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

Optimal. Leaf size=288 \[ \frac {x \left (3 a^2 f-2 a b e+b^2 d\right )}{b^4}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4 \left (a+b x^3\right )}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{18 a^{2/3} b^{13/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{9 a^{2/3} b^{13/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{3 \sqrt {3} a^{2/3} b^{13/3}}+\frac {x^4 (b e-2 a f)}{4 b^3}+\frac {f x^7}{7 b^2} \]

[Out]

(3*a^2*f-2*a*b*e+b^2*d)*x/b^4+1/4*(-2*a*f+b*e)*x^4/b^3+1/7*f*x^7/b^2-1/3*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/b^4/
(b*x^3+a)+1/9*(-10*a^3*f+7*a^2*b*e-4*a*b^2*d+b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(2/3)/b^(13/3)-1/18*(-10*a^3*f+7*a
^2*b*e-4*a*b^2*d+b^3*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(2/3)/b^(13/3)-1/9*(-10*a^3*f+7*a^2*b*e-4*
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^(2/3)/b^(13/3)*3^(1/2)

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Rubi [A]  time = 0.33, antiderivative size = 288, 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 = {1828, 1887, 200, 31, 634, 617, 204, 628} \[ -\frac {x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^4 \left (a+b x^3\right )}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (7 a^2 b e-10 a^3 f-4 a b^2 d+b^3 c\right )}{18 a^{2/3} b^{13/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (7 a^2 b e-10 a^3 f-4 a b^2 d+b^3 c\right )}{9 a^{2/3} b^{13/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (7 a^2 b e-10 a^3 f-4 a b^2 d+b^3 c\right )}{3 \sqrt {3} a^{2/3} b^{13/3}}+\frac {x \left (3 a^2 f-2 a b e+b^2 d\right )}{b^4}+\frac {x^4 (b e-2 a f)}{4 b^3}+\frac {f x^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b^2*d - 2*a*b*e + 3*a^2*f)*x)/b^4 + ((b*e - 2*a*f)*x^4)/(4*b^3) + (f*x^7)/(7*b^2) - ((b^3*c - a*b^2*d + a^2*
b*e - a^3*f)*x)/(3*b^4*(a + b*x^3)) - ((b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*
x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(2/3)*b^(13/3)) + ((b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*Log[a^(1/3)
+ b^(1/3)*x])/(9*a^(2/3)*b^(13/3)) - ((b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)
*x + b^(2/3)*x^2])/(18*a^(2/3)*b^(13/3))

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 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]

Rule 1887

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rubi steps

\begin {align*} \int \frac {x^3 \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}{3 b^4 \left (a+b x^3\right )}-\frac {\int \frac {-a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-3 a b \left (b^2 d-a b e+a^2 f\right ) x^3-3 a b^2 (b e-a f) x^6-3 a b^3 f x^9}{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}{3 b^4 \left (a+b x^3\right )}-\frac {\int \left (-3 a \left (b^2 d-2 a b e+3 a^2 f\right )-3 a b (b e-2 a f) x^3-3 a b^2 f x^6+\frac {-a b^3 c+4 a^2 b^2 d-7 a^3 b e+10 a^4 f}{a+b x^3}\right ) \, dx}{3 a b^4}\\ &=\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x}{b^4}+\frac {(b e-2 a f) x^4}{4 b^3}+\frac {f x^7}{7 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^4 \left (a+b x^3\right )}+\frac {\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) \int \frac {1}{a+b x^3} \, dx}{3 b^4}\\ &=\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x}{b^4}+\frac {(b e-2 a f) x^4}{4 b^3}+\frac {f x^7}{7 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^4 \left (a+b x^3\right )}+\frac {\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{2/3} b^4}+\frac {\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\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^{2/3} b^4}\\ &=\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x}{b^4}+\frac {(b e-2 a f) x^4}{4 b^3}+\frac {f x^7}{7 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^4 \left (a+b x^3\right )}+\frac {\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{13/3}}-\frac {\left (b^3 c-4 a b^2 d+7 a^2 b e-10 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^{2/3} b^{13/3}}+\frac {\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 \sqrt [3]{a} b^4}\\ &=\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x}{b^4}+\frac {(b e-2 a f) x^4}{4 b^3}+\frac {f x^7}{7 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^4 \left (a+b x^3\right )}+\frac {\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{13/3}}-\frac {\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{13/3}}+\frac {\left (b^3 c-4 a b^2 d+7 a^2 b e-10 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^{2/3} b^{13/3}}\\ &=\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x}{b^4}+\frac {(b e-2 a f) x^4}{4 b^3}+\frac {f x^7}{7 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^4 \left (a+b x^3\right )}-\frac {\left (b^3 c-4 a b^2 d+7 a^2 b e-10 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^{2/3} b^{13/3}}+\frac {\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{13/3}}-\frac {\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{13/3}}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 277, normalized size = 0.96 \[ \frac {252 \sqrt [3]{b} x \left (3 a^2 f-2 a b e+b^2 d\right )-\frac {84 \sqrt [3]{b} x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a+b x^3}+\frac {28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{a^{2/3}}+\frac {28 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (10 a^3 f-7 a^2 b e+4 a b^2 d-b^3 c\right )}{a^{2/3}}+\frac {14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (10 a^3 f-7 a^2 b e+4 a b^2 d-b^3 c\right )}{a^{2/3}}+63 b^{4/3} x^4 (b e-2 a f)+36 b^{7/3} f x^7}{252 b^{13/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(252*b^(1/3)*(b^2*d - 2*a*b*e + 3*a^2*f)*x + 63*b^(4/3)*(b*e - 2*a*f)*x^4 + 36*b^(7/3)*f*x^7 - (84*b^(1/3)*(b^
3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a + b*x^3) + (28*Sqrt[3]*(-(b^3*c) + 4*a*b^2*d - 7*a^2*b*e + 10*a^3*f)*Ar
cTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (28*(b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*Log[a^(1/3
) + b^(1/3)*x])/a^(2/3) + (14*(-(b^3*c) + 4*a*b^2*d - 7*a^2*b*e + 10*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2])/a^(2/3))/(252*b^(13/3))

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fricas [A]  time = 0.66, size = 946, normalized size = 3.28 \[ \left [\frac {36 \, a^{2} b^{4} f x^{10} + 9 \, {\left (7 \, a^{2} b^{4} e - 10 \, a^{3} b^{3} f\right )} x^{7} + 63 \, {\left (4 \, a^{2} b^{4} d - 7 \, a^{3} b^{3} e + 10 \, a^{4} b^{2} f\right )} x^{4} - 42 \, \sqrt {\frac {1}{3}} {\left (a^{2} b^{4} c - 4 \, a^{3} b^{3} d + 7 \, a^{4} b^{2} e - 10 \, a^{5} b f + {\left (a b^{5} c - 4 \, a^{2} b^{4} d + 7 \, a^{3} b^{3} e - 10 \, a^{4} b^{2} f\right )} x^{3}\right )} \sqrt {\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} + 3 \, \left (-a^{2} b\right )^{\frac {1}{3}} a x - a^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (-a^{2} b\right )^{\frac {2}{3}} x + \left (-a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - 14 \, {\left (a b^{3} c - 4 \, a^{2} b^{2} d + 7 \, a^{3} b e - 10 \, a^{4} f + {\left (b^{4} c - 4 \, a b^{3} d + 7 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{3}\right )} \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (-a^{2} b\right )^{\frac {2}{3}} x - \left (-a^{2} b\right )^{\frac {1}{3}} a\right ) + 28 \, {\left (a b^{3} c - 4 \, a^{2} b^{2} d + 7 \, a^{3} b e - 10 \, a^{4} f + {\left (b^{4} c - 4 \, a b^{3} d + 7 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{3}\right )} \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (-a^{2} b\right )^{\frac {2}{3}}\right ) - 84 \, {\left (a^{2} b^{4} c - 4 \, a^{3} b^{3} d + 7 \, a^{4} b^{2} e - 10 \, a^{5} b f\right )} x}{252 \, {\left (a^{2} b^{6} x^{3} + a^{3} b^{5}\right )}}, \frac {36 \, a^{2} b^{4} f x^{10} + 9 \, {\left (7 \, a^{2} b^{4} e - 10 \, a^{3} b^{3} f\right )} x^{7} + 63 \, {\left (4 \, a^{2} b^{4} d - 7 \, a^{3} b^{3} e + 10 \, a^{4} b^{2} f\right )} x^{4} + 84 \, \sqrt {\frac {1}{3}} {\left (a^{2} b^{4} c - 4 \, a^{3} b^{3} d + 7 \, a^{4} b^{2} e - 10 \, a^{5} b f + {\left (a b^{5} c - 4 \, a^{2} b^{4} d + 7 \, a^{3} b^{3} e - 10 \, a^{4} b^{2} f\right )} x^{3}\right )} \sqrt {-\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (-a^{2} b\right )^{\frac {2}{3}} x + \left (-a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 14 \, {\left (a b^{3} c - 4 \, a^{2} b^{2} d + 7 \, a^{3} b e - 10 \, a^{4} f + {\left (b^{4} c - 4 \, a b^{3} d + 7 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{3}\right )} \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (-a^{2} b\right )^{\frac {2}{3}} x - \left (-a^{2} b\right )^{\frac {1}{3}} a\right ) + 28 \, {\left (a b^{3} c - 4 \, a^{2} b^{2} d + 7 \, a^{3} b e - 10 \, a^{4} f + {\left (b^{4} c - 4 \, a b^{3} d + 7 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{3}\right )} \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (-a^{2} b\right )^{\frac {2}{3}}\right ) - 84 \, {\left (a^{2} b^{4} c - 4 \, a^{3} b^{3} d + 7 \, a^{4} b^{2} e - 10 \, a^{5} b f\right )} x}{252 \, {\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^3*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

[1/252*(36*a^2*b^4*f*x^10 + 9*(7*a^2*b^4*e - 10*a^3*b^3*f)*x^7 + 63*(4*a^2*b^4*d - 7*a^3*b^3*e + 10*a^4*b^2*f)
*x^4 - 42*sqrt(1/3)*(a^2*b^4*c - 4*a^3*b^3*d + 7*a^4*b^2*e - 10*a^5*b*f + (a*b^5*c - 4*a^2*b^4*d + 7*a^3*b^3*e
- 10*a^4*b^2*f)*x^3)*sqrt((-a^2*b)^(1/3)/b)*log((2*a*b*x^3 + 3*(-a^2*b)^(1/3)*a*x - a^2 - 3*sqrt(1/3)*(2*a*b*
x^2 + (-a^2*b)^(2/3)*x + (-a^2*b)^(1/3)*a)*sqrt((-a^2*b)^(1/3)/b))/(b*x^3 + a)) - 14*(a*b^3*c - 4*a^2*b^2*d +
7*a^3*b*e - 10*a^4*f + (b^4*c - 4*a*b^3*d + 7*a^2*b^2*e - 10*a^3*b*f)*x^3)*(-a^2*b)^(2/3)*log(a*b*x^2 - (-a^2*
b)^(2/3)*x - (-a^2*b)^(1/3)*a) + 28*(a*b^3*c - 4*a^2*b^2*d + 7*a^3*b*e - 10*a^4*f + (b^4*c - 4*a*b^3*d + 7*a^2
*b^2*e - 10*a^3*b*f)*x^3)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) - 84*(a^2*b^4*c - 4*a^3*b^3*d + 7*a^4*b^2
*e - 10*a^5*b*f)*x)/(a^2*b^6*x^3 + a^3*b^5), 1/252*(36*a^2*b^4*f*x^10 + 9*(7*a^2*b^4*e - 10*a^3*b^3*f)*x^7 + 6
3*(4*a^2*b^4*d - 7*a^3*b^3*e + 10*a^4*b^2*f)*x^4 + 84*sqrt(1/3)*(a^2*b^4*c - 4*a^3*b^3*d + 7*a^4*b^2*e - 10*a^
5*b*f + (a*b^5*c - 4*a^2*b^4*d + 7*a^3*b^3*e - 10*a^4*b^2*f)*x^3)*sqrt(-(-a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*
(-a^2*b)^(2/3)*x + (-a^2*b)^(1/3)*a)*sqrt(-(-a^2*b)^(1/3)/b)/a^2) - 14*(a*b^3*c - 4*a^2*b^2*d + 7*a^3*b*e - 10
*a^4*f + (b^4*c - 4*a*b^3*d + 7*a^2*b^2*e - 10*a^3*b*f)*x^3)*(-a^2*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (
-a^2*b)^(1/3)*a) + 28*(a*b^3*c - 4*a^2*b^2*d + 7*a^3*b*e - 10*a^4*f + (b^4*c - 4*a*b^3*d + 7*a^2*b^2*e - 10*a^
3*b*f)*x^3)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) - 84*(a^2*b^4*c - 4*a^3*b^3*d + 7*a^4*b^2*e - 10*a^5*b*
f)*x)/(a^2*b^6*x^3 + a^3*b^5)]

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giac [A]  time = 0.18, size = 295, normalized size = 1.02 \[ -\frac {\sqrt {3} {\left (b^{3} c - 4 \, a b^{2} d - 10 \, a^{3} f + 7 \, a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3}} - \frac {{\left (b^{3} c - 4 \, a b^{2} d - 10 \, a^{3} f + 7 \, a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3}} - \frac {{\left (b^{3} c - 4 \, a b^{2} d - 10 \, a^{3} f + 7 \, a^{2} b e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b^{4}} - \frac {b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{3 \, {\left (b x^{3} + a\right )} b^{4}} + \frac {4 \, b^{12} f x^{7} - 14 \, a b^{11} f x^{4} + 7 \, b^{12} x^{4} e + 28 \, b^{12} d x + 84 \, a^{2} b^{10} f x - 56 \, a b^{11} x e}{28 \, b^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(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 - 4*a*b^2*d - 10*a^3*f + 7*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/
((-a*b^2)^(2/3)*b^3) - 1/18*(b^3*c - 4*a*b^2*d - 10*a^3*f + 7*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3)
)/((-a*b^2)^(2/3)*b^3) - 1/9*(b^3*c - 4*a*b^2*d - 10*a^3*f + 7*a^2*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3))
)/(a*b^4) - 1/3*(b^3*c*x - a*b^2*d*x - a^3*f*x + a^2*b*x*e)/((b*x^3 + a)*b^4) + 1/28*(4*b^12*f*x^7 - 14*a*b^11
*f*x^4 + 7*b^12*x^4*e + 28*b^12*d*x + 84*a^2*b^10*f*x - 56*a*b^11*x*e)/b^14

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maple [B]  time = 0.05, size = 514, normalized size = 1.78 \[ \frac {f \,x^{7}}{7 b^{2}}-\frac {a f \,x^{4}}{2 b^{3}}+\frac {e \,x^{4}}{4 b^{2}}+\frac {a^{3} f x}{3 \left (b \,x^{3}+a \right ) b^{4}}-\frac {a^{2} e x}{3 \left (b \,x^{3}+a \right ) b^{3}}+\frac {a d x}{3 \left (b \,x^{3}+a \right ) b^{2}}-\frac {c x}{3 \left (b \,x^{3}+a \right ) b}-\frac {10 \sqrt {3}\, a^{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}} b^{5}}-\frac {10 a^{3} f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}+\frac {5 a^{3} 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 {2}{3}} b^{5}}+\frac {7 \sqrt {3}\, a^{2} 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}} b^{4}}+\frac {7 a^{2} e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}-\frac {7 a^{2} 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 {2}{3}} b^{4}}+\frac {3 a^{2} f x}{b^{4}}-\frac {4 \sqrt {3}\, a 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}} b^{3}}-\frac {4 a d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {2 a 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 {2}{3}} b^{3}}-\frac {2 a e x}{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 {2}{3}} b^{2}}+\frac {c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}-\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 {2}{3}} b^{2}}+\frac {d x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7/b^2*f*x^7-1/2/b^3*x^4*a*f+1/4/b^2*x^4*e+3/b^4*a^2*f*x-2/b^3*a*e*x+1/b^2*d*x+1/3/b^4*x/(b*x^3+a)*a^3*f-1/3/
b^3*x/(b*x^3+a)*a^2*e+1/3/b^2*x/(b*x^3+a)*a*d-1/3/b*x/(b*x^3+a)*c-10/9/b^5*a^3*f/(a/b)^(2/3)*ln(x+(a/b)^(1/3))
+5/9/b^5*a^3*f/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-10/9/b^5*a^3*f/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(
1/2)*(2/(a/b)^(1/3)*x-1))+7/9/b^4*a^2*e/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-7/18/b^4*a^2*e/(a/b)^(2/3)*ln(x^2-(a/b)^
(1/3)*x+(a/b)^(2/3))+7/9/b^4*a^2*e/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-4/9/b^3*a*d/(a/
b)^(2/3)*ln(x+(a/b)^(1/3))+2/9/b^3*a*d/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-4/9/b^3*a*d/(a/b)^(2/3)*3
^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/9/b^2*c/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/18/b^2*c/(a/b)^(2/3)*
ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/9/b^2*c/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))

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maxima [A]  time = 2.98, size = 270, normalized size = 0.94 \[ -\frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x}{3 \, {\left (b^{5} x^{3} + a b^{4}\right )}} + \frac {4 \, b^{2} f x^{7} + 7 \, {\left (b^{2} e - 2 \, a b f\right )} x^{4} + 28 \, {\left (b^{2} d - 2 \, a b e + 3 \, a^{2} f\right )} x}{28 \, b^{4}} + \frac {\sqrt {3} {\left (b^{3} c - 4 \, a b^{2} d + 7 \, a^{2} b e - 10 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{3} c - 4 \, a b^{2} d + 7 \, a^{2} b e - 10 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{3} c - 4 \, a b^{2} d + 7 \, a^{2} b e - 10 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(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/(b^5*x^3 + a*b^4) + 1/28*(4*b^2*f*x^7 + 7*(b^2*e - 2*a*b*f)*x^4 + 2
8*(b^2*d - 2*a*b*e + 3*a^2*f)*x)/b^4 + 1/9*sqrt(3)*(b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*arctan(1/3*sqrt(
3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^5*(a/b)^(2/3)) - 1/18*(b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*log(x^
2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^5*(a/b)^(2/3)) + 1/9*(b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*log(x + (a
/b)^(1/3))/(b^5*(a/b)^(2/3))

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mupad [B]  time = 0.31, size = 280, normalized size = 0.97 \[ x^4\,\left (\frac {e}{4\,b^2}-\frac {a\,f}{2\,b^3}\right )-x\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )-\frac {x\,\left (-\frac {f\,a^3}{3}+\frac {e\,a^2\,b}{3}-\frac {d\,a\,b^2}{3}+\frac {c\,b^3}{3}\right )}{b^5\,x^3+a\,b^4}+\frac {f\,x^7}{7\,b^2}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-10\,f\,a^3+7\,e\,a^2\,b-4\,d\,a\,b^2+c\,b^3\right )}{9\,a^{2/3}\,b^{13/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-10\,f\,a^3+7\,e\,a^2\,b-4\,d\,a\,b^2+c\,b^3\right )}{9\,a^{2/3}\,b^{13/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-10\,f\,a^3+7\,e\,a^2\,b-4\,d\,a\,b^2+c\,b^3\right )}{9\,a^{2/3}\,b^{13/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*(e/(4*b^2) - (a*f)/(2*b^3)) - x*((a^2*f)/b^4 - d/b^2 + (2*a*(e/b^2 - (2*a*f)/b^3))/b) - (x*((b^3*c)/3 - (a
^3*f)/3 - (a*b^2*d)/3 + (a^2*b*e)/3))/(a*b^4 + b^5*x^3) + (f*x^7)/(7*b^2) + (log(b^(1/3)*x + a^(1/3))*(b^3*c -
10*a^3*f - 4*a*b^2*d + 7*a^2*b*e))/(9*a^(2/3)*b^(13/3)) + (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 - 10*a^3*f - 4*a*b^2*d + 7*a^2*b*e))/(9*a^(2/3)*b^(13/3)) - (log(3^(1/2)*a^(1/3)*1
i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(b^3*c - 10*a^3*f - 4*a*b^2*d + 7*a^2*b*e))/(9*a^(2/3)*b^(13
/3))

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sympy [A]  time = 12.90, size = 401, normalized size = 1.39 \[ x^{4} \left (- \frac {a f}{2 b^{3}} + \frac {e}{4 b^{2}}\right ) + x \left (\frac {3 a^{2} f}{b^{4}} - \frac {2 a e}{b^{3}} + \frac {d}{b^{2}}\right ) + \frac {x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 a b^{4} + 3 b^{5} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a^{2} b^{13} + 1000 a^{9} f^{3} - 2100 a^{8} b e f^{2} + 1200 a^{7} b^{2} d f^{2} + 1470 a^{7} b^{2} e^{2} f - 300 a^{6} b^{3} c f^{2} - 1680 a^{6} b^{3} d e f - 343 a^{6} b^{3} e^{3} + 420 a^{5} b^{4} c e f + 480 a^{5} b^{4} d^{2} f + 588 a^{5} b^{4} d e^{2} - 240 a^{4} b^{5} c d f - 147 a^{4} b^{5} c e^{2} - 336 a^{4} b^{5} d^{2} e + 30 a^{3} b^{6} c^{2} f + 168 a^{3} b^{6} c d e + 64 a^{3} b^{6} d^{3} - 21 a^{2} b^{7} c^{2} e - 48 a^{2} b^{7} c d^{2} + 12 a b^{8} c^{2} d - b^{9} c^{3}, \left (t \mapsto t \log {\left (- \frac {9 t a b^{4}}{10 a^{3} f - 7 a^{2} b e + 4 a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac {f x^{7}}{7 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**4*(-a*f/(2*b**3) + e/(4*b**2)) + x*(3*a**2*f/b**4 - 2*a*e/b**3 + d/b**2) + x*(a**3*f - a**2*b*e + a*b**2*d
- b**3*c)/(3*a*b**4 + 3*b**5*x**3) + RootSum(729*_t**3*a**2*b**13 + 1000*a**9*f**3 - 2100*a**8*b*e*f**2 + 1200
*a**7*b**2*d*f**2 + 1470*a**7*b**2*e**2*f - 300*a**6*b**3*c*f**2 - 1680*a**6*b**3*d*e*f - 343*a**6*b**3*e**3 +
420*a**5*b**4*c*e*f + 480*a**5*b**4*d**2*f + 588*a**5*b**4*d*e**2 - 240*a**4*b**5*c*d*f - 147*a**4*b**5*c*e**
2 - 336*a**4*b**5*d**2*e + 30*a**3*b**6*c**2*f + 168*a**3*b**6*c*d*e + 64*a**3*b**6*d**3 - 21*a**2*b**7*c**2*e
- 48*a**2*b**7*c*d**2 + 12*a*b**8*c**2*d - b**9*c**3, Lambda(_t, _t*log(-9*_t*a*b**4/(10*a**3*f - 7*a**2*b*e
+ 4*a*b**2*d - b**3*c) + x))) + f*x**7/(7*b**2)

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