3.295 \(\int \frac{c+d x^3+e x^6+f x^9}{x^2 (a+b x^3)^3} \, dx\)

Optimal. Leaf size=303 \[ -\frac{x^2 \left (-a^2 b e+4 a^3 f-2 a b^2 d+5 b^3 c\right )}{9 a^3 b^2 \left (a+b x^3\right )}-\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^2 b e-5 a^3 f-2 a b^2 d+14 b^3 c\right )}{54 a^{10/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e-5 a^3 f-2 a b^2 d+14 b^3 c\right )}{27 a^{10/3} b^{8/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^2 b e-5 a^3 f-2 a b^2 d+14 b^3 c\right )}{9 \sqrt{3} a^{10/3} b^{8/3}}-\frac{c}{a^3 x} \]

[Out]

-(c/(a^3*x)) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a^2*b^2*(a + b*x^3)^2) - ((5*b^3*c - 2*a*b^2*d - a
^2*b*e + 4*a^3*f)*x^2)/(9*a^3*b^2*(a + b*x^3)) + ((14*b^3*c - 2*a*b^2*d - a^2*b*e - 5*a^3*f)*ArcTan[(a^(1/3) -
2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(10/3)*b^(8/3)) + ((14*b^3*c - 2*a*b^2*d - a^2*b*e - 5*a^3*f)*L
og[a^(1/3) + b^(1/3)*x])/(27*a^(10/3)*b^(8/3)) - ((14*b^3*c - 2*a*b^2*d - a^2*b*e - 5*a^3*f)*Log[a^(2/3) - a^(
1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(10/3)*b^(8/3))

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Rubi [A]  time = 0.339039, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1829, 1484, 453, 292, 31, 634, 617, 204, 628} \[ -\frac{x^2 \left (-a^2 b e+4 a^3 f-2 a b^2 d+5 b^3 c\right )}{9 a^3 b^2 \left (a+b x^3\right )}-\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^2 b e-5 a^3 f-2 a b^2 d+14 b^3 c\right )}{54 a^{10/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e-5 a^3 f-2 a b^2 d+14 b^3 c\right )}{27 a^{10/3} b^{8/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^2 b e-5 a^3 f-2 a b^2 d+14 b^3 c\right )}{9 \sqrt{3} a^{10/3} b^{8/3}}-\frac{c}{a^3 x} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^2*(a + b*x^3)^3),x]

[Out]

-(c/(a^3*x)) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a^2*b^2*(a + b*x^3)^2) - ((5*b^3*c - 2*a*b^2*d - a
^2*b*e + 4*a^3*f)*x^2)/(9*a^3*b^2*(a + b*x^3)) + ((14*b^3*c - 2*a*b^2*d - a^2*b*e - 5*a^3*f)*ArcTan[(a^(1/3) -
2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(10/3)*b^(8/3)) + ((14*b^3*c - 2*a*b^2*d - a^2*b*e - 5*a^3*f)*L
og[a^(1/3) + b^(1/3)*x])/(27*a^(10/3)*b^(8/3)) - ((14*b^3*c - 2*a*b^2*d - a^2*b*e - 5*a^3*f)*Log[a^(2/3) - a^(
1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(10/3)*b^(8/3))

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 1484

Int[(x_)^(m_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_), x_Symbol] :> S
imp[((-d)^((m - Mod[m, n])/n - 1)*(c*d^2 - b*d*e + a*e^2)^p*x^(Mod[m, n] + 1)*(d + e*x^n)^(q + 1))/(n*e^(2*p +
(m - Mod[m, n])/n)*(q + 1)), x] + Dist[(-d)^((m - Mod[m, n])/n - 1)/(n*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^n)^
(q + 1)*ExpandToSum[Together[(1*(n*(-d)^(-((m - Mod[m, n])/n) + 1)*e^(2*p)*(q + 1)*(a + b*x^n + c*x^(2*n))^p -
((c*d^2 - b*d*e + a*e^2)^p/(e^((m - Mod[m, n])/n)*x^(m - Mod[m, n])))*(d*(Mod[m, n] + 1) + e*(Mod[m, n] + n*(
q + 1) + 1)*x^n)))/(d + e*x^n)], x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0
] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m, 0]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

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^2 \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^2}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\int \frac{-6 b^3 c+2 b \left (\frac{2 b^3 c}{a}-2 b^2 d-a b e+a^2 f\right ) x^3-6 a b^2 f x^6}{x^2 \left (a+b x^3\right )^2} \, dx}{6 a b^3}\\ &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) x^2}{9 a^3 b^2 \left (a+b x^3\right )}+\frac{\int \frac{18 a b^5 c-2 b^3 \left (5 b^3 c-2 a b^2 d-a^2 b e-5 a^3 f\right ) x^3}{x^2 \left (a+b x^3\right )} \, dx}{18 a^3 b^5}\\ &=-\frac{c}{a^3 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) x^2}{9 a^3 b^2 \left (a+b x^3\right )}-\frac{\left (14 b^3 c-2 a b^2 d-a^2 b e-5 a^3 f\right ) \int \frac{x}{a+b x^3} \, dx}{9 a^3 b^2}\\ &=-\frac{c}{a^3 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) x^2}{9 a^3 b^2 \left (a+b x^3\right )}+\frac{\left (14 b^3 c-2 a b^2 d-a^2 b e-5 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{10/3} b^{7/3}}-\frac{\left (14 b^3 c-2 a b^2 d-a^2 b e-5 a^3 f\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^{10/3} b^{7/3}}\\ &=-\frac{c}{a^3 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) x^2}{9 a^3 b^2 \left (a+b x^3\right )}+\frac{\left (14 b^3 c-2 a b^2 d-a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{8/3}}-\frac{\left (14 b^3 c-2 a b^2 d-a^2 b e-5 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^{10/3} b^{8/3}}-\frac{\left (14 b^3 c-2 a b^2 d-a^2 b e-5 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^3 b^{7/3}}\\ &=-\frac{c}{a^3 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) x^2}{9 a^3 b^2 \left (a+b x^3\right )}+\frac{\left (14 b^3 c-2 a b^2 d-a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{8/3}}-\frac{\left (14 b^3 c-2 a b^2 d-a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} b^{8/3}}-\frac{\left (14 b^3 c-2 a b^2 d-a^2 b e-5 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^{10/3} b^{8/3}}\\ &=-\frac{c}{a^3 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) x^2}{9 a^3 b^2 \left (a+b x^3\right )}+\frac{\left (14 b^3 c-2 a b^2 d-a^2 b e-5 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^{10/3} b^{8/3}}+\frac{\left (14 b^3 c-2 a b^2 d-a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{8/3}}-\frac{\left (14 b^3 c-2 a b^2 d-a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} b^{8/3}}\\ \end{align*}

Mathematica [A]  time = 0.222532, size = 286, normalized size = 0.94 \[ \frac{-\frac{6 \sqrt [3]{a} x^2 \left (-a^2 b e+4 a^3 f-2 a b^2 d+5 b^3 c\right )}{b^2 \left (a+b x^3\right )}+\frac{9 a^{4/3} x^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{b^2 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+5 a^3 f+2 a b^2 d-14 b^3 c\right )}{b^{8/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+5 a^3 f+2 a b^2 d-14 b^3 c\right )}{b^{8/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^2 b e-5 a^3 f-2 a b^2 d+14 b^3 c\right )}{b^{8/3}}-\frac{54 \sqrt [3]{a} c}{x}}{54 a^{10/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^2*(a + b*x^3)^3),x]

[Out]

((-54*a^(1/3)*c)/x + (9*a^(4/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(b^2*(a + b*x^3)^2) - (6*a^(1/3)*(
5*b^3*c - 2*a*b^2*d - a^2*b*e + 4*a^3*f)*x^2)/(b^2*(a + b*x^3)) + (2*Sqrt[3]*(14*b^3*c - 2*a*b^2*d - a^2*b*e -
5*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(8/3) - (2*(-14*b^3*c + 2*a*b^2*d + a^2*b*e + 5*a^3*f
)*Log[a^(1/3) + b^(1/3)*x])/b^(8/3) + ((-14*b^3*c + 2*a*b^2*d + a^2*b*e + 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/
3)*x + b^(2/3)*x^2])/b^(8/3))/(54*a^(10/3))

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Maple [B]  time = 0.014, size = 547, 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((f*x^9+e*x^6+d*x^3+c)/x^2/(b*x^3+a)^3,x)

[Out]

-4/9/(b*x^3+a)^2/b*x^5*f+1/9/a/(b*x^3+a)^2*x^5*e+2/9/a^2/(b*x^3+a)^2*b*x^5*d-5/9/a^3/(b*x^3+a)^2*b^2*x^5*c-5/1
8*a/(b*x^3+a)^2/b^2*x^2*f-1/18/(b*x^3+a)^2/b*x^2*e+7/18*d/a*x^2/(b*x^3+a)^2-13/18/a^2/(b*x^3+a)^2*b*x^2*c-5/27
/b^3/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*f-1/27/a/b^2/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*e-2/27*d/a^2/b/(1/b*a)^(
1/3)*ln(x+(1/b*a)^(1/3))+14/27/a^3/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*c+5/54/b^3/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(
1/3)*x+(1/b*a)^(2/3))*f+1/54/a/b^2/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*e+1/27*d/a^2/b/(1/b*a)^
(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-7/27/a^3/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c+5/2
7/b^3*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*f+1/27/a/b^2*3^(1/2)/(1/b*a)^(1/3)*arcta
n(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*e+2/27*d/a^2*3^(1/2)/b/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*
x-1))-14/27/a^3*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c-c/a^3/x

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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^2/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.50544, size = 2591, normalized size = 8.55 \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^2/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

[-1/54*(54*a^3*b^4*c + 6*(14*a*b^6*c - 2*a^2*b^5*d - a^3*b^4*e + 4*a^4*b^3*f)*x^6 + 3*(49*a^2*b^5*c - 7*a^3*b^
4*d + a^4*b^3*e + 5*a^5*b^2*f)*x^3 + 3*sqrt(1/3)*((14*a*b^6*c - 2*a^2*b^5*d - a^3*b^4*e - 5*a^4*b^3*f)*x^7 + 2
*(14*a^2*b^5*c - 2*a^3*b^4*d - a^4*b^3*e - 5*a^5*b^2*f)*x^4 + (14*a^3*b^4*c - 2*a^4*b^3*d - a^5*b^2*e - 5*a^6*
b*f)*x)*sqrt((-a*b^2)^(1/3)/a)*log((2*b^2*x^3 - a*b + 3*sqrt(1/3)*(a*b*x + 2*(-a*b^2)^(2/3)*x^2 + (-a*b^2)^(1/
3)*a)*sqrt((-a*b^2)^(1/3)/a) - 3*(-a*b^2)^(2/3)*x)/(b*x^3 + a)) + ((14*b^5*c - 2*a*b^4*d - a^2*b^3*e - 5*a^3*b
^2*f)*x^7 + 2*(14*a*b^4*c - 2*a^2*b^3*d - a^3*b^2*e - 5*a^4*b*f)*x^4 + (14*a^2*b^3*c - 2*a^3*b^2*d - a^4*b*e -
5*a^5*f)*x)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 2*((14*b^5*c - 2*a*b^4*d - a^
2*b^3*e - 5*a^3*b^2*f)*x^7 + 2*(14*a*b^4*c - 2*a^2*b^3*d - a^3*b^2*e - 5*a^4*b*f)*x^4 + (14*a^2*b^3*c - 2*a^3*
b^2*d - a^4*b*e - 5*a^5*f)*x)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^4*b^6*x^7 + 2*a^5*b^5*x^4 + a^6*b^4
*x), -1/54*(54*a^3*b^4*c + 6*(14*a*b^6*c - 2*a^2*b^5*d - a^3*b^4*e + 4*a^4*b^3*f)*x^6 + 3*(49*a^2*b^5*c - 7*a^
3*b^4*d + a^4*b^3*e + 5*a^5*b^2*f)*x^3 + 6*sqrt(1/3)*((14*a*b^6*c - 2*a^2*b^5*d - a^3*b^4*e - 5*a^4*b^3*f)*x^7
+ 2*(14*a^2*b^5*c - 2*a^3*b^4*d - a^4*b^3*e - 5*a^5*b^2*f)*x^4 + (14*a^3*b^4*c - 2*a^4*b^3*d - a^5*b^2*e - 5*
a^6*b*f)*x)*sqrt(-(-a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*b*x + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3)/a)/b) + ((1
4*b^5*c - 2*a*b^4*d - a^2*b^3*e - 5*a^3*b^2*f)*x^7 + 2*(14*a*b^4*c - 2*a^2*b^3*d - a^3*b^2*e - 5*a^4*b*f)*x^4
+ (14*a^2*b^3*c - 2*a^3*b^2*d - a^4*b*e - 5*a^5*f)*x)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^
2)^(2/3)) - 2*((14*b^5*c - 2*a*b^4*d - a^2*b^3*e - 5*a^3*b^2*f)*x^7 + 2*(14*a*b^4*c - 2*a^2*b^3*d - a^3*b^2*e
- 5*a^4*b*f)*x^4 + (14*a^2*b^3*c - 2*a^3*b^2*d - a^4*b*e - 5*a^5*f)*x)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)
))/(a^4*b^6*x^7 + 2*a^5*b^5*x^4 + a^6*b^4*x)]

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.11206, size = 525, normalized size = 1.73 \begin{align*} -\frac{c}{a^{3} x} + \frac{{\left (14 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 2 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{2} b \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^{4} b^{2}} + \frac{\sqrt{3}{\left (14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f - \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^{4} b^{4}} - \frac{10 \, b^{4} c x^{5} - 4 \, a b^{3} d x^{5} + 8 \, a^{3} b f x^{5} - 2 \, a^{2} b^{2} x^{5} e + 13 \, a b^{3} c x^{2} - 7 \, a^{2} b^{2} d x^{2} + 5 \, a^{4} f x^{2} + a^{3} b x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{3} b^{2}} - \frac{{\left (14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f - \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^{4} b^{4}} \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^2/(b*x^3+a)^3,x, algorithm="giac")

[Out]

-c/(a^3*x) + 1/27*(14*b^3*c*(-a/b)^(1/3) - 2*a*b^2*d*(-a/b)^(1/3) - 5*a^3*f*(-a/b)^(1/3) - a^2*b*(-a/b)^(1/3)*
e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^4*b^2) + 1/27*sqrt(3)*(14*(-a*b^2)^(2/3)*b^3*c - 2*(-a*b^2)^(2/3
)*a*b^2*d - 5*(-a*b^2)^(2/3)*a^3*f - (-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^4*b^4) - 1/18*(10*b^4*c*x^5 - 4*a*b^3*d*x^5 + 8*a^3*b*f*x^5 - 2*a^2*b^2*x^5*e + 13*a*b^3*c*x^2 - 7*a^2
*b^2*d*x^2 + 5*a^4*f*x^2 + a^3*b*x^2*e)/((b*x^3 + a)^2*a^3*b^2) - 1/54*(14*(-a*b^2)^(2/3)*b^3*c - 2*(-a*b^2)^(
2/3)*a*b^2*d - 5*(-a*b^2)^(2/3)*a^3*f - (-a*b^2)^(2/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*
b^4)