3.666 \(\int x^5 \sqrt [3]{a+b x^2} \, dx\)

Optimal. Leaf size=59 \[ \frac{3 a^2 \left (a+b x^2\right )^{4/3}}{8 b^3}+\frac{3 \left (a+b x^2\right )^{10/3}}{20 b^3}-\frac{3 a \left (a+b x^2\right )^{7/3}}{7 b^3} \]

[Out]

(3*a^2*(a + b*x^2)^(4/3))/(8*b^3) - (3*a*(a + b*x^2)^(7/3))/(7*b^3) + (3*(a + b*x^2)^(10/3))/(20*b^3)

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Rubi [A]  time = 0.034617, 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} \[ \frac{3 a^2 \left (a+b x^2\right )^{4/3}}{8 b^3}+\frac{3 \left (a+b x^2\right )^{10/3}}{20 b^3}-\frac{3 a \left (a+b x^2\right )^{7/3}}{7 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^2)^(1/3),x]

[Out]

(3*a^2*(a + b*x^2)^(4/3))/(8*b^3) - (3*a*(a + b*x^2)^(7/3))/(7*b^3) + (3*(a + b*x^2)^(10/3))/(20*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 x^5 \sqrt [3]{a+b x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 \sqrt [3]{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2 \sqrt [3]{a+b x}}{b^2}-\frac{2 a (a+b x)^{4/3}}{b^2}+\frac{(a+b x)^{7/3}}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac{3 a^2 \left (a+b x^2\right )^{4/3}}{8 b^3}-\frac{3 a \left (a+b x^2\right )^{7/3}}{7 b^3}+\frac{3 \left (a+b x^2\right )^{10/3}}{20 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0174325, size = 39, normalized size = 0.66 \[ \frac{3 \left (a+b x^2\right )^{4/3} \left (9 a^2-12 a b x^2+14 b^2 x^4\right )}{280 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^2)^(1/3),x]

[Out]

(3*(a + b*x^2)^(4/3)*(9*a^2 - 12*a*b*x^2 + 14*b^2*x^4))/(280*b^3)

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Maple [A]  time = 0.003, size = 36, normalized size = 0.6 \begin{align*}{\frac{42\,{b}^{2}{x}^{4}-36\,ab{x}^{2}+27\,{a}^{2}}{280\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x^2+a)^(1/3),x)

[Out]

3/280*(b*x^2+a)^(4/3)*(14*b^2*x^4-12*a*b*x^2+9*a^2)/b^3

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Maxima [A]  time = 2.3167, size = 63, normalized size = 1.07 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}}}{20 \, b^{3}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a}{7 \, b^{3}} + \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{2}}{8 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^2+a)^(1/3),x, algorithm="maxima")

[Out]

3/20*(b*x^2 + a)^(10/3)/b^3 - 3/7*(b*x^2 + a)^(7/3)*a/b^3 + 3/8*(b*x^2 + a)^(4/3)*a^2/b^3

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Fricas [A]  time = 1.46205, size = 105, normalized size = 1.78 \begin{align*} \frac{3 \,{\left (14 \, b^{3} x^{6} + 2 \, a b^{2} x^{4} - 3 \, a^{2} b x^{2} + 9 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{280 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^2+a)^(1/3),x, algorithm="fricas")

[Out]

3/280*(14*b^3*x^6 + 2*a*b^2*x^4 - 3*a^2*b*x^2 + 9*a^3)*(b*x^2 + a)^(1/3)/b^3

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Sympy [B]  time = 1.7159, size = 700, normalized size = 11.86 \begin{align*} \frac{27 a^{\frac{34}{3}} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} - \frac{27 a^{\frac{34}{3}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac{72 a^{\frac{31}{3}} b x^{2} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} - \frac{81 a^{\frac{31}{3}} b x^{2}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac{60 a^{\frac{28}{3}} b^{2} x^{4} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} - \frac{81 a^{\frac{28}{3}} b^{2} x^{4}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac{60 a^{\frac{25}{3}} b^{3} x^{6} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} - \frac{27 a^{\frac{25}{3}} b^{3} x^{6}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac{135 a^{\frac{22}{3}} b^{4} x^{8} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac{132 a^{\frac{19}{3}} b^{5} x^{10} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac{42 a^{\frac{16}{3}} b^{6} x^{12} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 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)**(1/3),x)

[Out]

27*a**(34/3)*(1 + b*x**2/a)**(1/3)/(280*a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**6*b**5*x**4 + 280*a**5*b**6*x*
*6) - 27*a**(34/3)/(280*a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**6*b**5*x**4 + 280*a**5*b**6*x**6) + 72*a**(31/
3)*b*x**2*(1 + b*x**2/a)**(1/3)/(280*a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**6*b**5*x**4 + 280*a**5*b**6*x**6)
- 81*a**(31/3)*b*x**2/(280*a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**6*b**5*x**4 + 280*a**5*b**6*x**6) + 60*a**
(28/3)*b**2*x**4*(1 + b*x**2/a)**(1/3)/(280*a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**6*b**5*x**4 + 280*a**5*b**
6*x**6) - 81*a**(28/3)*b**2*x**4/(280*a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**6*b**5*x**4 + 280*a**5*b**6*x**6
) + 60*a**(25/3)*b**3*x**6*(1 + b*x**2/a)**(1/3)/(280*a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**6*b**5*x**4 + 28
0*a**5*b**6*x**6) - 27*a**(25/3)*b**3*x**6/(280*a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**6*b**5*x**4 + 280*a**5
*b**6*x**6) + 135*a**(22/3)*b**4*x**8*(1 + b*x**2/a)**(1/3)/(280*a**8*b**3 + 840*a**7*b**4*x**2 + 840*a**6*b**
5*x**4 + 280*a**5*b**6*x**6) + 132*a**(19/3)*b**5*x**10*(1 + b*x**2/a)**(1/3)/(280*a**8*b**3 + 840*a**7*b**4*x
**2 + 840*a**6*b**5*x**4 + 280*a**5*b**6*x**6) + 42*a**(16/3)*b**6*x**12*(1 + b*x**2/a)**(1/3)/(280*a**8*b**3
+ 840*a**7*b**4*x**2 + 840*a**6*b**5*x**4 + 280*a**5*b**6*x**6)

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Giac [A]  time = 2.66339, size = 58, normalized size = 0.98 \begin{align*} \frac{3 \,{\left (14 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} - 40 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{2}\right )}}{280 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^2+a)^(1/3),x, algorithm="giac")

[Out]

3/280*(14*(b*x^2 + a)^(10/3) - 40*(b*x^2 + a)^(7/3)*a + 35*(b*x^2 + a)^(4/3)*a^2)/b^3