3.88 \(\int \frac{x^{11} (A+B x^3)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=107 \[ -\frac{a^2 (3 A b-4 a B)}{3 b^5 \left (a+b x^3\right )}+\frac{a^3 (A b-a B)}{6 b^5 \left (a+b x^3\right )^2}+\frac{x^3 (A b-3 a B)}{3 b^4}-\frac{a (A b-2 a B) \log \left (a+b x^3\right )}{b^5}+\frac{B x^6}{6 b^3} \]

[Out]

((A*b - 3*a*B)*x^3)/(3*b^4) + (B*x^6)/(6*b^3) + (a^3*(A*b - a*B))/(6*b^5*(a + b*x^3)^2) - (a^2*(3*A*b - 4*a*B)
)/(3*b^5*(a + b*x^3)) - (a*(A*b - 2*a*B)*Log[a + b*x^3])/b^5

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Rubi [A]  time = 0.140116, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 77} \[ -\frac{a^2 (3 A b-4 a B)}{3 b^5 \left (a+b x^3\right )}+\frac{a^3 (A b-a B)}{6 b^5 \left (a+b x^3\right )^2}+\frac{x^3 (A b-3 a B)}{3 b^4}-\frac{a (A b-2 a B) \log \left (a+b x^3\right )}{b^5}+\frac{B x^6}{6 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^11*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

((A*b - 3*a*B)*x^3)/(3*b^4) + (B*x^6)/(6*b^3) + (a^3*(A*b - a*B))/(6*b^5*(a + b*x^3)^2) - (a^2*(3*A*b - 4*a*B)
)/(3*b^5*(a + b*x^3)) - (a*(A*b - 2*a*B)*Log[a + b*x^3])/b^5

Rule 446

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{x^{11} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3 (A+B x)}{(a+b x)^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{A b-3 a B}{b^4}+\frac{B x}{b^3}+\frac{a^3 (-A b+a B)}{b^4 (a+b x)^3}-\frac{a^2 (-3 A b+4 a B)}{b^4 (a+b x)^2}+\frac{3 a (-A b+2 a B)}{b^4 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{(A b-3 a B) x^3}{3 b^4}+\frac{B x^6}{6 b^3}+\frac{a^3 (A b-a B)}{6 b^5 \left (a+b x^3\right )^2}-\frac{a^2 (3 A b-4 a B)}{3 b^5 \left (a+b x^3\right )}-\frac{a (A b-2 a B) \log \left (a+b x^3\right )}{b^5}\\ \end{align*}

Mathematica [A]  time = 0.0651171, size = 94, normalized size = 0.88 \[ \frac{\frac{2 a^2 (4 a B-3 A b)}{a+b x^3}+\frac{a^3 (A b-a B)}{\left (a+b x^3\right )^2}+2 b x^3 (A b-3 a B)+6 a (2 a B-A b) \log \left (a+b x^3\right )+b^2 B x^6}{6 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^11*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

(2*b*(A*b - 3*a*B)*x^3 + b^2*B*x^6 + (a^3*(A*b - a*B))/(a + b*x^3)^2 + (2*a^2*(-3*A*b + 4*a*B))/(a + b*x^3) +
6*a*(-(A*b) + 2*a*B)*Log[a + b*x^3])/(6*b^5)

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Maple [A]  time = 0.012, size = 134, normalized size = 1.3 \begin{align*}{\frac{B{x}^{6}}{6\,{b}^{3}}}+{\frac{A{x}^{3}}{3\,{b}^{3}}}-{\frac{B{x}^{3}a}{{b}^{4}}}-{\frac{A{a}^{2}}{{b}^{4} \left ( b{x}^{3}+a \right ) }}+{\frac{4\,B{a}^{3}}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }}-{\frac{a\ln \left ( b{x}^{3}+a \right ) A}{{b}^{4}}}+2\,{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) B}{{b}^{5}}}+{\frac{{a}^{3}A}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{B{a}^{4}}{6\,{b}^{5} \left ( b{x}^{3}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(B*x^3+A)/(b*x^3+a)^3,x)

[Out]

1/6*B*x^6/b^3+1/3/b^3*A*x^3-1/b^4*B*x^3*a-a^2/b^4/(b*x^3+a)*A+4/3*a^3/b^5/(b*x^3+a)*B-a/b^4*ln(b*x^3+a)*A+2*a^
2/b^5*ln(b*x^3+a)*B+1/6*a^3/b^4/(b*x^3+a)^2*A-1/6*a^4/b^5/(b*x^3+a)^2*B

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Maxima [A]  time = 0.957149, size = 155, normalized size = 1.45 \begin{align*} \frac{7 \, B a^{4} - 5 \, A a^{3} b + 2 \,{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{3}}{6 \,{\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} + \frac{B b x^{6} - 2 \,{\left (3 \, B a - A b\right )} x^{3}}{6 \, b^{4}} + \frac{{\left (2 \, B a^{2} - A a b\right )} \log \left (b x^{3} + a\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

1/6*(7*B*a^4 - 5*A*a^3*b + 2*(4*B*a^3*b - 3*A*a^2*b^2)*x^3)/(b^7*x^6 + 2*a*b^6*x^3 + a^2*b^5) + 1/6*(B*b*x^6 -
2*(3*B*a - A*b)*x^3)/b^4 + (2*B*a^2 - A*a*b)*log(b*x^3 + a)/b^5

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Fricas [A]  time = 1.43394, size = 362, normalized size = 3.38 \begin{align*} \frac{B b^{4} x^{12} - 2 \,{\left (2 \, B a b^{3} - A b^{4}\right )} x^{9} -{\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6} + 7 \, B a^{4} - 5 \, A a^{3} b + 2 \,{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{3} + 6 \,{\left ({\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{6} + 2 \, B a^{4} - A a^{3} b + 2 \,{\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/6*(B*b^4*x^12 - 2*(2*B*a*b^3 - A*b^4)*x^9 - (11*B*a^2*b^2 - 4*A*a*b^3)*x^6 + 7*B*a^4 - 5*A*a^3*b + 2*(B*a^3*
b - 2*A*a^2*b^2)*x^3 + 6*((2*B*a^2*b^2 - A*a*b^3)*x^6 + 2*B*a^4 - A*a^3*b + 2*(2*B*a^3*b - A*a^2*b^2)*x^3)*log
(b*x^3 + a))/(b^7*x^6 + 2*a*b^6*x^3 + a^2*b^5)

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Sympy [A]  time = 3.09166, size = 112, normalized size = 1.05 \begin{align*} \frac{B x^{6}}{6 b^{3}} + \frac{a \left (- A b + 2 B a\right ) \log{\left (a + b x^{3} \right )}}{b^{5}} + \frac{- 5 A a^{3} b + 7 B a^{4} + x^{3} \left (- 6 A a^{2} b^{2} + 8 B a^{3} b\right )}{6 a^{2} b^{5} + 12 a b^{6} x^{3} + 6 b^{7} x^{6}} - \frac{x^{3} \left (- A b + 3 B a\right )}{3 b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(B*x**3+A)/(b*x**3+a)**3,x)

[Out]

B*x**6/(6*b**3) + a*(-A*b + 2*B*a)*log(a + b*x**3)/b**5 + (-5*A*a**3*b + 7*B*a**4 + x**3*(-6*A*a**2*b**2 + 8*B
*a**3*b))/(6*a**2*b**5 + 12*a*b**6*x**3 + 6*b**7*x**6) - x**3*(-A*b + 3*B*a)/(3*b**4)

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Giac [A]  time = 1.11545, size = 177, normalized size = 1.65 \begin{align*} \frac{{\left (2 \, B a^{2} - A a b\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{b^{5}} + \frac{B b^{3} x^{6} - 6 \, B a b^{2} x^{3} + 2 \, A b^{3} x^{3}}{6 \, b^{6}} - \frac{18 \, B a^{2} b^{2} x^{6} - 9 \, A a b^{3} x^{6} + 28 \, B a^{3} b x^{3} - 12 \, A a^{2} b^{2} x^{3} + 11 \, B a^{4} - 4 \, A a^{3} b}{6 \,{\left (b x^{3} + a\right )}^{2} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

(2*B*a^2 - A*a*b)*log(abs(b*x^3 + a))/b^5 + 1/6*(B*b^3*x^6 - 6*B*a*b^2*x^3 + 2*A*b^3*x^3)/b^6 - 1/6*(18*B*a^2*
b^2*x^6 - 9*A*a*b^3*x^6 + 28*B*a^3*b*x^3 - 12*A*a^2*b^2*x^3 + 11*B*a^4 - 4*A*a^3*b)/((b*x^3 + a)^2*b^5)