3.727 \(\int \frac{x^5}{(a+b x^2)^{4/3}} \, dx\)

Optimal. Leaf size=59 \[ -\frac{3 a^2}{2 b^3 \sqrt [3]{a+b x^2}}-\frac{3 a \left (a+b x^2\right )^{2/3}}{2 b^3}+\frac{3 \left (a+b x^2\right )^{5/3}}{10 b^3} \]

[Out]

(-3*a^2)/(2*b^3*(a + b*x^2)^(1/3)) - (3*a*(a + b*x^2)^(2/3))/(2*b^3) + (3*(a + b*x^2)^(5/3))/(10*b^3)

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Rubi [A]  time = 0.0332049, antiderivative size = 59, 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} \[ -\frac{3 a^2}{2 b^3 \sqrt [3]{a+b x^2}}-\frac{3 a \left (a+b x^2\right )^{2/3}}{2 b^3}+\frac{3 \left (a+b x^2\right )^{5/3}}{10 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x^2)^(4/3),x]

[Out]

(-3*a^2)/(2*b^3*(a + b*x^2)^(1/3)) - (3*a*(a + b*x^2)^(2/3))/(2*b^3) + (3*(a + b*x^2)^(5/3))/(10*b^3)

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

Mathematica [A]  time = 0.015648, size = 38, normalized size = 0.64 \[ \frac{3 \left (-9 a^2-3 a b x^2+b^2 x^4\right )}{10 b^3 \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x^2)^(4/3),x]

[Out]

(3*(-9*a^2 - 3*a*b*x^2 + b^2*x^4))/(10*b^3*(a + b*x^2)^(1/3))

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Maple [A]  time = 0.005, size = 36, normalized size = 0.6 \begin{align*} -{\frac{-3\,{b}^{2}{x}^{4}+9\,ab{x}^{2}+27\,{a}^{2}}{10\,{b}^{3}}{\frac{1}{\sqrt [3]{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^2+a)^(4/3),x)

[Out]

-3/10/(b*x^2+a)^(1/3)*(-b^2*x^4+3*a*b*x^2+9*a^2)/b^3

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Maxima [A]  time = 1.65963, size = 63, normalized size = 1.07 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}}}{10 \, b^{3}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a}{2 \, b^{3}} - \frac{3 \, a^{2}}{2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^(4/3),x, algorithm="maxima")

[Out]

3/10*(b*x^2 + a)^(5/3)/b^3 - 3/2*(b*x^2 + a)^(2/3)*a/b^3 - 3/2*a^2/((b*x^2 + a)^(1/3)*b^3)

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Fricas [A]  time = 1.74528, size = 97, normalized size = 1.64 \begin{align*} \frac{3 \,{\left (b^{2} x^{4} - 3 \, a b x^{2} - 9 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{10 \,{\left (b^{4} x^{2} + a b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^(4/3),x, algorithm="fricas")

[Out]

3/10*(b^2*x^4 - 3*a*b*x^2 - 9*a^2)*(b*x^2 + a)^(2/3)/(b^4*x^2 + a*b^3)

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Sympy [B]  time = 1.63656, size = 561, normalized size = 9.51 \begin{align*} - \frac{27 a^{\frac{29}{3}} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} + \frac{27 a^{\frac{29}{3}}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} - \frac{63 a^{\frac{26}{3}} b x^{2} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} + \frac{81 a^{\frac{26}{3}} b x^{2}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} - \frac{42 a^{\frac{23}{3}} b^{2} x^{4} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} + \frac{81 a^{\frac{23}{3}} b^{2} x^{4}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} - \frac{3 a^{\frac{20}{3}} b^{3} x^{6} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} + \frac{27 a^{\frac{20}{3}} b^{3} x^{6}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} + \frac{3 a^{\frac{17}{3}} b^{4} x^{8} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{10 a^{8} b^{3} + 30 a^{7} b^{4} x^{2} + 30 a^{6} b^{5} x^{4} + 10 a^{5} b^{6} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x**2+a)**(4/3),x)

[Out]

-27*a**(29/3)*(1 + b*x**2/a)**(2/3)/(10*a**8*b**3 + 30*a**7*b**4*x**2 + 30*a**6*b**5*x**4 + 10*a**5*b**6*x**6)
+ 27*a**(29/3)/(10*a**8*b**3 + 30*a**7*b**4*x**2 + 30*a**6*b**5*x**4 + 10*a**5*b**6*x**6) - 63*a**(26/3)*b*x*
*2*(1 + b*x**2/a)**(2/3)/(10*a**8*b**3 + 30*a**7*b**4*x**2 + 30*a**6*b**5*x**4 + 10*a**5*b**6*x**6) + 81*a**(2
6/3)*b*x**2/(10*a**8*b**3 + 30*a**7*b**4*x**2 + 30*a**6*b**5*x**4 + 10*a**5*b**6*x**6) - 42*a**(23/3)*b**2*x**
4*(1 + b*x**2/a)**(2/3)/(10*a**8*b**3 + 30*a**7*b**4*x**2 + 30*a**6*b**5*x**4 + 10*a**5*b**6*x**6) + 81*a**(23
/3)*b**2*x**4/(10*a**8*b**3 + 30*a**7*b**4*x**2 + 30*a**6*b**5*x**4 + 10*a**5*b**6*x**6) - 3*a**(20/3)*b**3*x*
*6*(1 + b*x**2/a)**(2/3)/(10*a**8*b**3 + 30*a**7*b**4*x**2 + 30*a**6*b**5*x**4 + 10*a**5*b**6*x**6) + 27*a**(2
0/3)*b**3*x**6/(10*a**8*b**3 + 30*a**7*b**4*x**2 + 30*a**6*b**5*x**4 + 10*a**5*b**6*x**6) + 3*a**(17/3)*b**4*x
**8*(1 + b*x**2/a)**(2/3)/(10*a**8*b**3 + 30*a**7*b**4*x**2 + 30*a**6*b**5*x**4 + 10*a**5*b**6*x**6)

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Giac [A]  time = 2.76319, size = 55, normalized size = 0.93 \begin{align*} \frac{3 \,{\left ({\left (b x^{2} + a\right )}^{\frac{5}{3}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a - \frac{5 \, a^{2}}{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}\right )}}{10 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^(4/3),x, algorithm="giac")

[Out]

3/10*((b*x^2 + a)^(5/3) - 5*(b*x^2 + a)^(2/3)*a - 5*a^2/(b*x^2 + a)^(1/3))/b^3