3.704 \(\int \frac{x^3}{\sqrt [3]{a+b x^2}} \, dx\)

Optimal. Leaf size=38 \[ \frac{3 \left (a+b x^2\right )^{5/3}}{10 b^2}-\frac{3 a \left (a+b x^2\right )^{2/3}}{4 b^2} \]

[Out]

(-3*a*(a + b*x^2)^(2/3))/(4*b^2) + (3*(a + b*x^2)^(5/3))/(10*b^2)

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Rubi [A]  time = 0.0225681, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{3 \left (a+b x^2\right )^{5/3}}{10 b^2}-\frac{3 a \left (a+b x^2\right )^{2/3}}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^2)^(1/3),x]

[Out]

(-3*a*(a + b*x^2)^(2/3))/(4*b^2) + (3*(a + b*x^2)^(5/3))/(10*b^2)

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^3}{\sqrt [3]{a+b x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\sqrt [3]{a+b x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b \sqrt [3]{a+b x}}+\frac{(a+b x)^{2/3}}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 a \left (a+b x^2\right )^{2/3}}{4 b^2}+\frac{3 \left (a+b x^2\right )^{5/3}}{10 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0117262, size = 28, normalized size = 0.74 \[ \frac{3 \left (a+b x^2\right )^{2/3} \left (2 b x^2-3 a\right )}{20 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^2)^(1/3),x]

[Out]

(3*(a + b*x^2)^(2/3)*(-3*a + 2*b*x^2))/(20*b^2)

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Maple [A]  time = 0.004, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-6\,b{x}^{2}+9\,a}{20\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^2+a)^(1/3),x)

[Out]

-3/20*(b*x^2+a)^(2/3)*(-2*b*x^2+3*a)/b^2

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Maxima [A]  time = 1.94062, size = 41, normalized size = 1.08 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}}}{10 \, b^{2}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a}{4 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^(1/3),x, algorithm="maxima")

[Out]

3/10*(b*x^2 + a)^(5/3)/b^2 - 3/4*(b*x^2 + a)^(2/3)*a/b^2

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Fricas [A]  time = 1.69788, size = 59, normalized size = 1.55 \begin{align*} \frac{3 \,{\left (2 \, b x^{2} - 3 \, a\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{20 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^(1/3),x, algorithm="fricas")

[Out]

3/20*(2*b*x^2 - 3*a)*(b*x^2 + a)^(2/3)/b^2

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Sympy [B]  time = 1.00957, size = 178, normalized size = 4.68 \begin{align*} - \frac{9 a^{\frac{11}{3}} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{20 a^{2} b^{2} + 20 a b^{3} x^{2}} + \frac{9 a^{\frac{11}{3}}}{20 a^{2} b^{2} + 20 a b^{3} x^{2}} - \frac{3 a^{\frac{8}{3}} b x^{2} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{20 a^{2} b^{2} + 20 a b^{3} x^{2}} + \frac{9 a^{\frac{8}{3}} b x^{2}}{20 a^{2} b^{2} + 20 a b^{3} x^{2}} + \frac{6 a^{\frac{5}{3}} b^{2} x^{4} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{20 a^{2} b^{2} + 20 a b^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**2+a)**(1/3),x)

[Out]

-9*a**(11/3)*(1 + b*x**2/a)**(2/3)/(20*a**2*b**2 + 20*a*b**3*x**2) + 9*a**(11/3)/(20*a**2*b**2 + 20*a*b**3*x**
2) - 3*a**(8/3)*b*x**2*(1 + b*x**2/a)**(2/3)/(20*a**2*b**2 + 20*a*b**3*x**2) + 9*a**(8/3)*b*x**2/(20*a**2*b**2
+ 20*a*b**3*x**2) + 6*a**(5/3)*b**2*x**4*(1 + b*x**2/a)**(2/3)/(20*a**2*b**2 + 20*a*b**3*x**2)

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Giac [A]  time = 2.58322, size = 39, normalized size = 1.03 \begin{align*} \frac{3 \,{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a\right )}}{20 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^(1/3),x, algorithm="giac")

[Out]

3/20*(2*(b*x^2 + a)^(5/3) - 5*(b*x^2 + a)^(2/3)*a)/b^2