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

[Out]

-1/7*c/a^2/x^7+1/4*(-a*d+2*b*c)/a^3/x^4+(-a^2*e+2*a*b*d-3*b^2*c)/a^4/x-1/3*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x^2/
a^4/(b*x^3+a)+1/9*(-a^3*f+4*a^2*b*e-7*a*b^2*d+10*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(13/3)/b^(2/3)-1/18*(-a^3*f+4*
a^2*b*e-7*a*b^2*d+10*b^3*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(13/3)/b^(2/3)+1/9*(-a^3*f+4*a^2*b*e-7
*a*b^2*d+10*b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(13/3)/b^(2/3)*3^(1/2)

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Rubi [A]  time = 0.38, antiderivative size = 297, normalized size of antiderivative = 1.00, 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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 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 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.26, size = 281, normalized size = 0.95 \[ \frac {-\frac {63 a^{4/3} (a d-2 b c)}{x^4}-\frac {36 a^{7/3} c}{x^7}-\frac {252 \sqrt [3]{a} \left (a^2 e-2 a b d+3 b^2 c\right )}{x}+\frac {84 \sqrt [3]{a} x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a+b x^3}+\frac {28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+4 a^2 b e-7 a b^2 d+10 b^3 c\right )}{b^{2/3}}+\frac {28 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^3 (-f)+4 a^2 b e-7 a b^2 d+10 b^3 c\right )}{b^{2/3}}+\frac {14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-4 a^2 b e+7 a b^2 d-10 b^3 c\right )}{b^{2/3}}}{252 a^{13/3}} \]

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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fricas [A]  time = 0.68, size = 982, normalized size = 3.31 \[ \text {result too large to display} \]

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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giac [A]  time = 0.23, size = 333, normalized size = 1.12 \[ -\frac {\sqrt {3} {\left (10 \, b^{3} c - 7 \, a b^{2} d - a^{3} f + 4 \, a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{4}} + \frac {{\left (10 \, b^{3} c - 7 \, a b^{2} d - a^{3} f + 4 \, a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{4}} + \frac {{\left (10 \, b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 7 \, a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 4 \, a^{2} b \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{5}} - \frac {b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e}{3 \, {\left (b x^{3} + a\right )} a^{4}} - \frac {84 \, b^{2} c x^{6} - 56 \, a b d x^{6} + 28 \, a^{2} x^{6} e - 14 \, a b c x^{3} + 7 \, a^{2} d x^{3} + 4 \, a^{2} c}{28 \, a^{4} x^{7}} \]

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*sqrt(3)*(10*b^3*c - 7*a*b^2*d - a^3*f + 4*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/
((-a*b^2)^(1/3)*a^4) + 1/18*(10*b^3*c - 7*a*b^2*d - a^3*f + 4*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3)
)/((-a*b^2)^(1/3)*a^4) + 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) - 1/28*(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)

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maple [B]  time = 0.07, size = 529, normalized size = 1.78 \[ \frac {f \,x^{2}}{3 \left (b \,x^{3}+a \right ) a}-\frac {b e \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {b^{2} d \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{3}}-\frac {b^{3} c \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{4}}+\frac {\sqrt {3}\, f \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}-\frac {f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}+\frac {f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}-\frac {4 \sqrt {3}\, e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}+\frac {4 e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}-\frac {2 e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}+\frac {7 \sqrt {3}\, b d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {7 b d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\frac {7 b d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {10 \sqrt {3}\, b^{2} c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}+\frac {10 b^{2} c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}-\frac {5 b^{2} c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}-\frac {e}{a^{2} x}+\frac {2 b d}{a^{3} x}-\frac {3 b^{2} c}{a^{4} x}-\frac {d}{4 a^{2} x^{4}}+\frac {b c}{2 a^{3} x^{4}}-\frac {c}{7 a^{2} x^{7}} \]

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/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-2/9/a^2*e/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-4/9/a^2*e*3^(1/2)/(a
/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-7/9/a^3*b*d/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+7/18/a^3*b*d/(a/b)
^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+7/9/a^3*b*d*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1
))+10/9/a^4*b^2*c/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-5/9/a^4*b^2*c/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-10
/9/a^4*b^2*c*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/9/a*f/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3
))+1/18/a*f/b/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/9/a*f*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(
2/(a/b)^(1/3)*x-1))-1/7/a^2*c/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 [A]  time = 3.05, size = 292, normalized size = 0.98 \[ -\frac {28 \, {\left (10 \, b^{3} c - 7 \, a b^{2} d + 4 \, a^{2} b e - a^{3} f\right )} x^{9} + 21 \, {\left (10 \, a b^{2} c - 7 \, a^{2} b d + 4 \, a^{3} e\right )} x^{6} + 12 \, a^{3} c - 3 \, {\left (10 \, a^{2} b c - 7 \, a^{3} d\right )} x^{3}}{84 \, {\left (a^{4} b x^{10} + a^{5} x^{7}\right )}} - \frac {\sqrt {3} {\left (10 \, b^{3} c - 7 \, a b^{2} d + 4 \, a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (10 \, b^{3} c - 7 \, a b^{2} d + 4 \, a^{2} b e - a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (10 \, b^{3} c - 7 \, a b^{2} d + 4 \, a^{2} b e - a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

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]

-1/84*(28*(10*b^3*c - 7*a*b^2*d + 4*a^2*b*e - a^3*f)*x^9 + 21*(10*a*b^2*c - 7*a^2*b*d + 4*a^3*e)*x^6 + 12*a^3*
c - 3*(10*a^2*b*c - 7*a^3*d)*x^3)/(a^4*b*x^10 + a^5*x^7) - 1/9*sqrt(3)*(10*b^3*c - 7*a*b^2*d + 4*a^2*b*e - a^3
*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^4*b*(a/b)^(1/3)) - 1/18*(10*b^3*c - 7*a*b^2*d + 4*a
^2*b*e - a^3*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^4*b*(a/b)^(1/3)) + 1/9*(10*b^3*c - 7*a*b^2*d + 4*a^2
*b*e - a^3*f)*log(x + (a/b)^(1/3))/(a^4*b*(a/b)^(1/3))

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mupad [B]  time = 5.18, size = 274, normalized size = 0.92 \[ \frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+4\,e\,a^2\,b-7\,d\,a\,b^2+10\,c\,b^3\right )}{9\,a^{13/3}\,b^{2/3}}-\frac {\frac {c}{7\,a}+\frac {x^9\,\left (-f\,a^3+4\,e\,a^2\,b-7\,d\,a\,b^2+10\,c\,b^3\right )}{3\,a^4}+\frac {x^3\,\left (7\,a\,d-10\,b\,c\right )}{28\,a^2}+\frac {x^6\,\left (4\,e\,a^2-7\,d\,a\,b+10\,c\,b^2\right )}{4\,a^3}}{b\,x^{10}+a\,x^7}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+4\,e\,a^2\,b-7\,d\,a\,b^2+10\,c\,b^3\right )}{9\,a^{13/3}\,b^{2/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+4\,e\,a^2\,b-7\,d\,a\,b^2+10\,c\,b^3\right )}{9\,a^{13/3}\,b^{2/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(b^(1/3)*x + a^(1/3))*(10*b^3*c - a^3*f - 7*a*b^2*d + 4*a^2*b*e))/(9*a^(13/3)*b^(2/3)) - (c/(7*a) + (x^9*(
10*b^3*c - a^3*f - 7*a*b^2*d + 4*a^2*b*e))/(3*a^4) + (x^3*(7*a*d - 10*b*c))/(28*a^2) + (x^6*(10*b^2*c + 4*a^2*
e - 7*a*b*d))/(4*a^3))/(a*x^7 + b*x^10) - (log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 + 1
/2)*(10*b^3*c - a^3*f - 7*a*b^2*d + 4*a^2*b*e))/(9*a^(13/3)*b^(2/3)) + (log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x +
a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(10*b^3*c - a^3*f - 7*a*b^2*d + 4*a^2*b*e))/(9*a^(13/3)*b^(2/3))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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