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

Optimal. Leaf size=336 \[ \frac {x \left (6 a^2 f-3 a b e+b^2 d\right )}{b^5}-\frac {x \left (-25 a^3 f+19 a^2 b e-13 a b^2 d+7 b^3 c\right )}{18 b^5 \left (a+b x^3\right )}+\frac {a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right )}{54 a^{2/3} b^{16/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right )}{27 a^{2/3} b^{16/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right )}{9 \sqrt {3} a^{2/3} b^{16/3}}+\frac {x^4 (b e-3 a f)}{4 b^4}+\frac {f x^7}{7 b^3} \]

[Out]

(6*a^2*f-3*a*b*e+b^2*d)*x/b^5+1/4*(-3*a*f+b*e)*x^4/b^4+1/7*f*x^7/b^3+1/6*a*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/b^
5/(b*x^3+a)^2-1/18*(-25*a^3*f+19*a^2*b*e-13*a*b^2*d+7*b^3*c)*x/b^5/(b*x^3+a)+1/27*(-65*a^3*f+35*a^2*b*e-14*a*b
^2*d+2*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(2/3)/b^(16/3)-1/54*(-65*a^3*f+35*a^2*b*e-14*a*b^2*d+2*b^3*c)*ln(a^(2/3)
-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(2/3)/b^(16/3)-1/27*(-65*a^3*f+35*a^2*b*e-14*a*b^2*d+2*b^3*c)*arctan(1/3*(a^
(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(2/3)/b^(16/3)*3^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.51, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1828, 1858, 1887, 200, 31, 634, 617, 204, 628} \[ -\frac {x \left (19 a^2 b e-25 a^3 f-13 a b^2 d+7 b^3 c\right )}{18 b^5 \left (a+b x^3\right )}+\frac {a x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (35 a^2 b e-65 a^3 f-14 a b^2 d+2 b^3 c\right )}{54 a^{2/3} b^{16/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (35 a^2 b e-65 a^3 f-14 a b^2 d+2 b^3 c\right )}{27 a^{2/3} b^{16/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (35 a^2 b e-65 a^3 f-14 a b^2 d+2 b^3 c\right )}{9 \sqrt {3} a^{2/3} b^{16/3}}+\frac {x \left (6 a^2 f-3 a b e+b^2 d\right )}{b^5}+\frac {x^4 (b e-3 a f)}{4 b^4}+\frac {f x^7}{7 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b^2*d - 3*a*b*e + 6*a^2*f)*x)/b^5 + ((b*e - 3*a*f)*x^4)/(4*b^4) + (f*x^7)/(7*b^3) + (a*(b^3*c - a*b^2*d + a^
2*b*e - a^3*f)*x)/(6*b^5*(a + b*x^3)^2) - ((7*b^3*c - 13*a*b^2*d + 19*a^2*b*e - 25*a^3*f)*x)/(18*b^5*(a + b*x^
3)) - ((2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sq
rt[3]*a^(2/3)*b^(16/3)) + ((2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(2/3
)*b^(16/3)) - ((2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/
(54*a^(2/3)*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 1858

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*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]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

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 )^3} \, dx &=\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac {\int \frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-6 a b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-6 a b^2 \left (b^2 d-a b e+a^2 f\right ) x^6-6 a b^3 (b e-a f) x^9-6 a b^4 f x^{12}}{\left (a+b x^3\right )^2} \, dx}{6 a b^5}\\ &=\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac {\left (7 b^3 c-13 a b^2 d+19 a^2 b e-25 a^3 f\right ) x}{18 b^5 \left (a+b x^3\right )}+\frac {\int \frac {2 a^2 b^4 \left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right )+18 a^2 b^5 \left (b^2 d-2 a b e+3 a^2 f\right ) x^3+18 a^2 b^6 (b e-2 a f) x^6+18 a^2 b^7 f x^9}{a+b x^3} \, dx}{18 a^2 b^9}\\ &=\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac {\left (7 b^3 c-13 a b^2 d+19 a^2 b e-25 a^3 f\right ) x}{18 b^5 \left (a+b x^3\right )}+\frac {\int \left (18 a^2 b^4 \left (b^2 d-3 a b e+6 a^2 f\right )+18 a^2 b^5 (b e-3 a f) x^3+18 a^2 b^6 f x^6-\frac {2 \left (-2 a^2 b^7 c+14 a^3 b^6 d-35 a^4 b^5 e+65 a^5 b^4 f\right )}{a+b x^3}\right ) \, dx}{18 a^2 b^9}\\ &=\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x}{b^5}+\frac {(b e-3 a f) x^4}{4 b^4}+\frac {f x^7}{7 b^3}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac {\left (7 b^3 c-13 a b^2 d+19 a^2 b e-25 a^3 f\right ) x}{18 b^5 \left (a+b x^3\right )}+\frac {\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \int \frac {1}{a+b x^3} \, dx}{9 b^5}\\ &=\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x}{b^5}+\frac {(b e-3 a f) x^4}{4 b^4}+\frac {f x^7}{7 b^3}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac {\left (7 b^3 c-13 a b^2 d+19 a^2 b e-25 a^3 f\right ) x}{18 b^5 \left (a+b x^3\right )}+\frac {\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{2/3} b^5}+\frac {\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 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}{27 a^{2/3} b^5}\\ &=\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x}{b^5}+\frac {(b e-3 a f) x^4}{4 b^4}+\frac {f x^7}{7 b^3}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac {\left (7 b^3 c-13 a b^2 d+19 a^2 b e-25 a^3 f\right ) x}{18 b^5 \left (a+b x^3\right )}+\frac {\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{16/3}}-\frac {\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{2/3} b^{16/3}}+\frac {\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 \sqrt [3]{a} b^5}\\ &=\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x}{b^5}+\frac {(b e-3 a f) x^4}{4 b^4}+\frac {f x^7}{7 b^3}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac {\left (7 b^3 c-13 a b^2 d+19 a^2 b e-25 a^3 f\right ) x}{18 b^5 \left (a+b x^3\right )}+\frac {\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{16/3}}-\frac {\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{2/3} b^{16/3}}+\frac {\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{2/3} b^{16/3}}\\ &=\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x}{b^5}+\frac {(b e-3 a f) x^4}{4 b^4}+\frac {f x^7}{7 b^3}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac {\left (7 b^3 c-13 a b^2 d+19 a^2 b e-25 a^3 f\right ) x}{18 b^5 \left (a+b x^3\right )}-\frac {\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{2/3} b^{16/3}}+\frac {\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{16/3}}-\frac {\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{2/3} b^{16/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.37, size = 323, normalized size = 0.96 \[ \frac {756 \sqrt [3]{b} x \left (6 a^2 f-3 a b e+b^2 d\right )-\frac {42 \sqrt [3]{b} x \left (-25 a^3 f+19 a^2 b e-13 a b^2 d+7 b^3 c\right )}{a+b x^3}+\frac {126 a \sqrt [3]{b} x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\left (a+b x^3\right )^2}+\frac {28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right )}{a^{2/3}}+\frac {28 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (65 a^3 f-35 a^2 b e+14 a b^2 d-2 b^3 c\right )}{a^{2/3}}+\frac {14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (65 a^3 f-35 a^2 b e+14 a b^2 d-2 b^3 c\right )}{a^{2/3}}+189 b^{4/3} x^4 (b e-3 a f)+108 b^{7/3} f x^7}{756 b^{16/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(756*b^(1/3)*(b^2*d - 3*a*b*e + 6*a^2*f)*x + 189*b^(4/3)*(b*e - 3*a*f)*x^4 + 108*b^(7/3)*f*x^7 + (126*a*b^(1/3
)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a + b*x^3)^2 - (42*b^(1/3)*(7*b^3*c - 13*a*b^2*d + 19*a^2*b*e - 25*a
^3*f)*x)/(a + b*x^3) + (28*Sqrt[3]*(-2*b^3*c + 14*a*b^2*d - 35*a^2*b*e + 65*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a
^(1/3))/Sqrt[3]])/a^(2/3) + (28*(2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(2/
3) + (14*(-2*b^3*c + 14*a*b^2*d - 35*a^2*b*e + 65*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/
3))/(756*b^(16/3))

________________________________________________________________________________________

fricas [B]  time = 0.84, size = 1318, normalized size = 3.92 \[ \text {result too large to display} \]

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)^3,x, algorithm="fricas")

[Out]

[1/756*(108*a^2*b^5*f*x^13 + 27*(7*a^2*b^5*e - 13*a^3*b^4*f)*x^10 + 54*(14*a^2*b^5*d - 35*a^3*b^4*e + 65*a^4*b
^3*f)*x^7 - 147*(2*a^2*b^5*c - 14*a^3*b^4*d + 35*a^4*b^3*e - 65*a^5*b^2*f)*x^4 - 42*sqrt(1/3)*(2*a^3*b^4*c - 1
4*a^4*b^3*d + 35*a^5*b^2*e - 65*a^6*b*f + (2*a*b^6*c - 14*a^2*b^5*d + 35*a^3*b^4*e - 65*a^4*b^3*f)*x^6 + 2*(2*
a^2*b^5*c - 14*a^3*b^4*d + 35*a^4*b^3*e - 65*a^5*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*((2*b^5*c - 14*a*b^4*d + 35*a^2*b^3*e - 65*a^3*b^2*f)*x^6 + 2*a^2*b^3*c - 14*a^3*b^2*d + 35*a^4*
b*e - 65*a^5*f + 2*(2*a*b^4*c - 14*a^2*b^3*d + 35*a^3*b^2*e - 65*a^4*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*((2*b^5*c - 14*a*b^4*d + 35*a^2*b^3*e - 65*a^3*b^2*f)*x^6 + 2*a^2*b^3*
c - 14*a^3*b^2*d + 35*a^4*b*e - 65*a^5*f + 2*(2*a*b^4*c - 14*a^2*b^3*d + 35*a^3*b^2*e - 65*a^4*b*f)*x^3)*(-a^2
*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) - 84*(2*a^3*b^4*c - 14*a^4*b^3*d + 35*a^5*b^2*e - 65*a^6*b*f)*x)/(a^2*b^
8*x^6 + 2*a^3*b^7*x^3 + a^4*b^6), 1/756*(108*a^2*b^5*f*x^13 + 27*(7*a^2*b^5*e - 13*a^3*b^4*f)*x^10 + 54*(14*a^
2*b^5*d - 35*a^3*b^4*e + 65*a^4*b^3*f)*x^7 - 147*(2*a^2*b^5*c - 14*a^3*b^4*d + 35*a^4*b^3*e - 65*a^5*b^2*f)*x^
4 + 84*sqrt(1/3)*(2*a^3*b^4*c - 14*a^4*b^3*d + 35*a^5*b^2*e - 65*a^6*b*f + (2*a*b^6*c - 14*a^2*b^5*d + 35*a^3*
b^4*e - 65*a^4*b^3*f)*x^6 + 2*(2*a^2*b^5*c - 14*a^3*b^4*d + 35*a^4*b^3*e - 65*a^5*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*((2*b^5*c -
14*a*b^4*d + 35*a^2*b^3*e - 65*a^3*b^2*f)*x^6 + 2*a^2*b^3*c - 14*a^3*b^2*d + 35*a^4*b*e - 65*a^5*f + 2*(2*a*b
^4*c - 14*a^2*b^3*d + 35*a^3*b^2*e - 65*a^4*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*((2*b^5*c - 14*a*b^4*d + 35*a^2*b^3*e - 65*a^3*b^2*f)*x^6 + 2*a^2*b^3*c - 14*a^3*b^2*d + 35*a^4
*b*e - 65*a^5*f + 2*(2*a*b^4*c - 14*a^2*b^3*d + 35*a^3*b^2*e - 65*a^4*b*f)*x^3)*(-a^2*b)^(2/3)*log(a*b*x + (-a
^2*b)^(2/3)) - 84*(2*a^3*b^4*c - 14*a^4*b^3*d + 35*a^5*b^2*e - 65*a^6*b*f)*x)/(a^2*b^8*x^6 + 2*a^3*b^7*x^3 + a
^4*b^6)]

________________________________________________________________________________________

giac [A]  time = 0.20, size = 345, normalized size = 1.03 \[ -\frac {\sqrt {3} {\left (2 \, b^{3} c - 14 \, a b^{2} d - 65 \, a^{3} f + 35 \, a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{4}} - \frac {{\left (2 \, b^{3} c - 14 \, a b^{2} d - 65 \, a^{3} f + 35 \, a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{4}} - \frac {{\left (2 \, b^{3} c - 14 \, a b^{2} d - 65 \, a^{3} f + 35 \, a^{2} b e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a b^{5}} - \frac {7 \, b^{4} c x^{4} - 13 \, a b^{3} d x^{4} - 25 \, a^{3} b f x^{4} + 19 \, a^{2} b^{2} x^{4} e + 4 \, a b^{3} c x - 10 \, a^{2} b^{2} d x - 22 \, a^{4} f x + 16 \, a^{3} b x e}{18 \, {\left (b x^{3} + a\right )}^{2} b^{5}} + \frac {4 \, b^{18} f x^{7} - 21 \, a b^{17} f x^{4} + 7 \, b^{18} x^{4} e + 28 \, b^{18} d x + 168 \, a^{2} b^{16} f x - 84 \, a b^{17} x e}{28 \, b^{21}} \]

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)^3,x, algorithm="giac")

[Out]

-1/27*sqrt(3)*(2*b^3*c - 14*a*b^2*d - 65*a^3*f + 35*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^4) - 1/54*(2*b^3*c - 14*a*b^2*d - 65*a^3*f + 35*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a
/b)^(2/3))/((-a*b^2)^(2/3)*b^4) - 1/27*(2*b^3*c - 14*a*b^2*d - 65*a^3*f + 35*a^2*b*e)*(-a/b)^(1/3)*log(abs(x -
(-a/b)^(1/3)))/(a*b^5) - 1/18*(7*b^4*c*x^4 - 13*a*b^3*d*x^4 - 25*a^3*b*f*x^4 + 19*a^2*b^2*x^4*e + 4*a*b^3*c*x
- 10*a^2*b^2*d*x - 22*a^4*f*x + 16*a^3*b*x*e)/((b*x^3 + a)^2*b^5) + 1/28*(4*b^18*f*x^7 - 21*a*b^17*f*x^4 + 7*
b^18*x^4*e + 28*b^18*d*x + 168*a^2*b^16*f*x - 84*a*b^17*x*e)/b^21

________________________________________________________________________________________

maple [B]  time = 0.05, size = 596, normalized size = 1.77 \[ \frac {f \,x^{7}}{7 b^{3}}+\frac {25 a^{3} f \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} b^{4}}-\frac {19 a^{2} e \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} b^{3}}+\frac {13 a d \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} b^{2}}-\frac {7 c \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} b}-\frac {3 a f \,x^{4}}{4 b^{4}}+\frac {e \,x^{4}}{4 b^{3}}+\frac {11 a^{4} f x}{9 \left (b \,x^{3}+a \right )^{2} b^{5}}-\frac {8 a^{3} e x}{9 \left (b \,x^{3}+a \right )^{2} b^{4}}+\frac {5 a^{2} d x}{9 \left (b \,x^{3}+a \right )^{2} b^{3}}-\frac {2 a c x}{9 \left (b \,x^{3}+a \right )^{2} b^{2}}-\frac {65 \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 )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{6}}-\frac {65 a^{3} f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{6}}+\frac {65 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 )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{6}}+\frac {35 \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 )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}+\frac {35 a^{2} e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}-\frac {35 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 )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}+\frac {6 a^{2} f x}{b^{5}}-\frac {14 \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 )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}-\frac {14 a d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}+\frac {7 a d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}-\frac {3 a e x}{b^{4}}+\frac {2 \sqrt {3}\, c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {2 c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {d x}{b^{3}} \]

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)^3,x)

[Out]

65/54/b^6*a^3*f/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+35/27/b^5*a^2*e/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-14
/27/b^4*a*d/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-3/b^4*a*e*x-3/4/b^4*x^4*a*f+2/27/b^3*c/(a/b)^(2/3)*ln(x+(a/b)^(1/3))
-7/18/b/(b*x^3+a)^2*x^4*c+6/b^5*a^2*f*x-1/27/b^3*c/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-8/9/b^4/(b*x^
3+a)^2*a^3*e*x+1/7/b^3*f*x^7+13/18/b^2/(b*x^3+a)^2*x^4*a*d-35/54/b^5*a^2*e/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a
/b)^(2/3))-2/9/b^2/(b*x^3+a)^2*a*c*x+11/9/b^5/(b*x^3+a)^2*a^4*f*x+1/4/b^3*x^4*e+7/27/b^4*a*d/(a/b)^(2/3)*ln(x^
2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/b^3*d*x+2/27/b^3*c/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-
65/27/b^6*a^3*f/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-65/27/b^6*a^3*f/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^
(1/3)*x-1))+35/27/b^5*a^2*e/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-14/27/b^4*a*d/(a/b)^(2
/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+25/18/b^4/(b*x^3+a)^2*x^4*a^3*f-19/18/b^3/(b*x^3+a)^2*x^4*
a^2*e+5/9/b^3/(b*x^3+a)^2*a^2*d*x

________________________________________________________________________________________

maxima [A]  time = 3.04, size = 326, normalized size = 0.97 \[ -\frac {{\left (7 \, b^{4} c - 13 \, a b^{3} d + 19 \, a^{2} b^{2} e - 25 \, a^{3} b f\right )} x^{4} + 2 \, {\left (2 \, a b^{3} c - 5 \, a^{2} b^{2} d + 8 \, a^{3} b e - 11 \, a^{4} f\right )} x}{18 \, {\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} + \frac {4 \, b^{2} f x^{7} + 7 \, {\left (b^{2} e - 3 \, a b f\right )} x^{4} + 28 \, {\left (b^{2} d - 3 \, a b e + 6 \, a^{2} f\right )} x}{28 \, b^{5}} + \frac {\sqrt {3} {\left (2 \, b^{3} c - 14 \, a b^{2} d + 35 \, a^{2} b e - 65 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, b^{3} c - 14 \, a b^{2} d + 35 \, a^{2} b e - 65 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, b^{3} c - 14 \, a b^{2} d + 35 \, a^{2} b e - 65 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

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)^3,x, algorithm="maxima")

[Out]

-1/18*((7*b^4*c - 13*a*b^3*d + 19*a^2*b^2*e - 25*a^3*b*f)*x^4 + 2*(2*a*b^3*c - 5*a^2*b^2*d + 8*a^3*b*e - 11*a^
4*f)*x)/(b^7*x^6 + 2*a*b^6*x^3 + a^2*b^5) + 1/28*(4*b^2*f*x^7 + 7*(b^2*e - 3*a*b*f)*x^4 + 28*(b^2*d - 3*a*b*e
+ 6*a^2*f)*x)/b^5 + 1/27*sqrt(3)*(2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b
)^(1/3))/(a/b)^(1/3))/(b^6*(a/b)^(2/3)) - 1/54*(2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*log(x^2 - x*(a/b
)^(1/3) + (a/b)^(2/3))/(b^6*(a/b)^(2/3)) + 1/27*(2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*log(x + (a/b)^(
1/3))/(b^6*(a/b)^(2/3))

________________________________________________________________________________________

mupad [B]  time = 5.30, size = 335, normalized size = 1.00 \[ x^4\,\left (\frac {e}{4\,b^3}-\frac {3\,a\,f}{4\,b^4}\right )-x\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )-\frac {x^4\,\left (-\frac {25\,f\,a^3\,b}{18}+\frac {19\,e\,a^2\,b^2}{18}-\frac {13\,d\,a\,b^3}{18}+\frac {7\,c\,b^4}{18}\right )-x\,\left (\frac {11\,f\,a^4}{9}-\frac {8\,e\,a^3\,b}{9}+\frac {5\,d\,a^2\,b^2}{9}-\frac {2\,c\,a\,b^3}{9}\right )}{a^2\,b^5+2\,a\,b^6\,x^3+b^7\,x^6}+\frac {f\,x^7}{7\,b^3}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-65\,f\,a^3+35\,e\,a^2\,b-14\,d\,a\,b^2+2\,c\,b^3\right )}{27\,a^{2/3}\,b^{16/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-65\,f\,a^3+35\,e\,a^2\,b-14\,d\,a\,b^2+2\,c\,b^3\right )}{27\,a^{2/3}\,b^{16/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-65\,f\,a^3+35\,e\,a^2\,b-14\,d\,a\,b^2+2\,c\,b^3\right )}{27\,a^{2/3}\,b^{16/3}} \]

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)^3,x)

[Out]

x^4*(e/(4*b^3) - (3*a*f)/(4*b^4)) - x*((3*a^2*f)/b^5 - d/b^3 + (3*a*(e/b^3 - (3*a*f)/b^4))/b) - (x^4*((7*b^4*c
)/18 + (19*a^2*b^2*e)/18 - (13*a*b^3*d)/18 - (25*a^3*b*f)/18) - x*((11*a^4*f)/9 + (5*a^2*b^2*d)/9 - (2*a*b^3*c
)/9 - (8*a^3*b*e)/9))/(a^2*b^5 + b^7*x^6 + 2*a*b^6*x^3) + (f*x^7)/(7*b^3) + (log(b^(1/3)*x + a^(1/3))*(2*b^3*c
- 65*a^3*f - 14*a*b^2*d + 35*a^2*b*e))/(27*a^(2/3)*b^(16/3)) + (log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3
))*((3^(1/2)*1i)/2 - 1/2)*(2*b^3*c - 65*a^3*f - 14*a*b^2*d + 35*a^2*b*e))/(27*a^(2/3)*b^(16/3)) - (log(3^(1/2)
*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(2*b^3*c - 65*a^3*f - 14*a*b^2*d + 35*a^2*b*e))/(2
7*a^(2/3)*b^(16/3))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________