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

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

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Rubi [A]  time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {444, 37} \begin {gather*} -\frac {\left (A+B x^3\right )^2}{6 \left (a+b x^3\right )^2 (A b-a B)} \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.02, size = 30, normalized size = 0.94 \begin {gather*} -\frac {B \left (a+2 b x^3\right )+A b}{6 b^2 \left (a+b x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.69, size = 42, normalized size = 1.31 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.18, size = 28, normalized size = 0.88 \begin {gather*} -\frac {2 \, B b x^{3} + B a + A b}{6 \, {\left (b x^{3} + a\right )}^{2} b^{2}} \end {gather*}

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)

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maple [A]  time = 0.05, size = 39, normalized size = 1.22 \begin {gather*} -\frac {B}{3 \left (b \,x^{3}+a \right ) b^{2}}-\frac {A b -B a}{6 \left (b \,x^{3}+a \right )^{2} b^{2}} \end {gather*}

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]

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

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maxima [A]  time = 0.46, size = 42, normalized size = 1.31 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 2.33, size = 44, normalized size = 1.38 \begin {gather*} -\frac {\frac {A\,b+B\,a}{6\,b^2}+\frac {B\,x^3}{3\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.82, size = 42, normalized size = 1.31 \begin {gather*} \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 {gather*}

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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