3.303 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{14} (a+b x^3)^3} \, dx\)

Optimal. Leaf size=424 \[ -\frac{b^2 x^2 \left (11 a^2 b e-8 a^3 f-14 a b^2 d+17 b^3 c\right )}{9 a^7 \left (a+b x^3\right )}-\frac{b^2 x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^6 \left (a+b x^3\right )^2}+\frac{3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{4 a^6 x^4}-\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (65 a^2 b e-35 a^3 f-104 a b^2 d+152 b^3 c\right )}{54 a^{22/3}}-\frac{b \left (6 a^2 b e-3 a^3 f-10 a b^2 d+15 b^3 c\right )}{a^7 x}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (65 a^2 b e-35 a^3 f-104 a b^2 d+152 b^3 c\right )}{27 a^{22/3}}+\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (65 a^2 b e-35 a^3 f-104 a b^2 d+152 b^3 c\right )}{9 \sqrt{3} a^{22/3}}-\frac{a^2 e-3 a b d+6 b^2 c}{7 a^5 x^7}+\frac{3 b c-a d}{10 a^4 x^{10}}-\frac{c}{13 a^3 x^{13}} \]

[Out]

-c/(13*a^3*x^13) + (3*b*c - a*d)/(10*a^4*x^10) - (6*b^2*c - 3*a*b*d + a^2*e)/(7*a^5*x^7) + (10*b^3*c - 6*a*b^2
*d + 3*a^2*b*e - a^3*f)/(4*a^6*x^4) - (b*(15*b^3*c - 10*a*b^2*d + 6*a^2*b*e - 3*a^3*f))/(a^7*x) - (b^2*(b^3*c
- a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a^6*(a + b*x^3)^2) - (b^2*(17*b^3*c - 14*a*b^2*d + 11*a^2*b*e - 8*a^3*f)*
x^2)/(9*a^7*(a + b*x^3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*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^(22/3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*b*e - 35*a^3*f)*L
og[a^(1/3) + b^(1/3)*x])/(27*a^(22/3)) - (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*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^(22/3))

________________________________________________________________________________________

Rubi [A]  time = 0.853138, antiderivative size = 424, 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, 292, 31, 634, 617, 204, 628} \[ -\frac{b^2 x^2 \left (11 a^2 b e-8 a^3 f-14 a b^2 d+17 b^3 c\right )}{9 a^7 \left (a+b x^3\right )}-\frac{b^2 x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^6 \left (a+b x^3\right )^2}+\frac{3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{4 a^6 x^4}-\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (65 a^2 b e-35 a^3 f-104 a b^2 d+152 b^3 c\right )}{54 a^{22/3}}-\frac{b \left (6 a^2 b e-3 a^3 f-10 a b^2 d+15 b^3 c\right )}{a^7 x}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (65 a^2 b e-35 a^3 f-104 a b^2 d+152 b^3 c\right )}{27 a^{22/3}}+\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (65 a^2 b e-35 a^3 f-104 a b^2 d+152 b^3 c\right )}{9 \sqrt{3} a^{22/3}}-\frac{a^2 e-3 a b d+6 b^2 c}{7 a^5 x^7}+\frac{3 b c-a d}{10 a^4 x^{10}}-\frac{c}{13 a^3 x^{13}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^14*(a + b*x^3)^3),x]

[Out]

-c/(13*a^3*x^13) + (3*b*c - a*d)/(10*a^4*x^10) - (6*b^2*c - 3*a*b*d + a^2*e)/(7*a^5*x^7) + (10*b^3*c - 6*a*b^2
*d + 3*a^2*b*e - a^3*f)/(4*a^6*x^4) - (b*(15*b^3*c - 10*a*b^2*d + 6*a^2*b*e - 3*a^3*f))/(a^7*x) - (b^2*(b^3*c
- a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a^6*(a + b*x^3)^2) - (b^2*(17*b^3*c - 14*a*b^2*d + 11*a^2*b*e - 8*a^3*f)*
x^2)/(9*a^7*(a + b*x^3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*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^(22/3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*b*e - 35*a^3*f)*L
og[a^(1/3) + b^(1/3)*x])/(27*a^(22/3)) - (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*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^(22/3))

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 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(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^{14} \left (a+b x^3\right )^3} \, dx &=-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \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{6 b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{12}}{a^4}+\frac{4 b^5 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{15}}{a^5}}{x^{14} \left (a+b x^3\right )^2} \, dx}{6 a b^3}\\ &=-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac{b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac{\int \frac{18 b^8 c-18 b^8 \left (\frac{2 b c}{a}-d\right ) x^3+18 b^8 \left (\frac{3 b^2 c}{a^2}-\frac{2 b d}{a}+e\right ) x^6-18 b^8 \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{18 b^9 \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right ) x^{12}}{a^4}-\frac{2 b^{10} \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^{15}}{a^5}}{x^{14} \left (a+b x^3\right )} \, dx}{18 a^2 b^8}\\ &=-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac{b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac{\int \left (\frac{18 b^8 c}{a x^{14}}+\frac{18 b^8 (-3 b c+a d)}{a^2 x^{11}}+\frac{18 b^8 \left (6 b^2 c-3 a b d+a^2 e\right )}{a^3 x^8}+\frac{18 b^8 \left (-10 b^3 c+6 a b^2 d-3 a^2 b e+a^3 f\right )}{a^4 x^5}-\frac{18 b^9 \left (-15 b^3 c+10 a b^2 d-6 a^2 b e+3 a^3 f\right )}{a^5 x^2}+\frac{2 b^{10} \left (-152 b^3 c+104 a b^2 d-65 a^2 b e+35 a^3 f\right ) x}{a^5 \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^8}\\ &=-\frac{c}{13 a^3 x^{13}}+\frac{3 b c-a d}{10 a^4 x^{10}}-\frac{6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}-\frac{b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac{b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}-\frac{\left (b^2 \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right )\right ) \int \frac{x}{a+b x^3} \, dx}{9 a^7}\\ &=-\frac{c}{13 a^3 x^{13}}+\frac{3 b c-a d}{10 a^4 x^{10}}-\frac{6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}-\frac{b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac{b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac{\left (b^{5/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{22/3}}-\frac{\left (b^{5/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right )\right ) \int \frac{\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^{22/3}}\\ &=-\frac{c}{13 a^3 x^{13}}+\frac{3 b c-a d}{10 a^4 x^{10}}-\frac{6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}-\frac{b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac{b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac{b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{22/3}}-\frac{\left (b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 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^{22/3}}-\frac{\left (b^{5/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 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^7}\\ &=-\frac{c}{13 a^3 x^{13}}+\frac{3 b c-a d}{10 a^4 x^{10}}-\frac{6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}-\frac{b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac{b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac{b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{22/3}}-\frac{b^{4/3} \left (152 b^3 c-104 a b^2 d+65 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^{22/3}}-\frac{\left (b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 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^{22/3}}\\ &=-\frac{c}{13 a^3 x^{13}}+\frac{3 b c-a d}{10 a^4 x^{10}}-\frac{6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}-\frac{b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac{b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac{b^{4/3} \left (152 b^3 c-104 a b^2 d+65 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^{22/3}}+\frac{b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{22/3}}-\frac{b^{4/3} \left (152 b^3 c-104 a b^2 d+65 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^{22/3}}\\ \end{align*}

Mathematica [A]  time = 0.465972, size = 419, normalized size = 0.99 \[ \frac{b^2 x^2 \left (-11 a^2 b e+8 a^3 f+14 a b^2 d-17 b^3 c\right )}{9 a^7 \left (a+b x^3\right )}+\frac{b^2 x^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{6 a^6 \left (a+b x^3\right )^2}+\frac{3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{4 a^6 x^4}+\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-65 a^2 b e+35 a^3 f+104 a b^2 d-152 b^3 c\right )}{54 a^{22/3}}+\frac{b \left (-6 a^2 b e+3 a^3 f+10 a b^2 d-15 b^3 c\right )}{a^7 x}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (65 a^2 b e-35 a^3 f-104 a b^2 d+152 b^3 c\right )}{27 a^{22/3}}+\frac{b^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (65 a^2 b e-35 a^3 f-104 a b^2 d+152 b^3 c\right )}{9 \sqrt{3} a^{22/3}}-\frac{a^2 e-3 a b d+6 b^2 c}{7 a^5 x^7}+\frac{3 b c-a d}{10 a^4 x^{10}}-\frac{c}{13 a^3 x^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^14*(a + b*x^3)^3),x]

[Out]

-c/(13*a^3*x^13) + (3*b*c - a*d)/(10*a^4*x^10) - (6*b^2*c - 3*a*b*d + a^2*e)/(7*a^5*x^7) + (10*b^3*c - 6*a*b^2
*d + 3*a^2*b*e - a^3*f)/(4*a^6*x^4) + (b*(-15*b^3*c + 10*a*b^2*d - 6*a^2*b*e + 3*a^3*f))/(a^7*x) + (b^2*(-(b^3
*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(6*a^6*(a + b*x^3)^2) + (b^2*(-17*b^3*c + 14*a*b^2*d - 11*a^2*b*e + 8*a^
3*f)*x^2)/(9*a^7*(a + b*x^3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*b*e - 35*a^3*f)*ArcTan[(1 - (2*b^(1
/3)*x)/a^(1/3))/Sqrt[3]])/(9*Sqrt[3]*a^(22/3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*b*e - 35*a^3*f)*Lo
g[a^(1/3) + b^(1/3)*x])/(27*a^(22/3)) + (b^(4/3)*(-152*b^3*c + 104*a*b^2*d - 65*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^(22/3))

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Maple [A]  time = 0.022, size = 716, 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^14/(b*x^3+a)^3,x)

[Out]

8/9*b^3/a^4/(b*x^3+a)^2*x^5*f-11/9*b^4/a^5/(b*x^3+a)^2*x^5*e+14/9*b^5/a^6/(b*x^3+a)^2*x^5*d+3/10/a^4/x^10*b*c+
3/7/a^4/x^7*b*d-6/7/a^5/x^7*b^2*c+3/4/a^4/x^4*b*e-3/2/a^5/x^4*b^2*d+5/2/a^6/x^4*b^3*c+3*b/a^4/x*f-6*b^2/a^5/x*
e+10*b^3/a^6/x*d-15*b^4/a^7/x*c+104/27*b^3/a^6*d*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1
))-152/27*b^4/a^7*c*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-1/13*c/a^3/x^13-1/10/a^3/x
^10*d-1/7/a^3/x^7*e-1/4/a^3/x^4*f-104/27*b^3/a^6*d/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+52/27*b^3/a^6*d/(1/b*a)^(
1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+152/27*b^4/a^7*c/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-76/27*b^4/a^7*c/
(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-35/27*b/a^4*f/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+35/54*b/a^
4*f/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+65/27*b^2/a^5*e/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-65/5
4*b^2/a^5*e/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-37/18*b^5/a^6/(b*x^3+a)^2*x^2*c-25/18*b^3/a^4/
(b*x^3+a)^2*x^2*e+31/18*b^4/a^5/(b*x^3+a)^2*x^2*d-17/9*b^6/a^7/(b*x^3+a)^2*x^5*c+19/18*b^2/a^3/(b*x^3+a)^2*x^2
*f+35/27*b/a^4*f*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-65/27*b^2/a^5*e*3^(1/2)/(1/b*
a)^(1/3)*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^14/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.68566, size = 1638, normalized size = 3.86 \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^14/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

-1/49140*(5460*(152*b^6*c - 104*a*b^5*d + 65*a^2*b^4*e - 35*a^3*b^3*f)*x^18 + 9555*(152*a*b^5*c - 104*a^2*b^4*
d + 65*a^3*b^3*e - 35*a^4*b^2*f)*x^15 + 3510*(152*a^2*b^4*c - 104*a^3*b^3*d + 65*a^4*b^2*e - 35*a^5*b*f)*x^12
- 351*(152*a^3*b^3*c - 104*a^4*b^2*d + 65*a^5*b*e - 35*a^6*f)*x^9 + 3780*a^6*c + 108*(152*a^4*b^2*c - 104*a^5*
b*d + 65*a^6*e)*x^6 - 378*(19*a^5*b*c - 13*a^6*d)*x^3 + 1820*sqrt(3)*((152*b^6*c - 104*a*b^5*d + 65*a^2*b^4*e
- 35*a^3*b^3*f)*x^19 + 2*(152*a*b^5*c - 104*a^2*b^4*d + 65*a^3*b^3*e - 35*a^4*b^2*f)*x^16 + (152*a^2*b^4*c - 1
04*a^3*b^3*d + 65*a^4*b^2*e - 35*a^5*b*f)*x^13)*(-b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(-b/a)^(1/3) + 1/3*sqrt(3))
- 910*((152*b^6*c - 104*a*b^5*d + 65*a^2*b^4*e - 35*a^3*b^3*f)*x^19 + 2*(152*a*b^5*c - 104*a^2*b^4*d + 65*a^3*
b^3*e - 35*a^4*b^2*f)*x^16 + (152*a^2*b^4*c - 104*a^3*b^3*d + 65*a^4*b^2*e - 35*a^5*b*f)*x^13)*(-b/a)^(1/3)*lo
g(b*x^2 - a*x*(-b/a)^(2/3) - a*(-b/a)^(1/3)) + 1820*((152*b^6*c - 104*a*b^5*d + 65*a^2*b^4*e - 35*a^3*b^3*f)*x
^19 + 2*(152*a*b^5*c - 104*a^2*b^4*d + 65*a^3*b^3*e - 35*a^4*b^2*f)*x^16 + (152*a^2*b^4*c - 104*a^3*b^3*d + 65
*a^4*b^2*e - 35*a^5*b*f)*x^13)*(-b/a)^(1/3)*log(b*x + a*(-b/a)^(2/3)))/(a^7*b^2*x^19 + 2*a^8*b*x^16 + a^9*x^13
)

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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**14/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.1053, size = 717, normalized size = 1.69 \begin{align*} \frac{\sqrt{3}{\left (152 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 104 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 35 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 65 \, \left (-a b^{2}\right )^{\frac{2}{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^{8}} + \frac{{\left (152 \, b^{5} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 104 \, a b^{4} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 35 \, a^{3} b^{2} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 65 \, a^{2} b^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} 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^{8}} - \frac{{\left (152 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 104 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 35 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 65 \, \left (-a b^{2}\right )^{\frac{2}{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^{8}} - \frac{34 \, b^{6} c x^{5} - 28 \, a b^{5} d x^{5} - 16 \, a^{3} b^{3} f x^{5} + 22 \, a^{2} b^{4} x^{5} e + 37 \, a b^{5} c x^{2} - 31 \, a^{2} b^{4} d x^{2} - 19 \, a^{4} b^{2} f x^{2} + 25 \, a^{3} b^{3} x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{7}} - \frac{27300 \, b^{4} c x^{12} - 18200 \, a b^{3} d x^{12} - 5460 \, a^{3} b f x^{12} + 10920 \, a^{2} b^{2} x^{12} e - 4550 \, a b^{3} c x^{9} + 2730 \, a^{2} b^{2} d x^{9} + 455 \, a^{4} f x^{9} - 1365 \, a^{3} b x^{9} e + 1560 \, a^{2} b^{2} c x^{6} - 780 \, a^{3} b d x^{6} + 260 \, a^{4} x^{6} e - 546 \, a^{3} b c x^{3} + 182 \, a^{4} d x^{3} + 140 \, a^{4} c}{1820 \, a^{7} x^{13}} \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^14/(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/27*sqrt(3)*(152*(-a*b^2)^(2/3)*b^3*c - 104*(-a*b^2)^(2/3)*a*b^2*d - 35*(-a*b^2)^(2/3)*a^3*f + 65*(-a*b^2)^(2
/3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/a^8 + 1/27*(152*b^5*c*(-a/b)^(1/3) - 104*a*
b^4*d*(-a/b)^(1/3) - 35*a^3*b^2*f*(-a/b)^(1/3) + 65*a^2*b^3*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1
/3)))/a^8 - 1/54*(152*(-a*b^2)^(2/3)*b^3*c - 104*(-a*b^2)^(2/3)*a*b^2*d - 35*(-a*b^2)^(2/3)*a^3*f + 65*(-a*b^2
)^(2/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/a^8 - 1/18*(34*b^6*c*x^5 - 28*a*b^5*d*x^5 - 16*a^3*b
^3*f*x^5 + 22*a^2*b^4*x^5*e + 37*a*b^5*c*x^2 - 31*a^2*b^4*d*x^2 - 19*a^4*b^2*f*x^2 + 25*a^3*b^3*x^2*e)/((b*x^3
+ a)^2*a^7) - 1/1820*(27300*b^4*c*x^12 - 18200*a*b^3*d*x^12 - 5460*a^3*b*f*x^12 + 10920*a^2*b^2*x^12*e - 4550
*a*b^3*c*x^9 + 2730*a^2*b^2*d*x^9 + 455*a^4*f*x^9 - 1365*a^3*b*x^9*e + 1560*a^2*b^2*c*x^6 - 780*a^3*b*d*x^6 +
260*a^4*x^6*e - 546*a^3*b*c*x^3 + 182*a^4*d*x^3 + 140*a^4*c)/(a^7*x^13)