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

Optimal. Leaf size=307 \[ \frac{x \left (13 a^2 b e-19 a^3 f-7 a b^2 d+b^3 c\right )}{18 a b^4 \left (a+b x^3\right )}-\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^2 b e+35 a^3 f+2 a b^2 d+b^3 c\right )}{54 a^{5/3} b^{13/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-14 a^2 b e+35 a^3 f+2 a b^2 d+b^3 c\right )}{27 a^{5/3} b^{13/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-14 a^2 b e+35 a^3 f+2 a b^2 d+b^3 c\right )}{9 \sqrt{3} a^{5/3} b^{13/3}}+\frac{x (b e-3 a f)}{b^4}+\frac{f x^4}{4 b^3} \]

[Out]

((b*e - 3*a*f)*x)/b^4 + (f*x^4)/(4*b^3) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*b^4*(a + b*x^3)^2) + ((b^
3*c - 7*a*b^2*d + 13*a^2*b*e - 19*a^3*f)*x)/(18*a*b^4*(a + b*x^3)) - ((b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3
*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(5/3)*b^(13/3)) + ((b^3*c + 2*a*b^2*d - 14
*a^2*b*e + 35*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(5/3)*b^(13/3)) - ((b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a
^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(5/3)*b^(13/3))

________________________________________________________________________________________

Rubi [A]  time = 0.411702, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1828, 1858, 1411, 388, 200, 31, 634, 617, 204, 628} \[ \frac{x \left (13 a^2 b e-19 a^3 f-7 a b^2 d+b^3 c\right )}{18 a b^4 \left (a+b x^3\right )}-\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^2 b e+35 a^3 f+2 a b^2 d+b^3 c\right )}{54 a^{5/3} b^{13/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-14 a^2 b e+35 a^3 f+2 a b^2 d+b^3 c\right )}{27 a^{5/3} b^{13/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-14 a^2 b e+35 a^3 f+2 a b^2 d+b^3 c\right )}{9 \sqrt{3} a^{5/3} b^{13/3}}+\frac{x (b e-3 a f)}{b^4}+\frac{f x^4}{4 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*e - 3*a*f)*x)/b^4 + (f*x^4)/(4*b^3) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*b^4*(a + b*x^3)^2) + ((b^
3*c - 7*a*b^2*d + 13*a^2*b*e - 19*a^3*f)*x)/(18*a*b^4*(a + b*x^3)) - ((b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3
*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(5/3)*b^(13/3)) + ((b^3*c + 2*a*b^2*d - 14
*a^2*b*e + 35*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(5/3)*b^(13/3)) - ((b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a
^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(5/3)*b^(13/3))

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 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 1411

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Simp[(c*x^(n + 1)*
(d + e*x^n)^(q + 1))/(e*(n*(q + 2) + 1)), x] + Dist[1/(e*(n*(q + 2) + 1)), Int[(d + e*x^n)^q*(a*e*(n*(q + 2) +
1) - (c*d*(n + 1) - b*e*(n*(q + 2) + 1))*x^n), x], x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && N
eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 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 )}{\left (a+b x^3\right )^3} \, dx &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^4 \left (a+b x^3\right )^2}-\frac{\int \frac{-a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-6 a b \left (b^2 d-a b e+a^2 f\right ) x^3-6 a b^2 (b e-a f) x^6-6 a b^3 f x^9}{\left (a+b x^3\right )^2} \, dx}{6 a b^4}\\ &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^4 \left (a+b x^3\right )^2}+\frac{\left (b^3 c-7 a b^2 d+13 a^2 b e-19 a^3 f\right ) x}{18 a b^4 \left (a+b x^3\right )}+\frac{\int \frac{2 a b^3 \left (b^3 c+2 a b^2 d-5 a^2 b e+8 a^3 f\right )+18 a^2 b^4 (b e-2 a f) x^3+18 a^2 b^5 f x^6}{a+b x^3} \, dx}{18 a^2 b^7}\\ &=\frac{f x^4}{4 b^3}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^4 \left (a+b x^3\right )^2}+\frac{\left (b^3 c-7 a b^2 d+13 a^2 b e-19 a^3 f\right ) x}{18 a b^4 \left (a+b x^3\right )}+\frac{\int \frac{8 a b^4 \left (b^3 c+2 a b^2 d-5 a^2 b e+8 a^3 f\right )-\left (72 a^3 b^5 f-72 a^2 b^5 (b e-2 a f)\right ) x^3}{a+b x^3} \, dx}{72 a^2 b^8}\\ &=\frac{(b e-3 a f) x}{b^4}+\frac{f x^4}{4 b^3}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^4 \left (a+b x^3\right )^2}+\frac{\left (b^3 c-7 a b^2 d+13 a^2 b e-19 a^3 f\right ) x}{18 a b^4 \left (a+b x^3\right )}+\frac{\left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\right ) \int \frac{1}{a+b x^3} \, dx}{9 a b^4}\\ &=\frac{(b e-3 a f) x}{b^4}+\frac{f x^4}{4 b^3}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^4 \left (a+b x^3\right )^2}+\frac{\left (b^3 c-7 a b^2 d+13 a^2 b e-19 a^3 f\right ) x}{18 a b^4 \left (a+b x^3\right )}+\frac{\left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{5/3} b^4}+\frac{\left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\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}{27 a^{5/3} b^4}\\ &=\frac{(b e-3 a f) x}{b^4}+\frac{f x^4}{4 b^3}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^4 \left (a+b x^3\right )^2}+\frac{\left (b^3 c-7 a b^2 d+13 a^2 b e-19 a^3 f\right ) x}{18 a b^4 \left (a+b x^3\right )}+\frac{\left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{13/3}}-\frac{\left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\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^{5/3} b^{13/3}}+\frac{\left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{4/3} b^4}\\ &=\frac{(b e-3 a f) x}{b^4}+\frac{f x^4}{4 b^3}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^4 \left (a+b x^3\right )^2}+\frac{\left (b^3 c-7 a b^2 d+13 a^2 b e-19 a^3 f\right ) x}{18 a b^4 \left (a+b x^3\right )}+\frac{\left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{13/3}}-\frac{\left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{13/3}}+\frac{\left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\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^{5/3} b^{13/3}}\\ &=\frac{(b e-3 a f) x}{b^4}+\frac{f x^4}{4 b^3}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^4 \left (a+b x^3\right )^2}+\frac{\left (b^3 c-7 a b^2 d+13 a^2 b e-19 a^3 f\right ) x}{18 a b^4 \left (a+b x^3\right )}-\frac{\left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{13/3}}+\frac{\left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{13/3}}-\frac{\left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{13/3}}\\ \end{align*}

Mathematica [A]  time = 0.201663, size = 294, normalized size = 0.96 \[ \frac{\frac{6 \sqrt [3]{b} x \left (13 a^2 b e-19 a^3 f-7 a b^2 d+b^3 c\right )}{a \left (a+b x^3\right )}-\frac{18 \sqrt [3]{b} x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\left (a+b x^3\right )^2}-\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^2 b e+35 a^3 f+2 a b^2 d+b^3 c\right )}{a^{5/3}}+\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-14 a^2 b e+35 a^3 f+2 a b^2 d+b^3 c\right )}{a^{5/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-14 a^2 b e+35 a^3 f+2 a b^2 d+b^3 c\right )}{a^{5/3}}+108 \sqrt [3]{b} x (b e-3 a f)+27 b^{4/3} f x^4}{108 b^{13/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(108*b^(1/3)*(b*e - 3*a*f)*x + 27*b^(4/3)*f*x^4 - (18*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a + b*x^
3)^2 + (6*b^(1/3)*(b^3*c - 7*a*b^2*d + 13*a^2*b*e - 19*a^3*f)*x)/(a*(a + b*x^3)) - (4*Sqrt[3]*(b^3*c + 2*a*b^2
*d - 14*a^2*b*e + 35*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (4*(b^3*c + 2*a*b^2*d - 14*
a^2*b*e + 35*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) - (2*(b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f)*Log[a^(
2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(108*b^(13/3))

________________________________________________________________________________________

Maple [B]  time = 0.011, size = 561, 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)^3,x)

[Out]

1/4*f*x^4/b^3-3/b^4*a*f*x+1/b^3*x*e-19/18/b^3/(b*x^3+a)^2*x^4*a^2*f+13/18/b^2/(b*x^3+a)^2*x^4*a*e-7/18/b/(b*x^
3+a)^2*x^4*d+1/18/(b*x^3+a)^2/a*x^4*c-8/9/b^4/(b*x^3+a)^2*a^3*f*x+5/9/b^3/(b*x^3+a)^2*a^2*e*x-2/9/b^2/(b*x^3+a
)^2*a*d*x-1/9/b/(b*x^3+a)^2*c*x+35/27/b^5*a^2/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*f-14/27/b^4*a/(1/b*a)^(2/3)*ln
(x+(1/b*a)^(1/3))*e+2/27/b^3/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*d+1/27/b^2/a/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*
c-35/54/b^5*a^2/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*f+7/27/b^4*a/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^
(1/3)*x+(1/b*a)^(2/3))*e-1/27/b^3/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*d-1/54/b^2/a/(1/b*a)^(2/
3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c+35/27/b^5*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-14/27/b^4*a/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*e+2/27/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+1/27/b^2/a/(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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.60752, size = 2716, normalized size = 8.85 \begin{align*} \text{result too large to display} \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)^3,x, algorithm="fricas")

[Out]

[1/108*(27*a^3*b^4*f*x^10 + 54*(2*a^3*b^4*e - 5*a^4*b^3*f)*x^7 + 3*(2*a^2*b^5*c - 14*a^3*b^4*d + 98*a^4*b^3*e
- 245*a^5*b^2*f)*x^4 + 6*sqrt(1/3)*(a^3*b^4*c + 2*a^4*b^3*d - 14*a^5*b^2*e + 35*a^6*b*f + (a*b^6*c + 2*a^2*b^5
*d - 14*a^3*b^4*e + 35*a^4*b^3*f)*x^6 + 2*(a^2*b^5*c + 2*a^3*b^4*d - 14*a^4*b^3*e + 35*a^5*b^2*f)*x^3)*sqrt(-(
a^2*b)^(1/3)/b)*log((2*a*b*x^3 - 3*(a^2*b)^(1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a^2*b)^(2/3)*x - (a^2*b
)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b))/(b*x^3 + a)) - 2*((b^5*c + 2*a*b^4*d - 14*a^2*b^3*e + 35*a^3*b^2*f)*x^6 + a
^2*b^3*c + 2*a^3*b^2*d - 14*a^4*b*e + 35*a^5*f + 2*(a*b^4*c + 2*a^2*b^3*d - 14*a^3*b^2*e + 35*a^4*b*f)*x^3)*(a
^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 4*((b^5*c + 2*a*b^4*d - 14*a^2*b^3*e + 35*a^3*b
^2*f)*x^6 + a^2*b^3*c + 2*a^3*b^2*d - 14*a^4*b*e + 35*a^5*f + 2*(a*b^4*c + 2*a^2*b^3*d - 14*a^3*b^2*e + 35*a^4
*b*f)*x^3)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)) - 12*(a^3*b^4*c + 2*a^4*b^3*d - 14*a^5*b^2*e + 35*a^6*b*f)
*x)/(a^3*b^7*x^6 + 2*a^4*b^6*x^3 + a^5*b^5), 1/108*(27*a^3*b^4*f*x^10 + 54*(2*a^3*b^4*e - 5*a^4*b^3*f)*x^7 + 3
*(2*a^2*b^5*c - 14*a^3*b^4*d + 98*a^4*b^3*e - 245*a^5*b^2*f)*x^4 + 12*sqrt(1/3)*(a^3*b^4*c + 2*a^4*b^3*d - 14*
a^5*b^2*e + 35*a^6*b*f + (a*b^6*c + 2*a^2*b^5*d - 14*a^3*b^4*e + 35*a^4*b^3*f)*x^6 + 2*(a^2*b^5*c + 2*a^3*b^4*
d - 14*a^4*b^3*e + 35*a^5*b^2*f)*x^3)*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*x - (a^2*b)^(1/3
)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - 2*((b^5*c + 2*a*b^4*d - 14*a^2*b^3*e + 35*a^3*b^2*f)*x^6 + a^2*b^3*c + 2*a^3
*b^2*d - 14*a^4*b*e + 35*a^5*f + 2*(a*b^4*c + 2*a^2*b^3*d - 14*a^3*b^2*e + 35*a^4*b*f)*x^3)*(a^2*b)^(2/3)*log(
a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 4*((b^5*c + 2*a*b^4*d - 14*a^2*b^3*e + 35*a^3*b^2*f)*x^6 + a^2*
b^3*c + 2*a^3*b^2*d - 14*a^4*b*e + 35*a^5*f + 2*(a*b^4*c + 2*a^2*b^3*d - 14*a^3*b^2*e + 35*a^4*b*f)*x^3)*(a^2*
b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)) - 12*(a^3*b^4*c + 2*a^4*b^3*d - 14*a^5*b^2*e + 35*a^6*b*f)*x)/(a^3*b^7*x^6
+ 2*a^4*b^6*x^3 + a^5*b^5)]

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

[Out]

Timed out

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Giac [A]  time = 1.09215, size = 495, normalized size = 1.61 \begin{align*} -\frac{{\left (b^{3} c + 2 \, a b^{2} d + 35 \, a^{3} f - 14 \, a^{2} b e\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^{2} b^{4}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f - 14 \, \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 )}{27 \, a^{2} b^{5}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f - 14 \, \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 )}{54 \, a^{2} b^{5}} + \frac{b^{4} c x^{4} - 7 \, a b^{3} d x^{4} - 19 \, a^{3} b f x^{4} + 13 \, a^{2} b^{2} x^{4} e - 2 \, a b^{3} c x - 4 \, a^{2} b^{2} d x - 16 \, a^{4} f x + 10 \, a^{3} b x e}{18 \,{\left (b x^{3} + a\right )}^{2} a b^{4}} + \frac{b^{9} f x^{4} - 12 \, a b^{8} f x + 4 \, b^{9} x e}{4 \, b^{12}} \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)^3,x, algorithm="giac")

[Out]

-1/27*(b^3*c + 2*a*b^2*d + 35*a^3*f - 14*a^2*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^4) + 1/27*sqr
t(3)*((-a*b^2)^(1/3)*b^3*c + 2*(-a*b^2)^(1/3)*a*b^2*d + 35*(-a*b^2)^(1/3)*a^3*f - 14*(-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))/(a^2*b^5) + 1/54*((-a*b^2)^(1/3)*b^3*c + 2*(-a*b^2)^(1/3)
*a*b^2*d + 35*(-a*b^2)^(1/3)*a^3*f - 14*(-a*b^2)^(1/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*
b^5) + 1/18*(b^4*c*x^4 - 7*a*b^3*d*x^4 - 19*a^3*b*f*x^4 + 13*a^2*b^2*x^4*e - 2*a*b^3*c*x - 4*a^2*b^2*d*x - 16*
a^4*f*x + 10*a^3*b*x*e)/((b*x^3 + a)^2*a*b^4) + 1/4*(b^9*f*x^4 - 12*a*b^8*f*x + 4*b^9*x*e)/b^12