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

Optimal. Leaf size=265 \[ -\frac {c}{a^2 x}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{18 a^{7/3} b^{8/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{9 a^{7/3} b^{8/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{3 \sqrt {3} a^{7/3} b^{8/3}}+\frac {f x^2}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.25, antiderivative size = 265, 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, 1488, 292, 31, 634, 617, 204, 628} \[ -\frac {x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-2 a^2 b e+5 a^3 f-a b^2 d+4 b^3 c\right )}{18 a^{7/3} b^{8/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^2 b e+5 a^3 f-a b^2 d+4 b^3 c\right )}{9 a^{7/3} b^{8/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-2 a^2 b e+5 a^3 f-a b^2 d+4 b^3 c\right )}{3 \sqrt {3} a^{7/3} b^{8/3}}-\frac {c}{a^2 x}+\frac {f x^2}{2 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c/(a^2*x)) + (f*x^2)/(2*b^2) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*a^2*b^2*(a + b*x^3)) + ((4*b^3*c
- a*b^2*d - 2*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^(7/3)*b^(8/3
)) + ((4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(7/3)*b^(8/3)) - ((4*b^3*c - a*
b^2*d - 2*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^(7/3)*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 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 1488

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Sy
mbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e,
f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 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 )^2} \, dx &=-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^2 b^2 \left (a+b x^3\right )}-\frac {\int \frac {-3 b^3 c+b \left (\frac {b^3 c}{a}-b^2 d-2 a b e+2 a^2 f\right ) x^3-3 a b^2 f x^6}{x^2 \left (a+b x^3\right )} \, dx}{3 a b^3}\\ &=-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^2 b^2 \left (a+b x^3\right )}-\frac {\int \left (-\frac {3 b^3 c}{a x^2}-3 a b f x+\frac {b \left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) x}{a \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac {c}{a^2 x}+\frac {f x^2}{2 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^2 b^2 \left (a+b x^3\right )}-\frac {\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \int \frac {x}{a+b x^3} \, dx}{3 a^2 b^2}\\ &=-\frac {c}{a^2 x}+\frac {f x^2}{2 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^2 b^2 \left (a+b x^3\right )}+\frac {\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{7/3} b^{7/3}}-\frac {\left (4 b^3 c-a b^2 d-2 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}{9 a^{7/3} b^{7/3}}\\ &=-\frac {c}{a^2 x}+\frac {f x^2}{2 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^2 b^2 \left (a+b x^3\right )}+\frac {\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} b^{8/3}}-\frac {\left (4 b^3 c-a b^2 d-2 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}{18 a^{7/3} b^{8/3}}-\frac {\left (4 b^3 c-a b^2 d-2 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}{6 a^2 b^{7/3}}\\ &=-\frac {c}{a^2 x}+\frac {f x^2}{2 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^2 b^2 \left (a+b x^3\right )}+\frac {\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} b^{8/3}}-\frac {\left (4 b^3 c-a b^2 d-2 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^{7/3} b^{8/3}}-\frac {\left (4 b^3 c-a b^2 d-2 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 )}{3 a^{7/3} b^{8/3}}\\ &=-\frac {c}{a^2 x}+\frac {f x^2}{2 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^2 b^2 \left (a+b x^3\right )}+\frac {\left (4 b^3 c-a b^2 d-2 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^{7/3} b^{8/3}}+\frac {\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} b^{8/3}}-\frac {\left (4 b^3 c-a b^2 d-2 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^{7/3} b^{8/3}}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 255, normalized size = 0.96 \[ \frac {1}{18} \left (-\frac {18 c}{a^2 x}+\frac {6 x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^2 b^2 \left (a+b x^3\right )}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{a^{7/3} b^{8/3}}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{a^{7/3} b^{8/3}}+\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{a^{7/3} b^{8/3}}+\frac {9 f x^2}{b^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-18*c)/(a^2*x) + (9*f*x^2)/b^2 + (6*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(a^2*b^2*(a + b*x^3)) + (2*S
qrt[3]*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(a^(7/3)*b^(8/3)
) + (2*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(a^(7/3)*b^(8/3)) - ((4*b^3*c - a*b
^2*d - 2*a^2*b*e + 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(a^(7/3)*b^(8/3)))/18

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fricas [A]  time = 0.80, size = 860, normalized size = 3.25 \[ \left [\frac {9 \, a^{3} b^{3} f x^{6} - 18 \, a^{2} b^{4} c - 3 \, {\left (8 \, a b^{5} c - 2 \, a^{2} b^{4} d + 2 \, a^{3} b^{3} e - 5 \, a^{4} b^{2} f\right )} x^{3} + 3 \, \sqrt {\frac {1}{3}} {\left ({\left (4 \, a b^{5} c - a^{2} b^{4} d - 2 \, a^{3} b^{3} e + 5 \, a^{4} b^{2} f\right )} x^{4} + {\left (4 \, a^{2} b^{4} c - a^{3} b^{3} d - 2 \, a^{4} b^{2} e + 5 \, a^{5} b f\right )} x\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b - 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (a b^{2}\right )^{\frac {2}{3}} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) - {\left ({\left (4 \, b^{4} c - a b^{3} d - 2 \, a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{4} + {\left (4 \, a b^{3} c - a^{2} b^{2} d - 2 \, a^{3} b e + 5 \, a^{4} f\right )} x\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} b x + \left (a b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, {\left ({\left (4 \, b^{4} c - a b^{3} d - 2 \, a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{4} + {\left (4 \, a b^{3} c - a^{2} b^{2} d - 2 \, a^{3} b e + 5 \, a^{4} f\right )} x\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b x + \left (a b^{2}\right )^{\frac {1}{3}}\right )}{18 \, {\left (a^{3} b^{5} x^{4} + a^{4} b^{4} x\right )}}, \frac {9 \, a^{3} b^{3} f x^{6} - 18 \, a^{2} b^{4} c - 3 \, {\left (8 \, a b^{5} c - 2 \, a^{2} b^{4} d + 2 \, a^{3} b^{3} e - 5 \, a^{4} b^{2} f\right )} x^{3} + 6 \, \sqrt {\frac {1}{3}} {\left ({\left (4 \, a b^{5} c - a^{2} b^{4} d - 2 \, a^{3} b^{3} e + 5 \, a^{4} b^{2} f\right )} x^{4} + {\left (4 \, a^{2} b^{4} c - a^{3} b^{3} d - 2 \, a^{4} b^{2} e + 5 \, a^{5} b f\right )} x\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x - \left (a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) - {\left ({\left (4 \, b^{4} c - a b^{3} d - 2 \, a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{4} + {\left (4 \, a b^{3} c - a^{2} b^{2} d - 2 \, a^{3} b e + 5 \, a^{4} f\right )} x\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} b x + \left (a b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, {\left ({\left (4 \, b^{4} c - a b^{3} d - 2 \, a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{4} + {\left (4 \, a b^{3} c - a^{2} b^{2} d - 2 \, a^{3} b e + 5 \, a^{4} f\right )} x\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b x + \left (a b^{2}\right )^{\frac {1}{3}}\right )}{18 \, {\left (a^{3} b^{5} x^{4} + a^{4} b^{4} x\right )}}\right ] \]

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)^2,x, algorithm="fricas")

[Out]

[1/18*(9*a^3*b^3*f*x^6 - 18*a^2*b^4*c - 3*(8*a*b^5*c - 2*a^2*b^4*d + 2*a^3*b^3*e - 5*a^4*b^2*f)*x^3 + 3*sqrt(1
/3)*((4*a*b^5*c - a^2*b^4*d - 2*a^3*b^3*e + 5*a^4*b^2*f)*x^4 + (4*a^2*b^4*c - a^3*b^3*d - 2*a^4*b^2*e + 5*a^5*
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)) - ((4*b^4*c - a*b^3*d - 2*a^2*b^2*e + 5*a^3*b*f)*
x^4 + (4*a*b^3*c - a^2*b^2*d - 2*a^3*b*e + 5*a^4*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*((4*b^4*c - a*b^3*d - 2*a^2*b^2*e + 5*a^3*b*f)*x^4 + (4*a*b^3*c - a^2*b^2*d - 2*a^3*b*e + 5*a^4*f)
*x)*(a*b^2)^(2/3)*log(b*x + (a*b^2)^(1/3)))/(a^3*b^5*x^4 + a^4*b^4*x), 1/18*(9*a^3*b^3*f*x^6 - 18*a^2*b^4*c -
3*(8*a*b^5*c - 2*a^2*b^4*d + 2*a^3*b^3*e - 5*a^4*b^2*f)*x^3 + 6*sqrt(1/3)*((4*a*b^5*c - a^2*b^4*d - 2*a^3*b^3*
e + 5*a^4*b^2*f)*x^4 + (4*a^2*b^4*c - a^3*b^3*d - 2*a^4*b^2*e + 5*a^5*b*f)*x)*sqrt((a*b^2)^(1/3)/a)*arctan(-sq
rt(1/3)*(2*b*x - (a*b^2)^(1/3))*sqrt((a*b^2)^(1/3)/a)/b) - ((4*b^4*c - a*b^3*d - 2*a^2*b^2*e + 5*a^3*b*f)*x^4
+ (4*a*b^3*c - a^2*b^2*d - 2*a^3*b*e + 5*a^4*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*((4*b^4*c - a*b^3*d - 2*a^2*b^2*e + 5*a^3*b*f)*x^4 + (4*a*b^3*c - a^2*b^2*d - 2*a^3*b*e + 5*a^4*f)*x)*
(a*b^2)^(2/3)*log(b*x + (a*b^2)^(1/3)))/(a^3*b^5*x^4 + a^4*b^4*x)]

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giac [A]  time = 0.19, size = 305, normalized size = 1.15 \[ \frac {f x^{2}}{2 \, b^{2}} - \frac {\sqrt {3} {\left (4 \, b^{3} c - a b^{2} d + 5 \, a^{3} f - 2 \, a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2}} + \frac {{\left (4 \, b^{3} c - a b^{2} d + 5 \, a^{3} f - 2 \, a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2}} + \frac {{\left (4 \, b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{2} b \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{3} b^{2}} - \frac {4 \, b^{3} c x^{3} - a b^{2} d x^{3} - a^{3} f x^{3} + a^{2} b x^{3} e + 3 \, a b^{2} c}{3 \, {\left (b x^{4} + a x\right )} a^{2} b^{2}} \]

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)^2,x, algorithm="giac")

[Out]

1/2*f*x^2/b^2 - 1/9*sqrt(3)*(4*b^3*c - a*b^2*d + 5*a^3*f - 2*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^2*b^2) + 1/18*(4*b^3*c - a*b^2*d + 5*a^3*f - 2*a^2*b*e)*log(x^2 + x*(-a/b)^(1/
3) + (-a/b)^(2/3))/((-a*b^2)^(1/3)*a^2*b^2) + 1/9*(4*b^3*c*(-a/b)^(1/3) - a*b^2*d*(-a/b)^(1/3) + 5*a^3*f*(-a/b
)^(1/3) - 2*a^2*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*b^2) - 1/3*(4*b^3*c*x^3 - a*b^2
*d*x^3 - a^3*f*x^3 + a^2*b*x^3*e + 3*a*b^2*c)/((b*x^4 + a*x)*a^2*b^2)

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maple [B]  time = 0.06, size = 474, normalized size = 1.79 \[ \frac {a f \,x^{2}}{3 \left (b \,x^{3}+a \right ) b^{2}}+\frac {d \,x^{2}}{3 \left (b \,x^{3}+a \right ) a}-\frac {b c \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{2}}-\frac {e \,x^{2}}{3 \left (b \,x^{3}+a \right ) b}+\frac {f \,x^{2}}{2 b^{2}}-\frac {5 \sqrt {3}\, a 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}} b^{3}}+\frac {5 a f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}-\frac {5 a 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 {1}{3}} b^{3}}+\frac {\sqrt {3}\, 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 b}-\frac {d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a 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 )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}-\frac {4 \sqrt {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^{2}}+\frac {4 c \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 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 {1}{3}} a^{2}}+\frac {2 \sqrt {3}\, 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}} b^{2}}-\frac {2 e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} 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 )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}-\frac {c}{a^{2} 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)^2,x)

[Out]

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

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maxima [A]  time = 3.01, size = 258, normalized size = 0.97 \[ \frac {f x^{2}}{2 \, b^{2}} - \frac {3 \, a b^{2} c + {\left (4 \, b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3}}{3 \, {\left (a^{2} b^{3} x^{4} + a^{3} b^{2} x\right )}} - \frac {\sqrt {3} {\left (4 \, b^{3} c - a b^{2} d - 2 \, a^{2} b e + 5 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (4 \, b^{3} c - a b^{2} d - 2 \, a^{2} b e + 5 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (4 \, b^{3} c - a b^{2} d - 2 \, a^{2} b e + 5 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

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)^2,x, algorithm="maxima")

[Out]

1/2*f*x^2/b^2 - 1/3*(3*a*b^2*c + (4*b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^3)/(a^2*b^3*x^4 + a^3*b^2*x) - 1/9*sq
rt(3)*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^2*b^3*(
a/b)^(1/3)) - 1/18*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b^3*(
a/b)^(1/3)) + 1/9*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*log(x + (a/b)^(1/3))/(a^2*b^3*(a/b)^(1/3))

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mupad [B]  time = 5.39, size = 244, normalized size = 0.92 \[ \frac {f\,x^2}{2\,b^2}-\frac {\frac {x^3\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+4\,c\,b^3\right )}{3\,a^2}+\frac {b^2\,c}{a}}{b^3\,x^4+a\,b^2\,x}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (5\,f\,a^3-2\,e\,a^2\,b-d\,a\,b^2+4\,c\,b^3\right )}{9\,a^{7/3}\,b^{8/3}}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,f\,a^3-2\,e\,a^2\,b-d\,a\,b^2+4\,c\,b^3\right )}{9\,a^{7/3}\,b^{8/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,f\,a^3-2\,e\,a^2\,b-d\,a\,b^2+4\,c\,b^3\right )}{9\,a^{7/3}\,b^{8/3}} \]

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)^2),x)

[Out]

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

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sympy [A]  time = 32.22, size = 457, normalized size = 1.72 \[ \frac {- 3 a b^{2} c + x^{3} \left (a^{3} f - a^{2} b e + a b^{2} d - 4 b^{3} c\right )}{3 a^{3} b^{2} x + 3 a^{2} b^{3} x^{4}} + \operatorname {RootSum} {\left (729 t^{3} a^{7} b^{8} - 125 a^{9} f^{3} + 150 a^{8} b e f^{2} + 75 a^{7} b^{2} d f^{2} - 60 a^{7} b^{2} e^{2} f - 300 a^{6} b^{3} c f^{2} - 60 a^{6} b^{3} d e f + 8 a^{6} b^{3} e^{3} + 240 a^{5} b^{4} c e f - 15 a^{5} b^{4} d^{2} f + 12 a^{5} b^{4} d e^{2} + 120 a^{4} b^{5} c d f - 48 a^{4} b^{5} c e^{2} + 6 a^{4} b^{5} d^{2} e - 240 a^{3} b^{6} c^{2} f - 48 a^{3} b^{6} c d e + a^{3} b^{6} d^{3} + 96 a^{2} b^{7} c^{2} e - 12 a^{2} b^{7} c d^{2} + 48 a b^{8} c^{2} d - 64 b^{9} c^{3}, \left (t \mapsto t \log {\left (\frac {81 t^{2} a^{5} b^{5}}{25 a^{6} f^{2} - 20 a^{5} b e f - 10 a^{4} b^{2} d f + 4 a^{4} b^{2} e^{2} + 40 a^{3} b^{3} c f + 4 a^{3} b^{3} d e - 16 a^{2} b^{4} c e + a^{2} b^{4} d^{2} - 8 a b^{5} c d + 16 b^{6} c^{2}} + x \right )} \right )\right )} + \frac {f x^{2}}{2 b^{2}} \]

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)**2,x)

[Out]

(-3*a*b**2*c + x**3*(a**3*f - a**2*b*e + a*b**2*d - 4*b**3*c))/(3*a**3*b**2*x + 3*a**2*b**3*x**4) + RootSum(72
9*_t**3*a**7*b**8 - 125*a**9*f**3 + 150*a**8*b*e*f**2 + 75*a**7*b**2*d*f**2 - 60*a**7*b**2*e**2*f - 300*a**6*b
**3*c*f**2 - 60*a**6*b**3*d*e*f + 8*a**6*b**3*e**3 + 240*a**5*b**4*c*e*f - 15*a**5*b**4*d**2*f + 12*a**5*b**4*
d*e**2 + 120*a**4*b**5*c*d*f - 48*a**4*b**5*c*e**2 + 6*a**4*b**5*d**2*e - 240*a**3*b**6*c**2*f - 48*a**3*b**6*
c*d*e + a**3*b**6*d**3 + 96*a**2*b**7*c**2*e - 12*a**2*b**7*c*d**2 + 48*a*b**8*c**2*d - 64*b**9*c**3, Lambda(_
t, _t*log(81*_t**2*a**5*b**5/(25*a**6*f**2 - 20*a**5*b*e*f - 10*a**4*b**2*d*f + 4*a**4*b**2*e**2 + 40*a**3*b**
3*c*f + 4*a**3*b**3*d*e - 16*a**2*b**4*c*e + a**2*b**4*d**2 - 8*a*b**5*c*d + 16*b**6*c**2) + x))) + f*x**2/(2*
b**2)

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