*x^6 + 12*b^5*g*x^5 + 15*b^5*f*x^4 - 20*a*b^4*h*x^3 + 20*b^5*x^3*e + 30*b^5*d*x^2
- 30*a*b^4*g*x^2 + 60*b^5*c*x - 60*a*b^4*f*x)/b^6 + 1/3*(a*b^12*d*(-a/b)^(1/3) - a^2*b^11*g*(-a/b)^(1/3) + a*
b^12*c - a^2*b^11*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^13)