3.261 \(\int \frac{x^7 (c+d x^3+e x^6+f x^9)}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=335 \[ \frac{x^2 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{2 b^5}+\frac{a x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (11 a^2 b e-14 a^3 f-8 a b^2 d+5 b^3 c\right )}{18 b^{17/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (11 a^2 b e-14 a^3 f-8 a b^2 d+5 b^3 c\right )}{9 b^{17/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (11 a^2 b e-14 a^3 f-8 a b^2 d+5 b^3 c\right )}{3 \sqrt{3} b^{17/3}}+\frac{x^5 \left (3 a^2 f-2 a b e+b^2 d\right )}{5 b^4}+\frac{x^8 (b e-2 a f)}{8 b^3}+\frac{f x^{11}}{11 b^2} \]

[Out]

((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^2)/(2*b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^5)/(5*b^4) + ((b*e -
2*a*f)*x^8)/(8*b^3) + (f*x^11)/(11*b^2) + (a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*b^5*(a + b*x^3)) + (a
^(2/3)*(5*b^3*c - 8*a*b^2*d + 11*a^2*b*e - 14*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqr
t[3]*b^(17/3)) + (a^(2/3)*(5*b^3*c - 8*a*b^2*d + 11*a^2*b*e - 14*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(17/3))
- (a^(2/3)*(5*b^3*c - 8*a*b^2*d + 11*a^2*b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*
b^(17/3))

________________________________________________________________________________________

Rubi [A]  time = 0.705022, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ \frac{x^2 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{2 b^5}+\frac{a x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (11 a^2 b e-14 a^3 f-8 a b^2 d+5 b^3 c\right )}{18 b^{17/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (11 a^2 b e-14 a^3 f-8 a b^2 d+5 b^3 c\right )}{9 b^{17/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (11 a^2 b e-14 a^3 f-8 a b^2 d+5 b^3 c\right )}{3 \sqrt{3} b^{17/3}}+\frac{x^5 \left (3 a^2 f-2 a b e+b^2 d\right )}{5 b^4}+\frac{x^8 (b e-2 a f)}{8 b^3}+\frac{f x^{11}}{11 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^7*(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^2)/(2*b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^5)/(5*b^4) + ((b*e -
2*a*f)*x^8)/(8*b^3) + (f*x^11)/(11*b^2) + (a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*b^5*(a + b*x^3)) + (a
^(2/3)*(5*b^3*c - 8*a*b^2*d + 11*a^2*b*e - 14*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqr
t[3]*b^(17/3)) + (a^(2/3)*(5*b^3*c - 8*a*b^2*d + 11*a^2*b*e - 14*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(17/3))
- (a^(2/3)*(5*b^3*c - 8*a*b^2*d + 11*a^2*b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*
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 )^2} \, dx &=\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac{\int \frac{2 a^2 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x-3 a b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4-3 a b^3 \left (b^2 d-a b e+a^2 f\right ) x^7-3 a b^4 (b e-a f) x^{10}-3 a b^5 f x^{13}}{a+b x^3} \, dx}{3 a b^6}\\ &=\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\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 )-3 a b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-3 a b^3 \left (b^2 d-a b e+a^2 f\right ) x^6-3 a b^4 (b e-a f) x^9-3 a b^5 f x^{12}\right )}{a+b x^3} \, dx}{3 a b^6}\\ &=\frac{f x^{11}}{11 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac{\int \frac{x \left (22 a^2 b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-33 a b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-33 a b^4 \left (b^2 d-a b e+a^2 f\right ) x^6-33 a b^5 (b e-2 a f) x^9\right )}{a+b x^3} \, dx}{33 a b^7}\\ &=\frac{(b e-2 a f) x^8}{8 b^3}+\frac{f x^{11}}{11 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac{\int \frac{x \left (176 a^2 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-264 a b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-264 a b^5 \left (b^2 d-2 a b e+3 a^2 f\right ) x^6\right )}{a+b x^3} \, dx}{264 a b^8}\\ &=\frac{(b e-2 a f) x^8}{8 b^3}+\frac{f x^{11}}{11 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac{\int \left (-264 a b^3 \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x-264 a b^4 \left (b^2 d-2 a b e+3 a^2 f\right ) x^4-\frac{88 \left (-5 a^2 b^6 c+8 a^3 b^5 d-11 a^4 b^4 e+14 a^5 b^3 f\right ) x}{a+b x^3}\right ) \, dx}{264 a b^8}\\ &=\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac{(b e-2 a f) x^8}{8 b^3}+\frac{f x^{11}}{11 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac{\left (a \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )\right ) \int \frac{x}{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^2}{2 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac{(b e-2 a f) x^8}{8 b^3}+\frac{f x^{11}}{11 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac{\left (a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 b^{16/3}}-\frac{\left (a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )\right ) \int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 b^{16/3}}\\ &=\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac{(b e-2 a f) x^8}{8 b^3}+\frac{f x^{11}}{11 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac{a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{17/3}}-\frac{\left (a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 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^{17/3}}-\frac{\left (a \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 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^{16/3}}\\ &=\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac{(b e-2 a f) x^8}{8 b^3}+\frac{f x^{11}}{11 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac{a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{17/3}}-\frac{a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{17/3}}-\frac{\left (a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 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^{17/3}}\\ &=\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac{(b e-2 a f) x^8}{8 b^3}+\frac{f x^{11}}{11 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac{a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 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^{17/3}}+\frac{a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{17/3}}-\frac{a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{17/3}}\\ \end{align*}

Mathematica [A]  time = 0.165613, size = 319, normalized size = 0.95 \[ \frac{1980 b^{2/3} x^2 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )+\frac{1320 a b^{2/3} x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a+b x^3}+220 a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-11 a^2 b e+14 a^3 f+8 a b^2 d-5 b^3 c\right )-440 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-11 a^2 b e+14 a^3 f+8 a b^2 d-5 b^3 c\right )-440 \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-11 a^2 b e+14 a^3 f+8 a b^2 d-5 b^3 c\right )+792 b^{5/3} x^5 \left (3 a^2 f-2 a b e+b^2 d\right )+495 b^{8/3} x^8 (b e-2 a f)+360 b^{11/3} f x^{11}}{3960 b^{17/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^7*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

(1980*b^(2/3)*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^2 + 792*b^(5/3)*(b^2*d - 2*a*b*e + 3*a^2*f)*x^5 + 49
5*b^(8/3)*(b*e - 2*a*f)*x^8 + 360*b^(11/3)*f*x^11 + (1320*a*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(
a + b*x^3) - 440*Sqrt[3]*a^(2/3)*(-5*b^3*c + 8*a*b^2*d - 11*a^2*b*e + 14*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1
/3))/Sqrt[3]] - 440*a^(2/3)*(-5*b^3*c + 8*a*b^2*d - 11*a^2*b*e + 14*a^3*f)*Log[a^(1/3) + b^(1/3)*x] + 220*a^(2
/3)*(-5*b^3*c + 8*a*b^2*d - 11*a^2*b*e + 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(3960*b^(17
/3))

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 584, normalized size = 1.7 \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)^2,x)

[Out]

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

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.29785, size = 1057, normalized size = 3.16 \begin{align*} \frac{360 \, b^{4} f x^{14} + 45 \,{\left (11 \, b^{4} e - 14 \, a b^{3} f\right )} x^{11} + 99 \,{\left (8 \, b^{4} d - 11 \, a b^{3} e + 14 \, a^{2} b^{2} f\right )} x^{8} + 396 \,{\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{5} + 660 \,{\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f\right )} x^{2} - 440 \, \sqrt{3}{\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f +{\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{3}\right )} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} + \sqrt{3} a}{3 \, a}\right ) + 220 \,{\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f +{\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{3}\right )} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} - a \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - 440 \,{\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f +{\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{3}\right )} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right )}{3960 \,{\left (b^{6} x^{3} + a b^{5}\right )}} \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)^2,x, algorithm="fricas")

[Out]

1/3960*(360*b^4*f*x^14 + 45*(11*b^4*e - 14*a*b^3*f)*x^11 + 99*(8*b^4*d - 11*a*b^3*e + 14*a^2*b^2*f)*x^8 + 396*
(5*b^4*c - 8*a*b^3*d + 11*a^2*b^2*e - 14*a^3*b*f)*x^5 + 660*(5*a*b^3*c - 8*a^2*b^2*d + 11*a^3*b*e - 14*a^4*f)*
x^2 - 440*sqrt(3)*(5*a*b^3*c - 8*a^2*b^2*d + 11*a^3*b*e - 14*a^4*f + (5*b^4*c - 8*a*b^3*d + 11*a^2*b^2*e - 14*
a^3*b*f)*x^3)*(-a^2/b^2)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(-a^2/b^2)^(1/3) + sqrt(3)*a)/a) + 220*(5*a*b^3*c - 8
*a^2*b^2*d + 11*a^3*b*e - 14*a^4*f + (5*b^4*c - 8*a*b^3*d + 11*a^2*b^2*e - 14*a^3*b*f)*x^3)*(-a^2/b^2)^(1/3)*l
og(a*x^2 - b*x*(-a^2/b^2)^(2/3) - a*(-a^2/b^2)^(1/3)) - 440*(5*a*b^3*c - 8*a^2*b^2*d + 11*a^3*b*e - 14*a^4*f +
(5*b^4*c - 8*a*b^3*d + 11*a^2*b^2*e - 14*a^3*b*f)*x^3)*(-a^2/b^2)^(1/3)*log(a*x + b*(-a^2/b^2)^(2/3)))/(b^6*x
^3 + a*b^5)

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Sympy [A]  time = 37.4541, size = 530, normalized size = 1.58 \begin{align*} - \frac{x^{2} \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^{17} + 2744 a^{11} f^{3} - 6468 a^{10} b e f^{2} + 4704 a^{9} b^{2} d f^{2} + 5082 a^{9} b^{2} e^{2} f - 2940 a^{8} b^{3} c f^{2} - 7392 a^{8} b^{3} d e f - 1331 a^{8} b^{3} e^{3} + 4620 a^{7} b^{4} c e f + 2688 a^{7} b^{4} d^{2} f + 2904 a^{7} b^{4} d e^{2} - 3360 a^{6} b^{5} c d f - 1815 a^{6} b^{5} c e^{2} - 2112 a^{6} b^{5} d^{2} e + 1050 a^{5} b^{6} c^{2} f + 2640 a^{5} b^{6} c d e + 512 a^{5} b^{6} d^{3} - 825 a^{4} b^{7} c^{2} e - 960 a^{4} b^{7} c d^{2} + 600 a^{3} b^{8} c^{2} d - 125 a^{2} b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} b^{11}}{196 a^{7} f^{2} - 308 a^{6} b e f + 224 a^{5} b^{2} d f + 121 a^{5} b^{2} e^{2} - 140 a^{4} b^{3} c f - 176 a^{4} b^{3} d e + 110 a^{3} b^{4} c e + 64 a^{3} b^{4} d^{2} - 80 a^{2} b^{5} c d + 25 a b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{11}}{11 b^{2}} - \frac{x^{8} \left (2 a f - b e\right )}{8 b^{3}} + \frac{x^{5} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{5 b^{4}} - \frac{x^{2} \left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c\right )}{2 b^{5}} \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)**2,x)

[Out]

-x**2*(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**17 + 2744*a
**11*f**3 - 6468*a**10*b*e*f**2 + 4704*a**9*b**2*d*f**2 + 5082*a**9*b**2*e**2*f - 2940*a**8*b**3*c*f**2 - 7392
*a**8*b**3*d*e*f - 1331*a**8*b**3*e**3 + 4620*a**7*b**4*c*e*f + 2688*a**7*b**4*d**2*f + 2904*a**7*b**4*d*e**2
- 3360*a**6*b**5*c*d*f - 1815*a**6*b**5*c*e**2 - 2112*a**6*b**5*d**2*e + 1050*a**5*b**6*c**2*f + 2640*a**5*b**
6*c*d*e + 512*a**5*b**6*d**3 - 825*a**4*b**7*c**2*e - 960*a**4*b**7*c*d**2 + 600*a**3*b**8*c**2*d - 125*a**2*b
**9*c**3, Lambda(_t, _t*log(81*_t**2*b**11/(196*a**7*f**2 - 308*a**6*b*e*f + 224*a**5*b**2*d*f + 121*a**5*b**2
*e**2 - 140*a**4*b**3*c*f - 176*a**4*b**3*d*e + 110*a**3*b**4*c*e + 64*a**3*b**4*d**2 - 80*a**2*b**5*c*d + 25*
a*b**6*c**2) + x))) + f*x**11/(11*b**2) - x**8*(2*a*f - b*e)/(8*b**3) + x**5*(3*a**2*f - 2*a*b*e + b**2*d)/(5*
b**4) - x**2*(4*a**3*f - 3*a**2*b*e + 2*a*b**2*d - b**3*c)/(2*b**5)

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Giac [A]  time = 1.08749, size = 597, normalized size = 1.78 \begin{align*} \frac{{\left (5 \, a b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 8 \, a^{2} b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 14 \, a^{4} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 11 \, a^{3} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{5}} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 11 \, \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 )}{9 \, b^{7}} + \frac{a b^{3} c x^{2} - a^{2} b^{2} d x^{2} - a^{4} f x^{2} + a^{3} b x^{2} e}{3 \,{\left (b x^{3} + a\right )} b^{5}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 11 \, \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 )}{18 \, b^{7}} + \frac{40 \, b^{20} f x^{11} - 110 \, a b^{19} f x^{8} + 55 \, b^{20} x^{8} e + 88 \, b^{20} d x^{5} + 264 \, a^{2} b^{18} f x^{5} - 176 \, a b^{19} x^{5} e + 220 \, b^{20} c x^{2} - 440 \, a b^{19} d x^{2} - 880 \, a^{3} b^{17} f x^{2} + 660 \, a^{2} b^{18} x^{2} e}{440 \, b^{22}} \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)^2,x, algorithm="giac")

[Out]

1/9*(5*a*b^3*c*(-a/b)^(1/3) - 8*a^2*b^2*d*(-a/b)^(1/3) - 14*a^4*f*(-a/b)^(1/3) + 11*a^3*b*(-a/b)^(1/3)*e)*(-a/
b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^5) + 1/9*sqrt(3)*(5*(-a*b^2)^(2/3)*b^3*c - 8*(-a*b^2)^(2/3)*a*b^2*d -
14*(-a*b^2)^(2/3)*a^3*f + 11*(-a*b^2)^(2/3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^
7 + 1/3*(a*b^3*c*x^2 - a^2*b^2*d*x^2 - a^4*f*x^2 + a^3*b*x^2*e)/((b*x^3 + a)*b^5) - 1/18*(5*(-a*b^2)^(2/3)*b^3
*c - 8*(-a*b^2)^(2/3)*a*b^2*d - 14*(-a*b^2)^(2/3)*a^3*f + 11*(-a*b^2)^(2/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3)
+ (-a/b)^(2/3))/b^7 + 1/440*(40*b^20*f*x^11 - 110*a*b^19*f*x^8 + 55*b^20*x^8*e + 88*b^20*d*x^5 + 264*a^2*b^18*
f*x^5 - 176*a*b^19*x^5*e + 220*b^20*c*x^2 - 440*a*b^19*d*x^2 - 880*a^3*b^17*f*x^2 + 660*a^2*b^18*x^2*e)/b^22