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]

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

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Rubi [A]  time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.74, size = 46, normalized size = 0.78 \[ \frac {3 \, {\left (14 \, b^{3} x^{6} + 2 \, a b^{2} x^{4} - 3 \, a^{2} b x^{2} + 9 \, a^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}}}{280 \, b^{3}} \]

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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giac [A]  time = 1.03, size = 43, normalized size = 0.73 \[ \frac {3 \, {\left (14 \, {\left (b x^{2} + a\right )}^{\frac {10}{3}} - 40 \, {\left (b x^{2} + a\right )}^{\frac {7}{3}} a + 35 \, {\left (b x^{2} + a\right )}^{\frac {4}{3}} a^{2}\right )}}{280 \, b^{3}} \]

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

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maple [A]  time = 0.01, size = 36, normalized size = 0.61 \[ \frac {3 \left (b \,x^{2}+a \right )^{\frac {4}{3}} \left (14 b^{2} x^{4}-12 a b \,x^{2}+9 a^{2}\right )}{280 b^{3}} \]

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 = 1.34, size = 47, normalized size = 0.80 \[ \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {10}{3}}}{20 \, b^{3}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{3}} a}{7 \, b^{3}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {4}{3}} a^{2}}{8 \, b^{3}} \]

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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mupad [B]  time = 4.80, size = 44, normalized size = 0.75 \[ {\left (b\,x^2+a\right )}^{1/3}\,\left (\frac {3\,x^6}{20}+\frac {27\,a^3}{280\,b^3}+\frac {3\,a\,x^4}{140\,b}-\frac {9\,a^2\,x^2}{280\,b^2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*x^2)^(1/3),x)

[Out]

(a + b*x^2)^(1/3)*((3*x^6)/20 + (27*a^3)/(280*b^3) + (3*a*x^4)/(140*b) - (9*a^2*x^2)/(280*b^2))

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sympy [B]  time = 1.88, size = 700, normalized size = 11.86 \[ \frac {27 a^{\frac {34}{3}} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} - \frac {27 a^{\frac {34}{3}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac {72 a^{\frac {31}{3}} b x^{2} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} - \frac {81 a^{\frac {31}{3}} b x^{2}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac {60 a^{\frac {28}{3}} b^{2} x^{4} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} - \frac {81 a^{\frac {28}{3}} b^{2} x^{4}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac {60 a^{\frac {25}{3}} b^{3} x^{6} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} - \frac {27 a^{\frac {25}{3}} b^{3} x^{6}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac {135 a^{\frac {22}{3}} b^{4} x^{8} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac {132 a^{\frac {19}{3}} b^{5} x^{10} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} + \frac {42 a^{\frac {16}{3}} b^{6} x^{12} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{280 a^{8} b^{3} + 840 a^{7} b^{4} x^{2} + 840 a^{6} b^{5} x^{4} + 280 a^{5} b^{6} x^{6}} \]

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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