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

Optimal. Leaf size=369 \[ \frac{x^4 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{4 b^5}-\frac{a^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (13 a^2 b e-16 a^3 f-10 a b^2 d+7 b^3 c\right )}{18 b^{19/3}}-\frac{a x \left (4 a^2 b e-5 a^3 f-3 a b^2 d+2 b^3 c\right )}{b^6}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (13 a^2 b e-16 a^3 f-10 a b^2 d+7 b^3 c\right )}{9 b^{19/3}}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (13 a^2 b e-16 a^3 f-10 a b^2 d+7 b^3 c\right )}{3 \sqrt{3} b^{19/3}}+\frac{x^7 \left (3 a^2 f-2 a b e+b^2 d\right )}{7 b^4}+\frac{x^{10} (b e-2 a f)}{10 b^3}+\frac{f x^{13}}{13 b^2} \]

[Out]

-((a*(2*b^3*c - 3*a*b^2*d + 4*a^2*b*e - 5*a^3*f)*x)/b^6) + ((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^4)/(4*
b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^7)/(7*b^4) + ((b*e - 2*a*f)*x^10)/(10*b^3) + (f*x^13)/(13*b^2) - (a^2*(b
^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(3*b^6*(a + b*x^3)) - (a^(4/3)*(7*b^3*c - 10*a*b^2*d + 13*a^2*b*e - 16*a^
3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*b^(19/3)) + (a^(4/3)*(7*b^3*c - 10*a*b^2*d
+ 13*a^2*b*e - 16*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(19/3)) - (a^(4/3)*(7*b^3*c - 10*a*b^2*d + 13*a^2*b*e
- 16*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*b^(19/3))

________________________________________________________________________________________

Rubi [A]  time = 0.467051, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {1828, 1887, 200, 31, 634, 617, 204, 628} \[ \frac{x^4 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{4 b^5}-\frac{a^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (13 a^2 b e-16 a^3 f-10 a b^2 d+7 b^3 c\right )}{18 b^{19/3}}-\frac{a x \left (4 a^2 b e-5 a^3 f-3 a b^2 d+2 b^3 c\right )}{b^6}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (13 a^2 b e-16 a^3 f-10 a b^2 d+7 b^3 c\right )}{9 b^{19/3}}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (13 a^2 b e-16 a^3 f-10 a b^2 d+7 b^3 c\right )}{3 \sqrt{3} b^{19/3}}+\frac{x^7 \left (3 a^2 f-2 a b e+b^2 d\right )}{7 b^4}+\frac{x^{10} (b e-2 a f)}{10 b^3}+\frac{f x^{13}}{13 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*(2*b^3*c - 3*a*b^2*d + 4*a^2*b*e - 5*a^3*f)*x)/b^6) + ((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^4)/(4*
b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^7)/(7*b^4) + ((b*e - 2*a*f)*x^10)/(10*b^3) + (f*x^13)/(13*b^2) - (a^2*(b
^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(3*b^6*(a + b*x^3)) - (a^(4/3)*(7*b^3*c - 10*a*b^2*d + 13*a^2*b*e - 16*a^
3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*b^(19/3)) + (a^(4/3)*(7*b^3*c - 10*a*b^2*d
+ 13*a^2*b*e - 16*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(19/3)) - (a^(4/3)*(7*b^3*c - 10*a*b^2*d + 13*a^2*b*e
- 16*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*b^(19/3))

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 1887

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

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{x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx &=-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}-\frac{\int \frac{-a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )+3 a^2 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-3 a b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6-3 a b^3 \left (b^2 d-a b e+a^2 f\right ) x^9-3 a b^4 (b e-a f) x^{12}-3 a b^5 f x^{15}}{a+b x^3} \, dx}{3 a b^6}\\ &=-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}-\frac{\int \left (3 a^2 \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right )-3 a b \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^3-3 a b^2 \left (b^2 d-2 a b e+3 a^2 f\right ) x^6-3 a b^3 (b e-2 a f) x^9-3 a b^4 f x^{12}+\frac{-7 a^3 b^3 c+10 a^4 b^2 d-13 a^5 b e+16 a^6 f}{a+b x^3}\right ) \, dx}{3 a b^6}\\ &=-\frac{a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) x}{b^6}+\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^4}{4 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^7}{7 b^4}+\frac{(b e-2 a f) x^{10}}{10 b^3}+\frac{f x^{13}}{13 b^2}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}+\frac{\left (a^2 \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right )\right ) \int \frac{1}{a+b x^3} \, dx}{3 b^6}\\ &=-\frac{a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) x}{b^6}+\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^4}{4 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^7}{7 b^4}+\frac{(b e-2 a f) x^{10}}{10 b^3}+\frac{f x^{13}}{13 b^2}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}+\frac{\left (a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 b^6}+\frac{\left (a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right )\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}{9 b^6}\\ &=-\frac{a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) x}{b^6}+\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^4}{4 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^7}{7 b^4}+\frac{(b e-2 a f) x^{10}}{10 b^3}+\frac{f x^{13}}{13 b^2}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}+\frac{a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{19/3}}-\frac{\left (a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 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^{19/3}}+\frac{\left (a^{5/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 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^6}\\ &=-\frac{a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) x}{b^6}+\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^4}{4 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^7}{7 b^4}+\frac{(b e-2 a f) x^{10}}{10 b^3}+\frac{f x^{13}}{13 b^2}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}+\frac{a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{19/3}}-\frac{a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{19/3}}+\frac{\left (a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 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^{19/3}}\\ &=-\frac{a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) x}{b^6}+\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^4}{4 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^7}{7 b^4}+\frac{(b e-2 a f) x^{10}}{10 b^3}+\frac{f x^{13}}{13 b^2}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}-\frac{a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 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^{19/3}}+\frac{a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{19/3}}-\frac{a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{19/3}}\\ \end{align*}

Mathematica [A]  time = 0.352521, size = 364, normalized size = 0.99 \[ \frac{x^4 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{4 b^5}+\frac{a^2 x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{3 b^6 \left (a+b x^3\right )}+\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-13 a^2 b e+16 a^3 f+10 a b^2 d-7 b^3 c\right )}{18 b^{19/3}}+\frac{a x \left (-4 a^2 b e+5 a^3 f+3 a b^2 d-2 b^3 c\right )}{b^6}-\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-13 a^2 b e+16 a^3 f+10 a b^2 d-7 b^3 c\right )}{9 b^{19/3}}+\frac{a^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-13 a^2 b e+16 a^3 f+10 a b^2 d-7 b^3 c\right )}{3 \sqrt{3} b^{19/3}}+\frac{x^7 \left (3 a^2 f-2 a b e+b^2 d\right )}{7 b^4}+\frac{x^{10} (b e-2 a f)}{10 b^3}+\frac{f x^{13}}{13 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(-2*b^3*c + 3*a*b^2*d - 4*a^2*b*e + 5*a^3*f)*x)/b^6 + ((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^4)/(4*b^
5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^7)/(7*b^4) + ((b*e - 2*a*f)*x^10)/(10*b^3) + (f*x^13)/(13*b^2) + (a^2*(-(b
^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(3*b^6*(a + b*x^3)) + (a^(4/3)*(-7*b^3*c + 10*a*b^2*d - 13*a^2*b*e + 16*
a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(3*Sqrt[3]*b^(19/3)) - (a^(4/3)*(-7*b^3*c + 10*a*b^2*d - 1
3*a^2*b*e + 16*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(19/3)) + (a^(4/3)*(-7*b^3*c + 10*a*b^2*d - 13*a^2*b*e +
16*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*b^(19/3))

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Maple [A]  time = 0.01, size = 622, 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^9*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x)

[Out]

7/9*a^2/b^4*c/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-16/9*a^5/b^7*f/(1/b*a)^(2/3)*3^(
1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+13/9*a^4/b^6*e/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b
*a)^(1/3)*x-1))+1/10/b^2*x^10*e+1/7/b^2*x^7*d+1/4/b^2*x^4*c-10/9*a^3/b^5*d/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^
(1/2)*(2/(1/b*a)^(1/3)*x-1))-7/18*a^2/b^4*c/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-1/3*a^4/b^5*x/
(b*x^3+a)*e+1/3*a^3/b^4*x/(b*x^3+a)*d-1/3*a^2/b^3*x/(b*x^3+a)*c+1/3*a^5/b^6*x/(b*x^3+a)*f+7/9*a^2/b^4*c/(1/b*a
)^(2/3)*ln(x+(1/b*a)^(1/3))+13/9*a^4/b^6*e/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-13/18*a^4/b^6*e/(1/b*a)^(2/3)*ln(
x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-10/9*a^3/b^5*d/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))+5/9*a^3/b^5*d/(1/b*a)^(2/3
)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+8/9*a^5/b^7*f/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-16/9
*a^5/b^7*f/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))+1/13*f*x^13/b^2-1/5/b^3*x^10*a*f+3/7/b^4*x^7*a^2*f-2/7/b^3*x^7*a*
e-1/b^5*x^4*a^3*f+3/4/b^4*x^4*a^2*e-1/2/b^3*x^4*a*d+5/b^6*a^4*f*x-4/b^5*a^3*e*x+3/b^4*a^2*d*x-2/b^3*a*c*x

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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^9*(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 = 1.41347, size = 1145, normalized size = 3.1 \begin{align*} \frac{1260 \, b^{5} f x^{16} + 126 \,{\left (13 \, b^{5} e - 16 \, a b^{4} f\right )} x^{13} + 234 \,{\left (10 \, b^{5} d - 13 \, a b^{4} e + 16 \, a^{2} b^{3} f\right )} x^{10} + 585 \,{\left (7 \, b^{5} c - 10 \, a b^{4} d + 13 \, a^{2} b^{3} e - 16 \, a^{3} b^{2} f\right )} x^{7} - 4095 \,{\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{4} - 1820 \, \sqrt{3}{\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f +{\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (-\frac{a}{b}\right )^{\frac{2}{3}} - \sqrt{3} a}{3 \, a}\right ) + 910 \,{\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f +{\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 1820 \,{\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f +{\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 5460 \,{\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f\right )} x}{16380 \,{\left (b^{7} x^{3} + a b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

1/16380*(1260*b^5*f*x^16 + 126*(13*b^5*e - 16*a*b^4*f)*x^13 + 234*(10*b^5*d - 13*a*b^4*e + 16*a^2*b^3*f)*x^10
+ 585*(7*b^5*c - 10*a*b^4*d + 13*a^2*b^3*e - 16*a^3*b^2*f)*x^7 - 4095*(7*a*b^4*c - 10*a^2*b^3*d + 13*a^3*b^2*e
- 16*a^4*b*f)*x^4 - 1820*sqrt(3)*(7*a^2*b^3*c - 10*a^3*b^2*d + 13*a^4*b*e - 16*a^5*f + (7*a*b^4*c - 10*a^2*b^
3*d + 13*a^3*b^2*e - 16*a^4*b*f)*x^3)*(-a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(-a/b)^(2/3) - sqrt(3)*a)/a) + 91
0*(7*a^2*b^3*c - 10*a^3*b^2*d + 13*a^4*b*e - 16*a^5*f + (7*a*b^4*c - 10*a^2*b^3*d + 13*a^3*b^2*e - 16*a^4*b*f)
*x^3)*(-a/b)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3)) - 1820*(7*a^2*b^3*c - 10*a^3*b^2*d + 13*a^4*b*e -
16*a^5*f + (7*a*b^4*c - 10*a^2*b^3*d + 13*a^3*b^2*e - 16*a^4*b*f)*x^3)*(-a/b)^(1/3)*log(x - (-a/b)^(1/3)) - 54
60*(7*a^2*b^3*c - 10*a^3*b^2*d + 13*a^4*b*e - 16*a^5*f)*x)/(b^7*x^3 + a*b^6)

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Sympy [A]  time = 10.5078, size = 490, normalized size = 1.33 \begin{align*} \frac{x \left (a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c\right )}{3 a b^{6} + 3 b^{7} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} b^{19} + 4096 a^{13} f^{3} - 9984 a^{12} b e f^{2} + 7680 a^{11} b^{2} d f^{2} + 8112 a^{11} b^{2} e^{2} f - 5376 a^{10} b^{3} c f^{2} - 12480 a^{10} b^{3} d e f - 2197 a^{10} b^{3} e^{3} + 8736 a^{9} b^{4} c e f + 4800 a^{9} b^{4} d^{2} f + 5070 a^{9} b^{4} d e^{2} - 6720 a^{8} b^{5} c d f - 3549 a^{8} b^{5} c e^{2} - 3900 a^{8} b^{5} d^{2} e + 2352 a^{7} b^{6} c^{2} f + 5460 a^{7} b^{6} c d e + 1000 a^{7} b^{6} d^{3} - 1911 a^{6} b^{7} c^{2} e - 2100 a^{6} b^{7} c d^{2} + 1470 a^{5} b^{8} c^{2} d - 343 a^{4} b^{9} c^{3}, \left ( t \mapsto t \log{\left (- \frac{9 t b^{6}}{16 a^{4} f - 13 a^{3} b e + 10 a^{2} b^{2} d - 7 a b^{3} c} + x \right )} \right )\right )} + \frac{f x^{13}}{13 b^{2}} - \frac{x^{10} \left (2 a f - b e\right )}{10 b^{3}} + \frac{x^{7} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{7 b^{4}} - \frac{x^{4} \left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c\right )}{4 b^{5}} + \frac{x \left (5 a^{4} f - 4 a^{3} b e + 3 a^{2} b^{2} d - 2 a b^{3} c\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)

[Out]

x*(a**5*f - a**4*b*e + a**3*b**2*d - a**2*b**3*c)/(3*a*b**6 + 3*b**7*x**3) + RootSum(729*_t**3*b**19 + 4096*a*
*13*f**3 - 9984*a**12*b*e*f**2 + 7680*a**11*b**2*d*f**2 + 8112*a**11*b**2*e**2*f - 5376*a**10*b**3*c*f**2 - 12
480*a**10*b**3*d*e*f - 2197*a**10*b**3*e**3 + 8736*a**9*b**4*c*e*f + 4800*a**9*b**4*d**2*f + 5070*a**9*b**4*d*
e**2 - 6720*a**8*b**5*c*d*f - 3549*a**8*b**5*c*e**2 - 3900*a**8*b**5*d**2*e + 2352*a**7*b**6*c**2*f + 5460*a**
7*b**6*c*d*e + 1000*a**7*b**6*d**3 - 1911*a**6*b**7*c**2*e - 2100*a**6*b**7*c*d**2 + 1470*a**5*b**8*c**2*d - 3
43*a**4*b**9*c**3, Lambda(_t, _t*log(-9*_t*b**6/(16*a**4*f - 13*a**3*b*e + 10*a**2*b**2*d - 7*a*b**3*c) + x)))
+ f*x**13/(13*b**2) - x**10*(2*a*f - b*e)/(10*b**3) + x**7*(3*a**2*f - 2*a*b*e + b**2*d)/(7*b**4) - x**4*(4*a
**3*f - 3*a**2*b*e + 2*a*b**2*d - b**3*c)/(4*b**5) + x*(5*a**4*f - 4*a**3*b*e + 3*a**2*b**2*d - 2*a*b**3*c)/b*
*6

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Giac [A]  time = 1.07843, size = 609, normalized size = 1.65 \begin{align*} \frac{\sqrt{3}{\left (7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - 10 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} d - 16 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} f + 13 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} 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{{\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d - 16 \, a^{5} f + 13 \, a^{4} 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 )}{9 \, a b^{6}} + \frac{{\left (7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - 10 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} d - 16 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} f + 13 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} 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{a^{2} b^{3} c x - a^{3} b^{2} d x - a^{5} f x + a^{4} b x e}{3 \,{\left (b x^{3} + a\right )} b^{6}} + \frac{140 \, b^{24} f x^{13} - 364 \, a b^{23} f x^{10} + 182 \, b^{24} x^{10} e + 260 \, b^{24} d x^{7} + 780 \, a^{2} b^{22} f x^{7} - 520 \, a b^{23} x^{7} e + 455 \, b^{24} c x^{4} - 910 \, a b^{23} d x^{4} - 1820 \, a^{3} b^{21} f x^{4} + 1365 \, a^{2} b^{22} x^{4} e - 3640 \, a b^{23} c x + 5460 \, a^{2} b^{22} d x + 9100 \, a^{4} b^{20} f x - 7280 \, a^{3} b^{21} x e}{1820 \, b^{26}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="giac")

[Out]

1/9*sqrt(3)*(7*(-a*b^2)^(1/3)*a*b^3*c - 10*(-a*b^2)^(1/3)*a^2*b^2*d - 16*(-a*b^2)^(1/3)*a^4*f + 13*(-a*b^2)^(1
/3)*a^3*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^7 - 1/9*(7*a^2*b^3*c - 10*a^3*b^2*d - 16*
a^5*f + 13*a^4*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^6) + 1/18*(7*(-a*b^2)^(1/3)*a*b^3*c - 10*(-a*
b^2)^(1/3)*a^2*b^2*d - 16*(-a*b^2)^(1/3)*a^4*f + 13*(-a*b^2)^(1/3)*a^3*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^
(2/3))/b^7 - 1/3*(a^2*b^3*c*x - a^3*b^2*d*x - a^5*f*x + a^4*b*x*e)/((b*x^3 + a)*b^6) + 1/1820*(140*b^24*f*x^13
- 364*a*b^23*f*x^10 + 182*b^24*x^10*e + 260*b^24*d*x^7 + 780*a^2*b^22*f*x^7 - 520*a*b^23*x^7*e + 455*b^24*c*x
^4 - 910*a*b^23*d*x^4 - 1820*a^3*b^21*f*x^4 + 1365*a^2*b^22*x^4*e - 3640*a*b^23*c*x + 5460*a^2*b^22*d*x + 9100
*a^4*b^20*f*x - 7280*a^3*b^21*x*e)/b^26