3.237 \(\int \frac{x^3 (c+d x^3+e x^6+f x^9)}{a+b x^3} \, dx\)

Optimal. Leaf size=274 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^{13/3}}+\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^4}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^{13/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} b^{13/3}}+\frac{x^4 \left (a^2 f-a b e+b^2 d\right )}{4 b^3}+\frac{x^7 (b e-a f)}{7 b^2}+\frac{f x^{10}}{10 b} \]

[Out]

((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^4 + ((b^2*d - a*b*e + a^2*f)*x^4)/(4*b^3) + ((b*e - a*f)*x^7)/(7*b^2
) + (f*x^10)/(10*b) + (a^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(
1/3))])/(Sqrt[3]*b^(13/3)) - (a^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(13/3
)) + (a^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(13/3))

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Rubi [A]  time = 0.26722, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {1836, 1488, 200, 31, 634, 617, 204, 628} \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^{13/3}}+\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^4}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^{13/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} b^{13/3}}+\frac{x^4 \left (a^2 f-a b e+b^2 d\right )}{4 b^3}+\frac{x^7 (b e-a f)}{7 b^2}+\frac{f x^{10}}{10 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^4 + ((b^2*d - a*b*e + a^2*f)*x^4)/(4*b^3) + ((b*e - a*f)*x^7)/(7*b^2
) + (f*x^10)/(10*b) + (a^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(
1/3))])/(Sqrt[3]*b^(13/3)) - (a^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(13/3
)) + (a^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(13/3))

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 1488

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Sy
mbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e,
f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

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{x^3 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx &=\frac{f x^{10}}{10 b}+\frac{\int \frac{x^3 \left (10 b c+10 b d x^3+10 (b e-a f) x^6\right )}{a+b x^3} \, dx}{10 b}\\ &=\frac{f x^{10}}{10 b}+\frac{\int \left (\frac{10 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{b^3}+\frac{10 \left (b^2 d-a b e+a^2 f\right ) x^3}{b^2}+\frac{10 (b e-a f) x^6}{b}+\frac{10 \left (-a b^3 c+a^2 b^2 d-a^3 b e+a^4 f\right )}{b^3 \left (a+b x^3\right )}\right ) \, dx}{10 b}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^4}{4 b^3}+\frac{(b e-a f) x^7}{7 b^2}+\frac{f x^{10}}{10 b}-\frac{\left (a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{1}{a+b x^3} \, dx}{b^4}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^4}{4 b^3}+\frac{(b e-a f) x^7}{7 b^2}+\frac{f x^{10}}{10 b}-\frac{\left (\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^4}-\frac{\left (\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 b^4}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^4}{4 b^3}+\frac{(b e-a f) x^7}{7 b^2}+\frac{f x^{10}}{10 b}-\frac{\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{13/3}}+\frac{\left (\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\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^{13/3}}-\frac{\left (a^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b^4}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^4}{4 b^3}+\frac{(b e-a f) x^7}{7 b^2}+\frac{f x^{10}}{10 b}-\frac{\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{13/3}}+\frac{\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{13/3}}-\frac{\left (\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\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^{13/3}}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^4}{4 b^3}+\frac{(b e-a f) x^7}{7 b^2}+\frac{f x^{10}}{10 b}+\frac{\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{13/3}}-\frac{\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{13/3}}+\frac{\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{13/3}}\\ \end{align*}

Mathematica [A]  time = 0.0911862, size = 264, normalized size = 0.96 \[ \frac{-70 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )+420 \sqrt [3]{b} x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )+140 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )-140 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )+105 b^{4/3} x^4 \left (a^2 f-a b e+b^2 d\right )+60 b^{7/3} x^7 (b e-a f)+42 b^{10/3} f x^{10}}{420 b^{13/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

(420*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x + 105*b^(4/3)*(b^2*d - a*b*e + a^2*f)*x^4 + 60*b^(7/3)*(b*e
- a*f)*x^7 + 42*b^(10/3)*f*x^10 - 140*Sqrt[3]*a^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(1 - (2*b
^(1/3)*x)/a^(1/3))/Sqrt[3]] + 140*a^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x] - 70
*a^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(420*b^(13/3))

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Maple [B]  time = 0.003, size = 492, 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(x^3*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x)

[Out]

1/10*f*x^10/b-1/7/b^2*x^7*a*f+1/7/b*x^7*e+1/4/b^3*x^4*a^2*f-1/4/b^2*x^4*a*e+1/4/b*x^4*d-1/b^4*a^3*f*x+1/b^3*a^
2*e*x-1/b^2*a*d*x+c*x/b+1/3*a^4/b^5/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*f-1/3*a^3/b^4/(1/b*a)^(2/3)*ln(x+(1/b*a)
^(1/3))*e+1/3*a^2/b^3/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*d-1/3*a/b^2/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*c-1/6*a^
4/b^5/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*f+1/6*a^3/b^4/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(
1/b*a)^(2/3))*e-1/6*a^2/b^3/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*d+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*a^4/b^5/(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^3/b^4/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*e+1/3*a^2/b^3/(1/b*a)^(2/3)*3^
(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*d-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

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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*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.29716, size = 566, normalized size = 2.07 \begin{align*} \frac{42 \, b^{3} f x^{10} + 60 \,{\left (b^{3} e - a b^{2} f\right )} x^{7} + 105 \,{\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{4} - 140 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (\frac{a}{b}\right )^{\frac{2}{3}} - \sqrt{3} a}{3 \, a}\right ) + 70 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 140 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 420 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x}{420 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="fricas")

[Out]

1/420*(42*b^3*f*x^10 + 60*(b^3*e - a*b^2*f)*x^7 + 105*(b^3*d - a*b^2*e + a^2*b*f)*x^4 - 140*sqrt(3)*(b^3*c - a
*b^2*d + a^2*b*e - a^3*f)*(a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(a/b)^(2/3) - sqrt(3)*a)/a) + 70*(b^3*c - a*b^
2*d + a^2*b*e - a^3*f)*(a/b)^(1/3)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3)) - 140*(b^3*c - a*b^2*d + a^2*b*e - a
^3*f)*(a/b)^(1/3)*log(x + (a/b)^(1/3)) + 420*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^4

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Sympy [A]  time = 1.71881, size = 371, normalized size = 1.35 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} b^{13} - a^{10} f^{3} + 3 a^{9} b e f^{2} - 3 a^{8} b^{2} d f^{2} - 3 a^{8} b^{2} e^{2} f + 3 a^{7} b^{3} c f^{2} + 6 a^{7} b^{3} d e f + a^{7} b^{3} e^{3} - 6 a^{6} b^{4} c e f - 3 a^{6} b^{4} d^{2} f - 3 a^{6} b^{4} d e^{2} + 6 a^{5} b^{5} c d f + 3 a^{5} b^{5} c e^{2} + 3 a^{5} b^{5} d^{2} e - 3 a^{4} b^{6} c^{2} f - 6 a^{4} b^{6} c d e - a^{4} b^{6} d^{3} + 3 a^{3} b^{7} c^{2} e + 3 a^{3} b^{7} c d^{2} - 3 a^{2} b^{8} c^{2} d + a b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{3 t b^{4}}{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac{f x^{10}}{10 b} - \frac{x^{7} \left (a f - b e\right )}{7 b^{2}} + \frac{x^{4} \left (a^{2} f - a b e + b^{2} d\right )}{4 b^{3}} - \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)

[Out]

RootSum(27*_t**3*b**13 - a**10*f**3 + 3*a**9*b*e*f**2 - 3*a**8*b**2*d*f**2 - 3*a**8*b**2*e**2*f + 3*a**7*b**3*
c*f**2 + 6*a**7*b**3*d*e*f + a**7*b**3*e**3 - 6*a**6*b**4*c*e*f - 3*a**6*b**4*d**2*f - 3*a**6*b**4*d*e**2 + 6*
a**5*b**5*c*d*f + 3*a**5*b**5*c*e**2 + 3*a**5*b**5*d**2*e - 3*a**4*b**6*c**2*f - 6*a**4*b**6*c*d*e - a**4*b**6
*d**3 + 3*a**3*b**7*c**2*e + 3*a**3*b**7*c*d**2 - 3*a**2*b**8*c**2*d + a*b**9*c**3, Lambda(_t, _t*log(3*_t*b**
4/(a**3*f - a**2*b*e + a*b**2*d - b**3*c) + x))) + f*x**10/(10*b) - x**7*(a*f - b*e)/(7*b**2) + x**4*(a**2*f -
a*b*e + b**2*d)/(4*b**3) - x*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/b**4

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Giac [A]  time = 1.08937, size = 467, normalized size = 1.7 \begin{align*} -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} 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 )}{3 \, b^{5}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} 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 )}{6 \, b^{5}} + \frac{{\left (a b^{9} c - a^{2} b^{8} d - a^{4} b^{6} f + a^{3} b^{7} e\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^{10}} + \frac{14 \, b^{9} f x^{10} - 20 \, a b^{8} f x^{7} + 20 \, b^{9} x^{7} e + 35 \, b^{9} d x^{4} + 35 \, a^{2} b^{7} f x^{4} - 35 \, a b^{8} x^{4} e + 140 \, b^{9} c x - 140 \, a b^{8} d x - 140 \, a^{3} b^{6} f x + 140 \, a^{2} b^{7} x e}{140 \, b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*((-a*b^2)^(1/3)*b^3*c - (-a*b^2)^(1/3)*a*b^2*d - (-a*b^2)^(1/3)*a^3*f + (-a*b^2)^(1/3)*a^2*b*e)*a
rctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^5 - 1/6*((-a*b^2)^(1/3)*b^3*c - (-a*b^2)^(1/3)*a*b^2*d
- (-a*b^2)^(1/3)*a^3*f + (-a*b^2)^(1/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^5 + 1/3*(a*b^9*c -
a^2*b^8*d - a^4*b^6*f + a^3*b^7*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^10) + 1/140*(14*b^9*f*x^10 -
20*a*b^8*f*x^7 + 20*b^9*x^7*e + 35*b^9*d*x^4 + 35*a^2*b^7*f*x^4 - 35*a*b^8*x^4*e + 140*b^9*c*x - 140*a*b^8*d*x
- 140*a^3*b^6*f*x + 140*a^2*b^7*x*e)/b^10