3.291 \(\int \frac{x^4 (c+d x^3+e x^6+f x^9)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=316 \[ \frac{x^2 \left (7 a^2 b e-10 a^3 f-4 a b^2 d+b^3 c\right )}{9 a b^4 \left (a+b x^3\right )}-\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-20 a^2 b e+44 a^3 f+5 a b^2 d+b^3 c\right )}{54 a^{4/3} b^{14/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^2 b e+44 a^3 f+5 a b^2 d+b^3 c\right )}{27 a^{4/3} b^{14/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-20 a^2 b e+44 a^3 f+5 a b^2 d+b^3 c\right )}{9 \sqrt{3} a^{4/3} b^{14/3}}+\frac{x^2 (b e-3 a f)}{2 b^4}+\frac{f x^5}{5 b^3} \]

[Out]

((b*e - 3*a*f)*x^2)/(2*b^4) + (f*x^5)/(5*b^3) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*b^4*(a + b*x^3)^2
) + ((b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*x^2)/(9*a*b^4*(a + b*x^3)) - ((b^3*c + 5*a*b^2*d - 20*a^2*b*e
+ 44*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(4/3)*b^(14/3)) - ((b^3*c + 5*a*b^
2*d - 20*a^2*b*e + 44*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(4/3)*b^(14/3)) + ((b^3*c + 5*a*b^2*d - 20*a^2*b*
e + 44*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(4/3)*b^(14/3))

________________________________________________________________________________________

Rubi [A]  time = 0.504646, antiderivative size = 316, 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, 1594, 1488, 292, 31, 634, 617, 204, 628} \[ \frac{x^2 \left (7 a^2 b e-10 a^3 f-4 a b^2 d+b^3 c\right )}{9 a b^4 \left (a+b x^3\right )}-\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-20 a^2 b e+44 a^3 f+5 a b^2 d+b^3 c\right )}{54 a^{4/3} b^{14/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^2 b e+44 a^3 f+5 a b^2 d+b^3 c\right )}{27 a^{4/3} b^{14/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-20 a^2 b e+44 a^3 f+5 a b^2 d+b^3 c\right )}{9 \sqrt{3} a^{4/3} b^{14/3}}+\frac{x^2 (b e-3 a f)}{2 b^4}+\frac{f x^5}{5 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

((b*e - 3*a*f)*x^2)/(2*b^4) + (f*x^5)/(5*b^3) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*b^4*(a + b*x^3)^2
) + ((b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*x^2)/(9*a*b^4*(a + b*x^3)) - ((b^3*c + 5*a*b^2*d - 20*a^2*b*e
+ 44*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(4/3)*b^(14/3)) - ((b^3*c + 5*a*b^
2*d - 20*a^2*b*e + 44*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(4/3)*b^(14/3)) + ((b^3*c + 5*a*b^2*d - 20*a^2*b*
e + 44*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(4/3)*b^(14/3))

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 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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^4 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^4 \left (a+b x^3\right )^2}-\frac{\int \frac{-2 a b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x-6 a b^2 \left (b^2 d-a b e+a^2 f\right ) x^4-6 a b^3 (b e-a f) x^7-6 a b^4 f x^{10}}{\left (a+b x^3\right )^2} \, dx}{6 a b^5}\\ &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^4 \left (a+b x^3\right )^2}-\frac{\int \frac{x \left (-2 a b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-6 a b^2 \left (b^2 d-a b e+a^2 f\right ) x^3-6 a b^3 (b e-a f) x^6-6 a b^4 f x^9\right )}{\left (a+b x^3\right )^2} \, dx}{6 a b^5}\\ &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^4 \left (a+b x^3\right )^2}+\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) x^2}{9 a b^4 \left (a+b x^3\right )}+\frac{\int \frac{2 a b^5 \left (b^3 c+5 a b^2 d-11 a^2 b e+17 a^3 f\right ) x+18 a^2 b^6 (b e-2 a f) x^4+18 a^2 b^7 f x^7}{a+b x^3} \, dx}{18 a^2 b^9}\\ &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^4 \left (a+b x^3\right )^2}+\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) x^2}{9 a b^4 \left (a+b x^3\right )}+\frac{\int \frac{x \left (2 a b^5 \left (b^3 c+5 a b^2 d-11 a^2 b e+17 a^3 f\right )+18 a^2 b^6 (b e-2 a f) x^3+18 a^2 b^7 f x^6\right )}{a+b x^3} \, dx}{18 a^2 b^9}\\ &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^4 \left (a+b x^3\right )^2}+\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) x^2}{9 a b^4 \left (a+b x^3\right )}+\frac{\int \left (18 a^2 b^5 (b e-3 a f) x+18 a^2 b^6 f x^4+\frac{2 \left (a b^8 c+5 a^2 b^7 d-20 a^3 b^6 e+44 a^4 b^5 f\right ) x}{a+b x^3}\right ) \, dx}{18 a^2 b^9}\\ &=\frac{(b e-3 a f) x^2}{2 b^4}+\frac{f x^5}{5 b^3}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^4 \left (a+b x^3\right )^2}+\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) x^2}{9 a b^4 \left (a+b x^3\right )}+\frac{\left (b^3 c+5 a b^2 d-20 a^2 b e+44 a^3 f\right ) \int \frac{x}{a+b x^3} \, dx}{9 a b^4}\\ &=\frac{(b e-3 a f) x^2}{2 b^4}+\frac{f x^5}{5 b^3}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^4 \left (a+b x^3\right )^2}+\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) x^2}{9 a b^4 \left (a+b x^3\right )}-\frac{\left (b^3 c+5 a b^2 d-20 a^2 b e+44 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{4/3} b^{13/3}}+\frac{\left (b^3 c+5 a b^2 d-20 a^2 b e+44 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 a^{4/3} b^{13/3}}\\ &=\frac{(b e-3 a f) x^2}{2 b^4}+\frac{f x^5}{5 b^3}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^4 \left (a+b x^3\right )^2}+\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) x^2}{9 a b^4 \left (a+b x^3\right )}-\frac{\left (b^3 c+5 a b^2 d-20 a^2 b e+44 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{14/3}}+\frac{\left (b^3 c+5 a b^2 d-20 a^2 b e+44 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^{4/3} b^{14/3}}+\frac{\left (b^3 c+5 a b^2 d-20 a^2 b e+44 a^3 f\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a b^{13/3}}\\ &=\frac{(b e-3 a f) x^2}{2 b^4}+\frac{f x^5}{5 b^3}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^4 \left (a+b x^3\right )^2}+\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) x^2}{9 a b^4 \left (a+b x^3\right )}-\frac{\left (b^3 c+5 a b^2 d-20 a^2 b e+44 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{14/3}}+\frac{\left (b^3 c+5 a b^2 d-20 a^2 b e+44 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{14/3}}+\frac{\left (b^3 c+5 a b^2 d-20 a^2 b e+44 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^{4/3} b^{14/3}}\\ &=\frac{(b e-3 a f) x^2}{2 b^4}+\frac{f x^5}{5 b^3}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^4 \left (a+b x^3\right )^2}+\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) x^2}{9 a b^4 \left (a+b x^3\right )}-\frac{\left (b^3 c+5 a b^2 d-20 a^2 b e+44 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^{4/3} b^{14/3}}-\frac{\left (b^3 c+5 a b^2 d-20 a^2 b e+44 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{14/3}}+\frac{\left (b^3 c+5 a b^2 d-20 a^2 b e+44 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{14/3}}\\ \end{align*}

Mathematica [A]  time = 0.205988, size = 300, normalized size = 0.95 \[ \frac{\frac{30 b^{2/3} x^2 \left (7 a^2 b e-10 a^3 f-4 a b^2 d+b^3 c\right )}{a \left (a+b x^3\right )}-\frac{45 b^{2/3} x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\left (a+b x^3\right )^2}+\frac{5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-20 a^2 b e+44 a^3 f+5 a b^2 d+b^3 c\right )}{a^{4/3}}-\frac{10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^2 b e+44 a^3 f+5 a b^2 d+b^3 c\right )}{a^{4/3}}-\frac{10 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-20 a^2 b e+44 a^3 f+5 a b^2 d+b^3 c\right )}{a^{4/3}}+135 b^{2/3} x^2 (b e-3 a f)+54 b^{5/3} f x^5}{270 b^{14/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

(135*b^(2/3)*(b*e - 3*a*f)*x^2 + 54*b^(5/3)*f*x^5 - (45*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a +
b*x^3)^2 + (30*b^(2/3)*(b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*x^2)/(a*(a + b*x^3)) - (10*Sqrt[3]*(b^3*c +
5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(4/3) - (10*(b^3*c + 5*a*b^2
*d - 20*a^2*b*e + 44*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + (5*(b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)
*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(4/3))/(270*b^(14/3))

________________________________________________________________________________________

Maple [B]  time = 0.011, size = 574, 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^4*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)

[Out]

1/5*f*x^5/b^3-3/2/b^4*x^2*a*f+1/2/b^3*x^2*e-10/9/b^3/(b*x^3+a)^2*a^2*x^5*f+7/9/b^2/(b*x^3+a)^2*a*x^5*e-4/9/b/(
b*x^3+a)^2*x^5*d+1/9/(b*x^3+a)^2/a*x^5*c-17/18/b^4/(b*x^3+a)^2*x^2*a^3*f+11/18/b^3/(b*x^3+a)^2*x^2*a^2*e-5/18/
b^2/(b*x^3+a)^2*x^2*a*d-1/18/b/(b*x^3+a)^2*x^2*c-44/27/b^5*a^2/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*f+20/27/b^4*a
/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*e-5/27/b^3/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*d-1/27/b^2/a/(1/b*a)^(1/3)*ln(
x+(1/b*a)^(1/3))*c+22/27/b^5*a^2/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*f-10/27/b^4*a/(1/b*a)^(1/
3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*e+5/54/b^3/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*d+1/54
/b^2/a/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c+44/27/b^5*a^2*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^
(1/2)*(2/(1/b*a)^(1/3)*x-1))*f-20/27/b^4*a*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*e+5
/27/b^3*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*d+1/27/b^2/a*3^(1/2)/(1/b*a)^(1/3)*arc
tan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c

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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^4*(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 = 1.5351, size = 2723, normalized size = 8.62 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

[1/270*(54*a^2*b^5*f*x^11 + 27*(5*a^2*b^5*e - 11*a^3*b^4*f)*x^8 + 6*(5*a*b^6*c - 20*a^2*b^5*d + 80*a^3*b^4*e -
176*a^4*b^3*f)*x^5 - 15*(a^2*b^5*c + 5*a^3*b^4*d - 20*a^4*b^3*e + 44*a^5*b^2*f)*x^2 + 15*sqrt(1/3)*(a^3*b^4*c
+ 5*a^4*b^3*d - 20*a^5*b^2*e + 44*a^6*b*f + (a*b^6*c + 5*a^2*b^5*d - 20*a^3*b^4*e + 44*a^4*b^3*f)*x^6 + 2*(a^
2*b^5*c + 5*a^3*b^4*d - 20*a^4*b^3*e + 44*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)) + 5*((b^5*c + 5*a*b^4*d - 20*a^2*b^3*e + 44*a^3*b^2*f)*x^6 + a^2*b^3*c + 5*a^3*b^2*d - 20*a^4*b*e + 44*a^5
*f + 2*(a*b^4*c + 5*a^2*b^3*d - 20*a^3*b^2*e + 44*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)) - 10*((b^5*c + 5*a*b^4*d - 20*a^2*b^3*e + 44*a^3*b^2*f)*x^6 + a^2*b^3*c + 5*a^3*b^2*d - 20
*a^4*b*e + 44*a^5*f + 2*(a*b^4*c + 5*a^2*b^3*d - 20*a^3*b^2*e + 44*a^4*b*f)*x^3)*(-a*b^2)^(2/3)*log(b*x - (-a*
b^2)^(1/3)))/(a^2*b^8*x^6 + 2*a^3*b^7*x^3 + a^4*b^6), 1/270*(54*a^2*b^5*f*x^11 + 27*(5*a^2*b^5*e - 11*a^3*b^4*
f)*x^8 + 6*(5*a*b^6*c - 20*a^2*b^5*d + 80*a^3*b^4*e - 176*a^4*b^3*f)*x^5 - 15*(a^2*b^5*c + 5*a^3*b^4*d - 20*a^
4*b^3*e + 44*a^5*b^2*f)*x^2 + 30*sqrt(1/3)*(a^3*b^4*c + 5*a^4*b^3*d - 20*a^5*b^2*e + 44*a^6*b*f + (a*b^6*c + 5
*a^2*b^5*d - 20*a^3*b^4*e + 44*a^4*b^3*f)*x^6 + 2*(a^2*b^5*c + 5*a^3*b^4*d - 20*a^4*b^3*e + 44*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) + 5*((b^5*c + 5*
a*b^4*d - 20*a^2*b^3*e + 44*a^3*b^2*f)*x^6 + a^2*b^3*c + 5*a^3*b^2*d - 20*a^4*b*e + 44*a^5*f + 2*(a*b^4*c + 5*
a^2*b^3*d - 20*a^3*b^2*e + 44*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))
- 10*((b^5*c + 5*a*b^4*d - 20*a^2*b^3*e + 44*a^3*b^2*f)*x^6 + a^2*b^3*c + 5*a^3*b^2*d - 20*a^4*b*e + 44*a^5*f
+ 2*(a*b^4*c + 5*a^2*b^3*d - 20*a^3*b^2*e + 44*a^4*b*f)*x^3)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^2*b^
8*x^6 + 2*a^3*b^7*x^3 + a^4*b^6)]

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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**4*(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 = 1.09331, size = 558, normalized size = 1.77 \begin{align*} -\frac{{\left (b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 44 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 20 \, 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^{2} b^{4}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 44 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f - 20 \, \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^{2} b^{6}} + \frac{2 \, b^{4} c x^{5} - 8 \, a b^{3} d x^{5} - 20 \, a^{3} b f x^{5} + 14 \, a^{2} b^{2} x^{5} e - a b^{3} c x^{2} - 5 \, a^{2} b^{2} d x^{2} - 17 \, a^{4} f x^{2} + 11 \, a^{3} b x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a b^{4}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 44 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f - 20 \, \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^{2} b^{6}} + \frac{2 \, b^{12} f x^{5} - 15 \, a b^{11} f x^{2} + 5 \, b^{12} x^{2} e}{10 \, b^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

-1/27*(b^3*c*(-a/b)^(1/3) + 5*a*b^2*d*(-a/b)^(1/3) + 44*a^3*f*(-a/b)^(1/3) - 20*a^2*b*(-a/b)^(1/3)*e)*(-a/b)^(
1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^4) - 1/27*sqrt(3)*((-a*b^2)^(2/3)*b^3*c + 5*(-a*b^2)^(2/3)*a*b^2*d + 44
*(-a*b^2)^(2/3)*a^3*f - 20*(-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^2*
b^6) + 1/18*(2*b^4*c*x^5 - 8*a*b^3*d*x^5 - 20*a^3*b*f*x^5 + 14*a^2*b^2*x^5*e - a*b^3*c*x^2 - 5*a^2*b^2*d*x^2 -
17*a^4*f*x^2 + 11*a^3*b*x^2*e)/((b*x^3 + a)^2*a*b^4) + 1/54*((-a*b^2)^(2/3)*b^3*c + 5*(-a*b^2)^(2/3)*a*b^2*d
+ 44*(-a*b^2)^(2/3)*a^3*f - 20*(-a*b^2)^(2/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b^6) + 1/
10*(2*b^12*f*x^5 - 15*a*b^11*f*x^2 + 5*b^12*x^2*e)/b^15