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

Optimal. Leaf size=328 \[ \frac {x^4 \left (3 a^2 f-2 a b e+b^2 d\right )}{4 b^4}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{9 b^{16/3}}+\frac {\sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{3 \sqrt {3} b^{16/3}}+\frac {a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}+\frac {x \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{b^5}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{18 b^{16/3}}+\frac {x^7 (b e-2 a f)}{7 b^3}+\frac {f x^{10}}{10 b^2} \]

[Out]

(-4*a^3*f+3*a^2*b*e-2*a*b^2*d+b^3*c)*x/b^5+1/4*(3*a^2*f-2*a*b*e+b^2*d)*x^4/b^4+1/7*(-2*a*f+b*e)*x^7/b^3+1/10*f
*x^10/b^2+1/3*a*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/b^5/(b*x^3+a)-1/9*a^(1/3)*(-13*a^3*f+10*a^2*b*e-7*a*b^2*d+4*b
^3*c)*ln(a^(1/3)+b^(1/3)*x)/b^(16/3)+1/18*a^(1/3)*(-13*a^3*f+10*a^2*b*e-7*a*b^2*d+4*b^3*c)*ln(a^(2/3)-a^(1/3)*
b^(1/3)*x+b^(2/3)*x^2)/b^(16/3)+1/9*a^(1/3)*(-13*a^3*f+10*a^2*b*e-7*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))/b^(16/3)*3^(1/2)

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Rubi [A]  time = 0.37, antiderivative size = 328, 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} \[ \frac {a x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (10 a^2 b e-13 a^3 f-7 a b^2 d+4 b^3 c\right )}{18 b^{16/3}}+\frac {x \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{b^5}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (10 a^2 b e-13 a^3 f-7 a b^2 d+4 b^3 c\right )}{9 b^{16/3}}+\frac {\sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (10 a^2 b e-13 a^3 f-7 a b^2 d+4 b^3 c\right )}{3 \sqrt {3} b^{16/3}}+\frac {x^4 \left (3 a^2 f-2 a b e+b^2 d\right )}{4 b^4}+\frac {x^7 (b e-2 a f)}{7 b^3}+\frac {f x^{10}}{10 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x)/b^5 + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^4)/(4*b^4) + ((b*e - 2*a*f)
*x^7)/(7*b^3) + (f*x^10)/(10*b^2) + (a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(3*b^5*(a + b*x^3)) + (a^(1/3)*(
4*b^3*c - 7*a*b^2*d + 10*a^2*b*e - 13*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*b^(
16/3)) - (a^(1/3)*(4*b^3*c - 7*a*b^2*d + 10*a^2*b*e - 13*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(16/3)) + (a^(1
/3)*(4*b^3*c - 7*a*b^2*d + 10*a^2*b*e - 13*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*b^(16/3)
)

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^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx &=\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^5 \left (a+b x^3\right )}-\frac {\int \frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-3 a b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-3 a b^2 \left (b^2 d-a b e+a^2 f\right ) x^6-3 a b^3 (b e-a f) x^9-3 a b^4 f x^{12}}{a+b x^3} \, dx}{3 a b^5}\\ &=\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^5 \left (a+b x^3\right )}-\frac {\int \left (-3 a \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right )-3 a b \left (b^2 d-2 a b e+3 a^2 f\right ) x^3-3 a b^2 (b e-2 a f) x^6-3 a b^3 f x^9+\frac {4 a^2 b^3 c-7 a^3 b^2 d+10 a^4 b e-13 a^5 f}{a+b x^3}\right ) \, dx}{3 a b^5}\\ &=\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x}{b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^4}{4 b^4}+\frac {(b e-2 a f) x^7}{7 b^3}+\frac {f x^{10}}{10 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^5 \left (a+b x^3\right )}-\frac {\left (a \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right )\right ) \int \frac {1}{a+b x^3} \, dx}{3 b^5}\\ &=\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x}{b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^4}{4 b^4}+\frac {(b e-2 a f) x^7}{7 b^3}+\frac {f x^{10}}{10 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^5 \left (a+b x^3\right )}-\frac {\left (\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 b^5}-\frac {\left (\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right )\right ) \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 b^5}\\ &=\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x}{b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^4}{4 b^4}+\frac {(b e-2 a f) x^7}{7 b^3}+\frac {f x^{10}}{10 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^5 \left (a+b x^3\right )}-\frac {\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{16/3}}+\frac {\left (\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right )\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 b^{16/3}}-\frac {\left (a^{2/3} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^5}\\ &=\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x}{b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^4}{4 b^4}+\frac {(b e-2 a f) x^7}{7 b^3}+\frac {f x^{10}}{10 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^5 \left (a+b x^3\right )}-\frac {\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{16/3}}+\frac {\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{16/3}}-\frac {\left (\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 b^{16/3}}\\ &=\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x}{b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^4}{4 b^4}+\frac {(b e-2 a f) x^7}{7 b^3}+\frac {f x^{10}}{10 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^5 \left (a+b x^3\right )}+\frac {\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 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} b^{16/3}}-\frac {\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{16/3}}+\frac {\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{16/3}}\\ \end {align*}

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Mathematica [A]  time = 0.30, size = 315, normalized size = 0.96 \[ \frac {315 b^{4/3} x^4 \left (3 a^2 f-2 a b e+b^2 d\right )+\frac {420 a \sqrt [3]{b} x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a+b x^3}+1260 \sqrt [3]{b} x \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )+140 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (13 a^3 f-10 a^2 b e+7 a b^2 d-4 b^3 c\right )-140 \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (13 a^3 f-10 a^2 b e+7 a b^2 d-4 b^3 c\right )-70 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (13 a^3 f-10 a^2 b e+7 a b^2 d-4 b^3 c\right )+180 b^{7/3} x^7 (b e-2 a f)+126 b^{10/3} f x^{10}}{1260 b^{16/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1260*b^(1/3)*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x + 315*b^(4/3)*(b^2*d - 2*a*b*e + 3*a^2*f)*x^4 + 180*
b^(7/3)*(b*e - 2*a*f)*x^7 + 126*b^(10/3)*f*x^10 + (420*a*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a + b
*x^3) - 140*Sqrt[3]*a^(1/3)*(-4*b^3*c + 7*a*b^2*d - 10*a^2*b*e + 13*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/
Sqrt[3]] + 140*a^(1/3)*(-4*b^3*c + 7*a*b^2*d - 10*a^2*b*e + 13*a^3*f)*Log[a^(1/3) + b^(1/3)*x] - 70*a^(1/3)*(-
4*b^3*c + 7*a*b^2*d - 10*a^2*b*e + 13*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1260*b^(16/3))

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fricas [A]  time = 0.63, size = 423, normalized size = 1.29 \[ \frac {126 \, b^{4} f x^{13} + 18 \, {\left (10 \, b^{4} e - 13 \, a b^{3} f\right )} x^{10} + 45 \, {\left (7 \, b^{4} d - 10 \, a b^{3} e + 13 \, a^{2} b^{2} f\right )} x^{7} + 315 \, {\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{4} - 140 \, \sqrt {3} {\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f + {\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{3}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 70 \, {\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f + {\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{3}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) - 140 \, {\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f + {\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{3}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 420 \, {\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f\right )} x}{1260 \, {\left (b^{6} x^{3} + a b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1260*(126*b^4*f*x^13 + 18*(10*b^4*e - 13*a*b^3*f)*x^10 + 45*(7*b^4*d - 10*a*b^3*e + 13*a^2*b^2*f)*x^7 + 315*
(4*b^4*c - 7*a*b^3*d + 10*a^2*b^2*e - 13*a^3*b*f)*x^4 - 140*sqrt(3)*(4*a*b^3*c - 7*a^2*b^2*d + 10*a^3*b*e - 13
*a^4*f + (4*b^4*c - 7*a*b^3*d + 10*a^2*b^2*e - 13*a^3*b*f)*x^3)*(a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(a/b)^(2
/3) - sqrt(3)*a)/a) + 70*(4*a*b^3*c - 7*a^2*b^2*d + 10*a^3*b*e - 13*a^4*f + (4*b^4*c - 7*a*b^3*d + 10*a^2*b^2*
e - 13*a^3*b*f)*x^3)*(a/b)^(1/3)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3)) - 140*(4*a*b^3*c - 7*a^2*b^2*d + 10*a^
3*b*e - 13*a^4*f + (4*b^4*c - 7*a*b^3*d + 10*a^2*b^2*e - 13*a^3*b*f)*x^3)*(a/b)^(1/3)*log(x + (a/b)^(1/3)) + 4
20*(4*a*b^3*c - 7*a^2*b^2*d + 10*a^3*b*e - 13*a^4*f)*x)/(b^6*x^3 + a*b^5)

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giac [A]  time = 0.18, size = 394, normalized size = 1.20 \[ -\frac {\sqrt {3} {\left (4 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d - 13 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f + 10 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{6}} + \frac {{\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d - 13 \, a^{4} f + 10 \, a^{3} b e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b^{5}} - \frac {{\left (4 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d - 13 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f + 10 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{6}} + \frac {a b^{3} c x - a^{2} b^{2} d x - a^{4} f x + a^{3} b x e}{3 \, {\left (b x^{3} + a\right )} b^{5}} + \frac {14 \, b^{18} f x^{10} - 40 \, a b^{17} f x^{7} + 20 \, b^{18} x^{7} e + 35 \, b^{18} d x^{4} + 105 \, a^{2} b^{16} f x^{4} - 70 \, a b^{17} x^{4} e + 140 \, b^{18} c x - 280 \, a b^{17} d x - 560 \, a^{3} b^{15} f x + 420 \, a^{2} b^{16} x e}{140 \, b^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*sqrt(3)*(4*(-a*b^2)^(1/3)*b^3*c - 7*(-a*b^2)^(1/3)*a*b^2*d - 13*(-a*b^2)^(1/3)*a^3*f + 10*(-a*b^2)^(1/3)*
a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^6 + 1/9*(4*a*b^3*c - 7*a^2*b^2*d - 13*a^4*f +
10*a^3*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^5) - 1/18*(4*(-a*b^2)^(1/3)*b^3*c - 7*(-a*b^2)^(1/3)
*a*b^2*d - 13*(-a*b^2)^(1/3)*a^3*f + 10*(-a*b^2)^(1/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^6 +
1/3*(a*b^3*c*x - a^2*b^2*d*x - a^4*f*x + a^3*b*x*e)/((b*x^3 + a)*b^5) + 1/140*(14*b^18*f*x^10 - 40*a*b^17*f*x
^7 + 20*b^18*x^7*e + 35*b^18*d*x^4 + 105*a^2*b^16*f*x^4 - 70*a*b^17*x^4*e + 140*b^18*c*x - 280*a*b^17*d*x - 56
0*a^3*b^15*f*x + 420*a^2*b^16*x*e)/b^20

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maple [B]  time = 0.06, size = 567, normalized size = 1.73 \[ \frac {f \,x^{10}}{10 b^{2}}-\frac {2 a f \,x^{7}}{7 b^{3}}+\frac {e \,x^{7}}{7 b^{2}}+\frac {3 a^{2} f \,x^{4}}{4 b^{4}}-\frac {a e \,x^{4}}{2 b^{3}}+\frac {d \,x^{4}}{4 b^{2}}-\frac {a^{4} f x}{3 \left (b \,x^{3}+a \right ) b^{5}}+\frac {a^{3} e x}{3 \left (b \,x^{3}+a \right ) b^{4}}-\frac {a^{2} d x}{3 \left (b \,x^{3}+a \right ) b^{3}}+\frac {a c x}{3 \left (b \,x^{3}+a \right ) b^{2}}+\frac {13 \sqrt {3}\, a^{4} 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^{6}}+\frac {13 a^{4} f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{6}}-\frac {13 a^{4} f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{6}}-\frac {10 \sqrt {3}\, a^{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 {2}{3}} b^{5}}-\frac {10 a^{3} e \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} e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}-\frac {4 a^{3} f x}{b^{5}}+\frac {7 \sqrt {3}\, a^{2} d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}+\frac {7 a^{2} d \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} d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}+\frac {3 a^{2} e x}{b^{4}}-\frac {4 \sqrt {3}\, a 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^{3}}-\frac {4 a c \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 c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {2 a d x}{b^{3}}+\frac {c x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/9*a/b^3*c/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+13/9*a^4/b^6*f/(a/b)^(2/3)*3^(1/2)*ar
ctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-10/9*a^3/b^5*e/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1
))+7/9*a^2/b^4*d/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/2/b^3*x^4*a*e-4/b^5*a^3*f*x+3/b
^4*a^2*e*x-2/b^3*a*d*x+3/4/b^4*x^4*a^2*f-2/7*a/b^3*f*x^7+2/9*a/b^3*c/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2
/3))-4/9*a/b^3*c/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+1/7/b^2*e*x^7-1/3*a^4/b^5*x/(b*x^3+a)*f+1/3*a^3/b^4*x/(b*x^3+a)
*e-1/3*a^2/b^3*x/(b*x^3+a)*d-13/18*a^4/b^6*f/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3*a/b^2*x/(b*x^3+
a)*c+13/9*a^4/b^6*f/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+1/4/b^2*x^4*d+1/b^2*c*x-10/9*a^3/b^5*e/(a/b)^(2/3)*ln(x+(a/b
)^(1/3))+5/9*a^3/b^5*e/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+7/9*a^2/b^4*d/(a/b)^(2/3)*ln(x+(a/b)^(1/3
))-7/18*a^2/b^4*d/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/10*f*x^10/b^2

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maxima [A]  time = 3.02, size = 321, normalized size = 0.98 \[ \frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x}{3 \, {\left (b^{6} x^{3} + a b^{5}\right )}} + \frac {14 \, b^{3} f x^{10} + 20 \, {\left (b^{3} e - 2 \, a b^{2} f\right )} x^{7} + 35 \, {\left (b^{3} d - 2 \, a b^{2} e + 3 \, a^{2} b f\right )} x^{4} + 140 \, {\left (b^{3} c - 2 \, a b^{2} d + 3 \, a^{2} b e - 4 \, a^{3} f\right )} x}{140 \, b^{5}} - \frac {\sqrt {3} {\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x/(b^6*x^3 + a*b^5) + 1/140*(14*b^3*f*x^10 + 20*(b^3*e - 2*a*b^2*f
)*x^7 + 35*(b^3*d - 2*a*b^2*e + 3*a^2*b*f)*x^4 + 140*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x)/b^5 - 1/9*sq
rt(3)*(4*a*b^3*c - 7*a^2*b^2*d + 10*a^3*b*e - 13*a^4*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b
^6*(a/b)^(2/3)) + 1/18*(4*a*b^3*c - 7*a^2*b^2*d + 10*a^3*b*e - 13*a^4*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3)
)/(b^6*(a/b)^(2/3)) - 1/9*(4*a*b^3*c - 7*a^2*b^2*d + 10*a^3*b*e - 13*a^4*f)*log(x + (a/b)^(1/3))/(b^6*(a/b)^(2
/3))

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mupad [B]  time = 5.20, size = 358, normalized size = 1.09 \[ x^7\,\left (\frac {e}{7\,b^2}-\frac {2\,a\,f}{7\,b^3}\right )+x\,\left (\frac {c}{b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b^2}+\frac {2\,a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b}\right )-x^4\,\left (\frac {a^2\,f}{4\,b^4}-\frac {d}{4\,b^2}+\frac {a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{2\,b}\right )-\frac {x\,\left (\frac {f\,a^4}{3}-\frac {e\,a^3\,b}{3}+\frac {d\,a^2\,b^2}{3}-\frac {c\,a\,b^3}{3}\right )}{b^6\,x^3+a\,b^5}+\frac {f\,x^{10}}{10\,b^2}-\frac {a^{1/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-13\,f\,a^3+10\,e\,a^2\,b-7\,d\,a\,b^2+4\,c\,b^3\right )}{9\,b^{16/3}}-\frac {a^{1/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-13\,f\,a^3+10\,e\,a^2\,b-7\,d\,a\,b^2+4\,c\,b^3\right )}{9\,b^{16/3}}+\frac {a^{1/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-13\,f\,a^3+10\,e\,a^2\,b-7\,d\,a\,b^2+4\,c\,b^3\right )}{9\,b^{16/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^7*(e/(7*b^2) - (2*a*f)/(7*b^3)) + x*(c/b^2 - (a^2*(e/b^2 - (2*a*f)/b^3))/b^2 + (2*a*((a^2*f)/b^4 - d/b^2 + (
2*a*(e/b^2 - (2*a*f)/b^3))/b))/b) - x^4*((a^2*f)/(4*b^4) - d/(4*b^2) + (a*(e/b^2 - (2*a*f)/b^3))/(2*b)) - (x*(
(a^4*f)/3 + (a^2*b^2*d)/3 - (a*b^3*c)/3 - (a^3*b*e)/3))/(a*b^5 + b^6*x^3) + (f*x^10)/(10*b^2) - (a^(1/3)*log(b
^(1/3)*x + a^(1/3))*(4*b^3*c - 13*a^3*f - 7*a*b^2*d + 10*a^2*b*e))/(9*b^(16/3)) - (a^(1/3)*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 - 13*a^3*f - 7*a*b^2*d + 10*a^2*b*e))/(9*b^(16/3)
) + (a^(1/3)*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 - 13*a^3*f - 7*a*
b^2*d + 10*a^2*b*e))/(9*b^(16/3))

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sympy [A]  time = 14.98, size = 449, normalized size = 1.37 \[ x^{7} \left (- \frac {2 a f}{7 b^{3}} + \frac {e}{7 b^{2}}\right ) + x^{4} \left (\frac {3 a^{2} f}{4 b^{4}} - \frac {a e}{2 b^{3}} + \frac {d}{4 b^{2}}\right ) + x \left (- \frac {4 a^{3} f}{b^{5}} + \frac {3 a^{2} e}{b^{4}} - \frac {2 a d}{b^{3}} + \frac {c}{b^{2}}\right ) + \frac {x \left (- a^{4} f + a^{3} b e - a^{2} b^{2} d + a b^{3} c\right )}{3 a b^{5} + 3 b^{6} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} b^{16} - 2197 a^{10} f^{3} + 5070 a^{9} b e f^{2} - 3549 a^{8} b^{2} d f^{2} - 3900 a^{8} b^{2} e^{2} f + 2028 a^{7} b^{3} c f^{2} + 5460 a^{7} b^{3} d e f + 1000 a^{7} b^{3} e^{3} - 3120 a^{6} b^{4} c e f - 1911 a^{6} b^{4} d^{2} f - 2100 a^{6} b^{4} d e^{2} + 2184 a^{5} b^{5} c d f + 1200 a^{5} b^{5} c e^{2} + 1470 a^{5} b^{5} d^{2} e - 624 a^{4} b^{6} c^{2} f - 1680 a^{4} b^{6} c d e - 343 a^{4} b^{6} d^{3} + 480 a^{3} b^{7} c^{2} e + 588 a^{3} b^{7} c d^{2} - 336 a^{2} b^{8} c^{2} d + 64 a b^{9} c^{3}, \left (t \mapsto t \log {\left (\frac {9 t b^{5}}{13 a^{3} f - 10 a^{2} b e + 7 a b^{2} d - 4 b^{3} c} + x \right )} \right )\right )} + \frac {f x^{10}}{10 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**7*(-2*a*f/(7*b**3) + e/(7*b**2)) + x**4*(3*a**2*f/(4*b**4) - a*e/(2*b**3) + d/(4*b**2)) + x*(-4*a**3*f/b**5
+ 3*a**2*e/b**4 - 2*a*d/b**3 + c/b**2) + x*(-a**4*f + a**3*b*e - a**2*b**2*d + a*b**3*c)/(3*a*b**5 + 3*b**6*x
**3) + RootSum(729*_t**3*b**16 - 2197*a**10*f**3 + 5070*a**9*b*e*f**2 - 3549*a**8*b**2*d*f**2 - 3900*a**8*b**2
*e**2*f + 2028*a**7*b**3*c*f**2 + 5460*a**7*b**3*d*e*f + 1000*a**7*b**3*e**3 - 3120*a**6*b**4*c*e*f - 1911*a**
6*b**4*d**2*f - 2100*a**6*b**4*d*e**2 + 2184*a**5*b**5*c*d*f + 1200*a**5*b**5*c*e**2 + 1470*a**5*b**5*d**2*e -
624*a**4*b**6*c**2*f - 1680*a**4*b**6*c*d*e - 343*a**4*b**6*d**3 + 480*a**3*b**7*c**2*e + 588*a**3*b**7*c*d**
2 - 336*a**2*b**8*c**2*d + 64*a*b**9*c**3, Lambda(_t, _t*log(9*_t*b**5/(13*a**3*f - 10*a**2*b*e + 7*a*b**2*d -
4*b**3*c) + x))) + f*x**10/(10*b**2)

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