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

Optimal. Leaf size=301 \[ \frac {x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{9 a^2 b^3 \left (a+b x^3\right )}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{54 a^{7/3} b^{11/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{27 a^{7/3} b^{11/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{9 \sqrt {3} a^{7/3} b^{11/3}}+\frac {f x^2}{2 b^3} \]

[Out]

1/2*f*x^2/b^3+1/6*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x^2/a/b^3/(b*x^3+a)^2+1/9*(7*a^3*f-4*a^2*b*e+a*b^2*d+2*b^3*c)
*x^2/a^2/b^3/(b*x^3+a)-1/27*(-20*a^3*f+5*a^2*b*e+a*b^2*d+2*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(7/3)/b^(11/3)+1/54*
(-20*a^3*f+5*a^2*b*e+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^(7/3)/b^(11/3)-1/27*(-20*a^3
*f+5*a^2*b*e+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^(7/3)/b^(11/3)*3^(1/2)

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Rubi [A]  time = 0.37, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1828, 1594, 1482, 459, 292, 31, 634, 617, 204, 628} \[ \frac {x^2 \left (-4 a^2 b e+7 a^3 f+a b^2 d+2 b^3 c\right )}{9 a^2 b^3 \left (a+b x^3\right )}+\frac {x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^2 b e-20 a^3 f+a b^2 d+2 b^3 c\right )}{54 a^{7/3} b^{11/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^2 b e-20 a^3 f+a b^2 d+2 b^3 c\right )}{27 a^{7/3} b^{11/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (5 a^2 b e-20 a^3 f+a b^2 d+2 b^3 c\right )}{9 \sqrt {3} a^{7/3} b^{11/3}}+\frac {f x^2}{2 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f*x^2)/(2*b^3) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a*b^3*(a + b*x^3)^2) + ((2*b^3*c + a*b^2*d - 4*
a^2*b*e + 7*a^3*f)*x^2)/(9*a^2*b^3*(a + b*x^3)) - ((2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3*f)*ArcTan[(a^(1/3)
- 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(7/3)*b^(11/3)) - ((2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3*f)*
Log[a^(1/3) + b^(1/3)*x])/(27*a^(7/3)*b^(11/3)) + ((2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3*f)*Log[a^(2/3) - a^
(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(7/3)*b^(11/3))

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 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 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 1482

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_), x_Symbol] :>
Simp[((-d)^((m - Mod[m, n])/n - 1)*(c*d^2 - b*d*e + a*e^2)^p*x^(Mod[m, n] + 1)*(d + e*x^n)^(q + 1))/(n*e^(2*p
+ (m - Mod[m, n])/n)*(q + 1)), x] + Dist[1/(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1)), Int[x^Mod[m, n]*(d + e*x^n
)^(q + 1)*ExpandToSum[Together[(1*(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1)*x^(m - Mod[m, n])*(a + b*x^n + c*x^(2
*n))^p - (-d)^((m - Mod[m, n])/n - 1)*(c*d^2 - b*d*e + a*e^2)^p*(d*(Mod[m, n] + 1) + e*(Mod[m, n] + n*(q + 1)
+ 1)*x^n)))/(d + e*x^n)], x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IG
tQ[n, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m, 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]

Rubi steps

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

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Mathematica [A]  time = 0.32, size = 284, normalized size = 0.94 \[ \frac {\frac {6 b^{2/3} x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{a^2 \left (a+b x^3\right )}+\frac {9 b^{2/3} x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a \left (a+b x^3\right )^2}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{a^{7/3}}-\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{a^{7/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{a^{7/3}}+27 b^{2/3} f x^2}{54 b^{11/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(27*b^(2/3)*f*x^2 + (9*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a*(a + b*x^3)^2) + (6*b^(2/3)*(2*b^3*
c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*x^2)/(a^2*(a + b*x^3)) - (2*Sqrt[3]*(2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3
*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(7/3) - (2*(2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3*f)*Log[a
^(1/3) + b^(1/3)*x])/a^(7/3) + ((2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b
^(2/3)*x^2])/a^(7/3))/(54*b^(11/3))

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fricas [B]  time = 0.79, size = 1158, normalized size = 3.85 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/54*(27*a^3*b^4*f*x^8 + 6*(2*a*b^6*c + a^2*b^5*d - 4*a^3*b^4*e + 16*a^4*b^3*f)*x^5 + 3*(7*a^2*b^5*c - a^3*b^
4*d - 5*a^4*b^3*e + 20*a^5*b^2*f)*x^2 - 3*sqrt(1/3)*(2*a^3*b^4*c + a^4*b^3*d + 5*a^5*b^2*e - 20*a^6*b*f + (2*a
*b^6*c + a^2*b^5*d + 5*a^3*b^4*e - 20*a^4*b^3*f)*x^6 + 2*(2*a^2*b^5*c + a^3*b^4*d + 5*a^4*b^3*e - 20*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)) + ((2*b^5*c + a*b^4*d + 5*a^2*b^3*e - 20*a^3*b^2*f
)*x^6 + 2*a^2*b^3*c + a^3*b^2*d + 5*a^4*b*e - 20*a^5*f + 2*(2*a*b^4*c + a^2*b^3*d + 5*a^3*b^2*e - 20*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)) - 2*((2*b^5*c + a*b^4*d + 5*a^2*b^3*e - 20
*a^3*b^2*f)*x^6 + 2*a^2*b^3*c + a^3*b^2*d + 5*a^4*b*e - 20*a^5*f + 2*(2*a*b^4*c + a^2*b^3*d + 5*a^3*b^2*e - 20
*a^4*b*f)*x^3)*(a*b^2)^(2/3)*log(b*x + (a*b^2)^(1/3)))/(a^3*b^7*x^6 + 2*a^4*b^6*x^3 + a^5*b^5), 1/54*(27*a^3*b
^4*f*x^8 + 6*(2*a*b^6*c + a^2*b^5*d - 4*a^3*b^4*e + 16*a^4*b^3*f)*x^5 + 3*(7*a^2*b^5*c - a^3*b^4*d - 5*a^4*b^3
*e + 20*a^5*b^2*f)*x^2 - 6*sqrt(1/3)*(2*a^3*b^4*c + a^4*b^3*d + 5*a^5*b^2*e - 20*a^6*b*f + (2*a*b^6*c + a^2*b^
5*d + 5*a^3*b^4*e - 20*a^4*b^3*f)*x^6 + 2*(2*a^2*b^5*c + a^3*b^4*d + 5*a^4*b^3*e - 20*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) + ((2*b^5*c + a*b^4*d + 5*a^2
*b^3*e - 20*a^3*b^2*f)*x^6 + 2*a^2*b^3*c + a^3*b^2*d + 5*a^4*b*e - 20*a^5*f + 2*(2*a*b^4*c + a^2*b^3*d + 5*a^3
*b^2*e - 20*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)) - 2*((2*b^5*c + a*b^4
*d + 5*a^2*b^3*e - 20*a^3*b^2*f)*x^6 + 2*a^2*b^3*c + a^3*b^2*d + 5*a^4*b*e - 20*a^5*f + 2*(2*a*b^4*c + a^2*b^3
*d + 5*a^3*b^2*e - 20*a^4*b*f)*x^3)*(a*b^2)^(2/3)*log(b*x + (a*b^2)^(1/3)))/(a^3*b^7*x^6 + 2*a^4*b^6*x^3 + a^5
*b^5)]

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giac [A]  time = 0.21, size = 339, normalized size = 1.13 \[ \frac {f x^{2}}{2 \, b^{3}} + \frac {\sqrt {3} {\left (2 \, b^{3} c + a b^{2} d - 20 \, a^{3} f + 5 \, a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{3}} - \frac {{\left (2 \, b^{3} c + a b^{2} d - 20 \, a^{3} f + 5 \, a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{3}} - \frac {{\left (2 \, b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 20 \, a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a^{2} b \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3} b^{3}} + \frac {4 \, b^{4} c x^{5} + 2 \, a b^{3} d x^{5} + 14 \, a^{3} b f x^{5} - 8 \, a^{2} b^{2} x^{5} e + 7 \, a b^{3} c x^{2} - a^{2} b^{2} d x^{2} + 11 \, a^{4} f x^{2} - 5 \, a^{3} b x^{2} e}{18 \, {\left (b x^{3} + a\right )}^{2} a^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*f*x^2/b^3 + 1/27*sqrt(3)*(2*b^3*c + a*b^2*d - 20*a^3*f + 5*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^2*b^3) - 1/54*(2*b^3*c + a*b^2*d - 20*a^3*f + 5*a^2*b*e)*log(x^2 + x*(-a/b)^
(1/3) + (-a/b)^(2/3))/((-a*b^2)^(1/3)*a^2*b^3) - 1/27*(2*b^3*c*(-a/b)^(1/3) + a*b^2*d*(-a/b)^(1/3) - 20*a^3*f*
(-a/b)^(1/3) + 5*a^2*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*b^3) + 1/18*(4*b^4*c*x^5 +
2*a*b^3*d*x^5 + 14*a^3*b*f*x^5 - 8*a^2*b^2*x^5*e + 7*a*b^3*c*x^2 - a^2*b^2*d*x^2 + 11*a^4*f*x^2 - 5*a^3*b*x^2
*e)/((b*x^3 + a)^2*a^2*b^3)

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maple [B]  time = 0.06, size = 550, normalized size = 1.83 \[ \frac {7 a f \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} b^{2}}+\frac {d \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a}+\frac {2 b c \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{2}}-\frac {4 e \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} b}+\frac {11 a^{2} f \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b^{3}}-\frac {5 a e \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b^{2}}+\frac {7 c \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a}-\frac {d \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b}+\frac {f \,x^{2}}{2 b^{3}}-\frac {20 \sqrt {3}\, a 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^{4}}+\frac {20 a f \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 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^{4}}+\frac {\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}} a \,b^{2}}-\frac {d \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 {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}} a \,b^{2}}+\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 {1}{3}} a^{2} b}-\frac {2 c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b}+\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 {1}{3}} a^{2} b}+\frac {5 \sqrt {3}\, 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^{3}}-\frac {5 e \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 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 {1}{3}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*f*x^2/b^3+7/9/b^2/(b*x^3+a)^2*a*x^5*f-4/9/b/(b*x^3+a)^2*x^5*e+1/9/(b*x^3+a)^2/a*x^5*d+2/9*b/(b*x^3+a)^2/a^
2*x^5*c+11/18/b^3/(b*x^3+a)^2*a^2*x^2*f-5/18/b^2/(b*x^3+a)^2*a*x^2*e-1/18/b/(b*x^3+a)^2*x^2*d+7/18/(b*x^3+a)^2
/a*x^2*c+20/27/b^4*a/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*f-5/27/b^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*e-1/27/b^2/a/(a/b)
^(1/3)*ln(x+(a/b)^(1/3))*d-2/27/b/a^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*c-10/27/b^4*a/(a/b)^(1/3)*ln(x^2-(a/b)^(1/
3)*x+(a/b)^(2/3))*f+5/54/b^3/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*e+1/54/b^2/a/(a/b)^(1/3)*ln(x^2-(a/
b)^(1/3)*x+(a/b)^(2/3))*d+1/27/b/a^2/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*c-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))*f+5/27/b^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3
)*x-1))*e+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))*d+2/27/b/a^2*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 = 2.94, size = 296, normalized size = 0.98 \[ \frac {2 \, {\left (2 \, b^{4} c + a b^{3} d - 4 \, a^{2} b^{2} e + 7 \, a^{3} b f\right )} x^{5} + {\left (7 \, a b^{3} c - a^{2} b^{2} d - 5 \, a^{3} b e + 11 \, a^{4} f\right )} x^{2}}{18 \, {\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}} + \frac {f x^{2}}{2 \, b^{3}} + \frac {\sqrt {3} {\left (2 \, b^{3} c + a b^{2} d + 5 \, a^{2} b e - 20 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (2 \, b^{3} c + a b^{2} d + 5 \, a^{2} b e - 20 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{2} b^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (2 \, b^{3} c + a b^{2} d + 5 \, a^{2} b e - 20 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{2} b^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.27, size = 280, normalized size = 0.93 \[ \frac {\frac {x^2\,\left (11\,f\,a^3-5\,e\,a^2\,b-d\,a\,b^2+7\,c\,b^3\right )}{18\,a}+\frac {x^5\,\left (7\,f\,a^3\,b-4\,e\,a^2\,b^2+d\,a\,b^3+2\,c\,b^4\right )}{9\,a^2}}{a^2\,b^3+2\,a\,b^4\,x^3+b^5\,x^6}+\frac {f\,x^2}{2\,b^3}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-20\,f\,a^3+5\,e\,a^2\,b+d\,a\,b^2+2\,c\,b^3\right )}{27\,a^{7/3}\,b^{11/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-20\,f\,a^3+5\,e\,a^2\,b+d\,a\,b^2+2\,c\,b^3\right )}{27\,a^{7/3}\,b^{11/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-20\,f\,a^3+5\,e\,a^2\,b+d\,a\,b^2+2\,c\,b^3\right )}{27\,a^{7/3}\,b^{11/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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