3.96 \(\int \frac{x^9 (A+B x^3)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=244 \[ \frac{7 \sqrt [3]{a} (2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{13/3}}+\frac{x^7 (2 A b-5 a B)}{9 a b^2 \left (a+b x^3\right )}-\frac{7 x^4 (2 A b-5 a B)}{36 a b^3}+\frac{7 x (2 A b-5 a B)}{9 b^4}-\frac{7 \sqrt [3]{a} (2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{13/3}}+\frac{7 \sqrt [3]{a} (2 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} b^{13/3}}+\frac{x^{10} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

[Out]

(7*(2*A*b - 5*a*B)*x)/(9*b^4) - (7*(2*A*b - 5*a*B)*x^4)/(36*a*b^3) + ((A*b - a*B)*x^10)/(6*a*b*(a + b*x^3)^2)
+ ((2*A*b - 5*a*B)*x^7)/(9*a*b^2*(a + b*x^3)) + (7*a^(1/3)*(2*A*b - 5*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqr
t[3]*a^(1/3))])/(9*Sqrt[3]*b^(13/3)) - (7*a^(1/3)*(2*A*b - 5*a*B)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(13/3)) + (7
*a^(1/3)*(2*A*b - 5*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(13/3))

________________________________________________________________________________________

Rubi [A]  time = 0.165375, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {457, 288, 302, 200, 31, 634, 617, 204, 628} \[ \frac{7 \sqrt [3]{a} (2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{13/3}}+\frac{x^7 (2 A b-5 a B)}{9 a b^2 \left (a+b x^3\right )}-\frac{7 x^4 (2 A b-5 a B)}{36 a b^3}+\frac{7 x (2 A b-5 a B)}{9 b^4}-\frac{7 \sqrt [3]{a} (2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{13/3}}+\frac{7 \sqrt [3]{a} (2 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} b^{13/3}}+\frac{x^{10} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^9*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

(7*(2*A*b - 5*a*B)*x)/(9*b^4) - (7*(2*A*b - 5*a*B)*x^4)/(36*a*b^3) + ((A*b - a*B)*x^10)/(6*a*b*(a + b*x^3)^2)
+ ((2*A*b - 5*a*B)*x^7)/(9*a*b^2*(a + b*x^3)) + (7*a^(1/3)*(2*A*b - 5*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqr
t[3]*a^(1/3))])/(9*Sqrt[3]*b^(13/3)) - (7*a^(1/3)*(2*A*b - 5*a*B)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(13/3)) + (7
*a^(1/3)*(2*A*b - 5*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(13/3))

Rule 457

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d
)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b*e*n*(p + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b
*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] &&
LeQ[-1, m, -(n*(p + 1))]))

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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 (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac{(A b-a B) x^{10}}{6 a b \left (a+b x^3\right )^2}+\frac{(-4 A b+10 a B) \int \frac{x^9}{\left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac{(A b-a B) x^{10}}{6 a b \left (a+b x^3\right )^2}+\frac{(2 A b-5 a B) x^7}{9 a b^2 \left (a+b x^3\right )}-\frac{(7 (2 A b-5 a B)) \int \frac{x^6}{a+b x^3} \, dx}{9 a b^2}\\ &=\frac{(A b-a B) x^{10}}{6 a b \left (a+b x^3\right )^2}+\frac{(2 A b-5 a B) x^7}{9 a b^2 \left (a+b x^3\right )}-\frac{(7 (2 A b-5 a B)) \int \left (-\frac{a}{b^2}+\frac{x^3}{b}+\frac{a^2}{b^2 \left (a+b x^3\right )}\right ) \, dx}{9 a b^2}\\ &=\frac{7 (2 A b-5 a B) x}{9 b^4}-\frac{7 (2 A b-5 a B) x^4}{36 a b^3}+\frac{(A b-a B) x^{10}}{6 a b \left (a+b x^3\right )^2}+\frac{(2 A b-5 a B) x^7}{9 a b^2 \left (a+b x^3\right )}-\frac{(7 a (2 A b-5 a B)) \int \frac{1}{a+b x^3} \, dx}{9 b^4}\\ &=\frac{7 (2 A b-5 a B) x}{9 b^4}-\frac{7 (2 A b-5 a B) x^4}{36 a b^3}+\frac{(A b-a B) x^{10}}{6 a b \left (a+b x^3\right )^2}+\frac{(2 A b-5 a B) x^7}{9 a b^2 \left (a+b x^3\right )}-\frac{\left (7 \sqrt [3]{a} (2 A b-5 a B)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 b^4}-\frac{\left (7 \sqrt [3]{a} (2 A b-5 a B)\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 b^4}\\ &=\frac{7 (2 A b-5 a B) x}{9 b^4}-\frac{7 (2 A b-5 a B) x^4}{36 a b^3}+\frac{(A b-a B) x^{10}}{6 a b \left (a+b x^3\right )^2}+\frac{(2 A b-5 a B) x^7}{9 a b^2 \left (a+b x^3\right )}-\frac{7 \sqrt [3]{a} (2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{13/3}}+\frac{\left (7 \sqrt [3]{a} (2 A b-5 a B)\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 b^{13/3}}-\frac{\left (7 a^{2/3} (2 A b-5 a B)\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 b^4}\\ &=\frac{7 (2 A b-5 a B) x}{9 b^4}-\frac{7 (2 A b-5 a B) x^4}{36 a b^3}+\frac{(A b-a B) x^{10}}{6 a b \left (a+b x^3\right )^2}+\frac{(2 A b-5 a B) x^7}{9 a b^2 \left (a+b x^3\right )}-\frac{7 \sqrt [3]{a} (2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{13/3}}+\frac{7 \sqrt [3]{a} (2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{13/3}}-\frac{\left (7 \sqrt [3]{a} (2 A b-5 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 b^{13/3}}\\ &=\frac{7 (2 A b-5 a B) x}{9 b^4}-\frac{7 (2 A b-5 a B) x^4}{36 a b^3}+\frac{(A b-a B) x^{10}}{6 a b \left (a+b x^3\right )^2}+\frac{(2 A b-5 a B) x^7}{9 a b^2 \left (a+b x^3\right )}+\frac{7 \sqrt [3]{a} (2 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} b^{13/3}}-\frac{7 \sqrt [3]{a} (2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{13/3}}+\frac{7 \sqrt [3]{a} (2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{13/3}}\\ \end{align*}

Mathematica [A]  time = 0.15317, size = 210, normalized size = 0.86 \[ \frac{-14 \sqrt [3]{a} (5 a B-2 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\frac{18 a^2 \sqrt [3]{b} x (a B-A b)}{\left (a+b x^3\right )^2}+\frac{6 a \sqrt [3]{b} x (13 A b-19 a B)}{a+b x^3}+108 \sqrt [3]{b} x (A b-3 a B)+28 \sqrt [3]{a} (5 a B-2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-28 \sqrt{3} \sqrt [3]{a} (5 a B-2 A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+27 b^{4/3} B x^4}{108 b^{13/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^9*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

(108*b^(1/3)*(A*b - 3*a*B)*x + 27*b^(4/3)*B*x^4 + (18*a^2*b^(1/3)*(-(A*b) + a*B)*x)/(a + b*x^3)^2 + (6*a*b^(1/
3)*(13*A*b - 19*a*B)*x)/(a + b*x^3) - 28*Sqrt[3]*a^(1/3)*(-2*A*b + 5*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/S
qrt[3]] + 28*a^(1/3)*(-2*A*b + 5*a*B)*Log[a^(1/3) + b^(1/3)*x] - 14*a^(1/3)*(-2*A*b + 5*a*B)*Log[a^(2/3) - a^(
1/3)*b^(1/3)*x + b^(2/3)*x^2])/(108*b^(13/3))

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 299, normalized size = 1.2 \begin{align*}{\frac{B{x}^{4}}{4\,{b}^{3}}}+{\frac{Ax}{{b}^{3}}}-3\,{\frac{Bax}{{b}^{4}}}+{\frac{13\,Aa{x}^{4}}{18\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{19\,{a}^{2}B{x}^{4}}{18\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{5\,{a}^{2}Ax}{9\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{8\,B{a}^{3}x}{9\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{14\,Aa}{27\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{7\,Aa}{27\,{b}^{4}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{14\,Aa\sqrt{3}}{27\,{b}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{35\,{a}^{2}B}{27\,{b}^{5}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{35\,{a}^{2}B}{54\,{b}^{5}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{35\,{a}^{2}B\sqrt{3}}{27\,{b}^{5}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*(B*x^3+A)/(b*x^3+a)^3,x)

[Out]

1/4/b^3*B*x^4+1/b^3*A*x-3/b^4*B*a*x+13/18*a/b^2/(b*x^3+a)^2*A*x^4-19/18*a^2/b^3/(b*x^3+a)^2*B*x^4+5/9*a^2/b^3/
(b*x^3+a)^2*A*x-8/9*a^3/b^4/(b*x^3+a)^2*B*x-14/27*a/b^4*A/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+7/27*a/b^4*A/(a/b)^(2/
3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-14/27*a/b^4*A/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))
+35/27*a^2/b^5*B/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-35/54*a^2/b^5*B/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+3
5/27*a^2/b^5*B/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))

________________________________________________________________________________________

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*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.51259, size = 779, normalized size = 3.19 \begin{align*} \frac{27 \, B b^{3} x^{10} - 54 \,{\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{7} - 147 \,{\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{4} - 28 \, \sqrt{3}{\left ({\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 5 \, B a^{3} - 2 \, A a^{2} b + 2 \,{\left (5 \, B a^{2} b - 2 \, A a b^{2}\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 ) + 14 \,{\left ({\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 5 \, B a^{3} - 2 \, A a^{2} b + 2 \,{\left (5 \, B a^{2} b - 2 \, A a b^{2}\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 ) - 28 \,{\left ({\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 5 \, B a^{3} - 2 \, A a^{2} b + 2 \,{\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{3}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 84 \,{\left (5 \, B a^{3} - 2 \, A a^{2} b\right )} x}{108 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/108*(27*B*b^3*x^10 - 54*(5*B*a*b^2 - 2*A*b^3)*x^7 - 147*(5*B*a^2*b - 2*A*a*b^2)*x^4 - 28*sqrt(3)*((5*B*a*b^2
- 2*A*b^3)*x^6 + 5*B*a^3 - 2*A*a^2*b + 2*(5*B*a^2*b - 2*A*a*b^2)*x^3)*(-a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*
(-a/b)^(2/3) - sqrt(3)*a)/a) + 14*((5*B*a*b^2 - 2*A*b^3)*x^6 + 5*B*a^3 - 2*A*a^2*b + 2*(5*B*a^2*b - 2*A*a*b^2)
*x^3)*(-a/b)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3)) - 28*((5*B*a*b^2 - 2*A*b^3)*x^6 + 5*B*a^3 - 2*A*a^
2*b + 2*(5*B*a^2*b - 2*A*a*b^2)*x^3)*(-a/b)^(1/3)*log(x - (-a/b)^(1/3)) - 84*(5*B*a^3 - 2*A*a^2*b)*x)/(b^6*x^6
+ 2*a*b^5*x^3 + a^2*b^4)

________________________________________________________________________________________

Sympy [A]  time = 2.60956, size = 162, normalized size = 0.66 \begin{align*} \frac{B x^{4}}{4 b^{3}} - \frac{x^{4} \left (- 13 A a b^{2} + 19 B a^{2} b\right ) + x \left (- 10 A a^{2} b + 16 B a^{3}\right )}{18 a^{2} b^{4} + 36 a b^{5} x^{3} + 18 b^{6} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} b^{13} + 2744 A^{3} a b^{3} - 20580 A^{2} B a^{2} b^{2} + 51450 A B^{2} a^{3} b - 42875 B^{3} a^{4}, \left ( t \mapsto t \log{\left (\frac{27 t b^{4}}{- 14 A b + 35 B a} + x \right )} \right )\right )} - \frac{x \left (- A b + 3 B a\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9*(B*x**3+A)/(b*x**3+a)**3,x)

[Out]

B*x**4/(4*b**3) - (x**4*(-13*A*a*b**2 + 19*B*a**2*b) + x*(-10*A*a**2*b + 16*B*a**3))/(18*a**2*b**4 + 36*a*b**5
*x**3 + 18*b**6*x**6) + RootSum(19683*_t**3*b**13 + 2744*A**3*a*b**3 - 20580*A**2*B*a**2*b**2 + 51450*A*B**2*a
**3*b - 42875*B**3*a**4, Lambda(_t, _t*log(27*_t*b**4/(-14*A*b + 35*B*a) + x))) - x*(-A*b + 3*B*a)/b**4

________________________________________________________________________________________

Giac [A]  time = 1.14003, size = 316, normalized size = 1.3 \begin{align*} \frac{7 \, \sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} A b\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 \, b^{5}} - \frac{7 \,{\left (5 \, B a^{2} - 2 \, A a b\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 b^{4}} + \frac{7 \,{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, b^{5}} - \frac{19 \, B a^{2} b x^{4} - 13 \, A a b^{2} x^{4} + 16 \, B a^{3} x - 10 \, A a^{2} b x}{18 \,{\left (b x^{3} + a\right )}^{2} b^{4}} + \frac{B b^{9} x^{4} - 12 \, B a b^{8} x + 4 \, A b^{9} x}{4 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

7/27*sqrt(3)*(5*(-a*b^2)^(1/3)*B*a - 2*(-a*b^2)^(1/3)*A*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3
))/b^5 - 7/27*(5*B*a^2 - 2*A*a*b)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^4) + 7/54*(5*(-a*b^2)^(1/3)*B*a
- 2*(-a*b^2)^(1/3)*A*b)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^5 - 1/18*(19*B*a^2*b*x^4 - 13*A*a*b^2*x^4
+ 16*B*a^3*x - 10*A*a^2*b*x)/((b*x^3 + a)^2*b^4) + 1/4*(B*b^9*x^4 - 12*B*a*b^8*x + 4*A*b^9*x)/b^12