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

Optimal. Leaf size=292 \[ \frac{x \left (-7 a^2 b e+13 a^3 f+a b^2 d+5 b^3 c\right )}{18 a^2 b^3 \left (a+b x^3\right )}+\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^2 b e-14 a^3 f+a b^2 d+5 b^3 c\right )}{54 a^{8/3} b^{10/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^2 b e-14 a^3 f+a b^2 d+5 b^3 c\right )}{27 a^{8/3} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^2 b e-14 a^3 f+a b^2 d+5 b^3 c\right )}{9 \sqrt{3} a^{8/3} b^{10/3}}+\frac{f x}{b^3} \]

[Out]

(f*x)/b^3 + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a*b^3*(a + b*x^3)^2) + ((5*b^3*c + a*b^2*d - 7*a^2*b*e
+ 13*a^3*f)*x)/(18*a^2*b^3*(a + b*x^3)) - ((5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1
/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(10/3)) + ((5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*Log[a^(1
/3) + b^(1/3)*x])/(27*a^(8/3)*b^(10/3)) - ((5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^
(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(10/3))

________________________________________________________________________________________

Rubi [A]  time = 0.307193, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1858, 1409, 388, 200, 31, 634, 617, 204, 628} \[ \frac{x \left (-7 a^2 b e+13 a^3 f+a b^2 d+5 b^3 c\right )}{18 a^2 b^3 \left (a+b x^3\right )}+\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^2 b e-14 a^3 f+a b^2 d+5 b^3 c\right )}{54 a^{8/3} b^{10/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^2 b e-14 a^3 f+a b^2 d+5 b^3 c\right )}{27 a^{8/3} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^2 b e-14 a^3 f+a b^2 d+5 b^3 c\right )}{9 \sqrt{3} a^{8/3} b^{10/3}}+\frac{f x}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f*x)/b^3 + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a*b^3*(a + b*x^3)^2) + ((5*b^3*c + a*b^2*d - 7*a^2*b*e
+ 13*a^3*f)*x)/(18*a^2*b^3*(a + b*x^3)) - ((5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1
/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(10/3)) + ((5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*Log[a^(1
/3) + b^(1/3)*x])/(27*a^(8/3)*b^(10/3)) - ((5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^
(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(10/3))

Rule 1858

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*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]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1409

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> -Simp[((c*d^2 - b*
d*e + a*e^2)*x*(d + e*x^n)^(q + 1))/(d*e^2*n*(q + 1)), x] + Dist[1/(n*(q + 1)*d*e^2), Int[(d + e*x^n)^(q + 1)*
Simp[c*d^2 - b*d*e + a*e^2*(n*(q + 1) + 1) + c*d*e*n*(q + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, n}, x] &
& EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[q, -1]

Rule 388

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

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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{c+d x^3+e x^6+f x^9}{\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}{6 a b^3 \left (a+b x^3\right )^2}-\frac{\int \frac{-5 b^3 c-a b^2 d+a^2 b e-a^3 f-6 a b (b e-a f) x^3-6 a b^2 f x^6}{\left (a+b x^3\right )^2} \, dx}{6 a b^3}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\left (5 b^3 c+a b^2 d-7 a^2 b e+13 a^3 f\right ) x}{18 a^2 b^3 \left (a+b x^3\right )}+\frac{\int \frac{2 b^2 \left (5 b^3 c+a b^2 d+2 a^2 b e-5 a^3 f\right )+18 a^2 b^3 f x^3}{a+b x^3} \, dx}{18 a^2 b^5}\\ &=\frac{f x}{b^3}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\left (5 b^3 c+a b^2 d-7 a^2 b e+13 a^3 f\right ) x}{18 a^2 b^3 \left (a+b x^3\right )}+\frac{\left (5 b^3 c+a b^2 d+2 a^2 b e-14 a^3 f\right ) \int \frac{1}{a+b x^3} \, dx}{9 a^2 b^3}\\ &=\frac{f x}{b^3}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\left (5 b^3 c+a b^2 d-7 a^2 b e+13 a^3 f\right ) x}{18 a^2 b^3 \left (a+b x^3\right )}+\frac{\left (5 b^3 c+a b^2 d+2 a^2 b e-14 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{8/3} b^3}+\frac{\left (5 b^3 c+a b^2 d+2 a^2 b e-14 a^3 f\right ) \int \frac{2 \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^{8/3} b^3}\\ &=\frac{f x}{b^3}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\left (5 b^3 c+a b^2 d-7 a^2 b e+13 a^3 f\right ) x}{18 a^2 b^3 \left (a+b x^3\right )}+\frac{\left (5 b^3 c+a b^2 d+2 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{10/3}}-\frac{\left (5 b^3 c+a b^2 d+2 a^2 b e-14 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^{8/3} b^{10/3}}+\frac{\left (5 b^3 c+a b^2 d+2 a^2 b e-14 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^{7/3} b^3}\\ &=\frac{f x}{b^3}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\left (5 b^3 c+a b^2 d-7 a^2 b e+13 a^3 f\right ) x}{18 a^2 b^3 \left (a+b x^3\right )}+\frac{\left (5 b^3 c+a b^2 d+2 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{10/3}}-\frac{\left (5 b^3 c+a b^2 d+2 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 )}{54 a^{8/3} b^{10/3}}+\frac{\left (5 b^3 c+a b^2 d+2 a^2 b e-14 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^{8/3} b^{10/3}}\\ &=\frac{f x}{b^3}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\left (5 b^3 c+a b^2 d-7 a^2 b e+13 a^3 f\right ) x}{18 a^2 b^3 \left (a+b x^3\right )}-\frac{\left (5 b^3 c+a b^2 d+2 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 )}{9 \sqrt{3} a^{8/3} b^{10/3}}+\frac{\left (5 b^3 c+a b^2 d+2 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{10/3}}-\frac{\left (5 b^3 c+a b^2 d+2 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 )}{54 a^{8/3} b^{10/3}}\\ \end{align*}

Mathematica [A]  time = 0.189997, size = 279, normalized size = 0.96 \[ \frac{\frac{3 \sqrt [3]{b} x \left (-7 a^2 b e+13 a^3 f+a b^2 d+5 b^3 c\right )}{a^2 \left (a+b x^3\right )}+\frac{9 \sqrt [3]{b} x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^2 b e-14 a^3 f+a b^2 d+5 b^3 c\right )}{a^{8/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^2 b e-14 a^3 f+a b^2 d+5 b^3 c\right )}{a^{8/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (2 a^2 b e-14 a^3 f+a b^2 d+5 b^3 c\right )}{a^{8/3}}+54 \sqrt [3]{b} f x}{54 b^{10/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(54*b^(1/3)*f*x + (9*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a*(a + b*x^3)^2) + (3*b^(1/3)*(5*b^3*c +
a*b^2*d - 7*a^2*b*e + 13*a^3*f)*x)/(a^2*(a + b*x^3)) - (2*Sqrt[3]*(5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*A
rcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(8/3) + (2*(5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*Log[a^(1/3
) + b^(1/3)*x])/a^(8/3) - ((5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3
)*x^2])/a^(8/3))/(54*b^(10/3))

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Maple [B]  time = 0.011, size = 539, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f*x/b^3+13/18/b^2/(b*x^3+a)^2*x^4*a*f-7/18/b/(b*x^3+a)^2*x^4*e+1/18/(b*x^3+a)^2/a*x^4*d+5/18*b/(b*x^3+a)^2/a^2
*x^4*c+5/9/b^3/(b*x^3+a)^2*a^2*f*x-2/9/b^2/(b*x^3+a)^2*a*e*x-1/9/b/(b*x^3+a)^2*d*x+4/9*c/a*x/(b*x^3+a)^2-14/27
/b^4*a/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*f+2/27/b^3/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*e+1/27/b^2/a/(1/b*a)^(2/
3)*ln(x+(1/b*a)^(1/3))*d+5/27*c/a^2/b/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))+7/27/b^4*a/(1/b*a)^(2/3)*ln(x^2-(1/b*a
)^(1/3)*x+(1/b*a)^(2/3))*f-1/27/b^3/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*e-1/54/b^2/a/(1/b*a)^(
2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*d-5/54*c/a^2/b/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-
14/27/b^4*a/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*f+2/27/b^3/(1/b*a)^(2/3)*3^(1/2)*a
rctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*e+1/27/b^2/a/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3
)*x-1))*d+5/27*c/a^2/b/(1/b*a)^(2/3)*3^(1/2)*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((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 = 1.49873, size = 2587, normalized size = 8.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/54*(54*a^4*b^3*f*x^7 + 3*(5*a^2*b^5*c + a^3*b^4*d - 7*a^4*b^3*e + 49*a^5*b^2*f)*x^4 - 3*sqrt(1/3)*(5*a^3*b^
4*c + a^4*b^3*d + 2*a^5*b^2*e - 14*a^6*b*f + (5*a*b^6*c + a^2*b^5*d + 2*a^3*b^4*e - 14*a^4*b^3*f)*x^6 + 2*(5*a
^2*b^5*c + a^3*b^4*d + 2*a^4*b^3*e - 14*a^5*b^2*f)*x^3)*sqrt((-a^2*b)^(1/3)/b)*log((2*a*b*x^3 + 3*(-a^2*b)^(1/
3)*a*x - a^2 - 3*sqrt(1/3)*(2*a*b*x^2 + (-a^2*b)^(2/3)*x + (-a^2*b)^(1/3)*a)*sqrt((-a^2*b)^(1/3)/b))/(b*x^3 +
a)) - ((5*b^5*c + a*b^4*d + 2*a^2*b^3*e - 14*a^3*b^2*f)*x^6 + 5*a^2*b^3*c + a^3*b^2*d + 2*a^4*b*e - 14*a^5*f +
2*(5*a*b^4*c + a^2*b^3*d + 2*a^3*b^2*e - 14*a^4*b*f)*x^3)*(-a^2*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (-a
^2*b)^(1/3)*a) + 2*((5*b^5*c + a*b^4*d + 2*a^2*b^3*e - 14*a^3*b^2*f)*x^6 + 5*a^2*b^3*c + a^3*b^2*d + 2*a^4*b*e
- 14*a^5*f + 2*(5*a*b^4*c + a^2*b^3*d + 2*a^3*b^2*e - 14*a^4*b*f)*x^3)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2
/3)) + 6*(4*a^3*b^4*c - a^4*b^3*d - 2*a^5*b^2*e + 14*a^6*b*f)*x)/(a^4*b^6*x^6 + 2*a^5*b^5*x^3 + a^6*b^4), 1/54
*(54*a^4*b^3*f*x^7 + 3*(5*a^2*b^5*c + a^3*b^4*d - 7*a^4*b^3*e + 49*a^5*b^2*f)*x^4 + 6*sqrt(1/3)*(5*a^3*b^4*c +
a^4*b^3*d + 2*a^5*b^2*e - 14*a^6*b*f + (5*a*b^6*c + a^2*b^5*d + 2*a^3*b^4*e - 14*a^4*b^3*f)*x^6 + 2*(5*a^2*b^
5*c + a^3*b^4*d + 2*a^4*b^3*e - 14*a^5*b^2*f)*x^3)*sqrt(-(-a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(-a^2*b)^(2/3)*
x + (-a^2*b)^(1/3)*a)*sqrt(-(-a^2*b)^(1/3)/b)/a^2) - ((5*b^5*c + a*b^4*d + 2*a^2*b^3*e - 14*a^3*b^2*f)*x^6 + 5
*a^2*b^3*c + a^3*b^2*d + 2*a^4*b*e - 14*a^5*f + 2*(5*a*b^4*c + a^2*b^3*d + 2*a^3*b^2*e - 14*a^4*b*f)*x^3)*(-a^
2*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a) + 2*((5*b^5*c + a*b^4*d + 2*a^2*b^3*e - 14*a^3*b
^2*f)*x^6 + 5*a^2*b^3*c + a^3*b^2*d + 2*a^4*b*e - 14*a^5*f + 2*(5*a*b^4*c + a^2*b^3*d + 2*a^3*b^2*e - 14*a^4*b
*f)*x^3)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) + 6*(4*a^3*b^4*c - a^4*b^3*d - 2*a^5*b^2*e + 14*a^6*b*f)*x
)/(a^4*b^6*x^6 + 2*a^5*b^5*x^3 + a^6*b^4)]

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Sympy [A]  time = 170.751, size = 418, normalized size = 1.43 \begin{align*} \frac{x^{4} \left (13 a^{3} b f - 7 a^{2} b^{2} e + a b^{3} d + 5 b^{4} c\right ) + x \left (10 a^{4} f - 4 a^{3} b e - 2 a^{2} b^{2} d + 8 a b^{3} c\right )}{18 a^{4} b^{3} + 36 a^{3} b^{4} x^{3} + 18 a^{2} b^{5} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a^{8} b^{10} + 2744 a^{9} f^{3} - 1176 a^{8} b e f^{2} - 588 a^{7} b^{2} d f^{2} + 168 a^{7} b^{2} e^{2} f - 2940 a^{6} b^{3} c f^{2} + 168 a^{6} b^{3} d e f - 8 a^{6} b^{3} e^{3} + 840 a^{5} b^{4} c e f + 42 a^{5} b^{4} d^{2} f - 12 a^{5} b^{4} d e^{2} + 420 a^{4} b^{5} c d f - 60 a^{4} b^{5} c e^{2} - 6 a^{4} b^{5} d^{2} e + 1050 a^{3} b^{6} c^{2} f - 60 a^{3} b^{6} c d e - a^{3} b^{6} d^{3} - 150 a^{2} b^{7} c^{2} e - 15 a^{2} b^{7} c d^{2} - 75 a b^{8} c^{2} d - 125 b^{9} c^{3}, \left ( t \mapsto t \log{\left (- \frac{27 t a^{3} b^{3}}{14 a^{3} f - 2 a^{2} b e - a b^{2} d - 5 b^{3} c} + x \right )} \right )\right )} + \frac{f x}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x**4*(13*a**3*b*f - 7*a**2*b**2*e + a*b**3*d + 5*b**4*c) + x*(10*a**4*f - 4*a**3*b*e - 2*a**2*b**2*d + 8*a*b*
*3*c))/(18*a**4*b**3 + 36*a**3*b**4*x**3 + 18*a**2*b**5*x**6) + RootSum(19683*_t**3*a**8*b**10 + 2744*a**9*f**
3 - 1176*a**8*b*e*f**2 - 588*a**7*b**2*d*f**2 + 168*a**7*b**2*e**2*f - 2940*a**6*b**3*c*f**2 + 168*a**6*b**3*d
*e*f - 8*a**6*b**3*e**3 + 840*a**5*b**4*c*e*f + 42*a**5*b**4*d**2*f - 12*a**5*b**4*d*e**2 + 420*a**4*b**5*c*d*
f - 60*a**4*b**5*c*e**2 - 6*a**4*b**5*d**2*e + 1050*a**3*b**6*c**2*f - 60*a**3*b**6*c*d*e - a**3*b**6*d**3 - 1
50*a**2*b**7*c**2*e - 15*a**2*b**7*c*d**2 - 75*a*b**8*c**2*d - 125*b**9*c**3, Lambda(_t, _t*log(-27*_t*a**3*b*
*3/(14*a**3*f - 2*a**2*b*e - a*b**2*d - 5*b**3*c) + x))) + f*x/b**3

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Giac [A]  time = 1.09674, size = 463, normalized size = 1.59 \begin{align*} \frac{f x}{b^{3}} - \frac{{\left (5 \, b^{3} c + a b^{2} d - 14 \, a^{3} f + 2 \, a^{2} b 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{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c + \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 2 \, \left (-a b^{2}\right )^{\frac{1}{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^{3} b^{4}} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c + \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 2 \, \left (-a b^{2}\right )^{\frac{1}{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^{3} b^{4}} + \frac{5 \, b^{4} c x^{4} + a b^{3} d x^{4} + 13 \, a^{3} b f x^{4} - 7 \, a^{2} b^{2} x^{4} e + 8 \, a b^{3} c x - 2 \, a^{2} b^{2} d x + 10 \, a^{4} f x - 4 \, a^{3} b x e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f*x/b^3 - 1/27*(5*b^3*c + a*b^2*d - 14*a^3*f + 2*a^2*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*b^3) +
1/27*sqrt(3)*(5*(-a*b^2)^(1/3)*b^3*c + (-a*b^2)^(1/3)*a*b^2*d - 14*(-a*b^2)^(1/3)*a^3*f + 2*(-a*b^2)^(1/3)*a^2
*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b^4) + 1/54*(5*(-a*b^2)^(1/3)*b^3*c + (-a*b^2
)^(1/3)*a*b^2*d - 14*(-a*b^2)^(1/3)*a^3*f + 2*(-a*b^2)^(1/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))
/(a^3*b^4) + 1/18*(5*b^4*c*x^4 + a*b^3*d*x^4 + 13*a^3*b*f*x^4 - 7*a^2*b^2*x^4*e + 8*a*b^3*c*x - 2*a^2*b^2*d*x
+ 10*a^4*f*x - 4*a^3*b*x*e)/((b*x^3 + a)^2*a^2*b^3)