3.302 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{12} (a+b x^3)^3} \, dx\)

Optimal. Leaf size=380 \[ \frac{b x \left (17 a^2 b e-11 a^3 f-23 a b^2 d+29 b^3 c\right )}{18 a^6 \left (a+b x^3\right )}+\frac{3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{2 a^6 x^2}+\frac{b x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2}-\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (44 a^2 b e-20 a^3 f-77 a b^2 d+119 b^3 c\right )}{54 a^{20/3}}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (44 a^2 b e-20 a^3 f-77 a b^2 d+119 b^3 c\right )}{27 a^{20/3}}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (44 a^2 b e-20 a^3 f-77 a b^2 d+119 b^3 c\right )}{9 \sqrt{3} a^{20/3}}-\frac{a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}+\frac{3 b c-a d}{8 a^4 x^8}-\frac{c}{11 a^3 x^{11}} \]

[Out]

-c/(11*a^3*x^11) + (3*b*c - a*d)/(8*a^4*x^8) - (6*b^2*c - 3*a*b*d + a^2*e)/(5*a^5*x^5) + (10*b^3*c - 6*a*b^2*d
+ 3*a^2*b*e - a^3*f)/(2*a^6*x^2) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a^5*(a + b*x^3)^2) + (b*(29*b
^3*c - 23*a*b^2*d + 17*a^2*b*e - 11*a^3*f)*x)/(18*a^6*(a + b*x^3)) - (b^(2/3)*(119*b^3*c - 77*a*b^2*d + 44*a^2
*b*e - 20*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(20/3)) + (b^(2/3)*(119*b^3*c
- 77*a*b^2*d + 44*a^2*b*e - 20*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(20/3)) - (b^(2/3)*(119*b^3*c - 77*a*b^
2*d + 44*a^2*b*e - 20*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(20/3))

________________________________________________________________________________________

Rubi [A]  time = 0.668966, antiderivative size = 380, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {1829, 1834, 200, 31, 634, 617, 204, 628} \[ \frac{b x \left (17 a^2 b e-11 a^3 f-23 a b^2 d+29 b^3 c\right )}{18 a^6 \left (a+b x^3\right )}+\frac{3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{2 a^6 x^2}+\frac{b x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2}-\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (44 a^2 b e-20 a^3 f-77 a b^2 d+119 b^3 c\right )}{54 a^{20/3}}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (44 a^2 b e-20 a^3 f-77 a b^2 d+119 b^3 c\right )}{27 a^{20/3}}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (44 a^2 b e-20 a^3 f-77 a b^2 d+119 b^3 c\right )}{9 \sqrt{3} a^{20/3}}-\frac{a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}+\frac{3 b c-a d}{8 a^4 x^8}-\frac{c}{11 a^3 x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^3),x]

[Out]

-c/(11*a^3*x^11) + (3*b*c - a*d)/(8*a^4*x^8) - (6*b^2*c - 3*a*b*d + a^2*e)/(5*a^5*x^5) + (10*b^3*c - 6*a*b^2*d
+ 3*a^2*b*e - a^3*f)/(2*a^6*x^2) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a^5*(a + b*x^3)^2) + (b*(29*b
^3*c - 23*a*b^2*d + 17*a^2*b*e - 11*a^3*f)*x)/(18*a^6*(a + b*x^3)) - (b^(2/3)*(119*b^3*c - 77*a*b^2*d + 44*a^2
*b*e - 20*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(20/3)) + (b^(2/3)*(119*b^3*c
- 77*a*b^2*d + 44*a^2*b*e - 20*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(20/3)) - (b^(2/3)*(119*b^3*c - 77*a*b^
2*d + 44*a^2*b*e - 20*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(20/3))

Rule 1829

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = Polynomi
alQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1
)*x^m*Pq, a + b*x^n, x], i}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[x^m*(a + b*x^n)^(p + 1)*Expand
ToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)*Coeff[R, x, i]*x^(i - m))/a, {i, 0, n - 1}], x], x], x] - S
imp[(x*R*(a + b*x^n)^(p + 1))/(a^2*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 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{c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^3} \, dx &=\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}-\frac{\int \frac{-6 b^3 c+6 b^3 \left (\frac{b c}{a}-d\right ) x^3-\frac{6 b^3 \left (b^2 c-a b d+a^2 e\right ) x^6}{a^2}+\frac{6 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}-\frac{5 b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{12}}{a^4}}{x^{12} \left (a+b x^3\right )^2} \, dx}{6 a b^3}\\ &=\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}+\frac{b \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x}{18 a^6 \left (a+b x^3\right )}+\frac{\int \frac{18 b^7 c-18 b^7 \left (\frac{2 b c}{a}-d\right ) x^3+18 b^7 \left (\frac{3 b^2 c}{a^2}-\frac{2 b d}{a}+e\right ) x^6-18 b^7 \left (\frac{4 b^3 c}{a^3}-\frac{3 b^2 d}{a^2}+\frac{2 b e}{a}-f\right ) x^9+\frac{2 b^8 \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x^{12}}{a^4}}{x^{12} \left (a+b x^3\right )} \, dx}{18 a^2 b^7}\\ &=\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}+\frac{b \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x}{18 a^6 \left (a+b x^3\right )}+\frac{\int \left (\frac{18 b^7 c}{a x^{12}}+\frac{18 b^7 (-3 b c+a d)}{a^2 x^9}+\frac{18 b^7 \left (6 b^2 c-3 a b d+a^2 e\right )}{a^3 x^6}+\frac{18 b^7 \left (-10 b^3 c+6 a b^2 d-3 a^2 b e+a^3 f\right )}{a^4 x^3}-\frac{2 b^8 \left (-119 b^3 c+77 a b^2 d-44 a^2 b e+20 a^3 f\right )}{a^4 \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^7}\\ &=-\frac{c}{11 a^3 x^{11}}+\frac{3 b c-a d}{8 a^4 x^8}-\frac{6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{2 a^6 x^2}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}+\frac{b \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x}{18 a^6 \left (a+b x^3\right )}+\frac{\left (b \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right )\right ) \int \frac{1}{a+b x^3} \, dx}{9 a^6}\\ &=-\frac{c}{11 a^3 x^{11}}+\frac{3 b c-a d}{8 a^4 x^8}-\frac{6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{2 a^6 x^2}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}+\frac{b \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x}{18 a^6 \left (a+b x^3\right )}+\frac{\left (b \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{20/3}}+\frac{\left (b \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 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}{27 a^{20/3}}\\ &=-\frac{c}{11 a^3 x^{11}}+\frac{3 b c-a d}{8 a^4 x^8}-\frac{6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{2 a^6 x^2}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}+\frac{b \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x}{18 a^6 \left (a+b x^3\right )}+\frac{b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{20/3}}-\frac{\left (b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 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}{54 a^{20/3}}+\frac{\left (b \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right )\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{19/3}}\\ &=-\frac{c}{11 a^3 x^{11}}+\frac{3 b c-a d}{8 a^4 x^8}-\frac{6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{2 a^6 x^2}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}+\frac{b \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x}{18 a^6 \left (a+b x^3\right )}+\frac{b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{20/3}}-\frac{b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{20/3}}+\frac{\left (b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 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 )}{9 a^{20/3}}\\ &=-\frac{c}{11 a^3 x^{11}}+\frac{3 b c-a d}{8 a^4 x^8}-\frac{6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{2 a^6 x^2}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}+\frac{b \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x}{18 a^6 \left (a+b x^3\right )}-\frac{b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 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^{20/3}}+\frac{b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{20/3}}-\frac{b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{20/3}}\\ \end{align*}

Mathematica [A]  time = 0.428823, size = 376, normalized size = 0.99 \[ \frac{b x \left (17 a^2 b e-11 a^3 f-23 a b^2 d+29 b^3 c\right )}{18 a^6 \left (a+b x^3\right )}+\frac{3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{2 a^6 x^2}+\frac{b x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2}+\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-44 a^2 b e+20 a^3 f+77 a b^2 d-119 b^3 c\right )}{54 a^{20/3}}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (44 a^2 b e-20 a^3 f-77 a b^2 d+119 b^3 c\right )}{27 a^{20/3}}+\frac{b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-44 a^2 b e+20 a^3 f+77 a b^2 d-119 b^3 c\right )}{9 \sqrt{3} a^{20/3}}-\frac{a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}+\frac{3 b c-a d}{8 a^4 x^8}-\frac{c}{11 a^3 x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^3),x]

[Out]

-c/(11*a^3*x^11) + (3*b*c - a*d)/(8*a^4*x^8) - (6*b^2*c - 3*a*b*d + a^2*e)/(5*a^5*x^5) + (10*b^3*c - 6*a*b^2*d
+ 3*a^2*b*e - a^3*f)/(2*a^6*x^2) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a^5*(a + b*x^3)^2) + (b*(29*b
^3*c - 23*a*b^2*d + 17*a^2*b*e - 11*a^3*f)*x)/(18*a^6*(a + b*x^3)) + (b^(2/3)*(-119*b^3*c + 77*a*b^2*d - 44*a^
2*b*e + 20*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(9*Sqrt[3]*a^(20/3)) + (b^(2/3)*(119*b^3*c - 77
*a*b^2*d + 44*a^2*b*e - 20*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(20/3)) + (b^(2/3)*(-119*b^3*c + 77*a*b^2*d
- 44*a^2*b*e + 20*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(20/3))

________________________________________________________________________________________

Maple [A]  time = 0.018, size = 651, 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((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^3,x)

[Out]

3/5/a^4/x^5*b*d-6/5/a^5/x^5*b^2*c+3/2/a^4/x^2*b*e-3/a^5/x^2*b^2*d+5/a^6/x^2*b^3*c-20/27/a^3*f/(1/b*a)^(2/3)*ln
(x+(1/b*a)^(1/3))+10/27/a^3*f/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+3/8/a^4/x^8*b*c-11/18*b^2/a^
3/(b*x^3+a)^2*x^4*f+17/18*b^3/a^4/(b*x^3+a)^2*x^4*e-1/11*c/a^3/x^11-23/18*b^4/a^5/(b*x^3+a)^2*x^4*d+29/18*b^5/
a^6/(b*x^3+a)^2*x^4*c-7/9*b/a^2/(b*x^3+a)^2*f*x-20/27/a^3*f/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a
)^(1/3)*x-1))+44/27*b/a^4*e/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-22/27*b/a^4*e/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)
*x+(1/b*a)^(2/3))-77/27*b^2/a^5*d/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))+77/54*b^2/a^5*d/(1/b*a)^(2/3)*ln(x^2-(1/b*
a)^(1/3)*x+(1/b*a)^(2/3))+119/27*b^3/a^6*c/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-119/54*b^3/a^6*c/(1/b*a)^(2/3)*ln
(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+10/9*b^2/a^3/(b*x^3+a)^2*e*x-13/9*b^3/a^4/(b*x^3+a)^2*d*x+16/9*b^4/a^5/(b*
x^3+a)^2*c*x-1/8/a^3/x^8*d-1/5/a^3/x^5*e-1/2/a^3/x^2*f-77/27*b^2/a^5*d/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2
)*(2/(1/b*a)^(1/3)*x-1))+119/27*b^3/a^6*c/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+44/2
7*b/a^4*e/(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((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.62499, size = 1543, normalized size = 4.06 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/11880*(660*(119*b^5*c - 77*a*b^4*d + 44*a^2*b^3*e - 20*a^3*b^2*f)*x^15 + 1056*(119*a*b^4*c - 77*a^2*b^3*d +
44*a^3*b^2*e - 20*a^4*b*f)*x^12 + 297*(119*a^2*b^3*c - 77*a^3*b^2*d + 44*a^4*b*e - 20*a^5*f)*x^9 - 54*(119*a^3
*b^2*c - 77*a^4*b*d + 44*a^5*e)*x^6 - 1080*a^5*c + 135*(17*a^4*b*c - 11*a^5*d)*x^3 - 440*sqrt(3)*((119*b^5*c -
77*a*b^4*d + 44*a^2*b^3*e - 20*a^3*b^2*f)*x^17 + 2*(119*a*b^4*c - 77*a^2*b^3*d + 44*a^3*b^2*e - 20*a^4*b*f)*x
^14 + (119*a^2*b^3*c - 77*a^3*b^2*d + 44*a^4*b*e - 20*a^5*f)*x^11)*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x*
(-b^2/a^2)^(2/3) - sqrt(3)*b)/b) + 220*((119*b^5*c - 77*a*b^4*d + 44*a^2*b^3*e - 20*a^3*b^2*f)*x^17 + 2*(119*a
*b^4*c - 77*a^2*b^3*d + 44*a^3*b^2*e - 20*a^4*b*f)*x^14 + (119*a^2*b^3*c - 77*a^3*b^2*d + 44*a^4*b*e - 20*a^5*
f)*x^11)*(-b^2/a^2)^(1/3)*log(b^2*x^2 + a*b*x*(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) - 440*((119*b^5*c - 77*
a*b^4*d + 44*a^2*b^3*e - 20*a^3*b^2*f)*x^17 + 2*(119*a*b^4*c - 77*a^2*b^3*d + 44*a^3*b^2*e - 20*a^4*b*f)*x^14
+ (119*a^2*b^3*c - 77*a^3*b^2*d + 44*a^4*b*e - 20*a^5*f)*x^11)*(-b^2/a^2)^(1/3)*log(b*x - a*(-b^2/a^2)^(1/3)))
/(a^6*b^2*x^17 + 2*a^7*b*x^14 + a^8*x^11)

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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((f*x**9+e*x**6+d*x**3+c)/x**12/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.07959, size = 594, normalized size = 1.56 \begin{align*} \frac{\sqrt{3}{\left (119 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 77 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 20 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 44 \, \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^{7}} - \frac{{\left (119 \, b^{4} c - 77 \, a b^{3} d - 20 \, a^{3} b f + 44 \, a^{2} b^{2} 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^{7}} + \frac{{\left (119 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 77 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 20 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 44 \, \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^{7}} + \frac{29 \, b^{5} c x^{4} - 23 \, a b^{4} d x^{4} - 11 \, a^{3} b^{2} f x^{4} + 17 \, a^{2} b^{3} x^{4} e + 32 \, a b^{4} c x - 26 \, a^{2} b^{3} d x - 14 \, a^{4} b f x + 20 \, a^{3} b^{2} x e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{6}} + \frac{2200 \, b^{3} c x^{9} - 1320 \, a b^{2} d x^{9} - 220 \, a^{3} f x^{9} + 660 \, a^{2} b x^{9} e - 528 \, a b^{2} c x^{6} + 264 \, a^{2} b d x^{6} - 88 \, a^{3} x^{6} e + 165 \, a^{2} b c x^{3} - 55 \, a^{3} d x^{3} - 40 \, a^{3} c}{440 \, a^{6} x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/27*sqrt(3)*(119*(-a*b^2)^(1/3)*b^3*c - 77*(-a*b^2)^(1/3)*a*b^2*d - 20*(-a*b^2)^(1/3)*a^3*f + 44*(-a*b^2)^(1/
3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/a^7 - 1/27*(119*b^4*c - 77*a*b^3*d - 20*a^3*
b*f + 44*a^2*b^2*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^7 + 1/54*(119*(-a*b^2)^(1/3)*b^3*c - 77*(-a*b^2)
^(1/3)*a*b^2*d - 20*(-a*b^2)^(1/3)*a^3*f + 44*(-a*b^2)^(1/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))
/a^7 + 1/18*(29*b^5*c*x^4 - 23*a*b^4*d*x^4 - 11*a^3*b^2*f*x^4 + 17*a^2*b^3*x^4*e + 32*a*b^4*c*x - 26*a^2*b^3*d
*x - 14*a^4*b*f*x + 20*a^3*b^2*x*e)/((b*x^3 + a)^2*a^6) + 1/440*(2200*b^3*c*x^9 - 1320*a*b^2*d*x^9 - 220*a^3*f
*x^9 + 660*a^2*b*x^9*e - 528*a*b^2*c*x^6 + 264*a^2*b*d*x^6 - 88*a^3*x^6*e + 165*a^2*b*c*x^3 - 55*a^3*d*x^3 - 4
0*a^3*c)/(a^6*x^11)