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

Optimal. Leaf size=301 \[ -\frac{x \left (-a^2 b e+7 a^3 f-5 a b^2 d+11 b^3 c\right )}{18 a^3 b^2 \left (a+b x^3\right )}-\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^2 b e-2 a^3 f-5 a b^2 d+20 b^3 c\right )}{54 a^{11/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e-2 a^3 f-5 a b^2 d+20 b^3 c\right )}{27 a^{11/3} b^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^2 b e-2 a^3 f-5 a b^2 d+20 b^3 c\right )}{9 \sqrt{3} a^{11/3} b^{7/3}}-\frac{c}{2 a^3 x^2} \]

[Out]

-c/(2*a^3*x^2) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a^2*b^2*(a + b*x^3)^2) - ((11*b^3*c - 5*a*b^2*d -
a^2*b*e + 7*a^3*f)*x)/(18*a^3*b^2*(a + b*x^3)) + ((20*b^3*c - 5*a*b^2*d - a^2*b*e - 2*a^3*f)*ArcTan[(a^(1/3) -
2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)*b^(7/3)) - ((20*b^3*c - 5*a*b^2*d - a^2*b*e - 2*a^3*f)*L
og[a^(1/3) + b^(1/3)*x])/(27*a^(11/3)*b^(7/3)) + ((20*b^3*c - 5*a*b^2*d - a^2*b*e - 2*a^3*f)*Log[a^(2/3) - a^(
1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(11/3)*b^(7/3))

________________________________________________________________________________________

Rubi [A]  time = 0.329154, antiderivative size = 301, 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, 200, 31, 634, 617, 204, 628} \[ -\frac{x \left (-a^2 b e+7 a^3 f-5 a b^2 d+11 b^3 c\right )}{18 a^3 b^2 \left (a+b x^3\right )}-\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^2 b e-2 a^3 f-5 a b^2 d+20 b^3 c\right )}{54 a^{11/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e-2 a^3 f-5 a b^2 d+20 b^3 c\right )}{27 a^{11/3} b^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^2 b e-2 a^3 f-5 a b^2 d+20 b^3 c\right )}{9 \sqrt{3} a^{11/3} b^{7/3}}-\frac{c}{2 a^3 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(2*a^3*x^2) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a^2*b^2*(a + b*x^3)^2) - ((11*b^3*c - 5*a*b^2*d -
a^2*b*e + 7*a^3*f)*x)/(18*a^3*b^2*(a + b*x^3)) + ((20*b^3*c - 5*a*b^2*d - a^2*b*e - 2*a^3*f)*ArcTan[(a^(1/3) -
2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)*b^(7/3)) - ((20*b^3*c - 5*a*b^2*d - a^2*b*e - 2*a^3*f)*L
og[a^(1/3) + b^(1/3)*x])/(27*a^(11/3)*b^(7/3)) + ((20*b^3*c - 5*a*b^2*d - a^2*b*e - 2*a^3*f)*Log[a^(2/3) - a^(
1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(11/3)*b^(7/3))

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 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^3 \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 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\int \frac{-6 b^3 c+b \left (\frac{5 b^3 c}{a}-5 b^2 d-a b e+a^2 f\right ) x^3-6 a b^2 f x^6}{x^3 \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}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\left (11 b^3 c-5 a b^2 d-a^2 b e+7 a^3 f\right ) x}{18 a^3 b^2 \left (a+b x^3\right )}+\frac{\int \frac{18 a b^5 c-2 b^3 \left (11 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) x^3}{x^3 \left (a+b x^3\right )} \, dx}{18 a^3 b^5}\\ &=-\frac{c}{2 a^3 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\left (11 b^3 c-5 a b^2 d-a^2 b e+7 a^3 f\right ) x}{18 a^3 b^2 \left (a+b x^3\right )}-\frac{\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \int \frac{1}{a+b x^3} \, dx}{9 a^3 b^2}\\ &=-\frac{c}{2 a^3 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\left (11 b^3 c-5 a b^2 d-a^2 b e+7 a^3 f\right ) x}{18 a^3 b^2 \left (a+b x^3\right )}-\frac{\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{11/3} b^2}-\frac{\left (20 b^3 c-5 a b^2 d-a^2 b e-2 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^{11/3} b^2}\\ &=-\frac{c}{2 a^3 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\left (11 b^3 c-5 a b^2 d-a^2 b e+7 a^3 f\right ) x}{18 a^3 b^2 \left (a+b x^3\right )}-\frac{\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} b^{7/3}}+\frac{\left (20 b^3 c-5 a b^2 d-a^2 b e-2 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^{11/3} b^{7/3}}-\frac{\left (20 b^3 c-5 a b^2 d-a^2 b e-2 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^{10/3} b^2}\\ &=-\frac{c}{2 a^3 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\left (11 b^3 c-5 a b^2 d-a^2 b e+7 a^3 f\right ) x}{18 a^3 b^2 \left (a+b x^3\right )}-\frac{\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} b^{7/3}}+\frac{\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} b^{7/3}}-\frac{\left (20 b^3 c-5 a b^2 d-a^2 b e-2 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^{11/3} b^{7/3}}\\ &=-\frac{c}{2 a^3 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\left (11 b^3 c-5 a b^2 d-a^2 b e+7 a^3 f\right ) x}{18 a^3 b^2 \left (a+b x^3\right )}+\frac{\left (20 b^3 c-5 a b^2 d-a^2 b e-2 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^{11/3} b^{7/3}}-\frac{\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} b^{7/3}}+\frac{\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} b^{7/3}}\\ \end{align*}

Mathematica [A]  time = 0.213489, size = 283, normalized size = 0.94 \[ \frac{\frac{9 a^{5/3} x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{b^2 \left (a+b x^3\right )^2}-\frac{3 a^{2/3} x \left (-a^2 b e+7 a^3 f-5 a b^2 d+11 b^3 c\right )}{b^2 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+2 a^3 f+5 a b^2 d-20 b^3 c\right )}{b^{7/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+2 a^3 f+5 a b^2 d-20 b^3 c\right )}{b^{7/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^2 b e-2 a^3 f-5 a b^2 d+20 b^3 c\right )}{b^{7/3}}-\frac{27 a^{2/3} c}{x^2}}{54 a^{11/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-27*a^(2/3)*c)/x^2 + (9*a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(b^2*(a + b*x^3)^2) - (3*a^(2/3)*(
11*b^3*c - 5*a*b^2*d - a^2*b*e + 7*a^3*f)*x)/(b^2*(a + b*x^3)) + (2*Sqrt[3]*(20*b^3*c - 5*a*b^2*d - a^2*b*e -
2*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(7/3) + (2*(-20*b^3*c + 5*a*b^2*d + a^2*b*e + 2*a^3*f)
*Log[a^(1/3) + b^(1/3)*x])/b^(7/3) - ((-20*b^3*c + 5*a*b^2*d + a^2*b*e + 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3
)*x + b^(2/3)*x^2])/b^(7/3))/(54*a^(11/3))

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Maple [B]  time = 0.013, size = 539, 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^3/(b*x^3+a)^3,x)

[Out]

-7/18/(b*x^3+a)^2/b*x^4*f+1/18/a/(b*x^3+a)^2*x^4*e+5/18/a^2/(b*x^3+a)^2*b*x^4*d-11/18/a^3/(b*x^3+a)^2*b^2*x^4*
c-2/9*a/(b*x^3+a)^2/b^2*x*f-1/9/(b*x^3+a)^2/b*x*e+4/9/a/(b*x^3+a)^2*x*d-7/9/a^2/(b*x^3+a)^2*b*x*c+2/27/b^3/(1/
b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*f+1/27/a/b^2/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*e+5/27/a^2/b/(1/b*a)^(2/3)*ln(x+
(1/b*a)^(1/3))*d-20/27/a^3/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*c-1/27/b^3/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(
1/b*a)^(2/3))*f-1/54/a/b^2/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*e-5/54/a^2/b/(1/b*a)^(2/3)*ln(x
^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*d+10/27/a^3/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c+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))*f+1/27/a/b^2/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3
^(1/2)*(2/(1/b*a)^(1/3)*x-1))*e+5/27/a^2/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*d-2
0/27/a^3/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c-1/2*c/a^3/x^2

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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^3/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.84402, size = 2618, normalized size = 8.7 \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^3/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

[-1/54*(27*a^4*b^3*c + 3*(20*a^2*b^5*c - 5*a^3*b^4*d - a^4*b^3*e + 7*a^5*b^2*f)*x^6 + 6*(16*a^3*b^4*c - 4*a^4*
b^3*d + a^5*b^2*e + 2*a^6*b*f)*x^3 + 3*sqrt(1/3)*((20*a*b^6*c - 5*a^2*b^5*d - a^3*b^4*e - 2*a^4*b^3*f)*x^8 + 2
*(20*a^2*b^5*c - 5*a^3*b^4*d - a^4*b^3*e - 2*a^5*b^2*f)*x^5 + (20*a^3*b^4*c - 5*a^4*b^3*d - a^5*b^2*e - 2*a^6*
b*f)*x^2)*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)) - ((20*b^5*c - 5*a*b^4*d - a^2*b^3*e - 2*a^3*
b^2*f)*x^8 + 2*(20*a*b^4*c - 5*a^2*b^3*d - a^3*b^2*e - 2*a^4*b*f)*x^5 + (20*a^2*b^3*c - 5*a^3*b^2*d - a^4*b*e
- 2*a^5*f)*x^2)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 2*((20*b^5*c - 5*a*b^4*d - a^
2*b^3*e - 2*a^3*b^2*f)*x^8 + 2*(20*a*b^4*c - 5*a^2*b^3*d - a^3*b^2*e - 2*a^4*b*f)*x^5 + (20*a^2*b^3*c - 5*a^3*
b^2*d - a^4*b*e - 2*a^5*f)*x^2)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^5*b^5*x^8 + 2*a^6*b^4*x^5 + a^7*b
^3*x^2), -1/54*(27*a^4*b^3*c + 3*(20*a^2*b^5*c - 5*a^3*b^4*d - a^4*b^3*e + 7*a^5*b^2*f)*x^6 + 6*(16*a^3*b^4*c
- 4*a^4*b^3*d + a^5*b^2*e + 2*a^6*b*f)*x^3 + 6*sqrt(1/3)*((20*a*b^6*c - 5*a^2*b^5*d - a^3*b^4*e - 2*a^4*b^3*f)
*x^8 + 2*(20*a^2*b^5*c - 5*a^3*b^4*d - a^4*b^3*e - 2*a^5*b^2*f)*x^5 + (20*a^3*b^4*c - 5*a^4*b^3*d - a^5*b^2*e
- 2*a^6*b*f)*x^2)*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) - ((20*b^5*c - 5*a*b^4*d - a^2*b^3*e - 2*a^3*b^2*f)*x^8 + 2*(20*a*b^4*c - 5*a^2*b^3*d - a^3*b^2*e
- 2*a^4*b*f)*x^5 + (20*a^2*b^3*c - 5*a^3*b^2*d - a^4*b*e - 2*a^5*f)*x^2)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(
2/3)*x + (a^2*b)^(1/3)*a) + 2*((20*b^5*c - 5*a*b^4*d - a^2*b^3*e - 2*a^3*b^2*f)*x^8 + 2*(20*a*b^4*c - 5*a^2*b^
3*d - a^3*b^2*e - 2*a^4*b*f)*x^5 + (20*a^2*b^3*c - 5*a^3*b^2*d - a^4*b*e - 2*a^5*f)*x^2)*(a^2*b)^(2/3)*log(a*b
*x + (a^2*b)^(2/3)))/(a^5*b^5*x^8 + 2*a^6*b^4*x^5 + a^7*b^3*x^2)]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.084, size = 486, normalized size = 1.61 \begin{align*} \frac{{\left (20 \, b^{3} c - 5 \, a b^{2} d - 2 \, a^{3} f - 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^{4} b^{2}} - \frac{\sqrt{3}{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f - \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^{4} b^{3}} - \frac{{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f - \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^{4} b^{3}} - \frac{20 \, b^{4} c x^{6} - 5 \, a b^{3} d x^{6} + 7 \, a^{3} b f x^{6} - a^{2} b^{2} x^{6} e + 32 \, a b^{3} c x^{3} - 8 \, a^{2} b^{2} d x^{3} + 4 \, a^{4} f x^{3} + 2 \, a^{3} b x^{3} e + 9 \, a^{2} b^{2} c}{18 \,{\left (b x^{4} + a x\right )}^{2} a^{3} b^{2}} \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^3/(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/27*(20*b^3*c - 5*a*b^2*d - 2*a^3*f - a^2*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^4*b^2) - 1/27*sqrt(
3)*(20*(-a*b^2)^(1/3)*b^3*c - 5*(-a*b^2)^(1/3)*a*b^2*d - 2*(-a*b^2)^(1/3)*a^3*f - (-a*b^2)^(1/3)*a^2*b*e)*arct
an(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b^3) - 1/54*(20*(-a*b^2)^(1/3)*b^3*c - 5*(-a*b^2)^(1/3)
*a*b^2*d - 2*(-a*b^2)^(1/3)*a^3*f - (-a*b^2)^(1/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b^3)
- 1/18*(20*b^4*c*x^6 - 5*a*b^3*d*x^6 + 7*a^3*b*f*x^6 - a^2*b^2*x^6*e + 32*a*b^3*c*x^3 - 8*a^2*b^2*d*x^3 + 4*a
^4*f*x^3 + 2*a^3*b*x^3*e + 9*a^2*b^2*c)/((b*x^4 + a*x)^2*a^3*b^2)