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

Optimal. Leaf size=271 \[ \frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right )}{18 a^{4/3} b^{11/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right )}{9 a^{4/3} b^{11/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right )}{3 \sqrt {3} a^{4/3} b^{11/3}}+\frac {x^2 (b e-2 a f)}{2 b^3}+\frac {f x^5}{5 b^2} \]

[Out]

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

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Rubi [A]  time = 0.29, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {1828, 1594, 1488, 292, 31, 634, 617, 204, 628} \[ \frac {x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^2 b e+8 a^3 f+2 a b^2 d+b^3 c\right )}{18 a^{4/3} b^{11/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^2 b e+8 a^3 f+2 a b^2 d+b^3 c\right )}{9 a^{4/3} b^{11/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-5 a^2 b e+8 a^3 f+2 a b^2 d+b^3 c\right )}{3 \sqrt {3} a^{4/3} b^{11/3}}+\frac {x^2 (b e-2 a f)}{2 b^3}+\frac {f x^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*e - 2*a*f)*x^2)/(2*b^3) + (f*x^5)/(5*b^2) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*a*b^3*(a + b*x^3)
) - ((b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a
^(4/3)*b^(11/3)) - ((b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(4/3)*b^(11/3)) +
((b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(4/3)*b^(11/
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 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1828

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = m + Expon[Pq, x]}, Module[{Q = Pol
ynomialQuotient[b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] +
1)*x^m*Pq, a + b*x^n, x]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[
a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q
- 1)/n] + 1)), x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 0]

Rubi steps

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

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Mathematica [A]  time = 0.19, size = 255, normalized size = 0.94 \[ \frac {\frac {30 b^{2/3} x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a \left (a+b x^3\right )}-\frac {10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right )}{a^{4/3}}-\frac {10 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right )}{a^{4/3}}+\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right )}{a^{4/3}}+45 b^{2/3} x^2 (b e-2 a f)+18 b^{5/3} f x^5}{90 b^{11/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(45*b^(2/3)*(b*e - 2*a*f)*x^2 + 18*b^(5/3)*f*x^5 + (30*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a*(a
+ b*x^3)) - (10*Sqrt[3]*(b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])
/a^(4/3) - (10*(b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + (5*(b^3*c + 2*a*b
^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(4/3))/(90*b^(11/3))

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fricas [A]  time = 0.80, size = 874, normalized size = 3.23 \[ \left [\frac {18 \, a^{2} b^{4} f x^{8} + 9 \, {\left (5 \, a^{2} b^{4} e - 8 \, a^{3} b^{3} f\right )} x^{5} + 15 \, {\left (2 \, a b^{5} c - 2 \, a^{2} b^{4} d + 5 \, a^{3} b^{3} e - 8 \, a^{4} b^{2} f\right )} x^{2} + 15 \, \sqrt {\frac {1}{3}} {\left (a^{2} b^{4} c + 2 \, a^{3} b^{3} d - 5 \, a^{4} b^{2} e + 8 \, a^{5} b f + {\left (a b^{5} c + 2 \, a^{2} b^{4} d - 5 \, a^{3} b^{3} e + 8 \, a^{4} b^{2} f\right )} x^{3}\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 ) + 5 \, {\left (a b^{3} c + 2 \, a^{2} b^{2} d - 5 \, a^{3} b e + 8 \, a^{4} f + {\left (b^{4} c + 2 \, a b^{3} d - 5 \, a^{2} b^{2} e + 8 \, a^{3} b f\right )} x^{3}\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 ) - 10 \, {\left (a b^{3} c + 2 \, a^{2} b^{2} d - 5 \, a^{3} b e + 8 \, a^{4} f + {\left (b^{4} c + 2 \, a b^{3} d - 5 \, a^{2} b^{2} e + 8 \, a^{3} b f\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{90 \, {\left (a^{2} b^{6} x^{3} + a^{3} b^{5}\right )}}, \frac {18 \, a^{2} b^{4} f x^{8} + 9 \, {\left (5 \, a^{2} b^{4} e - 8 \, a^{3} b^{3} f\right )} x^{5} + 15 \, {\left (2 \, a b^{5} c - 2 \, a^{2} b^{4} d + 5 \, a^{3} b^{3} e - 8 \, a^{4} b^{2} f\right )} x^{2} + 30 \, \sqrt {\frac {1}{3}} {\left (a^{2} b^{4} c + 2 \, a^{3} b^{3} d - 5 \, a^{4} b^{2} e + 8 \, a^{5} b f + {\left (a b^{5} c + 2 \, a^{2} b^{4} d - 5 \, a^{3} b^{3} e + 8 \, a^{4} b^{2} f\right )} x^{3}\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 ) + 5 \, {\left (a b^{3} c + 2 \, a^{2} b^{2} d - 5 \, a^{3} b e + 8 \, a^{4} f + {\left (b^{4} c + 2 \, a b^{3} d - 5 \, a^{2} b^{2} e + 8 \, a^{3} b f\right )} x^{3}\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 ) - 10 \, {\left (a b^{3} c + 2 \, a^{2} b^{2} d - 5 \, a^{3} b e + 8 \, a^{4} f + {\left (b^{4} c + 2 \, a b^{3} d - 5 \, a^{2} b^{2} e + 8 \, a^{3} b f\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{90 \, {\left (a^{2} b^{6} x^{3} + a^{3} b^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/90*(18*a^2*b^4*f*x^8 + 9*(5*a^2*b^4*e - 8*a^3*b^3*f)*x^5 + 15*(2*a*b^5*c - 2*a^2*b^4*d + 5*a^3*b^3*e - 8*a^
4*b^2*f)*x^2 + 15*sqrt(1/3)*(a^2*b^4*c + 2*a^3*b^3*d - 5*a^4*b^2*e + 8*a^5*b*f + (a*b^5*c + 2*a^2*b^4*d - 5*a^
3*b^3*e + 8*a^4*b^2*f)*x^3)*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)) + 5*(a*b^3*c + 2*a^2*b^2*d
- 5*a^3*b*e + 8*a^4*f + (b^4*c + 2*a*b^3*d - 5*a^2*b^2*e + 8*a^3*b*f)*x^3)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b
^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 10*(a*b^3*c + 2*a^2*b^2*d - 5*a^3*b*e + 8*a^4*f + (b^4*c + 2*a*b^3*d - 5*a^2
*b^2*e + 8*a^3*b*f)*x^3)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^2*b^6*x^3 + a^3*b^5), 1/90*(18*a^2*b^4*f
*x^8 + 9*(5*a^2*b^4*e - 8*a^3*b^3*f)*x^5 + 15*(2*a*b^5*c - 2*a^2*b^4*d + 5*a^3*b^3*e - 8*a^4*b^2*f)*x^2 + 30*s
qrt(1/3)*(a^2*b^4*c + 2*a^3*b^3*d - 5*a^4*b^2*e + 8*a^5*b*f + (a*b^5*c + 2*a^2*b^4*d - 5*a^3*b^3*e + 8*a^4*b^2
*f)*x^3)*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) + 5*(a*b
^3*c + 2*a^2*b^2*d - 5*a^3*b*e + 8*a^4*f + (b^4*c + 2*a*b^3*d - 5*a^2*b^2*e + 8*a^3*b*f)*x^3)*(-a*b^2)^(2/3)*l
og(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 10*(a*b^3*c + 2*a^2*b^2*d - 5*a^3*b*e + 8*a^4*f + (b^4*c +
2*a*b^3*d - 5*a^2*b^2*e + 8*a^3*b*f)*x^3)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^2*b^6*x^3 + a^3*b^5)]

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giac [A]  time = 0.22, size = 318, normalized size = 1.17 \[ \frac {\sqrt {3} {\left (b^{3} c + 2 \, a b^{2} d + 8 \, a^{3} f - 5 \, a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{3}} - \frac {{\left (b^{3} c + 2 \, a b^{2} d + 8 \, a^{3} f - 5 \, a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{3}} - \frac {{\left (b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 8 \, a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a^{2} b \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2} b^{3}} + \frac {b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e}{3 \, {\left (b x^{3} + a\right )} a b^{3}} + \frac {2 \, b^{8} f x^{5} - 10 \, a b^{7} f x^{2} + 5 \, b^{8} x^{2} e}{10 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*sqrt(3)*(b^3*c + 2*a*b^2*d + 8*a^3*f - 5*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*b^3) - 1/18*(b^3*c + 2*a*b^2*d + 8*a^3*f - 5*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))
/((-a*b^2)^(1/3)*a*b^3) - 1/9*(b^3*c*(-a/b)^(1/3) + 2*a*b^2*d*(-a/b)^(1/3) + 8*a^3*f*(-a/b)^(1/3) - 5*a^2*b*(-
a/b)^(1/3)*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^3) + 1/3*(b^3*c*x^2 - a*b^2*d*x^2 - a^3*f*x^2 + a
^2*b*x^2*e)/((b*x^3 + a)*a*b^3) + 1/10*(2*b^8*f*x^5 - 10*a*b^7*f*x^2 + 5*b^8*x^2*e)/b^10

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maple [B]  time = 0.05, size = 495, normalized size = 1.83 \[ \frac {f \,x^{5}}{5 b^{2}}-\frac {a^{2} f \,x^{2}}{3 \left (b \,x^{3}+a \right ) b^{3}}+\frac {a e \,x^{2}}{3 \left (b \,x^{3}+a \right ) b^{2}}+\frac {c \,x^{2}}{3 \left (b \,x^{3}+a \right ) a}-\frac {d \,x^{2}}{3 \left (b \,x^{3}+a \right ) b}-\frac {a f \,x^{2}}{b^{3}}+\frac {e \,x^{2}}{2 b^{2}}+\frac {8 \sqrt {3}\, a^{2} 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^{4}}-\frac {8 a^{2} f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}+\frac {4 a^{2} f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}-\frac {5 \sqrt {3}\, a 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^{3}}+\frac {5 a e \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 e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}+\frac {\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 b}-\frac {c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}+\frac {c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}+\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 )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}-\frac {2 d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}+\frac {d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.12, size = 259, normalized size = 0.96 \[ \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{2}}{3 \, {\left (a b^{4} x^{3} + a^{2} b^{3}\right )}} + \frac {2 \, b f x^{5} + 5 \, {\left (b e - 2 \, a f\right )} x^{2}}{10 \, b^{3}} + \frac {\sqrt {3} {\left (b^{3} c + 2 \, a b^{2} d - 5 \, a^{2} b e + 8 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (b^{3} c + 2 \, a b^{2} d - 5 \, a^{2} b e + 8 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (b^{3} c + 2 \, a b^{2} d - 5 \, a^{2} b e + 8 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2/(a*b^4*x^3 + a^2*b^3) + 1/10*(2*b*f*x^5 + 5*(b*e - 2*a*f)*x^2)/b^3
+ 1/9*sqrt(3)*(b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(
a*b^4*(a/b)^(1/3)) + 1/18*(b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*
b^4*(a/b)^(1/3)) - 1/9*(b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*log(x + (a/b)^(1/3))/(a*b^4*(a/b)^(1/3))

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mupad [B]  time = 5.23, size = 246, normalized size = 0.91 \[ x^2\,\left (\frac {e}{2\,b^2}-\frac {a\,f}{b^3}\right )+\frac {f\,x^5}{5\,b^2}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (8\,f\,a^3-5\,e\,a^2\,b+2\,d\,a\,b^2+c\,b^3\right )}{9\,a^{4/3}\,b^{11/3}}+\frac {x^2\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a\,\left (b^4\,x^3+a\,b^3\right )}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (8\,f\,a^3-5\,e\,a^2\,b+2\,d\,a\,b^2+c\,b^3\right )}{9\,a^{4/3}\,b^{11/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (8\,f\,a^3-5\,e\,a^2\,b+2\,d\,a\,b^2+c\,b^3\right )}{9\,a^{4/3}\,b^{11/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*(e/(2*b^2) - (a*f)/b^3) + (f*x^5)/(5*b^2) - (log(b^(1/3)*x + a^(1/3))*(b^3*c + 8*a^3*f + 2*a*b^2*d - 5*a^2
*b*e))/(9*a^(4/3)*b^(11/3)) + (x^2*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*a*(a*b^3 + b^4*x^3)) + (log(3^(1/2)
*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(b^3*c + 8*a^3*f + 2*a*b^2*d - 5*a^2*b*e))/(9*a^(4
/3)*b^(11/3)) - (log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(b^3*c + 8*a^3*f + 2*a
*b^2*d - 5*a^2*b*e))/(9*a^(4/3)*b^(11/3))

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sympy [A]  time = 22.48, size = 461, normalized size = 1.70 \[ x^{2} \left (- \frac {a f}{b^{3}} + \frac {e}{2 b^{2}}\right ) + \frac {x^{2} \left (- a^{3} f + a^{2} b e - a b^{2} d + b^{3} c\right )}{3 a^{2} b^{3} + 3 a b^{4} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a^{4} b^{11} + 512 a^{9} f^{3} - 960 a^{8} b e f^{2} + 384 a^{7} b^{2} d f^{2} + 600 a^{7} b^{2} e^{2} f + 192 a^{6} b^{3} c f^{2} - 480 a^{6} b^{3} d e f - 125 a^{6} b^{3} e^{3} - 240 a^{5} b^{4} c e f + 96 a^{5} b^{4} d^{2} f + 150 a^{5} b^{4} d e^{2} + 96 a^{4} b^{5} c d f + 75 a^{4} b^{5} c e^{2} - 60 a^{4} b^{5} d^{2} e + 24 a^{3} b^{6} c^{2} f - 60 a^{3} b^{6} c d e + 8 a^{3} b^{6} d^{3} - 15 a^{2} b^{7} c^{2} e + 12 a^{2} b^{7} c d^{2} + 6 a b^{8} c^{2} d + b^{9} c^{3}, \left (t \mapsto t \log {\left (\frac {81 t^{2} a^{3} b^{7}}{64 a^{6} f^{2} - 80 a^{5} b e f + 32 a^{4} b^{2} d f + 25 a^{4} b^{2} e^{2} + 16 a^{3} b^{3} c f - 20 a^{3} b^{3} d e - 10 a^{2} b^{4} c e + 4 a^{2} b^{4} d^{2} + 4 a b^{5} c d + b^{6} c^{2}} + x \right )} \right )\right )} + \frac {f x^{5}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*(-a*f/b**3 + e/(2*b**2)) + x**2*(-a**3*f + a**2*b*e - a*b**2*d + b**3*c)/(3*a**2*b**3 + 3*a*b**4*x**3) +
RootSum(729*_t**3*a**4*b**11 + 512*a**9*f**3 - 960*a**8*b*e*f**2 + 384*a**7*b**2*d*f**2 + 600*a**7*b**2*e**2*f
+ 192*a**6*b**3*c*f**2 - 480*a**6*b**3*d*e*f - 125*a**6*b**3*e**3 - 240*a**5*b**4*c*e*f + 96*a**5*b**4*d**2*f
+ 150*a**5*b**4*d*e**2 + 96*a**4*b**5*c*d*f + 75*a**4*b**5*c*e**2 - 60*a**4*b**5*d**2*e + 24*a**3*b**6*c**2*f
- 60*a**3*b**6*c*d*e + 8*a**3*b**6*d**3 - 15*a**2*b**7*c**2*e + 12*a**2*b**7*c*d**2 + 6*a*b**8*c**2*d + b**9*
c**3, Lambda(_t, _t*log(81*_t**2*a**3*b**7/(64*a**6*f**2 - 80*a**5*b*e*f + 32*a**4*b**2*d*f + 25*a**4*b**2*e**
2 + 16*a**3*b**3*c*f - 20*a**3*b**3*d*e - 10*a**2*b**4*c*e + 4*a**2*b**4*d**2 + 4*a*b**5*c*d + b**6*c**2) + x)
)) + f*x**5/(5*b**2)

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