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

Optimal. Leaf size=32 \[ -\frac{\left (A+B x^3\right )^2}{6 \left (a+b x^3\right )^2 (A b-a B)} \]

[Out]

-(A + B*x^3)^2/(6*(A*b - a*B)*(a + b*x^3)^2)

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Rubi [A]  time = 0.022282, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {444, 37} \[ -\frac{\left (A+B x^3\right )^2}{6 \left (a+b x^3\right )^2 (A b-a B)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A + B*x^3)^2/(6*(A*b - a*B)*(a + b*x^3)^2)

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0134919, size = 30, normalized size = 0.94 \[ -\frac{B \left (a+2 b x^3\right )+A b}{6 b^2 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A*b + B*(a + 2*b*x^3))/(6*b^2*(a + b*x^3)^2)

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Maple [A]  time = 0.008, size = 39, normalized size = 1.2 \begin{align*} -{\frac{B}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{Ab-Ba}{6\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(B*x^3+A)/(b*x^3+a)^3,x)

[Out]

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

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Maxima [A]  time = 0.963658, size = 57, normalized size = 1.78 \begin{align*} -\frac{2 \, B b x^{3} + B a + A b}{6 \,{\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.3627, size = 86, normalized size = 2.69 \begin{align*} -\frac{2 \, B b x^{3} + B a + A b}{6 \,{\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.11282, size = 42, normalized size = 1.31 \begin{align*} - \frac{A b + B a + 2 B b x^{3}}{6 a^{2} b^{2} + 12 a b^{3} x^{3} + 6 b^{4} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(B*x**3+A)/(b*x**3+a)**3,x)

[Out]

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

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Giac [A]  time = 1.10107, size = 38, normalized size = 1.19 \begin{align*} -\frac{2 \, B b x^{3} + B a + A b}{6 \,{\left (b x^{3} + a\right )}^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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