3.273 \(\int \frac {c+d x^3+e x^6+f x^9}{x^{11} (a+b x^3)^2} \, dx\)

Optimal. Leaf size=334 \[ \frac {2 b c-a d}{7 a^3 x^7}-\frac {c}{10 a^2 x^{10}}-\frac {a^2 e-2 a b d+3 b^2 c}{4 a^4 x^4}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right )}{9 a^{16/3}}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right )}{3 \sqrt {3} a^{16/3}}+\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right )}{18 a^{16/3}}+\frac {b x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac {a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{a^5 x} \]

[Out]

-1/10*c/a^2/x^10+1/7*(-a*d+2*b*c)/a^3/x^7+1/4*(-a^2*e+2*a*b*d-3*b^2*c)/a^4/x^4+(-a^3*f+2*a^2*b*e-3*a*b^2*d+4*b
^3*c)/a^5/x+1/3*b*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x^2/a^5/(b*x^3+a)-1/9*b^(1/3)*(-4*a^3*f+7*a^2*b*e-10*a*b^2*d+
13*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(16/3)+1/18*b^(1/3)*(-4*a^3*f+7*a^2*b*e-10*a*b^2*d+13*b^3*c)*ln(a^(2/3)-a^(1
/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(16/3)-1/9*b^(1/3)*(-4*a^3*f+7*a^2*b*e-10*a*b^2*d+13*b^3*c)*arctan(1/3*(a^(1/3)-2
*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(16/3)*3^(1/2)

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Rubi [A]  time = 0.46, antiderivative size = 334, 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} \[ \frac {b x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (7 a^2 b e-4 a^3 f-10 a b^2 d+13 b^3 c\right )}{18 a^{16/3}}+\frac {2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{a^5 x}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (7 a^2 b e-4 a^3 f-10 a b^2 d+13 b^3 c\right )}{9 a^{16/3}}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (7 a^2 b e-4 a^3 f-10 a b^2 d+13 b^3 c\right )}{3 \sqrt {3} a^{16/3}}-\frac {a^2 e-2 a b d+3 b^2 c}{4 a^4 x^4}+\frac {2 b c-a d}{7 a^3 x^7}-\frac {c}{10 a^2 x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)^2),x]

[Out]

-c/(10*a^2*x^10) + (2*b*c - a*d)/(7*a^3*x^7) - (3*b^2*c - 2*a*b*d + a^2*e)/(4*a^4*x^4) + (4*b^3*c - 3*a*b^2*d
+ 2*a^2*b*e - a^3*f)/(a^5*x) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*a^5*(a + b*x^3)) - (b^(1/3)*(13*
b^3*c - 10*a*b^2*d + 7*a^2*b*e - 4*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(16/
3)) - (b^(1/3)*(13*b^3*c - 10*a*b^2*d + 7*a^2*b*e - 4*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(16/3)) + (b^(1/3)
*(13*b^3*c - 10*a*b^2*d + 7*a^2*b*e - 4*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(16/3))

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^{11} \left (a+b x^3\right )^2} \, dx &=\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^5 \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 {3 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}-\frac {b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{12}}{a^4}}{x^{11} \left (a+b x^3\right )} \, dx}{3 a b^3}\\ &=\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^5 \left (a+b x^3\right )}-\frac {\int \left (-\frac {3 b^3 c}{a x^{11}}-\frac {3 b^3 (-2 b c+a d)}{a^2 x^8}-\frac {3 b^3 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^5}-\frac {3 b^3 \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{a^4 x^2}+\frac {b^4 \left (-13 b^3 c+10 a b^2 d-7 a^2 b e+4 a^3 f\right ) x}{a^4 \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac {c}{10 a^2 x^{10}}+\frac {2 b c-a d}{7 a^3 x^7}-\frac {3 b^2 c-2 a b d+a^2 e}{4 a^4 x^4}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^5 \left (a+b x^3\right )}+\frac {\left (b \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right )\right ) \int \frac {x}{a+b x^3} \, dx}{3 a^5}\\ &=-\frac {c}{10 a^2 x^{10}}+\frac {2 b c-a d}{7 a^3 x^7}-\frac {3 b^2 c-2 a b d+a^2 e}{4 a^4 x^4}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^5 \left (a+b x^3\right )}-\frac {\left (b^{2/3} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{16/3}}+\frac {\left (b^{2/3} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 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}{9 a^{16/3}}\\ &=-\frac {c}{10 a^2 x^{10}}+\frac {2 b c-a d}{7 a^3 x^7}-\frac {3 b^2 c-2 a b d+a^2 e}{4 a^4 x^4}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^5 \left (a+b x^3\right )}-\frac {\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{16/3}}+\frac {\left (\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 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}{18 a^{16/3}}+\frac {\left (b^{2/3} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^5}\\ &=-\frac {c}{10 a^2 x^{10}}+\frac {2 b c-a d}{7 a^3 x^7}-\frac {3 b^2 c-2 a b d+a^2 e}{4 a^4 x^4}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^5 \left (a+b x^3\right )}-\frac {\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{16/3}}+\frac {\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{16/3}}+\frac {\left (\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 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 )}{3 a^{16/3}}\\ &=-\frac {c}{10 a^2 x^{10}}+\frac {2 b c-a d}{7 a^3 x^7}-\frac {3 b^2 c-2 a b d+a^2 e}{4 a^4 x^4}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^5 \left (a+b x^3\right )}-\frac {\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 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^{16/3}}-\frac {\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{16/3}}+\frac {\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{16/3}}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 319, normalized size = 0.96 \[ \frac {-\frac {180 a^{7/3} (a d-2 b c)}{x^7}-\frac {126 a^{10/3} c}{x^{10}}-\frac {315 a^{4/3} \left (a^2 e-2 a b d+3 b^2 c\right )}{x^4}-\frac {420 \sqrt [3]{a} b x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a+b x^3}-\frac {1260 \sqrt [3]{a} \left (a^3 f-2 a^2 b e+3 a b^2 d-4 b^3 c\right )}{x}+140 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (4 a^3 f-7 a^2 b e+10 a b^2 d-13 b^3 c\right )-140 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right )+70 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right )}{1260 a^{16/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)^2),x]

[Out]

((-126*a^(10/3)*c)/x^10 - (180*a^(7/3)*(-2*b*c + a*d))/x^7 - (315*a^(4/3)*(3*b^2*c - 2*a*b*d + a^2*e))/x^4 - (
1260*a^(1/3)*(-4*b^3*c + 3*a*b^2*d - 2*a^2*b*e + a^3*f))/x - (420*a^(1/3)*b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^
3*f)*x^2)/(a + b*x^3) - 140*Sqrt[3]*b^(1/3)*(13*b^3*c - 10*a*b^2*d + 7*a^2*b*e - 4*a^3*f)*ArcTan[(1 - (2*b^(1/
3)*x)/a^(1/3))/Sqrt[3]] + 140*b^(1/3)*(-13*b^3*c + 10*a*b^2*d - 7*a^2*b*e + 4*a^3*f)*Log[a^(1/3) + b^(1/3)*x]
+ 70*b^(1/3)*(13*b^3*c - 10*a*b^2*d + 7*a^2*b*e - 4*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(12
60*a^(16/3))

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fricas [A]  time = 0.76, size = 442, normalized size = 1.32 \[ \frac {420 \, {\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{12} + 315 \, {\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{9} - 45 \, {\left (13 \, a^{2} b^{2} c - 10 \, a^{3} b d + 7 \, a^{4} e\right )} x^{6} - 126 \, a^{4} c + 18 \, {\left (13 \, a^{3} b c - 10 \, a^{4} d\right )} x^{3} + 140 \, \sqrt {3} {\left ({\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{13} + {\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{10}\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 70 \, {\left ({\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{13} + {\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{10}\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 140 \, {\left ({\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{13} + {\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{10}\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{1260 \, {\left (a^{5} b x^{13} + a^{6} x^{10}\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

1/1260*(420*(13*b^4*c - 10*a*b^3*d + 7*a^2*b^2*e - 4*a^3*b*f)*x^12 + 315*(13*a*b^3*c - 10*a^2*b^2*d + 7*a^3*b*
e - 4*a^4*f)*x^9 - 45*(13*a^2*b^2*c - 10*a^3*b*d + 7*a^4*e)*x^6 - 126*a^4*c + 18*(13*a^3*b*c - 10*a^4*d)*x^3 +
140*sqrt(3)*((13*b^4*c - 10*a*b^3*d + 7*a^2*b^2*e - 4*a^3*b*f)*x^13 + (13*a*b^3*c - 10*a^2*b^2*d + 7*a^3*b*e
- 4*a^4*f)*x^10)*(b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(b/a)^(1/3) - 1/3*sqrt(3)) + 70*((13*b^4*c - 10*a*b^3*d + 7*
a^2*b^2*e - 4*a^3*b*f)*x^13 + (13*a*b^3*c - 10*a^2*b^2*d + 7*a^3*b*e - 4*a^4*f)*x^10)*(b/a)^(1/3)*log(b*x^2 -
a*x*(b/a)^(2/3) + a*(b/a)^(1/3)) - 140*((13*b^4*c - 10*a*b^3*d + 7*a^2*b^2*e - 4*a^3*b*f)*x^13 + (13*a*b^3*c -
10*a^2*b^2*d + 7*a^3*b*e - 4*a^4*f)*x^10)*(b/a)^(1/3)*log(b*x + a*(b/a)^(2/3)))/(a^5*b*x^13 + a^6*x^10)

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giac [A]  time = 0.37, size = 437, normalized size = 1.31 \[ -\frac {{\left (13 \, b^{4} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 10 \, a b^{3} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 4 \, a^{3} b f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 7 \, a^{2} b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{6}} - \frac {\sqrt {3} {\left (13 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 10 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - 4 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{6} b} + \frac {b^{4} c x^{2} - a b^{3} d x^{2} - a^{3} b f x^{2} + a^{2} b^{2} x^{2} e}{3 \, {\left (b x^{3} + a\right )} a^{5}} + \frac {{\left (13 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 10 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - 4 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{6} b} + \frac {560 \, b^{3} c x^{9} - 420 \, a b^{2} d x^{9} - 140 \, a^{3} f x^{9} + 280 \, a^{2} b x^{9} e - 105 \, a b^{2} c x^{6} + 70 \, a^{2} b d x^{6} - 35 \, a^{3} x^{6} e + 40 \, a^{2} b c x^{3} - 20 \, a^{3} d x^{3} - 14 \, a^{3} c}{140 \, a^{5} x^{10}} \]

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)^2,x, algorithm="giac")

[Out]

-1/9*(13*b^4*c*(-a/b)^(1/3) - 10*a*b^3*d*(-a/b)^(1/3) - 4*a^3*b*f*(-a/b)^(1/3) + 7*a^2*b^2*(-a/b)^(1/3)*e)*(-a
/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^6 - 1/9*sqrt(3)*(13*(-a*b^2)^(2/3)*b^3*c - 10*(-a*b^2)^(2/3)*a*b^2*d -
4*(-a*b^2)^(2/3)*a^3*f + 7*(-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^6*
b) + 1/3*(b^4*c*x^2 - a*b^3*d*x^2 - a^3*b*f*x^2 + a^2*b^2*x^2*e)/((b*x^3 + a)*a^5) + 1/18*(13*(-a*b^2)^(2/3)*b
^3*c - 10*(-a*b^2)^(2/3)*a*b^2*d - 4*(-a*b^2)^(2/3)*a^3*f + 7*(-a*b^2)^(2/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3)
+ (-a/b)^(2/3))/(a^6*b) + 1/140*(560*b^3*c*x^9 - 420*a*b^2*d*x^9 - 140*a^3*f*x^9 + 280*a^2*b*x^9*e - 105*a*b^
2*c*x^6 + 70*a^2*b*d*x^6 - 35*a^3*x^6*e + 40*a^2*b*c*x^3 - 20*a^3*d*x^3 - 14*a^3*c)/(a^5*x^10)

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maple [A]  time = 0.06, size = 575, normalized size = 1.72 \[ -\frac {b f \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {b^{2} e \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{3}}-\frac {b^{3} d \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{4}}+\frac {b^{4} c \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{5}}-\frac {4 \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^{2}}+\frac {4 f \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 f \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 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^{3}}-\frac {7 b e \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 e \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} 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^{4}}+\frac {10 b^{2} d \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} d \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 {13 \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 )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{5}}-\frac {13 b^{3} c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{5}}+\frac {13 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 )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{5}}-\frac {f}{a^{2} x}+\frac {2 b e}{a^{3} x}-\frac {3 b^{2} d}{a^{4} x}+\frac {4 b^{3} c}{a^{5} x}-\frac {e}{4 a^{2} x^{4}}+\frac {b d}{2 a^{3} x^{4}}-\frac {3 b^{2} c}{4 a^{4} x^{4}}-\frac {d}{7 a^{2} x^{7}}+\frac {2 b c}{7 a^{3} x^{7}}-\frac {c}{10 a^{2} 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)^2,x)

[Out]

4/a^5/x*b^3*c+4/9/a^2*f/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-2/9/a^2*f/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+
2/7/a^3/x^7*b*c+1/2/a^3/x^4*b*d-3/4/a^4/x^4*b^2*c+2/a^3/x*b*e-3/a^4/x*b^2*d-1/3*b/a^2*x^2/(b*x^3+a)*f+1/3*b^2/
a^3*x^2/(b*x^3+a)*e-10/9*b^2/a^4*d*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+13/9*b^3/a^5*c*
3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+7/9*b/a^3*e*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)
*(2/(a/b)^(1/3)*x-1))-1/a^2/x*f-1/4/a^2/x^4*e-1/7/a^2/x^7*d-1/10*c/a^2/x^10-1/3*b^3/a^4*x^2/(b*x^3+a)*d+1/3*b^
4/a^5*x^2/(b*x^3+a)*c-4/9/a^2*f*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-7/9*b/a^3*e/(a/b)^
(1/3)*ln(x+(a/b)^(1/3))+7/18*b/a^3*e/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+10/9*b^2/a^4*d/(a/b)^(1/3)*
ln(x+(a/b)^(1/3))-5/9*b^2/a^4*d/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-13/9*b^3/a^5*c/(a/b)^(1/3)*ln(x+
(a/b)^(1/3))+13/18*b^3/a^5*c/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))

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maxima [A]  time = 3.13, size = 323, normalized size = 0.97 \[ \frac {140 \, {\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{12} + 105 \, {\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{9} - 15 \, {\left (13 \, a^{2} b^{2} c - 10 \, a^{3} b d + 7 \, a^{4} e\right )} x^{6} - 42 \, a^{4} c + 6 \, {\left (13 \, a^{3} b c - 10 \, a^{4} d\right )} x^{3}}{420 \, {\left (a^{5} b x^{13} + a^{6} x^{10}\right )}} + \frac {\sqrt {3} {\left (13 \, b^{3} c - 10 \, a b^{2} d + 7 \, a^{2} b e - 4 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (13 \, b^{3} c - 10 \, a b^{2} d + 7 \, a^{2} b e - 4 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (13 \, b^{3} c - 10 \, a b^{2} d + 7 \, a^{2} b e - 4 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^11/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

1/420*(140*(13*b^4*c - 10*a*b^3*d + 7*a^2*b^2*e - 4*a^3*b*f)*x^12 + 105*(13*a*b^3*c - 10*a^2*b^2*d + 7*a^3*b*e
- 4*a^4*f)*x^9 - 15*(13*a^2*b^2*c - 10*a^3*b*d + 7*a^4*e)*x^6 - 42*a^4*c + 6*(13*a^3*b*c - 10*a^4*d)*x^3)/(a^
5*b*x^13 + a^6*x^10) + 1/9*sqrt(3)*(13*b^3*c - 10*a*b^2*d + 7*a^2*b*e - 4*a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/
b)^(1/3))/(a/b)^(1/3))/(a^5*(a/b)^(1/3)) + 1/18*(13*b^3*c - 10*a*b^2*d + 7*a^2*b*e - 4*a^3*f)*log(x^2 - x*(a/b
)^(1/3) + (a/b)^(2/3))/(a^5*(a/b)^(1/3)) - 1/9*(13*b^3*c - 10*a*b^2*d + 7*a^2*b*e - 4*a^3*f)*log(x + (a/b)^(1/
3))/(a^5*(a/b)^(1/3))

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mupad [B]  time = 5.41, size = 310, normalized size = 0.93 \[ -\frac {\frac {c}{10\,a}-\frac {x^9\,\left (-4\,f\,a^3+7\,e\,a^2\,b-10\,d\,a\,b^2+13\,c\,b^3\right )}{4\,a^4}+\frac {x^3\,\left (10\,a\,d-13\,b\,c\right )}{70\,a^2}+\frac {x^6\,\left (7\,e\,a^2-10\,d\,a\,b+13\,c\,b^2\right )}{28\,a^3}-\frac {b\,x^{12}\,\left (-4\,f\,a^3+7\,e\,a^2\,b-10\,d\,a\,b^2+13\,c\,b^3\right )}{3\,a^5}}{b\,x^{13}+a\,x^{10}}-\frac {b^{1/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-4\,f\,a^3+7\,e\,a^2\,b-10\,d\,a\,b^2+13\,c\,b^3\right )}{9\,a^{16/3}}+\frac {b^{1/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-4\,f\,a^3+7\,e\,a^2\,b-10\,d\,a\,b^2+13\,c\,b^3\right )}{9\,a^{16/3}}-\frac {b^{1/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-4\,f\,a^3+7\,e\,a^2\,b-10\,d\,a\,b^2+13\,c\,b^3\right )}{9\,a^{16/3}} \]

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)^2),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)*(13*b^3*c - 4*a^3*f - 10*a*b^2
*d + 7*a^2*b*e))/(9*a^(16/3)) - (b^(1/3)*log(b^(1/3)*x + a^(1/3))*(13*b^3*c - 4*a^3*f - 10*a*b^2*d + 7*a^2*b*e
))/(9*a^(16/3)) - (c/(10*a) - (x^9*(13*b^3*c - 4*a^3*f - 10*a*b^2*d + 7*a^2*b*e))/(4*a^4) + (x^3*(10*a*d - 13*
b*c))/(70*a^2) + (x^6*(13*b^2*c + 7*a^2*e - 10*a*b*d))/(28*a^3) - (b*x^12*(13*b^3*c - 4*a^3*f - 10*a*b^2*d + 7
*a^2*b*e))/(3*a^5))/(a*x^10 + b*x^13) - (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)*(13*b^3*c - 4*a^3*f - 10*a*b^2*d + 7*a^2*b*e))/(9*a^(16/3))

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

[Out]

Timed out

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