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

[Out]

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

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Rubi [A]  time = 0.50, antiderivative size = 316, normalized size of antiderivative = 1.00, 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 31

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

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*}

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Mathematica [A]  time = 0.29, size = 300, normalized size = 0.95 \[ \frac {\frac {30 b^{2/3} x^2 \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{a \left (a+b x^3\right )}-\frac {45 b^{2/3} x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\left (a+b x^3\right )^2}-\frac {10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right )}{a^{4/3}}-\frac {10 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right )}{a^{4/3}}+\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right )}{a^{4/3}}+135 b^{2/3} x^2 (b e-3 a f)+54 b^{5/3} f x^5}{270 b^{14/3}} \]

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))

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fricas [B]  time = 0.60, size = 1224, normalized size = 3.87 \[ \text {result too large to display} \]

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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giac [A]  time = 0.20, size = 365, normalized size = 1.16 \[ \frac {\sqrt {3} {\left (b^{3} c + 5 \, a b^{2} d + 44 \, a^{3} f - 20 \, a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{4}} - \frac {{\left (b^{3} c + 5 \, a b^{2} d + 44 \, a^{3} f - 20 \, a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{4}} - \frac {{\left (b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 44 \, a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 20 \, a^{2} b \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{2} b^{4}} + \frac {2 \, b^{4} c x^{5} - 8 \, a b^{3} d x^{5} - 20 \, a^{3} b f x^{5} + 14 \, a^{2} b^{2} x^{5} e - a b^{3} c x^{2} - 5 \, a^{2} b^{2} d x^{2} - 17 \, a^{4} f x^{2} + 11 \, a^{3} b x^{2} e}{18 \, {\left (b x^{3} + a\right )}^{2} a b^{4}} + \frac {2 \, b^{12} f x^{5} - 15 \, a b^{11} f x^{2} + 5 \, b^{12} x^{2} e}{10 \, b^{15}} \]

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

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maple [B]  time = 0.06, size = 574, normalized size = 1.82 \[ -\frac {10 a^{2} f \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} b^{3}}+\frac {7 a e \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} b^{2}}+\frac {c \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a}-\frac {4 d \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} b}+\frac {f \,x^{5}}{5 b^{3}}-\frac {17 a^{3} f \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b^{4}}+\frac {11 a^{2} e \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b^{3}}-\frac {5 a d \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b^{2}}-\frac {c \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b}-\frac {3 a f \,x^{2}}{2 b^{4}}+\frac {e \,x^{2}}{2 b^{3}}+\frac {44 \sqrt {3}\, a^{2} 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 {1}{3}} b^{5}}-\frac {44 a^{2} f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}}+\frac {22 a^{2} f \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 {1}{3}} b^{5}}-\frac {20 \sqrt {3}\, a 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 {1}{3}} b^{4}}+\frac {20 a e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}-\frac {10 a e \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 {1}{3}} b^{4}}+\frac {\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 {1}{3}} a \,b^{2}}-\frac {c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{2}}+\frac {c \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 {1}{3}} a \,b^{2}}+\frac {5 \sqrt {3}\, 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 {1}{3}} b^{3}}-\frac {5 d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}+\frac {5 d \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 {1}{3}} b^{3}} \]

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/b^3*f*x^5-3/2/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/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*f+20/27/b^4*a/(a/
b)^(1/3)*ln(x+(a/b)^(1/3))*e-5/27/b^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*d-1/27/b^2/a/(a/b)^(1/3)*ln(x+(a/b)^(1/3))
*c+22/27/b^5*a^2/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*f-10/27/b^4*a/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+
(a/b)^(2/3))*e+5/54/b^3/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*d+1/54/b^2/a/(a/b)^(1/3)*ln(x^2-(a/b)^(1
/3)*x+(a/b)^(2/3))*c+44/27/b^5*a^2*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f-20/27/b^4*a*3
^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e+5/27/b^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(
2/(a/b)^(1/3)*x-1))*d+1/27/b^2/a*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c

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maxima [A]  time = 3.10, size = 311, normalized size = 0.98 \[ \frac {2 \, {\left (b^{4} c - 4 \, a b^{3} d + 7 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{5} - {\left (a b^{3} c + 5 \, a^{2} b^{2} d - 11 \, a^{3} b e + 17 \, a^{4} f\right )} x^{2}}{18 \, {\left (a b^{6} x^{6} + 2 \, a^{2} b^{5} x^{3} + a^{3} b^{4}\right )}} + \frac {2 \, b f x^{5} + 5 \, {\left (b e - 3 \, a f\right )} x^{2}}{10 \, b^{4}} + \frac {\sqrt {3} {\left (b^{3} c + 5 \, a b^{2} d - 20 \, a^{2} b e + 44 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (b^{3} c + 5 \, a b^{2} d - 20 \, a^{2} b e + 44 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (b^{3} c + 5 \, a b^{2} d - 20 \, a^{2} b e + 44 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

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]

1/18*(2*(b^4*c - 4*a*b^3*d + 7*a^2*b^2*e - 10*a^3*b*f)*x^5 - (a*b^3*c + 5*a^2*b^2*d - 11*a^3*b*e + 17*a^4*f)*x
^2)/(a*b^6*x^6 + 2*a^2*b^5*x^3 + a^3*b^4) + 1/10*(2*b*f*x^5 + 5*(b*e - 3*a*f)*x^2)/b^4 + 1/27*sqrt(3)*(b^3*c +
5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^5*(a/b)^(1/3)) +
1/54*(b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b^5*(a/b)^(1/3)) -
1/27*(b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*log(x + (a/b)^(1/3))/(a*b^5*(a/b)^(1/3))

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mupad [B]  time = 5.27, size = 295, normalized size = 0.93 \[ x^2\,\left (\frac {e}{2\,b^3}-\frac {3\,a\,f}{2\,b^4}\right )-\frac {x^2\,\left (\frac {17\,f\,a^3}{18}-\frac {11\,e\,a^2\,b}{18}+\frac {5\,d\,a\,b^2}{18}+\frac {c\,b^3}{18}\right )-\frac {x^5\,\left (-10\,f\,a^3\,b+7\,e\,a^2\,b^2-4\,d\,a\,b^3+c\,b^4\right )}{9\,a}}{a^2\,b^4+2\,a\,b^5\,x^3+b^6\,x^6}+\frac {f\,x^5}{5\,b^3}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (44\,f\,a^3-20\,e\,a^2\,b+5\,d\,a\,b^2+c\,b^3\right )}{27\,a^{4/3}\,b^{14/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (44\,f\,a^3-20\,e\,a^2\,b+5\,d\,a\,b^2+c\,b^3\right )}{27\,a^{4/3}\,b^{14/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (44\,f\,a^3-20\,e\,a^2\,b+5\,d\,a\,b^2+c\,b^3\right )}{27\,a^{4/3}\,b^{14/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*(e/(2*b^3) - (3*a*f)/(2*b^4)) - (x^2*((b^3*c)/18 + (17*a^3*f)/18 + (5*a*b^2*d)/18 - (11*a^2*b*e)/18) - (x^
5*(b^4*c + 7*a^2*b^2*e - 4*a*b^3*d - 10*a^3*b*f))/(9*a))/(a^2*b^4 + b^6*x^6 + 2*a*b^5*x^3) + (f*x^5)/(5*b^3) -
(log(b^(1/3)*x + a^(1/3))*(b^3*c + 44*a^3*f + 5*a*b^2*d - 20*a^2*b*e))/(27*a^(4/3)*b^(14/3)) + (log(3^(1/2)*a
^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(b^3*c + 44*a^3*f + 5*a*b^2*d - 20*a^2*b*e))/(27*a^(
4/3)*b^(14/3)) - (log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(b^3*c + 44*a^3*f + 5
*a*b^2*d - 20*a^2*b*e))/(27*a^(4/3)*b^(14/3))

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

[Out]

Timed out

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