3.246 \(\int \frac {c+d x^3+e x^6+f x^9}{x^{11} (a+b x^3)} \, dx\)

Optimal. Leaf size=277 \[ \frac {b c-a d}{7 a^2 x^7}-\frac {a^2 e-a b d+b^2 c}{4 a^3 x^4}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{13/3}}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} a^{13/3}}+\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{13/3}}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{a^4 x}-\frac {c}{10 a x^{10}} \]

[Out]

-1/10*c/a/x^10+1/7*(-a*d+b*c)/a^2/x^7+1/4*(-a^2*e+a*b*d-b^2*c)/a^3/x^4+(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)/a^4/x-1/
3*b^(1/3)*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(13/3)+1/6*b^(1/3)*(-a^3*f+a^2*b*e-a*b^2*d+b^
3*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(13/3)-1/3*b^(1/3)*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*arctan(1/3*
(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(13/3)*3^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.22, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {1834, 292, 31, 634, 617, 204, 628} \[ \frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{13/3}}+\frac {a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{a^4 x}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{13/3}}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt {3} a^{13/3}}-\frac {a^2 e-a b d+b^2 c}{4 a^3 x^4}+\frac {b c-a d}{7 a^2 x^7}-\frac {c}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)),x]

[Out]

-c/(10*a*x^10) + (b*c - a*d)/(7*a^2*x^7) - (b^2*c - a*b*d + a^2*e)/(4*a^3*x^4) + (b^3*c - a*b^2*d + a^2*b*e -
a^3*f)/(a^4*x) - (b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))
])/(Sqrt[3]*a^(13/3)) - (b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(13/3)) +
(b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(13/3))

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 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^{11} \left (a+b x^3\right )} \, dx &=\int \left (\frac {c}{a x^{11}}+\frac {-b c+a d}{a^2 x^8}+\frac {b^2 c-a b d+a^2 e}{a^3 x^5}+\frac {-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^2}-\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{a^4 \left (a+b x^3\right )}\right ) \, dx\\ &=-\frac {c}{10 a x^{10}}+\frac {b c-a d}{7 a^2 x^7}-\frac {b^2 c-a b d+a^2 e}{4 a^3 x^4}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}+\frac {\left (b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {x}{a+b x^3} \, dx}{a^4}\\ &=-\frac {c}{10 a x^{10}}+\frac {b c-a d}{7 a^2 x^7}-\frac {b^2 c-a b d+a^2 e}{4 a^3 x^4}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}-\frac {\left (b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{13/3}}+\frac {\left (b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-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}{3 a^{13/3}}\\ &=-\frac {c}{10 a x^{10}}+\frac {b c-a d}{7 a^2 x^7}-\frac {b^2 c-a b d+a^2 e}{4 a^3 x^4}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}-\frac {\sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{13/3}}+\frac {\left (\sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-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}{6 a^{13/3}}+\frac {\left (b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^4}\\ &=-\frac {c}{10 a x^{10}}+\frac {b c-a d}{7 a^2 x^7}-\frac {b^2 c-a b d+a^2 e}{4 a^3 x^4}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}-\frac {\sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{13/3}}+\frac {\sqrt [3]{b} \left (b^3 c-a b^2 d+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 )}{6 a^{13/3}}+\frac {\left (\sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-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 )}{a^{13/3}}\\ &=-\frac {c}{10 a x^{10}}+\frac {b c-a d}{7 a^2 x^7}-\frac {b^2 c-a b d+a^2 e}{4 a^3 x^4}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}-\frac {\sqrt [3]{b} \left (b^3 c-a b^2 d+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 )}{\sqrt {3} a^{13/3}}-\frac {\sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{13/3}}+\frac {\sqrt [3]{b} \left (b^3 c-a b^2 d+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 )}{6 a^{13/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 266, normalized size = 0.96 \[ \frac {\frac {60 a^{7/3} (b c-a d)}{x^7}-\frac {42 a^{10/3} c}{x^{10}}-\frac {105 a^{4/3} \left (a^2 e-a b d+b^2 c\right )}{x^4}+\frac {420 \sqrt [3]{a} \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{x}+140 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )-140 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )+70 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{420 a^{13/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)),x]

[Out]

((-42*a^(10/3)*c)/x^10 + (60*a^(7/3)*(b*c - a*d))/x^7 - (105*a^(4/3)*(b^2*c - a*b*d + a^2*e))/x^4 + (420*a^(1/
3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/x - 140*Sqrt[3]*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(1
- (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 140*b^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x
] + 70*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(420*a^(13/
3))

________________________________________________________________________________________

fricas [A]  time = 0.74, size = 262, normalized size = 0.95 \[ \frac {140 \, \sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{10} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 70 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{10} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 140 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{10} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 420 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} - 105 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{6} - 42 \, a^{3} c + 60 \, {\left (a^{2} b c - a^{3} d\right )} x^{3}}{420 \, a^{4} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^11/(b*x^3+a),x, algorithm="fricas")

[Out]

1/420*(140*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^10*(b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(b/a)^(1/3) - 1/3
*sqrt(3)) + 70*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^10*(b/a)^(1/3)*log(b*x^2 - a*x*(b/a)^(2/3) + a*(b/a)^(1/3
)) - 140*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^10*(b/a)^(1/3)*log(b*x + a*(b/a)^(2/3)) + 420*(b^3*c - a*b^2*d
+ a^2*b*e - a^3*f)*x^9 - 105*(a*b^2*c - a^2*b*d + a^3*e)*x^6 - 42*a^3*c + 60*(a^2*b*c - a^3*d)*x^3)/(a^4*x^10)

________________________________________________________________________________________

giac [A]  time = 0.19, size = 376, normalized size = 1.36 \[ -\frac {{\left (b^{4} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{3} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} b f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{5}} - \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{5} b} + \frac {{\left (\left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{5} b} + \frac {140 \, b^{3} c x^{9} - 140 \, a b^{2} d x^{9} - 140 \, a^{3} f x^{9} + 140 \, a^{2} b x^{9} e - 35 \, a b^{2} c x^{6} + 35 \, a^{2} b d x^{6} - 35 \, a^{3} x^{6} e + 20 \, a^{2} b c x^{3} - 20 \, a^{3} d x^{3} - 14 \, a^{3} c}{140 \, a^{4} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^11/(b*x^3+a),x, algorithm="giac")

[Out]

-1/3*(b^4*c*(-a/b)^(1/3) - a*b^3*d*(-a/b)^(1/3) - a^3*b*f*(-a/b)^(1/3) + a^2*b^2*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*
log(abs(x - (-a/b)^(1/3)))/a^5 - 1/3*sqrt(3)*((-a*b^2)^(2/3)*b^3*c - (-a*b^2)^(2/3)*a*b^2*d - (-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^5*b) + 1/6*((-a*b^2)^(
2/3)*b^3*c - (-a*b^2)^(2/3)*a*b^2*d - (-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^5*b) + 1/140*(140*b^3*c*x^9 - 140*a*b^2*d*x^9 - 140*a^3*f*x^9 + 140*a^2*b*x^9*e - 35*a*b^2*
c*x^6 + 35*a^2*b*d*x^6 - 35*a^3*x^6*e + 20*a^2*b*c*x^3 - 20*a^3*d*x^3 - 14*a^3*c)/(a^4*x^10)

________________________________________________________________________________________

maple [B]  time = 0.06, size = 491, normalized size = 1.77 \[ -\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 )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} a}+\frac {f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} a}-\frac {f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {1}{3}} a}+\frac {\sqrt {3}\, b e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}-\frac {b e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}+\frac {b e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}-\frac {\sqrt {3}\, b^{2} d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\frac {b^{2} d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {b^{2} d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\frac {\sqrt {3}\, b^{3} c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}-\frac {b^{3} c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}+\frac {b^{3} c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}-\frac {f}{a x}+\frac {b e}{a^{2} x}-\frac {b^{2} d}{a^{3} x}+\frac {b^{3} c}{a^{4} x}-\frac {e}{4 a \,x^{4}}+\frac {b d}{4 a^{2} x^{4}}-\frac {b^{2} c}{4 a^{3} x^{4}}-\frac {d}{7 a \,x^{7}}+\frac {b c}{7 a^{2} x^{7}}-\frac {c}{10 a \,x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^9+e*x^6+d*x^3+c)/x^11/(b*x^3+a),x)

[Out]

1/3/a/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*f-1/3/a^2*b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*e+1/3/a^3*b^2/(a/b)^(1/3)*ln(x+(
a/b)^(1/3))*d-1/3/a^4*b^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*c-1/6/a/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*
f+1/6/a^2*b/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*e-1/6/a^3*b^2/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)
^(2/3))*d+1/6/a^4*b^3/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*c-1/3/a*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(
1/2)*(2/(a/b)^(1/3)*x-1))*f+1/3/a^2*b*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e-1/3/a^3*b^
2*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d+1/3/a^4*b^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(
1/2)*(2/(a/b)^(1/3)*x-1))*c-1/10*c/a/x^10-1/7/a/x^7*d+1/7/a^2/x^7*b*c-1/4/a/x^4*e+1/4/a^2/x^4*b*d-1/4/a^3/x^4*
b^2*c-1/a/x*f+1/a^2/x*b*e-1/a^3/x*b^2*d+1/a^4/x*b^3*c

________________________________________________________________________________________

maxima [A]  time = 3.05, size = 260, normalized size = 0.94 \[ \frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {140 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} - 35 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{6} - 14 \, a^{3} c + 20 \, {\left (a^{2} b c - a^{3} d\right )} x^{3}}{140 \, a^{4} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^11/(b*x^3+a),x, algorithm="maxima")

[Out]

1/3*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^4*(a/b)
^(1/3)) + 1/6*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^4*(a/b)^(1/3)) - 1
/3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*log(x + (a/b)^(1/3))/(a^4*(a/b)^(1/3)) + 1/140*(140*(b^3*c - a*b^2*d +
a^2*b*e - a^3*f)*x^9 - 35*(a*b^2*c - a^2*b*d + a^3*e)*x^6 - 14*a^3*c + 20*(a^2*b*c - a^3*d)*x^3)/(a^4*x^10)

________________________________________________________________________________________

mupad [B]  time = 5.33, size = 253, normalized size = 0.91 \[ -\frac {\frac {c}{10\,a}-\frac {x^9\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^4}+\frac {x^3\,\left (a\,d-b\,c\right )}{7\,a^2}+\frac {x^6\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{4\,a^3}}{x^{10}}-\frac {b^{1/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{13/3}}+\frac {b^{1/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{13/3}}-\frac {b^{1/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{13/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)),x)

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________