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

Optimal. Leaf size=303 \[ -\frac {c}{a^3 x}-\frac {x^2 \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{9 a^3 b^2 \left (a+b x^3\right )}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f-a^2 b e-2 a b^2 d+14 b^3 c\right )}{54 a^{10/3} b^{8/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f-a^2 b e-2 a b^2 d+14 b^3 c\right )}{27 a^{10/3} b^{8/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-5 a^3 f-a^2 b e-2 a b^2 d+14 b^3 c\right )}{9 \sqrt {3} a^{10/3} b^{8/3}} \]

[Out]

-c/a^3/x-1/6*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x^2/a^2/b^2/(b*x^3+a)^2-1/9*(4*a^3*f-a^2*b*e-2*a*b^2*d+5*b^3*c)*x^
2/a^3/b^2/(b*x^3+a)+1/27*(-5*a^3*f-a^2*b*e-2*a*b^2*d+14*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(10/3)/b^(8/3)-1/54*(-5
*a^3*f-a^2*b*e-2*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^(10/3)/b^(8/3)+1/27*(-5*a^3*f-a
^2*b*e-2*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^(10/3)/b^(8/3)*3^(1/2)

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Rubi [A]  time = 0.34, antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1829, 1484, 453, 292, 31, 634, 617, 204, 628} \[ -\frac {x^2 \left (-a^2 b e+4 a^3 f-2 a b^2 d+5 b^3 c\right )}{9 a^3 b^2 \left (a+b x^3\right )}-\frac {x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^2 b e-5 a^3 f-2 a b^2 d+14 b^3 c\right )}{54 a^{10/3} b^{8/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e-5 a^3 f-2 a b^2 d+14 b^3 c\right )}{27 a^{10/3} b^{8/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-a^2 b e-5 a^3 f-2 a b^2 d+14 b^3 c\right )}{9 \sqrt {3} a^{10/3} b^{8/3}}-\frac {c}{a^3 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c/(a^3*x)) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a^2*b^2*(a + b*x^3)^2) - ((5*b^3*c - 2*a*b^2*d - a
^2*b*e + 4*a^3*f)*x^2)/(9*a^3*b^2*(a + b*x^3)) + ((14*b^3*c - 2*a*b^2*d - a^2*b*e - 5*a^3*f)*ArcTan[(a^(1/3) -
2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(10/3)*b^(8/3)) + ((14*b^3*c - 2*a*b^2*d - a^2*b*e - 5*a^3*f)*L
og[a^(1/3) + b^(1/3)*x])/(27*a^(10/3)*b^(8/3)) - ((14*b^3*c - 2*a*b^2*d - a^2*b*e - 5*a^3*f)*Log[a^(2/3) - a^(
1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(10/3)*b^(8/3))

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 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

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 1484

Int[(x_)^(m_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_), x_Symbol] :> S
imp[((-d)^((m - Mod[m, n])/n - 1)*(c*d^2 - b*d*e + a*e^2)^p*x^(Mod[m, n] + 1)*(d + e*x^n)^(q + 1))/(n*e^(2*p +
(m - Mod[m, n])/n)*(q + 1)), x] + Dist[(-d)^((m - Mod[m, n])/n - 1)/(n*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^n)^
(q + 1)*ExpandToSum[Together[(1*(n*(-d)^(-((m - Mod[m, n])/n) + 1)*e^(2*p)*(q + 1)*(a + b*x^n + c*x^(2*n))^p -
((c*d^2 - b*d*e + a*e^2)^p/(e^((m - Mod[m, n])/n)*x^(m - Mod[m, n])))*(d*(Mod[m, n] + 1) + e*(Mod[m, n] + n*(
q + 1) + 1)*x^n)))/(d + e*x^n)], x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0
] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m, 0]

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]

Rubi steps

\begin {align*} \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )^3} \, dx &=-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\int \frac {-6 b^3 c+2 b \left (\frac {2 b^3 c}{a}-2 b^2 d-a b e+a^2 f\right ) x^3-6 a b^2 f x^6}{x^2 \left (a+b x^3\right )^2} \, dx}{6 a b^3}\\ &=-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) x^2}{9 a^3 b^2 \left (a+b x^3\right )}+\frac {\int \frac {18 a b^5 c-2 b^3 \left (5 b^3 c-2 a b^2 d-a^2 b e-5 a^3 f\right ) x^3}{x^2 \left (a+b x^3\right )} \, dx}{18 a^3 b^5}\\ &=-\frac {c}{a^3 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) x^2}{9 a^3 b^2 \left (a+b x^3\right )}-\frac {\left (14 b^3 c-2 a b^2 d-a^2 b e-5 a^3 f\right ) \int \frac {x}{a+b x^3} \, dx}{9 a^3 b^2}\\ &=-\frac {c}{a^3 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) x^2}{9 a^3 b^2 \left (a+b x^3\right )}+\frac {\left (14 b^3 c-2 a b^2 d-a^2 b e-5 a^3 f\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{10/3} b^{7/3}}-\frac {\left (14 b^3 c-2 a b^2 d-a^2 b e-5 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}{27 a^{10/3} b^{7/3}}\\ &=-\frac {c}{a^3 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) x^2}{9 a^3 b^2 \left (a+b x^3\right )}+\frac {\left (14 b^3 c-2 a b^2 d-a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{8/3}}-\frac {\left (14 b^3 c-2 a b^2 d-a^2 b e-5 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}{54 a^{10/3} b^{8/3}}-\frac {\left (14 b^3 c-2 a b^2 d-a^2 b e-5 a^3 f\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^3 b^{7/3}}\\ &=-\frac {c}{a^3 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) x^2}{9 a^3 b^2 \left (a+b x^3\right )}+\frac {\left (14 b^3 c-2 a b^2 d-a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{8/3}}-\frac {\left (14 b^3 c-2 a b^2 d-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 )}{54 a^{10/3} b^{8/3}}-\frac {\left (14 b^3 c-2 a b^2 d-a^2 b e-5 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 )}{9 a^{10/3} b^{8/3}}\\ &=-\frac {c}{a^3 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) x^2}{9 a^3 b^2 \left (a+b x^3\right )}+\frac {\left (14 b^3 c-2 a b^2 d-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 )}{9 \sqrt {3} a^{10/3} b^{8/3}}+\frac {\left (14 b^3 c-2 a b^2 d-a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{8/3}}-\frac {\left (14 b^3 c-2 a b^2 d-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 )}{54 a^{10/3} b^{8/3}}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 286, normalized size = 0.94 \[ \frac {-\frac {6 \sqrt [3]{a} x^2 \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{b^2 \left (a+b x^3\right )}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^3 f+a^2 b e+2 a b^2 d-14 b^3 c\right )}{b^{8/3}}+\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (-5 a^3 f-a^2 b e-2 a b^2 d+14 b^3 c\right )}{b^{8/3}}+\frac {9 a^{4/3} x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b^2 \left (a+b x^3\right )^2}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^3 f+a^2 b e+2 a b^2 d-14 b^3 c\right )}{b^{8/3}}-\frac {54 \sqrt [3]{a} c}{x}}{54 a^{10/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-54*a^(1/3)*c)/x + (9*a^(4/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(b^2*(a + b*x^3)^2) - (6*a^(1/3)*(
5*b^3*c - 2*a*b^2*d - a^2*b*e + 4*a^3*f)*x^2)/(b^2*(a + b*x^3)) + (2*Sqrt[3]*(14*b^3*c - 2*a*b^2*d - a^2*b*e -
5*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(8/3) - (2*(-14*b^3*c + 2*a*b^2*d + a^2*b*e + 5*a^3*f
)*Log[a^(1/3) + b^(1/3)*x])/b^(8/3) + ((-14*b^3*c + 2*a*b^2*d + a^2*b*e + 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/
3)*x + b^(2/3)*x^2])/b^(8/3))/(54*a^(10/3))

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fricas [B]  time = 0.60, size = 1206, normalized size = 3.98 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/54*(54*a^3*b^4*c + 6*(14*a*b^6*c - 2*a^2*b^5*d - a^3*b^4*e + 4*a^4*b^3*f)*x^6 + 3*(49*a^2*b^5*c - 7*a^3*b^
4*d + a^4*b^3*e + 5*a^5*b^2*f)*x^3 + 3*sqrt(1/3)*((14*a*b^6*c - 2*a^2*b^5*d - a^3*b^4*e - 5*a^4*b^3*f)*x^7 + 2
*(14*a^2*b^5*c - 2*a^3*b^4*d - a^4*b^3*e - 5*a^5*b^2*f)*x^4 + (14*a^3*b^4*c - 2*a^4*b^3*d - a^5*b^2*e - 5*a^6*
b*f)*x)*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*b^5*c - 2*a*b^4*d - a^2*b^3*e - 5*a^3*b
^2*f)*x^7 + 2*(14*a*b^4*c - 2*a^2*b^3*d - a^3*b^2*e - 5*a^4*b*f)*x^4 + (14*a^2*b^3*c - 2*a^3*b^2*d - a^4*b*e -
5*a^5*f)*x)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 2*((14*b^5*c - 2*a*b^4*d - a^
2*b^3*e - 5*a^3*b^2*f)*x^7 + 2*(14*a*b^4*c - 2*a^2*b^3*d - a^3*b^2*e - 5*a^4*b*f)*x^4 + (14*a^2*b^3*c - 2*a^3*
b^2*d - a^4*b*e - 5*a^5*f)*x)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^4*b^6*x^7 + 2*a^5*b^5*x^4 + a^6*b^4
*x), -1/54*(54*a^3*b^4*c + 6*(14*a*b^6*c - 2*a^2*b^5*d - a^3*b^4*e + 4*a^4*b^3*f)*x^6 + 3*(49*a^2*b^5*c - 7*a^
3*b^4*d + a^4*b^3*e + 5*a^5*b^2*f)*x^3 + 6*sqrt(1/3)*((14*a*b^6*c - 2*a^2*b^5*d - a^3*b^4*e - 5*a^4*b^3*f)*x^7
+ 2*(14*a^2*b^5*c - 2*a^3*b^4*d - a^4*b^3*e - 5*a^5*b^2*f)*x^4 + (14*a^3*b^4*c - 2*a^4*b^3*d - a^5*b^2*e - 5*
a^6*b*f)*x)*sqrt(-(-a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*b*x + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3)/a)/b) + ((1
4*b^5*c - 2*a*b^4*d - a^2*b^3*e - 5*a^3*b^2*f)*x^7 + 2*(14*a*b^4*c - 2*a^2*b^3*d - a^3*b^2*e - 5*a^4*b*f)*x^4
+ (14*a^2*b^3*c - 2*a^3*b^2*d - a^4*b*e - 5*a^5*f)*x)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^
2)^(2/3)) - 2*((14*b^5*c - 2*a*b^4*d - a^2*b^3*e - 5*a^3*b^2*f)*x^7 + 2*(14*a*b^4*c - 2*a^2*b^3*d - a^3*b^2*e
- 5*a^4*b*f)*x^4 + (14*a^2*b^3*c - 2*a^3*b^2*d - a^4*b*e - 5*a^5*f)*x)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)
))/(a^4*b^6*x^7 + 2*a^5*b^5*x^4 + a^6*b^4*x)]

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giac [A]  time = 0.21, size = 341, normalized size = 1.13 \[ -\frac {\sqrt {3} {\left (14 \, b^{3} c - 2 \, a b^{2} d - 5 \, a^{3} f - a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b^{2}} - \frac {c}{a^{3} x} + \frac {{\left (14 \, b^{3} c - 2 \, a b^{2} d - 5 \, a^{3} f - a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b^{2}} + \frac {{\left (14 \, b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{4} b^{2}} - \frac {10 \, b^{4} c x^{5} - 4 \, a b^{3} d x^{5} + 8 \, a^{3} b f x^{5} - 2 \, a^{2} b^{2} x^{5} e + 13 \, a b^{3} c x^{2} - 7 \, a^{2} b^{2} d x^{2} + 5 \, a^{4} f x^{2} + a^{3} b x^{2} e}{18 \, {\left (b x^{3} + a\right )}^{2} a^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*sqrt(3)*(14*b^3*c - 2*a*b^2*d - 5*a^3*f - 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^3*b^2) - c/(a^3*x) + 1/54*(14*b^3*c - 2*a*b^2*d - 5*a^3*f - a^2*b*e)*log(x^2 + x*(-a/b)^(1/
3) + (-a/b)^(2/3))/((-a*b^2)^(1/3)*a^3*b^2) + 1/27*(14*b^3*c*(-a/b)^(1/3) - 2*a*b^2*d*(-a/b)^(1/3) - 5*a^3*f*(
-a/b)^(1/3) - a^2*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^4*b^2) - 1/18*(10*b^4*c*x^5 - 4
*a*b^3*d*x^5 + 8*a^3*b*f*x^5 - 2*a^2*b^2*x^5*e + 13*a*b^3*c*x^2 - 7*a^2*b^2*d*x^2 + 5*a^4*f*x^2 + a^3*b*x^2*e)
/((b*x^3 + a)^2*a^3*b^2)

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maple [B]  time = 0.12, size = 547, normalized size = 1.81 \[ \frac {e \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a}+\frac {2 b d \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{2}}-\frac {5 b^{2} c \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{3}}-\frac {4 f \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} b}-\frac {5 a f \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b^{2}}+\frac {7 d \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a}-\frac {13 b c \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a^{2}}-\frac {e \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b}+\frac {\sqrt {3}\, e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{2}}-\frac {e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{2}}+\frac {e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{2}}+\frac {2 \sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b}-\frac {2 d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b}+\frac {d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b}-\frac {14 \sqrt {3}\, c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\frac {14 c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {7 c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\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 )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}-\frac {5 f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}+\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 )}{54 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}-\frac {c}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/9/(b*x^3+a)^2/b*x^5*f+1/9/a/(b*x^3+a)^2*x^5*e+2/9/a^2/(b*x^3+a)^2*b*x^5*d-5/9/a^3/(b*x^3+a)^2*b^2*x^5*c-5/1
8*a/(b*x^3+a)^2/b^2*x^2*f-1/18/(b*x^3+a)^2/b*x^2*e+7/18/a/(b*x^3+a)^2*x^2*d-13/18/a^2/(b*x^3+a)^2*b*x^2*c-5/27
/b^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*f-1/27/a/b^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*e-2/27/a^2/b/(a/b)^(1/3)*ln(x+(a
/b)^(1/3))*d+14/27/a^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*c+5/54/b^3/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*
f+1/54/a/b^2/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*e+1/27/a^2/b/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)
^(2/3))*d-7/27/a^3/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*c+5/27/b^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(
1/2)*(2/(a/b)^(1/3)*x-1))*f+1/27/a/b^2*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e+2/27/a^2/
b*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d-14/27/a^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/
2)*(2/(a/b)^(1/3)*x-1))*c-1/a^3*c/x

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maxima [A]  time = 2.96, size = 300, normalized size = 0.99 \[ -\frac {2 \, {\left (14 \, b^{4} c - 2 \, a b^{3} d - a^{2} b^{2} e + 4 \, a^{3} b f\right )} x^{6} + 18 \, a^{2} b^{2} c + {\left (49 \, a b^{3} c - 7 \, a^{2} b^{2} d + a^{3} b e + 5 \, a^{4} f\right )} x^{3}}{18 \, {\left (a^{3} b^{4} x^{7} + 2 \, a^{4} b^{3} x^{4} + a^{5} b^{2} x\right )}} - \frac {\sqrt {3} {\left (14 \, b^{3} c - 2 \, a b^{2} d - a^{2} b e - 5 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{3} b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (14 \, b^{3} c - 2 \, a b^{2} d - a^{2} b e - 5 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{3} b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (14 \, b^{3} c - 2 \, a b^{2} d - a^{2} b e - 5 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(2*(14*b^4*c - 2*a*b^3*d - a^2*b^2*e + 4*a^3*b*f)*x^6 + 18*a^2*b^2*c + (49*a*b^3*c - 7*a^2*b^2*d + a^3*b
*e + 5*a^4*f)*x^3)/(a^3*b^4*x^7 + 2*a^4*b^3*x^4 + a^5*b^2*x) - 1/27*sqrt(3)*(14*b^3*c - 2*a*b^2*d - a^2*b*e -
5*a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^3*b^3*(a/b)^(1/3)) - 1/54*(14*b^3*c - 2*a*b^2*
d - a^2*b*e - 5*a^3*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*b^3*(a/b)^(1/3)) + 1/27*(14*b^3*c - 2*a*b^2
*d - a^2*b*e - 5*a^3*f)*log(x + (a/b)^(1/3))/(a^3*b^3*(a/b)^(1/3))

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mupad [B]  time = 5.20, size = 276, normalized size = 0.91 \[ -\frac {\frac {c}{a}+\frac {x^6\,\left (4\,f\,a^3-e\,a^2\,b-2\,d\,a\,b^2+14\,c\,b^3\right )}{9\,a^3\,b}+\frac {x^3\,\left (5\,f\,a^3+e\,a^2\,b-7\,d\,a\,b^2+49\,c\,b^3\right )}{18\,a^2\,b^2}}{a^2\,x+2\,a\,b\,x^4+b^2\,x^7}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (5\,f\,a^3+e\,a^2\,b+2\,d\,a\,b^2-14\,c\,b^3\right )}{27\,a^{10/3}\,b^{8/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,f\,a^3+e\,a^2\,b+2\,d\,a\,b^2-14\,c\,b^3\right )}{27\,a^{10/3}\,b^{8/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,f\,a^3+e\,a^2\,b+2\,d\,a\,b^2-14\,c\,b^3\right )}{27\,a^{10/3}\,b^{8/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(5*a^3*f - 14*b^3*c + 2*a*b^2*d + a^2*
b*e))/(27*a^(10/3)*b^(8/3)) - (log(b^(1/3)*x + a^(1/3))*(5*a^3*f - 14*b^3*c + 2*a*b^2*d + a^2*b*e))/(27*a^(10/
3)*b^(8/3)) - (c/a + (x^6*(14*b^3*c + 4*a^3*f - 2*a*b^2*d - a^2*b*e))/(9*a^3*b) + (x^3*(49*b^3*c + 5*a^3*f - 7
*a*b^2*d + a^2*b*e))/(18*a^2*b^2))/(a^2*x + b^2*x^7 + 2*a*b*x^4) - (log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(
1/3))*((3^(1/2)*1i)/2 - 1/2)*(5*a^3*f - 14*b^3*c + 2*a*b^2*d + a^2*b*e))/(27*a^(10/3)*b^(8/3))

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

[Out]

Timed out

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