3.289 \(\int \frac{x^7 (c+d x^3+e x^6+f x^9)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=345 \[ -\frac{x^2 \left (10 a^2 b e-13 a^3 f-7 a b^2 d+4 b^3 c\right )}{9 b^5 \left (a+b x^3\right )}+\frac{a x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (44 a^2 b e-77 a^3 f-20 a b^2 d+5 b^3 c\right )}{54 \sqrt [3]{a} b^{17/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (44 a^2 b e-77 a^3 f-20 a b^2 d+5 b^3 c\right )}{27 \sqrt [3]{a} b^{17/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (44 a^2 b e-77 a^3 f-20 a b^2 d+5 b^3 c\right )}{9 \sqrt{3} \sqrt [3]{a} b^{17/3}}+\frac{x^2 \left (6 a^2 f-3 a b e+b^2 d\right )}{2 b^5}+\frac{x^5 (b e-3 a f)}{5 b^4}+\frac{f x^8}{8 b^3} \]

[Out]

((b^2*d - 3*a*b*e + 6*a^2*f)*x^2)/(2*b^5) + ((b*e - 3*a*f)*x^5)/(5*b^4) + (f*x^8)/(8*b^3) + (a*(b^3*c - a*b^2*
d + a^2*b*e - a^3*f)*x^2)/(6*b^5*(a + b*x^3)^2) - ((4*b^3*c - 7*a*b^2*d + 10*a^2*b*e - 13*a^3*f)*x^2)/(9*b^5*(
a + b*x^3)) - ((5*b^3*c - 20*a*b^2*d + 44*a^2*b*e - 77*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))
])/(9*Sqrt[3]*a^(1/3)*b^(17/3)) - ((5*b^3*c - 20*a*b^2*d + 44*a^2*b*e - 77*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(2
7*a^(1/3)*b^(17/3)) + ((5*b^3*c - 20*a*b^2*d + 44*a^2*b*e - 77*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3
)*x^2])/(54*a^(1/3)*b^(17/3))

________________________________________________________________________________________

Rubi [A]  time = 0.763222, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1828, 1851, 1836, 1488, 292, 31, 634, 617, 204, 628} \[ -\frac{x^2 \left (10 a^2 b e-13 a^3 f-7 a b^2 d+4 b^3 c\right )}{9 b^5 \left (a+b x^3\right )}+\frac{a x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (44 a^2 b e-77 a^3 f-20 a b^2 d+5 b^3 c\right )}{54 \sqrt [3]{a} b^{17/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (44 a^2 b e-77 a^3 f-20 a b^2 d+5 b^3 c\right )}{27 \sqrt [3]{a} b^{17/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (44 a^2 b e-77 a^3 f-20 a b^2 d+5 b^3 c\right )}{9 \sqrt{3} \sqrt [3]{a} b^{17/3}}+\frac{x^2 \left (6 a^2 f-3 a b e+b^2 d\right )}{2 b^5}+\frac{x^5 (b e-3 a f)}{5 b^4}+\frac{f x^8}{8 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^7*(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^2)/(2*b^5) + ((b*e - 3*a*f)*x^5)/(5*b^4) + (f*x^8)/(8*b^3) + (a*(b^3*c - a*b^2*
d + a^2*b*e - a^3*f)*x^2)/(6*b^5*(a + b*x^3)^2) - ((4*b^3*c - 7*a*b^2*d + 10*a^2*b*e - 13*a^3*f)*x^2)/(9*b^5*(
a + b*x^3)) - ((5*b^3*c - 20*a*b^2*d + 44*a^2*b*e - 77*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))
])/(9*Sqrt[3]*a^(1/3)*b^(17/3)) - ((5*b^3*c - 20*a*b^2*d + 44*a^2*b*e - 77*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(2
7*a^(1/3)*b^(17/3)) + ((5*b^3*c - 20*a*b^2*d + 44*a^2*b*e - 77*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3
)*x^2])/(54*a^(1/3)*b^(17/3))

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 1851

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x^n)^p, x] /;
FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] &&  !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m
]]

Rule 1836

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq =
Coeff[Pq, x, q]}, Dist[1/(b*(m + q + n*p + 1)), Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(Pq - Pqq*x^q) - a
*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x] + Simp[(Pqq*(c*x)^(m + q - n + 1)*(a + b*x^n)^(p + 1)
)/(b*c^(q - n + 1)*(m + q + n*p + 1)), x]] /; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || Integ
erQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]

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 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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 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 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 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]

Rubi steps

\begin{align*} \int \frac{x^7 \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^2}{6 b^5 \left (a+b x^3\right )^2}-\frac{\int \frac{2 a^2 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x-6 a b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4-6 a b^3 \left (b^2 d-a b e+a^2 f\right ) x^7-6 a b^4 (b e-a f) x^{10}-6 a b^5 f x^{13}}{\left (a+b x^3\right )^2} \, dx}{6 a b^6}\\ &=\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^5 \left (a+b x^3\right )^2}-\frac{\int \frac{x \left (2 a^2 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-6 a b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-6 a b^3 \left (b^2 d-a b e+a^2 f\right ) x^6-6 a b^4 (b e-a f) x^9-6 a b^5 f x^{12}\right )}{\left (a+b x^3\right )^2} \, dx}{6 a b^6}\\ &=\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^5 \left (a+b x^3\right )^2}-\frac{\left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) x^2}{9 b^5 \left (a+b x^3\right )}+\frac{\int \frac{2 a^2 b^6 \left (5 b^3 c-11 a b^2 d+17 a^2 b e-23 a^3 f\right ) x+18 a^2 b^7 \left (b^2 d-2 a b e+3 a^2 f\right ) x^4+18 a^2 b^8 (b e-2 a f) x^7+18 a^2 b^9 f x^{10}}{a+b x^3} \, dx}{18 a^2 b^{11}}\\ &=\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^5 \left (a+b x^3\right )^2}-\frac{\left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) x^2}{9 b^5 \left (a+b x^3\right )}+\frac{\int \frac{x \left (2 a^2 b^6 \left (5 b^3 c-11 a b^2 d+17 a^2 b e-23 a^3 f\right )+18 a^2 b^7 \left (b^2 d-2 a b e+3 a^2 f\right ) x^3+18 a^2 b^8 (b e-2 a f) x^6+18 a^2 b^9 f x^9\right )}{a+b x^3} \, dx}{18 a^2 b^{11}}\\ &=\frac{f x^8}{8 b^3}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^5 \left (a+b x^3\right )^2}-\frac{\left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) x^2}{9 b^5 \left (a+b x^3\right )}+\frac{\int \frac{x \left (16 a^2 b^7 \left (5 b^3 c-11 a b^2 d+17 a^2 b e-23 a^3 f\right )+144 a^2 b^8 \left (b^2 d-2 a b e+3 a^2 f\right ) x^3+144 a^2 b^9 (b e-3 a f) x^6\right )}{a+b x^3} \, dx}{144 a^2 b^{12}}\\ &=\frac{f x^8}{8 b^3}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^5 \left (a+b x^3\right )^2}-\frac{\left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) x^2}{9 b^5 \left (a+b x^3\right )}+\frac{\int \left (144 a^2 b^7 \left (b^2 d-3 a b e+6 a^2 f\right ) x+144 a^2 b^8 (b e-3 a f) x^4-\frac{16 \left (-5 a^2 b^{10} c+20 a^3 b^9 d-44 a^4 b^8 e+77 a^5 b^7 f\right ) x}{a+b x^3}\right ) \, dx}{144 a^2 b^{12}}\\ &=\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^2}{2 b^5}+\frac{(b e-3 a f) x^5}{5 b^4}+\frac{f x^8}{8 b^3}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^5 \left (a+b x^3\right )^2}-\frac{\left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) x^2}{9 b^5 \left (a+b x^3\right )}+\frac{\left (5 b^3 c-20 a b^2 d+44 a^2 b e-77 a^3 f\right ) \int \frac{x}{a+b x^3} \, dx}{9 b^5}\\ &=\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^2}{2 b^5}+\frac{(b e-3 a f) x^5}{5 b^4}+\frac{f x^8}{8 b^3}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^5 \left (a+b x^3\right )^2}-\frac{\left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) x^2}{9 b^5 \left (a+b x^3\right )}-\frac{\left (5 b^3 c-20 a b^2 d+44 a^2 b e-77 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 \sqrt [3]{a} b^{16/3}}+\frac{\left (5 b^3 c-20 a b^2 d+44 a^2 b e-77 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}{27 \sqrt [3]{a} b^{16/3}}\\ &=\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^2}{2 b^5}+\frac{(b e-3 a f) x^5}{5 b^4}+\frac{f x^8}{8 b^3}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^5 \left (a+b x^3\right )^2}-\frac{\left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) x^2}{9 b^5 \left (a+b x^3\right )}-\frac{\left (5 b^3 c-20 a b^2 d+44 a^2 b e-77 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{17/3}}+\frac{\left (5 b^3 c-20 a b^2 d+44 a^2 b e-77 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 \sqrt [3]{a} b^{17/3}}+\frac{\left (5 b^3 c-20 a b^2 d+44 a^2 b e-77 a^3 f\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 b^{16/3}}\\ &=\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^2}{2 b^5}+\frac{(b e-3 a f) x^5}{5 b^4}+\frac{f x^8}{8 b^3}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^5 \left (a+b x^3\right )^2}-\frac{\left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) x^2}{9 b^5 \left (a+b x^3\right )}-\frac{\left (5 b^3 c-20 a b^2 d+44 a^2 b e-77 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{17/3}}+\frac{\left (5 b^3 c-20 a b^2 d+44 a^2 b e-77 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{17/3}}+\frac{\left (5 b^3 c-20 a b^2 d+44 a^2 b e-77 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 \sqrt [3]{a} b^{17/3}}\\ &=\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^2}{2 b^5}+\frac{(b e-3 a f) x^5}{5 b^4}+\frac{f x^8}{8 b^3}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^5 \left (a+b x^3\right )^2}-\frac{\left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) x^2}{9 b^5 \left (a+b x^3\right )}-\frac{\left (5 b^3 c-20 a b^2 d+44 a^2 b e-77 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} \sqrt [3]{a} b^{17/3}}-\frac{\left (5 b^3 c-20 a b^2 d+44 a^2 b e-77 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{17/3}}+\frac{\left (5 b^3 c-20 a b^2 d+44 a^2 b e-77 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{17/3}}\\ \end{align*}

Mathematica [A]  time = 0.223697, size = 329, normalized size = 0.95 \[ \frac{-\frac{120 b^{2/3} x^2 \left (10 a^2 b e-13 a^3 f-7 a b^2 d+4 b^3 c\right )}{a+b x^3}+\frac{180 a b^{2/3} x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\left (a+b x^3\right )^2}+\frac{20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (44 a^2 b e-77 a^3 f-20 a b^2 d+5 b^3 c\right )}{\sqrt [3]{a}}+\frac{40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-44 a^2 b e+77 a^3 f+20 a b^2 d-5 b^3 c\right )}{\sqrt [3]{a}}+\frac{40 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-44 a^2 b e+77 a^3 f+20 a b^2 d-5 b^3 c\right )}{\sqrt [3]{a}}+540 b^{2/3} x^2 \left (6 a^2 f-3 a b e+b^2 d\right )+216 b^{5/3} x^5 (b e-3 a f)+135 b^{8/3} f x^8}{1080 b^{17/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^7*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

(540*b^(2/3)*(b^2*d - 3*a*b*e + 6*a^2*f)*x^2 + 216*b^(5/3)*(b*e - 3*a*f)*x^5 + 135*b^(8/3)*f*x^8 + (180*a*b^(2
/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a + b*x^3)^2 - (120*b^(2/3)*(4*b^3*c - 7*a*b^2*d + 10*a^2*b*e -
13*a^3*f)*x^2)/(a + b*x^3) + (40*Sqrt[3]*(-5*b^3*c + 20*a*b^2*d - 44*a^2*b*e + 77*a^3*f)*ArcTan[(1 - (2*b^(1/3
)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (40*(-5*b^3*c + 20*a*b^2*d - 44*a^2*b*e + 77*a^3*f)*Log[a^(1/3) + b^(1/3)*x]
)/a^(1/3) + (20*(5*b^3*c - 20*a*b^2*d + 44*a^2*b*e - 77*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])
/a^(1/3))/(1080*b^(17/3))

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Maple [B]  time = 0.013, size = 611, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)

[Out]

-3/2/b^4*x^2*a*e-4/9/b/(b*x^3+a)^2*x^5*c-5/27/b^3*c/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+5/54/b^3*c/(1/b*a)^(1/3)
*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-3/5/b^4*x^5*a*f+3/b^5*x^2*a^2*f+1/8*f*x^8/b^3-5/18/b^2/(b*x^3+a)^2*x^2*
a*c+11/18/b^3/(b*x^3+a)^2*x^2*a^2*d+1/5/b^3*x^5*e+1/2/b^3*x^2*d-17/18/b^4/(b*x^3+a)^2*x^2*a^3*e+23/18/b^5/(b*x
^3+a)^2*x^2*a^4*f+13/9/b^4/(b*x^3+a)^2*x^5*f*a^3-10/9/b^3/(b*x^3+a)^2*x^5*a^2*e+7/9/b^2/(b*x^3+a)^2*x^5*a*d+5/
27/b^3*c*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-10/27/b^4*a*d/(1/b*a)^(1/3)*ln(x^2-(1
/b*a)^(1/3)*x+(1/b*a)^(2/3))+77/27/b^6*a^3*f/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-77/54/b^6*a^3*f/(1/b*a)^(1/3)*l
n(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-44/27/b^5*a^2*e/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+22/27/b^5*a^2*e/(1/b*a)
^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+20/27/b^4*a*d/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-77/27/b^6*a^3*f*3
^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+44/27/b^5*a^2*e*3^(1/2)/(1/b*a)^(1/3)*arctan(1/
3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-20/27/b^4*a*d*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.48138, size = 2915, normalized size = 8.45 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

[1/1080*(135*a*b^6*f*x^14 + 54*(4*a*b^6*e - 7*a^2*b^5*f)*x^11 + 27*(20*a*b^6*d - 44*a^2*b^5*e + 77*a^3*b^4*f)*
x^8 - 96*(5*a*b^6*c - 20*a^2*b^5*d + 44*a^3*b^4*e - 77*a^4*b^3*f)*x^5 - 60*(5*a^2*b^5*c - 20*a^3*b^4*d + 44*a^
4*b^3*e - 77*a^5*b^2*f)*x^2 - 60*sqrt(1/3)*(5*a^3*b^4*c - 20*a^4*b^3*d + 44*a^5*b^2*e - 77*a^6*b*f + (5*a*b^6*
c - 20*a^2*b^5*d + 44*a^3*b^4*e - 77*a^4*b^3*f)*x^6 + 2*(5*a^2*b^5*c - 20*a^3*b^4*d + 44*a^4*b^3*e - 77*a^5*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)) + 20*((5*b^5*c - 20*a*b^4*d + 44*a^2*b^3*e - 77
*a^3*b^2*f)*x^6 + 5*a^2*b^3*c - 20*a^3*b^2*d + 44*a^4*b*e - 77*a^5*f + 2*(5*a*b^4*c - 20*a^2*b^3*d + 44*a^3*b^
2*e - 77*a^4*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)) - 40*((5*b^5*c - 20*a*b^
4*d + 44*a^2*b^3*e - 77*a^3*b^2*f)*x^6 + 5*a^2*b^3*c - 20*a^3*b^2*d + 44*a^4*b*e - 77*a^5*f + 2*(5*a*b^4*c - 2
0*a^2*b^3*d + 44*a^3*b^2*e - 77*a^4*b*f)*x^3)*(a*b^2)^(2/3)*log(b*x + (a*b^2)^(1/3)))/(a*b^9*x^6 + 2*a^2*b^8*x
^3 + a^3*b^7), 1/1080*(135*a*b^6*f*x^14 + 54*(4*a*b^6*e - 7*a^2*b^5*f)*x^11 + 27*(20*a*b^6*d - 44*a^2*b^5*e +
77*a^3*b^4*f)*x^8 - 96*(5*a*b^6*c - 20*a^2*b^5*d + 44*a^3*b^4*e - 77*a^4*b^3*f)*x^5 - 60*(5*a^2*b^5*c - 20*a^3
*b^4*d + 44*a^4*b^3*e - 77*a^5*b^2*f)*x^2 - 120*sqrt(1/3)*(5*a^3*b^4*c - 20*a^4*b^3*d + 44*a^5*b^2*e - 77*a^6*
b*f + (5*a*b^6*c - 20*a^2*b^5*d + 44*a^3*b^4*e - 77*a^4*b^3*f)*x^6 + 2*(5*a^2*b^5*c - 20*a^3*b^4*d + 44*a^4*b^
3*e - 77*a^5*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) + 20*((5*b^5*c - 20*a*b^4*d + 44*a^2*b^3*e - 77*a^3*b^2*f)*x^6 + 5*a^2*b^3*c - 20*a^3*b^2*d + 44*a^4*b*e -
77*a^5*f + 2*(5*a*b^4*c - 20*a^2*b^3*d + 44*a^3*b^2*e - 77*a^4*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)) - 40*((5*b^5*c - 20*a*b^4*d + 44*a^2*b^3*e - 77*a^3*b^2*f)*x^6 + 5*a^2*b^3*c - 20*a
^3*b^2*d + 44*a^4*b*e - 77*a^5*f + 2*(5*a*b^4*c - 20*a^2*b^3*d + 44*a^3*b^2*e - 77*a^4*b*f)*x^3)*(a*b^2)^(2/3)
*log(b*x + (a*b^2)^(1/3)))/(a*b^9*x^6 + 2*a^2*b^8*x^3 + a^3*b^7)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.09243, size = 601, normalized size = 1.74 \begin{align*} -\frac{{\left (5 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 20 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 77 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 44 \, 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 )}{27 \, a b^{5}} - \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 20 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 77 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 44 \, \left (-a b^{2}\right )^{\frac{2}{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 )}{27 \, a b^{7}} - \frac{8 \, b^{4} c x^{5} - 14 \, a b^{3} d x^{5} - 26 \, a^{3} b f x^{5} + 20 \, a^{2} b^{2} x^{5} e + 5 \, a b^{3} c x^{2} - 11 \, a^{2} b^{2} d x^{2} - 23 \, a^{4} f x^{2} + 17 \, a^{3} b x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} b^{5}} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 20 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 77 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 44 \, \left (-a b^{2}\right )^{\frac{2}{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 )}{54 \, a b^{7}} + \frac{5 \, b^{21} f x^{8} - 24 \, a b^{20} f x^{5} + 8 \, b^{21} x^{5} e + 20 \, b^{21} d x^{2} + 120 \, a^{2} b^{19} f x^{2} - 60 \, a b^{20} x^{2} e}{40 \, b^{24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

-1/27*(5*b^3*c*(-a/b)^(1/3) - 20*a*b^2*d*(-a/b)^(1/3) - 77*a^3*f*(-a/b)^(1/3) + 44*a^2*b*(-a/b)^(1/3)*e)*(-a/b
)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^5) - 1/27*sqrt(3)*(5*(-a*b^2)^(2/3)*b^3*c - 20*(-a*b^2)^(2/3)*a*b^2*d
- 77*(-a*b^2)^(2/3)*a^3*f + 44*(-a*b^2)^(2/3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(
a*b^7) - 1/18*(8*b^4*c*x^5 - 14*a*b^3*d*x^5 - 26*a^3*b*f*x^5 + 20*a^2*b^2*x^5*e + 5*a*b^3*c*x^2 - 11*a^2*b^2*d
*x^2 - 23*a^4*f*x^2 + 17*a^3*b*x^2*e)/((b*x^3 + a)^2*b^5) + 1/54*(5*(-a*b^2)^(2/3)*b^3*c - 20*(-a*b^2)^(2/3)*a
*b^2*d - 77*(-a*b^2)^(2/3)*a^3*f + 44*(-a*b^2)^(2/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^7)
+ 1/40*(5*b^21*f*x^8 - 24*a*b^20*f*x^5 + 8*b^21*x^5*e + 20*b^21*d*x^2 + 120*a^2*b^19*f*x^2 - 60*a*b^20*x^2*e)
/b^24