3.71 \(\int \frac{x^9 (A+B x^3)}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=233 \[ -\frac{a^{4/3} (7 A b-10 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{13/3}}+\frac{a^{4/3} (7 A b-10 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{13/3}}-\frac{a^{4/3} (7 A b-10 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{13/3}}-\frac{x^7 (7 A b-10 a B)}{21 a b^2}+\frac{x^4 (7 A b-10 a B)}{12 b^3}-\frac{a x (7 A b-10 a B)}{3 b^4}+\frac{x^{10} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

-(a*(7*A*b - 10*a*B)*x)/(3*b^4) + ((7*A*b - 10*a*B)*x^4)/(12*b^3) - ((7*A*b - 10*a*B)*x^7)/(21*a*b^2) + ((A*b
- a*B)*x^10)/(3*a*b*(a + b*x^3)) - (a^(4/3)*(7*A*b - 10*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))]
)/(3*Sqrt[3]*b^(13/3)) + (a^(4/3)*(7*A*b - 10*a*B)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(13/3)) - (a^(4/3)*(7*A*b -
10*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*b^(13/3))

________________________________________________________________________________________

Rubi [A]  time = 0.138068, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {457, 302, 200, 31, 634, 617, 204, 628} \[ -\frac{a^{4/3} (7 A b-10 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{13/3}}+\frac{a^{4/3} (7 A b-10 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{13/3}}-\frac{a^{4/3} (7 A b-10 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{13/3}}-\frac{x^7 (7 A b-10 a B)}{21 a b^2}+\frac{x^4 (7 A b-10 a B)}{12 b^3}-\frac{a x (7 A b-10 a B)}{3 b^4}+\frac{x^{10} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x^9*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

-(a*(7*A*b - 10*a*B)*x)/(3*b^4) + ((7*A*b - 10*a*B)*x^4)/(12*b^3) - ((7*A*b - 10*a*B)*x^7)/(21*a*b^2) + ((A*b
- a*B)*x^10)/(3*a*b*(a + b*x^3)) - (a^(4/3)*(7*A*b - 10*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))]
)/(3*Sqrt[3]*b^(13/3)) + (a^(4/3)*(7*A*b - 10*a*B)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(13/3)) - (a^(4/3)*(7*A*b -
10*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*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 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 )^2} \, dx &=\frac{(A b-a B) x^{10}}{3 a b \left (a+b x^3\right )}+\frac{(-7 A b+10 a B) \int \frac{x^9}{a+b x^3} \, dx}{3 a b}\\ &=\frac{(A b-a B) x^{10}}{3 a b \left (a+b x^3\right )}+\frac{(-7 A b+10 a B) \int \left (\frac{a^2}{b^3}-\frac{a x^3}{b^2}+\frac{x^6}{b}-\frac{a^3}{b^3 \left (a+b x^3\right )}\right ) \, dx}{3 a b}\\ &=-\frac{a (7 A b-10 a B) x}{3 b^4}+\frac{(7 A b-10 a B) x^4}{12 b^3}-\frac{(7 A b-10 a B) x^7}{21 a b^2}+\frac{(A b-a B) x^{10}}{3 a b \left (a+b x^3\right )}+\frac{\left (a^2 (7 A b-10 a B)\right ) \int \frac{1}{a+b x^3} \, dx}{3 b^4}\\ &=-\frac{a (7 A b-10 a B) x}{3 b^4}+\frac{(7 A b-10 a B) x^4}{12 b^3}-\frac{(7 A b-10 a B) x^7}{21 a b^2}+\frac{(A b-a B) x^{10}}{3 a b \left (a+b x^3\right )}+\frac{\left (a^{4/3} (7 A b-10 a B)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 b^4}+\frac{\left (a^{4/3} (7 A b-10 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}{9 b^4}\\ &=-\frac{a (7 A b-10 a B) x}{3 b^4}+\frac{(7 A b-10 a B) x^4}{12 b^3}-\frac{(7 A b-10 a B) x^7}{21 a b^2}+\frac{(A b-a B) x^{10}}{3 a b \left (a+b x^3\right )}+\frac{a^{4/3} (7 A b-10 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{13/3}}-\frac{\left (a^{4/3} (7 A b-10 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}{18 b^{13/3}}+\frac{\left (a^{5/3} (7 A b-10 a B)\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^4}\\ &=-\frac{a (7 A b-10 a B) x}{3 b^4}+\frac{(7 A b-10 a B) x^4}{12 b^3}-\frac{(7 A b-10 a B) x^7}{21 a b^2}+\frac{(A b-a B) x^{10}}{3 a b \left (a+b x^3\right )}+\frac{a^{4/3} (7 A b-10 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{13/3}}-\frac{a^{4/3} (7 A b-10 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{13/3}}+\frac{\left (a^{4/3} (7 A b-10 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 )}{3 b^{13/3}}\\ &=-\frac{a (7 A b-10 a B) x}{3 b^4}+\frac{(7 A b-10 a B) x^4}{12 b^3}-\frac{(7 A b-10 a B) x^7}{21 a b^2}+\frac{(A b-a B) x^{10}}{3 a b \left (a+b x^3\right )}-\frac{a^{4/3} (7 A b-10 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{13/3}}+\frac{a^{4/3} (7 A b-10 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{13/3}}-\frac{a^{4/3} (7 A b-10 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{13/3}}\\ \end{align*}

Mathematica [A]  time = 0.13434, size = 203, normalized size = 0.87 \[ \frac{14 a^{4/3} (10 a B-7 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\frac{84 a^2 \sqrt [3]{b} x (a B-A b)}{a+b x^3}-28 a^{4/3} (10 a B-7 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+28 \sqrt{3} a^{4/3} (10 a B-7 A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+63 b^{4/3} x^4 (A b-2 a B)+252 a \sqrt [3]{b} x (3 a B-2 A b)+36 b^{7/3} B x^7}{252 b^{13/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^9*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

(252*a*b^(1/3)*(-2*A*b + 3*a*B)*x + 63*b^(4/3)*(A*b - 2*a*B)*x^4 + 36*b^(7/3)*B*x^7 + (84*a^2*b^(1/3)*(-(A*b)
+ a*B)*x)/(a + b*x^3) + 28*Sqrt[3]*a^(4/3)*(-7*A*b + 10*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 28*
a^(4/3)*(-7*A*b + 10*a*B)*Log[a^(1/3) + b^(1/3)*x] + 14*a^(4/3)*(-7*A*b + 10*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3
)*x + b^(2/3)*x^2])/(252*b^(13/3))

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Maple [A]  time = 0.017, size = 288, normalized size = 1.2 \begin{align*}{\frac{B{x}^{7}}{7\,{b}^{2}}}+{\frac{A{x}^{4}}{4\,{b}^{2}}}-{\frac{B{x}^{4}a}{2\,{b}^{3}}}-2\,{\frac{aAx}{{b}^{3}}}+3\,{\frac{{a}^{2}Bx}{{b}^{4}}}-{\frac{{a}^{2}Ax}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{{a}^{3}xB}{3\,{b}^{4} \left ( b{x}^{3}+a \right ) }}+{\frac{7\,A{a}^{2}}{9\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{7\,A{a}^{2}}{18\,{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{7\,A{a}^{2}\sqrt{3}}{9\,{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{10\,B{a}^{3}}{9\,{b}^{5}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,B{a}^{3}}{9\,{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{10\,B{a}^{3}\sqrt{3}}{9\,{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)^2,x)

[Out]

1/7/b^2*B*x^7+1/4/b^2*A*x^4-1/2/b^3*B*x^4*a-2/b^3*a*A*x+3/b^4*a^2*B*x-1/3*a^2/b^3*x/(b*x^3+a)*A+1/3*a^3/b^4*x/
(b*x^3+a)*B+7/9*a^2/b^4*A/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-7/18*a^2/b^4*A/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^
(2/3))+7/9*a^2/b^4*A/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-10/9*a^3/b^5*B/(a/b)^(2/3)*ln
(x+(a/b)^(1/3))+5/9*a^3/b^5*B/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-10/9*a^3/b^5*B/(a/b)^(2/3)*3^(1/2)
*arctan(1/3*3^(1/2)*(2/(a/b)^(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(x^9*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.80624, size = 636, normalized size = 2.73 \begin{align*} \frac{36 \, B b^{3} x^{10} - 9 \,{\left (10 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 63 \,{\left (10 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 28 \, \sqrt{3}{\left (10 \, B a^{3} - 7 \, A a^{2} b +{\left (10 \, B a^{2} b - 7 \, 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 (10 \, B a^{3} - 7 \, A a^{2} b +{\left (10 \, B a^{2} b - 7 \, 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 (10 \, B a^{3} - 7 \, A a^{2} b +{\left (10 \, B a^{2} b - 7 \, 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 (10 \, B a^{3} - 7 \, A a^{2} b\right )} x}{252 \,{\left (b^{5} x^{3} + a 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)^2,x, algorithm="fricas")

[Out]

1/252*(36*B*b^3*x^10 - 9*(10*B*a*b^2 - 7*A*b^3)*x^7 + 63*(10*B*a^2*b - 7*A*a*b^2)*x^4 - 28*sqrt(3)*(10*B*a^3 -
7*A*a^2*b + (10*B*a^2*b - 7*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*(10*B*a^3 - 7*A*a^2*b + (10*B*a^2*b - 7*A*a*b^2)*x^3)*(a/b)^(1/3)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3)) -
28*(10*B*a^3 - 7*A*a^2*b + (10*B*a^2*b - 7*A*a*b^2)*x^3)*(a/b)^(1/3)*log(x + (a/b)^(1/3)) + 84*(10*B*a^3 - 7*
A*a^2*b)*x)/(b^5*x^3 + a*b^4)

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Sympy [A]  time = 1.58968, size = 153, normalized size = 0.66 \begin{align*} \frac{B x^{7}}{7 b^{2}} + \frac{x \left (- A a^{2} b + B a^{3}\right )}{3 a b^{4} + 3 b^{5} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} b^{13} - 343 A^{3} a^{4} b^{3} + 1470 A^{2} B a^{5} b^{2} - 2100 A B^{2} a^{6} b + 1000 B^{3} a^{7}, \left ( t \mapsto t \log{\left (- \frac{9 t b^{4}}{- 7 A a b + 10 B a^{2}} + x \right )} \right )\right )} - \frac{x^{4} \left (- A b + 2 B a\right )}{4 b^{3}} + \frac{x \left (- 2 A a b + 3 B a^{2}\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)**2,x)

[Out]

B*x**7/(7*b**2) + x*(-A*a**2*b + B*a**3)/(3*a*b**4 + 3*b**5*x**3) + RootSum(729*_t**3*b**13 - 343*A**3*a**4*b*
*3 + 1470*A**2*B*a**5*b**2 - 2100*A*B**2*a**6*b + 1000*B**3*a**7, Lambda(_t, _t*log(-9*_t*b**4/(-7*A*a*b + 10*
B*a**2) + x))) - x**4*(-A*b + 2*B*a)/(4*b**3) + x*(-2*A*a*b + 3*B*a**2)/b**4

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Giac [A]  time = 1.12912, size = 329, normalized size = 1.41 \begin{align*} -\frac{\sqrt{3}{\left (10 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a^{2} - 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} A 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 )}{9 \, b^{5}} + \frac{{\left (10 \, B a^{3} - 7 \, A a^{2} b\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^{4}} - \frac{{\left (10 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a^{2} - 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} A 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 )}{18 \, b^{5}} + \frac{B a^{3} x - A a^{2} b x}{3 \,{\left (b x^{3} + a\right )} b^{4}} + \frac{4 \, B b^{12} x^{7} - 14 \, B a b^{11} x^{4} + 7 \, A b^{12} x^{4} + 84 \, B a^{2} b^{10} x - 56 \, A a b^{11} x}{28 \, b^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="giac")

[Out]

-1/9*sqrt(3)*(10*(-a*b^2)^(1/3)*B*a^2 - 7*(-a*b^2)^(1/3)*A*a*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)
^(1/3))/b^5 + 1/9*(10*B*a^3 - 7*A*a^2*b)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^4) - 1/18*(10*(-a*b^2)^(
1/3)*B*a^2 - 7*(-a*b^2)^(1/3)*A*a*b)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^5 + 1/3*(B*a^3*x - A*a^2*b*x)/
((b*x^3 + a)*b^4) + 1/28*(4*B*b^12*x^7 - 14*B*a*b^11*x^4 + 7*A*b^12*x^4 + 84*B*a^2*b^10*x - 56*A*a*b^11*x)/b^1
4