3.274 \(\int \frac {c+d x^3+e x^6+f x^9}{x^{12} (a+b x^3)^2} \, dx\)

Optimal. Leaf size=335 \[ \frac {2 b c-a d}{8 a^3 x^8}-\frac {c}{11 a^2 x^{11}}-\frac {a^2 e-2 a b d+3 b^2 c}{5 a^4 x^5}-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{18 a^{17/3}}+\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{9 a^{17/3}}-\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{3 \sqrt {3} a^{17/3}}+\frac {b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac {a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{2 a^5 x^2} \]

[Out]

-1/11*c/a^2/x^11+1/8*(-a*d+2*b*c)/a^3/x^8+1/5*(-a^2*e+2*a*b*d-3*b^2*c)/a^4/x^5+1/2*(-a^3*f+2*a^2*b*e-3*a*b^2*d
+4*b^3*c)/a^5/x^2+1/3*b*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/a^5/(b*x^3+a)+1/9*b^(2/3)*(-5*a^3*f+8*a^2*b*e-11*a*b^
2*d+14*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(17/3)-1/18*b^(2/3)*(-5*a^3*f+8*a^2*b*e-11*a*b^2*d+14*b^3*c)*ln(a^(2/3)-
a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(17/3)-1/9*b^(2/3)*(-5*a^3*f+8*a^2*b*e-11*a*b^2*d+14*b^3*c)*arctan(1/3*(a^(1/
3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(17/3)*3^(1/2)

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Rubi [A]  time = 0.43, antiderivative size = 335, 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, 200, 31, 634, 617, 204, 628} \[ \frac {b x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac {2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{2 a^5 x^2}-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (8 a^2 b e-5 a^3 f-11 a b^2 d+14 b^3 c\right )}{18 a^{17/3}}+\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (8 a^2 b e-5 a^3 f-11 a b^2 d+14 b^3 c\right )}{9 a^{17/3}}-\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (8 a^2 b e-5 a^3 f-11 a b^2 d+14 b^3 c\right )}{3 \sqrt {3} a^{17/3}}-\frac {a^2 e-2 a b d+3 b^2 c}{5 a^4 x^5}+\frac {2 b c-a d}{8 a^3 x^8}-\frac {c}{11 a^2 x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^2),x]

[Out]

-c/(11*a^2*x^11) + (2*b*c - a*d)/(8*a^3*x^8) - (3*b^2*c - 2*a*b*d + a^2*e)/(5*a^4*x^5) + (4*b^3*c - 3*a*b^2*d
+ 2*a^2*b*e - a^3*f)/(2*a^5*x^2) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(3*a^5*(a + b*x^3)) - (b^(2/3)*(1
4*b^3*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(1
7/3)) + (b^(2/3)*(14*b^3*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(17/3)) - (b^(2/
3)*(14*b^3*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(17/3))

Rule 31

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

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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 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 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^{12} \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}{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 {2 b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{12}}{a^4}}{x^{12} \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}{3 a^5 \left (a+b x^3\right )}-\frac {\int \left (-\frac {3 b^3 c}{a x^{12}}-\frac {3 b^3 (-2 b c+a d)}{a^2 x^9}-\frac {3 b^3 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^6}-\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^3}+\frac {b^4 \left (-14 b^3 c+11 a b^2 d-8 a^2 b e+5 a^3 f\right )}{a^4 \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac {c}{11 a^2 x^{11}}+\frac {2 b c-a d}{8 a^3 x^8}-\frac {3 b^2 c-2 a b d+a^2 e}{5 a^4 x^5}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{2 a^5 x^2}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^5 \left (a+b x^3\right )}+\frac {\left (b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right )\right ) \int \frac {1}{a+b x^3} \, dx}{3 a^5}\\ &=-\frac {c}{11 a^2 x^{11}}+\frac {2 b c-a d}{8 a^3 x^8}-\frac {3 b^2 c-2 a b d+a^2 e}{5 a^4 x^5}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{2 a^5 x^2}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^5 \left (a+b x^3\right )}+\frac {\left (b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{17/3}}+\frac {\left (b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right )\right ) \int \frac {2 \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^{17/3}}\\ &=-\frac {c}{11 a^2 x^{11}}+\frac {2 b c-a d}{8 a^3 x^8}-\frac {3 b^2 c-2 a b d+a^2 e}{5 a^4 x^5}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{2 a^5 x^2}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^5 \left (a+b x^3\right )}+\frac {b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{17/3}}-\frac {\left (b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 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^{17/3}}+\frac {\left (b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 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^{16/3}}\\ &=-\frac {c}{11 a^2 x^{11}}+\frac {2 b c-a d}{8 a^3 x^8}-\frac {3 b^2 c-2 a b d+a^2 e}{5 a^4 x^5}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{2 a^5 x^2}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^5 \left (a+b x^3\right )}+\frac {b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{17/3}}-\frac {b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{17/3}}+\frac {\left (b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 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^{17/3}}\\ &=-\frac {c}{11 a^2 x^{11}}+\frac {2 b c-a d}{8 a^3 x^8}-\frac {3 b^2 c-2 a b d+a^2 e}{5 a^4 x^5}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{2 a^5 x^2}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^5 \left (a+b x^3\right )}-\frac {b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 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^{17/3}}+\frac {b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{17/3}}-\frac {b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{17/3}}\\ \end {align*}

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Mathematica [A]  time = 0.33, size = 317, normalized size = 0.95 \[ \frac {-\frac {495 a^{8/3} (a d-2 b c)}{x^8}-\frac {360 a^{11/3} c}{x^{11}}-\frac {792 a^{5/3} \left (a^2 e-2 a b d+3 b^2 c\right )}{x^5}+440 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )-440 \sqrt {3} b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )-\frac {1320 a^{2/3} b x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a+b x^3}-\frac {1980 a^{2/3} \left (a^3 f-2 a^2 b e+3 a b^2 d-4 b^3 c\right )}{x^2}+220 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^3 f-8 a^2 b e+11 a b^2 d-14 b^3 c\right )}{3960 a^{17/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^2),x]

[Out]

((-360*a^(11/3)*c)/x^11 - (495*a^(8/3)*(-2*b*c + a*d))/x^8 - (792*a^(5/3)*(3*b^2*c - 2*a*b*d + a^2*e))/x^5 - (
1980*a^(2/3)*(-4*b^3*c + 3*a*b^2*d - 2*a^2*b*e + a^3*f))/x^2 - (1320*a^(2/3)*b*(-(b^3*c) + a*b^2*d - a^2*b*e +
a^3*f)*x)/(a + b*x^3) - 440*Sqrt[3]*b^(2/3)*(14*b^3*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*ArcTan[(1 - (2*b^(1
/3)*x)/a^(1/3))/Sqrt[3]] + 440*b^(2/3)*(14*b^3*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x]
+ 220*b^(2/3)*(-14*b^3*c + 11*a*b^2*d - 8*a^2*b*e + 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(
3960*a^(17/3))

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fricas [A]  time = 0.76, size = 475, normalized size = 1.42 \[ \frac {660 \, {\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{12} + 396 \, {\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{9} - 99 \, {\left (14 \, a^{2} b^{2} c - 11 \, a^{3} b d + 8 \, a^{4} e\right )} x^{6} - 360 \, a^{4} c + 45 \, {\left (14 \, a^{3} b c - 11 \, a^{4} d\right )} x^{3} - 440 \, \sqrt {3} {\left ({\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{14} + {\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{11}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 220 \, {\left ({\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{14} + {\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{11}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) - 440 \, {\left ({\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{14} + {\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{11}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right )}{3960 \, {\left (a^{5} b x^{14} + a^{6} x^{11}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

1/3960*(660*(14*b^4*c - 11*a*b^3*d + 8*a^2*b^2*e - 5*a^3*b*f)*x^12 + 396*(14*a*b^3*c - 11*a^2*b^2*d + 8*a^3*b*
e - 5*a^4*f)*x^9 - 99*(14*a^2*b^2*c - 11*a^3*b*d + 8*a^4*e)*x^6 - 360*a^4*c + 45*(14*a^3*b*c - 11*a^4*d)*x^3 -
440*sqrt(3)*((14*b^4*c - 11*a*b^3*d + 8*a^2*b^2*e - 5*a^3*b*f)*x^14 + (14*a*b^3*c - 11*a^2*b^2*d + 8*a^3*b*e
- 5*a^4*f)*x^11)*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x*(-b^2/a^2)^(2/3) - sqrt(3)*b)/b) + 220*((14*b^4*c
- 11*a*b^3*d + 8*a^2*b^2*e - 5*a^3*b*f)*x^14 + (14*a*b^3*c - 11*a^2*b^2*d + 8*a^3*b*e - 5*a^4*f)*x^11)*(-b^2/a
^2)^(1/3)*log(b^2*x^2 + a*b*x*(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) - 440*((14*b^4*c - 11*a*b^3*d + 8*a^2*b
^2*e - 5*a^3*b*f)*x^14 + (14*a*b^3*c - 11*a^2*b^2*d + 8*a^3*b*e - 5*a^4*f)*x^11)*(-b^2/a^2)^(1/3)*log(b*x - a*
(-b^2/a^2)^(1/3)))/(a^5*b*x^14 + a^6*x^11)

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giac [A]  time = 0.18, size = 391, normalized size = 1.17 \[ \frac {\sqrt {3} {\left (14 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 11 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f + 8 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{6}} - \frac {{\left (14 \, b^{4} c - 11 \, a b^{3} d - 5 \, a^{3} b f + 8 \, a^{2} b^{2} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{6}} + \frac {{\left (14 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 11 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f + 8 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{6}} + \frac {b^{4} c x - a b^{3} d x - a^{3} b f x + a^{2} b^{2} x e}{3 \, {\left (b x^{3} + a\right )} a^{5}} + \frac {880 \, b^{3} c x^{9} - 660 \, a b^{2} d x^{9} - 220 \, a^{3} f x^{9} + 440 \, a^{2} b x^{9} e - 264 \, a b^{2} c x^{6} + 176 \, a^{2} b d x^{6} - 88 \, a^{3} x^{6} e + 110 \, a^{2} b c x^{3} - 55 \, a^{3} d x^{3} - 40 \, a^{3} c}{440 \, a^{5} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^2,x, algorithm="giac")

[Out]

1/9*sqrt(3)*(14*(-a*b^2)^(1/3)*b^3*c - 11*(-a*b^2)^(1/3)*a*b^2*d - 5*(-a*b^2)^(1/3)*a^3*f + 8*(-a*b^2)^(1/3)*a
^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/a^6 - 1/9*(14*b^4*c - 11*a*b^3*d - 5*a^3*b*f + 8
*a^2*b^2*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^6 + 1/18*(14*(-a*b^2)^(1/3)*b^3*c - 11*(-a*b^2)^(1/3)*a*
b^2*d - 5*(-a*b^2)^(1/3)*a^3*f + 8*(-a*b^2)^(1/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/a^6 + 1/3*
(b^4*c*x - a*b^3*d*x - a^3*b*f*x + a^2*b^2*x*e)/((b*x^3 + a)*a^5) + 1/440*(880*b^3*c*x^9 - 660*a*b^2*d*x^9 - 2
20*a^3*f*x^9 + 440*a^2*b*x^9*e - 264*a*b^2*c*x^6 + 176*a^2*b*d*x^6 - 88*a^3*x^6*e + 110*a^2*b*c*x^3 - 55*a^3*d
*x^3 - 40*a^3*c)/(a^5*x^11)

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maple [A]  time = 0.06, size = 566, normalized size = 1.69 \[ -\frac {b f x}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {b^{2} e x}{3 \left (b \,x^{3}+a \right ) a^{3}}-\frac {b^{3} d x}{3 \left (b \,x^{3}+a \right ) a^{4}}+\frac {b^{4} c x}{3 \left (b \,x^{3}+a \right ) a^{5}}-\frac {5 \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 {2}{3}} a^{2}}-\frac {5 f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}+\frac {5 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 {2}{3}} a^{2}}+\frac {8 \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 {2}{3}} a^{3}}+\frac {8 b e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}-\frac {4 b 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 {2}{3}} a^{3}}-\frac {11 \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 {2}{3}} a^{4}}-\frac {11 b^{2} d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}+\frac {11 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 )}{18 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}+\frac {14 \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 {2}{3}} a^{5}}+\frac {14 b^{3} c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{5}}-\frac {7 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 )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{5}}-\frac {f}{2 a^{2} x^{2}}+\frac {b e}{a^{3} x^{2}}-\frac {3 b^{2} d}{2 a^{4} x^{2}}+\frac {2 b^{3} c}{a^{5} x^{2}}-\frac {e}{5 a^{2} x^{5}}+\frac {2 b d}{5 a^{3} x^{5}}-\frac {3 b^{2} c}{5 a^{4} x^{5}}-\frac {d}{8 a^{2} x^{8}}+\frac {b c}{4 a^{3} x^{8}}-\frac {c}{11 a^{2} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^2,x)

[Out]

5/18/a^2*f/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/a^3/x^2*b*e-3/2/a^4/x^2*b^2*d+2/a^5/x^2*b^3*c+1/4/a
^3/x^8*b*c+2/5/a^3/x^5*b*d-3/5/a^4/x^5*b^2*c-5/9/a^2*f/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/3*b/a^2*x/(b*x^3+a)*f+1
/3*b^2/a^3*x/(b*x^3+a)*e-1/3*b^3/a^4*x/(b*x^3+a)*d+11/18*b^2/a^4*d/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3
))+14/9*b^3/a^5*c/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-7/9*b^3/a^5*c/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/
3*b^4/a^5*x/(b*x^3+a)*c-1/5/a^2/x^5*e-1/2/a^2/x^2*f-1/8/a^2/x^8*d-5/9/a^2*f/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(
1/2)*(2/(a/b)^(1/3)*x-1))+8/9*b/a^3*e/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-4/9*b/a^3*e/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)
*x+(a/b)^(2/3))-11/9*b^2/a^4*d/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+8/9*b/a^3*e/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2
)*(2/(a/b)^(1/3)*x-1))-11/9*b^2/a^4*d/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+14/9*b^3/a^5
*c/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/11*c/a^2/x^11

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maxima [A]  time = 3.07, size = 323, normalized size = 0.96 \[ \frac {220 \, {\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{12} + 132 \, {\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{9} - 33 \, {\left (14 \, a^{2} b^{2} c - 11 \, a^{3} b d + 8 \, a^{4} e\right )} x^{6} - 120 \, a^{4} c + 15 \, {\left (14 \, a^{3} b c - 11 \, a^{4} d\right )} x^{3}}{1320 \, {\left (a^{5} b x^{14} + a^{6} x^{11}\right )}} + \frac {\sqrt {3} {\left (14 \, b^{3} c - 11 \, a b^{2} d + 8 \, a^{2} b e - 5 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (14 \, b^{3} c - 11 \, a b^{2} d + 8 \, a^{2} b e - 5 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (14 \, b^{3} c - 11 \, a b^{2} d + 8 \, a^{2} b e - 5 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

1/1320*(220*(14*b^4*c - 11*a*b^3*d + 8*a^2*b^2*e - 5*a^3*b*f)*x^12 + 132*(14*a*b^3*c - 11*a^2*b^2*d + 8*a^3*b*
e - 5*a^4*f)*x^9 - 33*(14*a^2*b^2*c - 11*a^3*b*d + 8*a^4*e)*x^6 - 120*a^4*c + 15*(14*a^3*b*c - 11*a^4*d)*x^3)/
(a^5*b*x^14 + a^6*x^11) + 1/9*sqrt(3)*(14*b^3*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*arctan(1/3*sqrt(3)*(2*x -
(a/b)^(1/3))/(a/b)^(1/3))/(a^5*(a/b)^(2/3)) - 1/18*(14*b^3*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*log(x^2 - x*(
a/b)^(1/3) + (a/b)^(2/3))/(a^5*(a/b)^(2/3)) + 1/9*(14*b^3*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*log(x + (a/b)^
(1/3))/(a^5*(a/b)^(2/3))

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mupad [B]  time = 5.12, size = 310, normalized size = 0.93 \[ \frac {b^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-5\,f\,a^3+8\,e\,a^2\,b-11\,d\,a\,b^2+14\,c\,b^3\right )}{9\,a^{17/3}}-\frac {\frac {c}{11\,a}-\frac {x^9\,\left (-5\,f\,a^3+8\,e\,a^2\,b-11\,d\,a\,b^2+14\,c\,b^3\right )}{10\,a^4}+\frac {x^3\,\left (11\,a\,d-14\,b\,c\right )}{88\,a^2}+\frac {x^6\,\left (8\,e\,a^2-11\,d\,a\,b+14\,c\,b^2\right )}{40\,a^3}-\frac {b\,x^{12}\,\left (-5\,f\,a^3+8\,e\,a^2\,b-11\,d\,a\,b^2+14\,c\,b^3\right )}{6\,a^5}}{b\,x^{14}+a\,x^{11}}+\frac {b^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-5\,f\,a^3+8\,e\,a^2\,b-11\,d\,a\,b^2+14\,c\,b^3\right )}{9\,a^{17/3}}-\frac {b^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-5\,f\,a^3+8\,e\,a^2\,b-11\,d\,a\,b^2+14\,c\,b^3\right )}{9\,a^{17/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^2),x)

[Out]

(b^(2/3)*log(b^(1/3)*x + a^(1/3))*(14*b^3*c - 5*a^3*f - 11*a*b^2*d + 8*a^2*b*e))/(9*a^(17/3)) - (c/(11*a) - (x
^9*(14*b^3*c - 5*a^3*f - 11*a*b^2*d + 8*a^2*b*e))/(10*a^4) + (x^3*(11*a*d - 14*b*c))/(88*a^2) + (x^6*(14*b^2*c
+ 8*a^2*e - 11*a*b*d))/(40*a^3) - (b*x^12*(14*b^3*c - 5*a^3*f - 11*a*b^2*d + 8*a^2*b*e))/(6*a^5))/(a*x^11 + b
*x^14) + (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)*(14*b^3*c - 5*a^3*f -
11*a*b^2*d + 8*a^2*b*e))/(9*a^(17/3)) - (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)*(14*b^3*c - 5*a^3*f - 11*a*b^2*d + 8*a^2*b*e))/(9*a^(17/3))

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

[Out]

Timed out

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