\(\int \frac {c+d x^3+e x^6+f x^9}{x^{11} (a+b x^3)^2} \, dx\) [273]

Optimal result

Integrand size = 30, antiderivative size = 334 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^2} \, dx=-\frac {c}{10 a^2 x^{10}}+\frac {2 b c-a d}{7 a^3 x^7}-\frac {3 b^2 c-2 a b d+a^2 e}{4 a^4 x^4}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^5 \left (a+b x^3\right )}-\frac {\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{16/3}}-\frac {\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{16/3}}+\frac {\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{16/3}} \]

[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)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1843, 1848, 298, 31, 648, 631, 210, 642} \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^2} \, dx=\frac {2 b c-a d}{7 a^3 x^7}-\frac {c}{10 a^2 x^{10}}-\frac {a^2 e-2 a b d+3 b^2 c}{4 a^4 x^4}-\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right )}{3 \sqrt {3} a^{16/3}}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right )}{9 a^{16/3}}+\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right )}{18 a^{16/3}}+\frac {b x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac {a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{a^5 x} \]

[In]

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

[Out]

-1/10*c/(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)*(1
3*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^(1
6/3)) - (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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 298

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 631

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 642

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 648

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 1843

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)/a)*Coeff[R, x, i]*x^(i - m), {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[P
q, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1848

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*} \text {integral}& = \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 ) \text {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*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^2} \, dx=\frac {-\frac {126 a^{10/3} c}{x^{10}}-\frac {180 a^{7/3} (-2 b c+a d)}{x^7}-\frac {315 a^{4/3} \left (3 b^2 c-2 a b d+a^2 e\right )}{x^4}-\frac {1260 \sqrt [3]{a} \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{x}-\frac {420 \sqrt [3]{a} b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x^2}{a+b x^3}-140 \sqrt {3} \sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+140 \sqrt [3]{b} \left (-13 b^3 c+10 a b^2 d-7 a^2 b e+4 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+70 \sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1260 a^{16/3}} \]

[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))

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.75

method	result	size
default	\(-\frac {c}{10 a^{2} x^{10}}-\frac {a d -2 b c}{7 a^{3} x^{7}}-\frac {a^{2} e -2 a b d +3 b^{2} c}{4 a^{4} x^{4}}-\frac {f \,a^{3}-2 a^{2} b e +3 a \,b^{2} d -4 b^{3} c}{a^{5} x}-\frac {b \left (\frac {\left (\frac {1}{3} f \,a^{3}-\frac {1}{3} a^{2} b e +\frac {1}{3} a \,b^{2} d -\frac {1}{3} b^{3} c \right ) x^{2}}{b \,x^{3}+a}+\left (\frac {4}{3} f \,a^{3}-\frac {7}{3} a^{2} b e +\frac {10}{3} a \,b^{2} d -\frac {13}{3} b^{3} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )\right )}{a^{5}}\)	\(251\)
risch	\(\frac {-\frac {b \left (4 f \,a^{3}-7 a^{2} b e +10 a \,b^{2} d -13 b^{3} c \right ) x^{12}}{3 a^{5}}-\frac {\left (4 f \,a^{3}-7 a^{2} b e +10 a \,b^{2} d -13 b^{3} c \right ) x^{9}}{4 a^{4}}-\frac {\left (7 a^{2} e -10 a b d +13 b^{2} c \right ) x^{6}}{28 a^{3}}-\frac {\left (10 a d -13 b c \right ) x^{3}}{70 a^{2}}-\frac {c}{10 a}}{x^{10} \left (b \,x^{3}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{16} \textit {\_Z}^{3}-64 a^{9} b \,f^{3}+336 a^{8} b^{2} e \,f^{2}-480 a^{7} b^{3} d \,f^{2}-588 a^{7} b^{3} e^{2} f +624 a^{6} b^{4} c \,f^{2}+1680 a^{6} b^{4} d e f +343 a^{6} b^{4} e^{3}-2184 a^{5} b^{5} c e f -1200 a^{5} b^{5} d^{2} f -1470 a^{5} b^{5} d \,e^{2}+3120 a^{4} b^{6} c d f +1911 a^{4} b^{6} c \,e^{2}+2100 a^{4} b^{6} d^{2} e -2028 a^{3} b^{7} c^{2} f -5460 a^{3} b^{7} c d e -1000 a^{3} b^{7} d^{3}+3549 a^{2} b^{8} c^{2} e +3900 a^{2} b^{8} c \,d^{2}-5070 a \,b^{9} c^{2} d +2197 b^{10} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{16}+192 a^{9} b \,f^{3}-1008 a^{8} b^{2} e \,f^{2}+1440 a^{7} b^{3} d \,f^{2}+1764 a^{7} b^{3} e^{2} f -1872 a^{6} b^{4} c \,f^{2}-5040 a^{6} b^{4} d e f -1029 a^{6} b^{4} e^{3}+6552 a^{5} b^{5} c e f +3600 a^{5} b^{5} d^{2} f +4410 a^{5} b^{5} d \,e^{2}-9360 a^{4} b^{6} c d f -5733 a^{4} b^{6} c \,e^{2}-6300 a^{4} b^{6} d^{2} e +6084 a^{3} b^{7} c^{2} f +16380 a^{3} b^{7} c d e +3000 a^{3} b^{7} d^{3}-10647 a^{2} b^{8} c^{2} e -11700 a^{2} b^{8} c \,d^{2}+15210 a \,b^{9} c^{2} d -6591 b^{10} c^{3}\right ) x +\left (-4 a^{14} f +7 a^{13} b e -10 a^{12} b^{2} d +13 a^{11} b^{3} c \right ) \textit {\_R}^{2}\right )\right )}{9}\)	\(651\)

[In]

int((f*x^9+e*x^6+d*x^3+c)/x^11/(b*x^3+a)^2,x,method=_RETURNVERBOSE)

[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/a^5*b*((1/3*f*a^3-1/3*a^2*b*e+1/3*a*b^2*d-1/3*b^3*c)*x^2/(b*x^3+a)+(4/3*f*a^3-7/3*a^2*b*e+10/3*a*b^
2*d-13/3*b^3*c)*(-1/3/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/6/b/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3*
3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))))

Fricas [A] (verification not implemented)

none

Time = 0.67 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.32 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^2} \, dx=\frac {420 \, {\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{12} + 315 \, {\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{9} - 45 \, {\left (13 \, a^{2} b^{2} c - 10 \, a^{3} b d + 7 \, a^{4} e\right )} x^{6} - 126 \, a^{4} c + 18 \, {\left (13 \, a^{3} b c - 10 \, a^{4} d\right )} x^{3} + 140 \, \sqrt {3} {\left ({\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{13} + {\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{10}\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 70 \, {\left ({\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{13} + {\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{10}\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 140 \, {\left ({\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{13} + {\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{10}\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{1260 \, {\left (a^{5} b x^{13} + a^{6} x^{10}\right )}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((f*x**9+e*x**6+d*x**3+c)/x**11/(b*x**3+a)**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^2} \, dx=\frac {140 \, {\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{12} + 105 \, {\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{9} - 15 \, {\left (13 \, a^{2} b^{2} c - 10 \, a^{3} b d + 7 \, a^{4} e\right )} x^{6} - 42 \, a^{4} c + 6 \, {\left (13 \, a^{3} b c - 10 \, a^{4} d\right )} x^{3}}{420 \, {\left (a^{5} b x^{13} + a^{6} x^{10}\right )}} + \frac {\sqrt {3} {\left (13 \, b^{3} c - 10 \, a b^{2} d + 7 \, a^{2} b e - 4 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (13 \, b^{3} c - 10 \, a b^{2} d + 7 \, a^{2} b e - 4 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (13 \, b^{3} c - 10 \, a b^{2} d + 7 \, a^{2} b e - 4 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.29 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^2} \, dx=-\frac {{\left (13 \, b^{4} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 10 \, a b^{3} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 7 \, a^{2} b^{2} e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 4 \, a^{3} b f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{6}} - \frac {\sqrt {3} {\left (13 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 10 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d + 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e - 4 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{6} b} + \frac {b^{4} c x^{2} - a b^{3} d x^{2} + a^{2} b^{2} e x^{2} - a^{3} b f x^{2}}{3 \, {\left (b x^{3} + a\right )} a^{5}} + \frac {{\left (13 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 10 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d + 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e - 4 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{6} b} + \frac {560 \, b^{3} c x^{9} - 420 \, a b^{2} d x^{9} + 280 \, a^{2} b e x^{9} - 140 \, a^{3} f x^{9} - 105 \, a b^{2} c x^{6} + 70 \, a^{2} b d x^{6} - 35 \, a^{3} e x^{6} + 40 \, a^{2} b c x^{3} - 20 \, a^{3} d x^{3} - 14 \, a^{3} c}{140 \, a^{5} x^{10}} \]

[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) + 7*a^2*b^2*e*(-a/b)^(1/3) - 4*a^3*b*f*(-a/b)^(1/3))*(-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 +
7*(-a*b^2)^(2/3)*a^2*b*e - 4*(-a*b^2)^(2/3)*a^3*f)*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^2*b^2*e*x^2 - a^3*b*f*x^2)/((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 + 7*(-a*b^2)^(2/3)*a^2*b*e - 4*(-a*b^2)^(2/3)*a^3*f)*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 + 280*a^2*b*e*x^9 - 140*a^3*f*x^9 - 105*a*b^
2*c*x^6 + 70*a^2*b*d*x^6 - 35*a^3*e*x^6 + 40*a^2*b*c*x^3 - 20*a^3*d*x^3 - 14*a^3*c)/(a^5*x^10)

Mupad [B] (verification not implemented)

Time = 9.80 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^2} \, dx=-\frac {\frac {c}{10\,a}-\frac {x^9\,\left (-4\,f\,a^3+7\,e\,a^2\,b-10\,d\,a\,b^2+13\,c\,b^3\right )}{4\,a^4}+\frac {x^3\,\left (10\,a\,d-13\,b\,c\right )}{70\,a^2}+\frac {x^6\,\left (7\,e\,a^2-10\,d\,a\,b+13\,c\,b^2\right )}{28\,a^3}-\frac {b\,x^{12}\,\left (-4\,f\,a^3+7\,e\,a^2\,b-10\,d\,a\,b^2+13\,c\,b^3\right )}{3\,a^5}}{b\,x^{13}+a\,x^{10}}-\frac {b^{1/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-4\,f\,a^3+7\,e\,a^2\,b-10\,d\,a\,b^2+13\,c\,b^3\right )}{9\,a^{16/3}}+\frac {b^{1/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-4\,f\,a^3+7\,e\,a^2\,b-10\,d\,a\,b^2+13\,c\,b^3\right )}{9\,a^{16/3}}-\frac {b^{1/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-4\,f\,a^3+7\,e\,a^2\,b-10\,d\,a\,b^2+13\,c\,b^3\right )}{9\,a^{16/3}} \]

[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))