3.271 \(\int \frac{c+d x^3+e x^6+f x^9}{x^8 (a+b x^3)^2} \, dx\)

Optimal. Leaf size=297 \[ -\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^4 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (4 a^2 b e+a^3 (-f)-7 a b^2 d+10 b^3 c\right )}{18 a^{13/3} b^{2/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (4 a^2 b e+a^3 (-f)-7 a b^2 d+10 b^3 c\right )}{9 a^{13/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (4 a^2 b e+a^3 (-f)-7 a b^2 d+10 b^3 c\right )}{3 \sqrt{3} a^{13/3} b^{2/3}}-\frac{a^2 e-2 a b d+3 b^2 c}{a^4 x}+\frac{2 b c-a d}{4 a^3 x^4}-\frac{c}{7 a^2 x^7} \]

[Out]

-c/(7*a^2*x^7) + (2*b*c - a*d)/(4*a^3*x^4) - (3*b^2*c - 2*a*b*d + a^2*e)/(a^4*x) - ((b^3*c - a*b^2*d + a^2*b*e
- a^3*f)*x^2)/(3*a^4*(a + b*x^3)) + ((10*b^3*c - 7*a*b^2*d + 4*a^2*b*e - a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x
)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(13/3)*b^(2/3)) + ((10*b^3*c - 7*a*b^2*d + 4*a^2*b*e - a^3*f)*Log[a^(1/3) +
b^(1/3)*x])/(9*a^(13/3)*b^(2/3)) - ((10*b^3*c - 7*a*b^2*d + 4*a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*
x + b^(2/3)*x^2])/(18*a^(13/3)*b^(2/3))

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Rubi [A]  time = 0.383572, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^4 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (4 a^2 b e+a^3 (-f)-7 a b^2 d+10 b^3 c\right )}{18 a^{13/3} b^{2/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (4 a^2 b e+a^3 (-f)-7 a b^2 d+10 b^3 c\right )}{9 a^{13/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (4 a^2 b e+a^3 (-f)-7 a b^2 d+10 b^3 c\right )}{3 \sqrt{3} a^{13/3} b^{2/3}}-\frac{a^2 e-2 a b d+3 b^2 c}{a^4 x}+\frac{2 b c-a d}{4 a^3 x^4}-\frac{c}{7 a^2 x^7} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^8*(a + b*x^3)^2),x]

[Out]

-c/(7*a^2*x^7) + (2*b*c - a*d)/(4*a^3*x^4) - (3*b^2*c - 2*a*b*d + a^2*e)/(a^4*x) - ((b^3*c - a*b^2*d + a^2*b*e
- a^3*f)*x^2)/(3*a^4*(a + b*x^3)) + ((10*b^3*c - 7*a*b^2*d + 4*a^2*b*e - a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x
)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(13/3)*b^(2/3)) + ((10*b^3*c - 7*a*b^2*d + 4*a^2*b*e - a^3*f)*Log[a^(1/3) +
b^(1/3)*x])/(9*a^(13/3)*b^(2/3)) - ((10*b^3*c - 7*a*b^2*d + 4*a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*
x + b^(2/3)*x^2])/(18*a^(13/3)*b^(2/3))

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^8 \left (a+b x^3\right )^2} \, dx &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^4 \left (a+b x^3\right )}-\frac{\int \frac{-3 b^3 c+3 b^3 \left (\frac{b c}{a}-d\right ) x^3-\frac{3 b^3 \left (b^2 c-a b d+a^2 e\right ) x^6}{a^2}+\frac{b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}}{x^8 \left (a+b x^3\right )} \, dx}{3 a b^3}\\ &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^4 \left (a+b x^3\right )}-\frac{\int \left (-\frac{3 b^3 c}{a x^8}-\frac{3 b^3 (-2 b c+a d)}{a^2 x^5}-\frac{3 b^3 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^2}-\frac{b^3 \left (-10 b^3 c+7 a b^2 d-4 a^2 b e+a^3 f\right ) x}{a^3 \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac{c}{7 a^2 x^7}+\frac{2 b c-a d}{4 a^3 x^4}-\frac{3 b^2 c-2 a b d+a^2 e}{a^4 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^4 \left (a+b x^3\right )}-\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-a^3 f\right ) \int \frac{x}{a+b x^3} \, dx}{3 a^4}\\ &=-\frac{c}{7 a^2 x^7}+\frac{2 b c-a d}{4 a^3 x^4}-\frac{3 b^2 c-2 a b d+a^2 e}{a^4 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^4 \left (a+b x^3\right )}+\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{13/3} \sqrt [3]{b}}-\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-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}{9 a^{13/3} \sqrt [3]{b}}\\ &=-\frac{c}{7 a^2 x^7}+\frac{2 b c-a d}{4 a^3 x^4}-\frac{3 b^2 c-2 a b d+a^2 e}{a^4 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^4 \left (a+b x^3\right )}+\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{13/3} b^{2/3}}-\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-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}{18 a^{13/3} b^{2/3}}-\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-a^3 f\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^4 \sqrt [3]{b}}\\ &=-\frac{c}{7 a^2 x^7}+\frac{2 b c-a d}{4 a^3 x^4}-\frac{3 b^2 c-2 a b d+a^2 e}{a^4 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^4 \left (a+b x^3\right )}+\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{13/3} b^{2/3}}-\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{13/3} b^{2/3}}-\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-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 )}{3 a^{13/3} b^{2/3}}\\ &=-\frac{c}{7 a^2 x^7}+\frac{2 b c-a d}{4 a^3 x^4}-\frac{3 b^2 c-2 a b d+a^2 e}{a^4 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^4 \left (a+b x^3\right )}+\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{13/3} b^{2/3}}+\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{13/3} b^{2/3}}-\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{13/3} b^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.179237, size = 281, normalized size = 0.95 \[ \frac{\frac{84 \sqrt [3]{a} x^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a+b x^3}+\frac{14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-4 a^2 b e+a^3 f+7 a b^2 d-10 b^3 c\right )}{b^{2/3}}+\frac{28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (4 a^2 b e+a^3 (-f)-7 a b^2 d+10 b^3 c\right )}{b^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (4 a^2 b e+a^3 (-f)-7 a b^2 d+10 b^3 c\right )}{b^{2/3}}-\frac{252 \sqrt [3]{a} \left (a^2 e-2 a b d+3 b^2 c\right )}{x}-\frac{63 a^{4/3} (a d-2 b c)}{x^4}-\frac{36 a^{7/3} c}{x^7}}{252 a^{13/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^8*(a + b*x^3)^2),x]

[Out]

((-36*a^(7/3)*c)/x^7 - (63*a^(4/3)*(-2*b*c + a*d))/x^4 - (252*a^(1/3)*(3*b^2*c - 2*a*b*d + a^2*e))/x + (84*a^(
1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(a + b*x^3) + (28*Sqrt[3]*(10*b^3*c - 7*a*b^2*d + 4*a^2*b*e -
a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (28*(10*b^3*c - 7*a*b^2*d + 4*a^2*b*e - a^3*f)*
Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (14*(-10*b^3*c + 7*a*b^2*d - 4*a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1
/3)*x + b^(2/3)*x^2])/b^(2/3))/(252*a^(13/3))

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Maple [B]  time = 0.015, size = 529, 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^8/(b*x^3+a)^2,x)

[Out]

1/3/a*x^2/(b*x^3+a)*f-1/3/a^2*x^2/(b*x^3+a)*b*e+1/3/a^3*x^2/(b*x^3+a)*b^2*d-1/3/a^4*x^2/(b*x^3+a)*b^3*c+4/9/a^
2*e/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-2/9/a^2*e/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-4/9/a^2*e*
3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-7/9/a^3*b*d/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+
7/18/a^3*b*d/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+7/9/a^3*b*d*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*
3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+10/9/a^4*b^2*c/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-5/9/a^4*b^2*c/(1/b*a)^(1/3)*ln
(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-10/9/a^4*b^2*c*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x
-1))-1/9/a*f/b/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+1/18/a*f/b/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3)
)+1/9/a*f*3^(1/2)/b/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-1/7*c/a^2/x^7-1/4/a^2/x^4*d+1/2/a^
3/x^4*b*c-e/a^2/x+2/a^3/x*b*d-3/a^4/x*b^2*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((f*x^9+e*x^6+d*x^3+c)/x^8/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.46461, size = 2159, normalized size = 7.27 \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^8/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

[-1/252*(84*(10*a*b^5*c - 7*a^2*b^4*d + 4*a^3*b^3*e - a^4*b^2*f)*x^9 + 36*a^4*b^2*c + 63*(10*a^2*b^4*c - 7*a^3
*b^3*d + 4*a^4*b^2*e)*x^6 - 9*(10*a^3*b^3*c - 7*a^4*b^2*d)*x^3 + 42*sqrt(1/3)*((10*a*b^5*c - 7*a^2*b^4*d + 4*a
^3*b^3*e - a^4*b^2*f)*x^10 + (10*a^2*b^4*c - 7*a^3*b^3*d + 4*a^4*b^2*e - a^5*b*f)*x^7)*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*((10*b^4*c - 7*a*b^3*d + 4*a^2*b^2*e - a^3*b*f)*x^10 + (10*a*b^3*c - 7*a
^2*b^2*d + 4*a^3*b*e - a^4*f)*x^7)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 28*((10
*b^4*c - 7*a*b^3*d + 4*a^2*b^2*e - a^3*b*f)*x^10 + (10*a*b^3*c - 7*a^2*b^2*d + 4*a^3*b*e - a^4*f)*x^7)*(-a*b^2
)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^5*b^3*x^10 + a^6*b^2*x^7), -1/252*(84*(10*a*b^5*c - 7*a^2*b^4*d + 4*a^3*
b^3*e - a^4*b^2*f)*x^9 + 36*a^4*b^2*c + 63*(10*a^2*b^4*c - 7*a^3*b^3*d + 4*a^4*b^2*e)*x^6 - 9*(10*a^3*b^3*c -
7*a^4*b^2*d)*x^3 + 84*sqrt(1/3)*((10*a*b^5*c - 7*a^2*b^4*d + 4*a^3*b^3*e - a^4*b^2*f)*x^10 + (10*a^2*b^4*c - 7
*a^3*b^3*d + 4*a^4*b^2*e - a^5*b*f)*x^7)*sqrt(-(-a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*b*x + (-a*b^2)^(1/3))*sqr
t(-(-a*b^2)^(1/3)/a)/b) + 14*((10*b^4*c - 7*a*b^3*d + 4*a^2*b^2*e - a^3*b*f)*x^10 + (10*a*b^3*c - 7*a^2*b^2*d
+ 4*a^3*b*e - a^4*f)*x^7)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 28*((10*b^4*c -
7*a*b^3*d + 4*a^2*b^2*e - a^3*b*f)*x^10 + (10*a*b^3*c - 7*a^2*b^2*d + 4*a^3*b*e - a^4*f)*x^7)*(-a*b^2)^(2/3)*l
og(b*x - (-a*b^2)^(1/3)))/(a^5*b^3*x^10 + a^6*b^2*x^7)]

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

[Out]

Timed out

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Giac [A]  time = 1.11736, size = 522, normalized size = 1.76 \begin{align*} \frac{{\left (10 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 7 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 4 \, 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 )}{9 \, a^{5}} - \frac{b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e}{3 \,{\left (b x^{3} + a\right )} a^{4}} + \frac{\sqrt{3}{\left (10 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 4 \, \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 )}{9 \, a^{5} b^{2}} - \frac{{\left (10 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 4 \, \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 )}{18 \, a^{5} b^{2}} - \frac{84 \, b^{2} c x^{6} - 56 \, a b d x^{6} + 28 \, a^{2} x^{6} e - 14 \, a b c x^{3} + 7 \, a^{2} d x^{3} + 4 \, a^{2} c}{28 \, a^{4} x^{7}} \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^8/(b*x^3+a)^2,x, algorithm="giac")

[Out]

1/9*(10*b^3*c*(-a/b)^(1/3) - 7*a*b^2*d*(-a/b)^(1/3) - a^3*f*(-a/b)^(1/3) + 4*a^2*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3
)*log(abs(x - (-a/b)^(1/3)))/a^5 - 1/3*(b^3*c*x^2 - a*b^2*d*x^2 - a^3*f*x^2 + a^2*b*x^2*e)/((b*x^3 + a)*a^4) +
1/9*sqrt(3)*(10*(-a*b^2)^(2/3)*b^3*c - 7*(-a*b^2)^(2/3)*a*b^2*d - (-a*b^2)^(2/3)*a^3*f + 4*(-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^5*b^2) - 1/18*(10*(-a*b^2)^(2/3)*b^3*c - 7*(-a*
b^2)^(2/3)*a*b^2*d - (-a*b^2)^(2/3)*a^3*f + 4*(-a*b^2)^(2/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))
/(a^5*b^2) - 1/28*(84*b^2*c*x^6 - 56*a*b*d*x^6 + 28*a^2*x^6*e - 14*a*b*c*x^3 + 7*a^2*d*x^3 + 4*a^2*c)/(a^4*x^7
)