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

Optimal. Leaf size=312 \[ \frac{x^4 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{4 b^4}-\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^{16/3}}-\frac{a x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^5}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^{16/3}}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} b^{16/3}}+\frac{x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}+\frac{x^{10} (b e-a f)}{10 b^2}+\frac{f x^{13}}{13 b} \]

[Out]

-((a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^5) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^4)/(4*b^4) + ((b^2*d
- a*b*e + a^2*f)*x^7)/(7*b^3) + ((b*e - a*f)*x^10)/(10*b^2) + (f*x^13)/(13*b) - (a^(4/3)*(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^(16/3)) + (a^(4/3)*(b^3*c - a*b^
2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(16/3)) - (a^(4/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*L
og[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(16/3))

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Rubi [A]  time = 0.297768, antiderivative size = 312, 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} \[ \frac{x^4 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{4 b^4}-\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^{16/3}}-\frac{a x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^5}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^{16/3}}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} b^{16/3}}+\frac{x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}+\frac{x^{10} (b e-a f)}{10 b^2}+\frac{f x^{13}}{13 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^5) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^4)/(4*b^4) + ((b^2*d
- a*b*e + a^2*f)*x^7)/(7*b^3) + ((b*e - a*f)*x^10)/(10*b^2) + (f*x^13)/(13*b) - (a^(4/3)*(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^(16/3)) + (a^(4/3)*(b^3*c - a*b^
2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(16/3)) - (a^(4/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*L
og[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(16/3))

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^6 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx &=\frac{f x^{13}}{13 b}+\frac{\int \frac{x^6 \left (13 b c+13 b d x^3+13 (b e-a f) x^6\right )}{a+b x^3} \, dx}{13 b}\\ &=\frac{f x^{13}}{13 b}+\frac{\int \left (-\frac{13 a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{b^4}+\frac{13 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3}{b^3}+\frac{13 \left (b^2 d-a b e+a^2 f\right ) x^6}{b^2}+\frac{13 (b e-a f) x^9}{b}-\frac{13 \left (-a^2 b^3 c+a^3 b^2 d-a^4 b e+a^5 f\right )}{b^4 \left (a+b x^3\right )}\right ) \, dx}{13 b}\\ &=-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4}{4 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^7}{7 b^3}+\frac{(b e-a f) x^{10}}{10 b^2}+\frac{f x^{13}}{13 b}+\frac{\left (a^2 \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^5}\\ &=-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4}{4 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^7}{7 b^3}+\frac{(b e-a f) x^{10}}{10 b^2}+\frac{f x^{13}}{13 b}+\frac{\left (a^{4/3} \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^5}+\frac{\left (a^{4/3} \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^5}\\ &=-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4}{4 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^7}{7 b^3}+\frac{(b e-a f) x^{10}}{10 b^2}+\frac{f x^{13}}{13 b}+\frac{a^{4/3} \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^{16/3}}-\frac{\left (a^{4/3} \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^{16/3}}+\frac{\left (a^{5/3} \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^5}\\ &=-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4}{4 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^7}{7 b^3}+\frac{(b e-a f) x^{10}}{10 b^2}+\frac{f x^{13}}{13 b}+\frac{a^{4/3} \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^{16/3}}-\frac{a^{4/3} \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^{16/3}}+\frac{\left (a^{4/3} \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^{16/3}}\\ &=-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4}{4 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^7}{7 b^3}+\frac{(b e-a f) x^{10}}{10 b^2}+\frac{f x^{13}}{13 b}-\frac{a^{4/3} \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^{16/3}}+\frac{a^{4/3} \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^{16/3}}-\frac{a^{4/3} \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^{16/3}}\\ \end{align*}

Mathematica [A]  time = 0.0934935, size = 306, normalized size = 0.98 \[ \frac{x^4 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{4 b^4}+\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{6 b^{16/3}}+\frac{a x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{b^5}-\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{3 b^{16/3}}+\frac{a^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{\sqrt{3} b^{16/3}}+\frac{x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}+\frac{x^{10} (b e-a f)}{10 b^2}+\frac{f x^{13}}{13 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/b^5 + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^4)/(4*b^4) + ((b^2*d
- a*b*e + a^2*f)*x^7)/(7*b^3) + ((b*e - a*f)*x^10)/(10*b^2) + (f*x^13)/(13*b) + (a^(4/3)*(-(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]])/(Sqrt[3]*b^(16/3)) - (a^(4/3)*(-(b^3*c) + a*b^
2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(16/3)) + (a^(4/3)*(-(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^(16/3))

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Maple [B]  time = 0.004, size = 544, 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^6*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x)

[Out]

-1/7/b^2*x^7*a*e-1/4/b^4*x^4*a^3*f-1/10/b^2*x^10*a*f+1/b^3*a^2*d*x-1/b^2*a*c*x+1/4/b^3*x^4*a^2*e-1/4/b^2*x^4*a
*d+1/b^5*a^4*f*x-1/b^4*a^3*e*x-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))*c+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))*d-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))*d+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))*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))*e-1/3*a^5/b^6/(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^4/b^5/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*e-1/3*a^3/b^4/(1/b*a)^(2/3)*l
n(x+(1/b*a)^(1/3))*d+1/3*a^2/b^3/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*c+1/6*a^5/b^6/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^
(1/3)*x+(1/b*a)^(2/3))*f-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))*e+1/10/b*x^10*e+1/7/b
*x^7*d+1/4/b*x^4*c-1/3*a^5/b^6/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*f+1/7/b^3*x^7*a^2*f+1/13*f*x^13/b

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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^6*(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 = 1.37597, size = 679, normalized size = 2.18 \begin{align*} \frac{420 \, b^{4} f x^{13} + 546 \,{\left (b^{4} e - a b^{3} f\right )} x^{10} + 780 \,{\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{7} + 1365 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{4} - 1820 \, \sqrt{3}{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} 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 ) + 910 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} 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 ) - 1820 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 5460 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x}{5460 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5460*(420*b^4*f*x^13 + 546*(b^4*e - a*b^3*f)*x^10 + 780*(b^4*d - a*b^3*e + a^2*b^2*f)*x^7 + 1365*(b^4*c - a*
b^3*d + a^2*b^2*e - a^3*b*f)*x^4 - 1820*sqrt(3)*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*(-a/b)^(1/3)*arctan(1/
3*(2*sqrt(3)*b*x*(-a/b)^(2/3) - sqrt(3)*a)/a) + 910*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*(-a/b)^(1/3)*log(x
^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3)) - 1820*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*(-a/b)^(1/3)*log(x - (-a/b)
^(1/3)) - 5460*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x)/b^5

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Sympy [A]  time = 1.82208, size = 411, normalized size = 1.32 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} b^{16} + a^{13} f^{3} - 3 a^{12} b e f^{2} + 3 a^{11} b^{2} d f^{2} + 3 a^{11} b^{2} e^{2} f - 3 a^{10} b^{3} c f^{2} - 6 a^{10} b^{3} d e f - a^{10} b^{3} e^{3} + 6 a^{9} b^{4} c e f + 3 a^{9} b^{4} d^{2} f + 3 a^{9} b^{4} d e^{2} - 6 a^{8} b^{5} c d f - 3 a^{8} b^{5} c e^{2} - 3 a^{8} b^{5} d^{2} e + 3 a^{7} b^{6} c^{2} f + 6 a^{7} b^{6} c d e + a^{7} b^{6} d^{3} - 3 a^{6} b^{7} c^{2} e - 3 a^{6} b^{7} c d^{2} + 3 a^{5} b^{8} c^{2} d - a^{4} b^{9} c^{3}, \left ( t \mapsto t \log{\left (- \frac{3 t b^{5}}{a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c} + x \right )} \right )\right )} + \frac{f x^{13}}{13 b} - \frac{x^{10} \left (a f - b e\right )}{10 b^{2}} + \frac{x^{7} \left (a^{2} f - a b e + b^{2} d\right )}{7 b^{3}} - \frac{x^{4} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{4 b^{4}} + \frac{x \left (a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(27*_t**3*b**16 + a**13*f**3 - 3*a**12*b*e*f**2 + 3*a**11*b**2*d*f**2 + 3*a**11*b**2*e**2*f - 3*a**10*b
**3*c*f**2 - 6*a**10*b**3*d*e*f - a**10*b**3*e**3 + 6*a**9*b**4*c*e*f + 3*a**9*b**4*d**2*f + 3*a**9*b**4*d*e**
2 - 6*a**8*b**5*c*d*f - 3*a**8*b**5*c*e**2 - 3*a**8*b**5*d**2*e + 3*a**7*b**6*c**2*f + 6*a**7*b**6*c*d*e + a**
7*b**6*d**3 - 3*a**6*b**7*c**2*e - 3*a**6*b**7*c*d**2 + 3*a**5*b**8*c**2*d - a**4*b**9*c**3, Lambda(_t, _t*log
(-3*_t*b**5/(a**4*f - a**3*b*e + a**2*b**2*d - a*b**3*c) + x))) + f*x**13/(13*b) - x**10*(a*f - b*e)/(10*b**2)
+ x**7*(a**2*f - a*b*e + b**2*d)/(7*b**3) - x**4*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(4*b**4) + x*(a**4*f
- a**3*b*e + a**2*b**2*d - a*b**3*c)/b**5

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Giac [A]  time = 1.07153, size = 541, normalized size = 1.73 \begin{align*} \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{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 )}{3 \, b^{6}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{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 )}{6 \, b^{6}} - \frac{{\left (a^{2} b^{11} c - a^{3} b^{10} d - a^{5} b^{8} f + a^{4} b^{9} 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^{13}} + \frac{140 \, b^{12} f x^{13} - 182 \, a b^{11} f x^{10} + 182 \, b^{12} x^{10} e + 260 \, b^{12} d x^{7} + 260 \, a^{2} b^{10} f x^{7} - 260 \, a b^{11} x^{7} e + 455 \, b^{12} c x^{4} - 455 \, a b^{11} d x^{4} - 455 \, a^{3} b^{9} f x^{4} + 455 \, a^{2} b^{10} x^{4} e - 1820 \, a b^{11} c x + 1820 \, a^{2} b^{10} d x + 1820 \, a^{4} b^{8} f x - 1820 \, a^{3} b^{9} x e}{1820 \, b^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(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)*a*b^3*c - (-a*b^2)^(1/3)*a^2*b^2*d - (-a*b^2)^(1/3)*a^4*f + (-a*b^2)^(1/3)*a^3*b*e
)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^6 + 1/6*((-a*b^2)^(1/3)*a*b^3*c - (-a*b^2)^(1/3)*a^2
*b^2*d - (-a*b^2)^(1/3)*a^4*f + (-a*b^2)^(1/3)*a^3*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^6 - 1/3*(a^
2*b^11*c - a^3*b^10*d - a^5*b^8*f + a^4*b^9*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^13) + 1/1820*(140*
b^12*f*x^13 - 182*a*b^11*f*x^10 + 182*b^12*x^10*e + 260*b^12*d*x^7 + 260*a^2*b^10*f*x^7 - 260*a*b^11*x^7*e + 4
55*b^12*c*x^4 - 455*a*b^11*d*x^4 - 455*a^3*b^9*f*x^4 + 455*a^2*b^10*x^4*e - 1820*a*b^11*c*x + 1820*a^2*b^10*d*
x + 1820*a^4*b^8*f*x - 1820*a^3*b^9*x*e)/b^13