3.8.15 \(\int \frac {x^5}{(a+b x^2)^{2/3}} \, dx\) [715]

Optimal. Leaf size=59 \[ \frac {3 a^2 \sqrt [3]{a+b x^2}}{2 b^3}-\frac {3 a \left (a+b x^2\right )^{4/3}}{4 b^3}+\frac {3 \left (a+b x^2\right )^{7/3}}{14 b^3} \]

[Out]

3/2*a^2*(b*x^2+a)^(1/3)/b^3-3/4*a*(b*x^2+a)^(4/3)/b^3+3/14*(b*x^2+a)^(7/3)/b^3

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Rubi [A]
time = 0.02, 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 = {272, 45} \begin {gather*} \frac {3 a^2 \sqrt [3]{a+b x^2}}{2 b^3}+\frac {3 \left (a+b x^2\right )^{7/3}}{14 b^3}-\frac {3 a \left (a+b x^2\right )^{4/3}}{4 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x^2)^(2/3),x]

[Out]

(3*a^2*(a + b*x^2)^(1/3))/(2*b^3) - (3*a*(a + b*x^2)^(4/3))/(4*b^3) + (3*(a + b*x^2)^(7/3))/(14*b^3)

Rule 45

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 272

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 \frac {x^5}{\left (a+b x^2\right )^{2/3}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(a+b x)^{2/3}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a^2}{b^2 (a+b x)^{2/3}}-\frac {2 a \sqrt [3]{a+b x}}{b^2}+\frac {(a+b x)^{4/3}}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac {3 a^2 \sqrt [3]{a+b x^2}}{2 b^3}-\frac {3 a \left (a+b x^2\right )^{4/3}}{4 b^3}+\frac {3 \left (a+b x^2\right )^{7/3}}{14 b^3}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 39, normalized size = 0.66 \begin {gather*} \frac {3 \sqrt [3]{a+b x^2} \left (9 a^2-3 a b x^2+2 b^2 x^4\right )}{28 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x^2)^(2/3),x]

[Out]

(3*(a + b*x^2)^(1/3)*(9*a^2 - 3*a*b*x^2 + 2*b^2*x^4))/(28*b^3)

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Maple [A]
time = 0.04, size = 36, normalized size = 0.61

method	result	size
gosper	\(\frac {3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} \left (2 b^{2} x^{4}-3 a b \,x^{2}+9 a^{2}\right )}{28 b^{3}}\)	\(36\)
trager	\(\frac {3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} \left (2 b^{2} x^{4}-3 a b \,x^{2}+9 a^{2}\right )}{28 b^{3}}\)	\(36\)
risch	\(\frac {3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} \left (2 b^{2} x^{4}-3 a b \,x^{2}+9 a^{2}\right )}{28 b^{3}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^2+a)^(2/3),x,method=_RETURNVERBOSE)

[Out]

3/28*(b*x^2+a)^(1/3)*(2*b^2*x^4-3*a*b*x^2+9*a^2)/b^3

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Maxima [A]
time = 0.29, size = 47, normalized size = 0.80 \begin {gather*} \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{3}}}{14 \, b^{3}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {4}{3}} a}{4 \, b^{3}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{2}}{2 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/14*(b*x^2 + a)^(7/3)/b^3 - 3/4*(b*x^2 + a)^(4/3)*a/b^3 + 3/2*(b*x^2 + a)^(1/3)*a^2/b^3

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Fricas [A]
time = 1.26, size = 35, normalized size = 0.59 \begin {gather*} \frac {3 \, {\left (2 \, b^{2} x^{4} - 3 \, a b x^{2} + 9 \, a^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}}}{28 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/28*(2*b^2*x^4 - 3*a*b*x^2 + 9*a^2)*(b*x^2 + a)^(1/3)/b^3

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (54) = 108\).
time = 0.87, size = 631, normalized size = 10.69 \begin {gather*} \frac {27 a^{\frac {31}{3}} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} - \frac {27 a^{\frac {31}{3}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac {72 a^{\frac {28}{3}} b x^{2} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} - \frac {81 a^{\frac {28}{3}} b x^{2}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac {60 a^{\frac {25}{3}} b^{2} x^{4} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} - \frac {81 a^{\frac {25}{3}} b^{2} x^{4}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac {18 a^{\frac {22}{3}} b^{3} x^{6} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} - \frac {27 a^{\frac {22}{3}} b^{3} x^{6}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac {9 a^{\frac {19}{3}} b^{4} x^{8} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac {6 a^{\frac {16}{3}} b^{5} x^{10} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27*a**(31/3)*(1 + b*x**2/a)**(1/3)/(28*a**8*b**3 + 84*a**7*b**4*x**2 + 84*a**6*b**5*x**4 + 28*a**5*b**6*x**6)
- 27*a**(31/3)/(28*a**8*b**3 + 84*a**7*b**4*x**2 + 84*a**6*b**5*x**4 + 28*a**5*b**6*x**6) + 72*a**(28/3)*b*x**
2*(1 + b*x**2/a)**(1/3)/(28*a**8*b**3 + 84*a**7*b**4*x**2 + 84*a**6*b**5*x**4 + 28*a**5*b**6*x**6) - 81*a**(28
/3)*b*x**2/(28*a**8*b**3 + 84*a**7*b**4*x**2 + 84*a**6*b**5*x**4 + 28*a**5*b**6*x**6) + 60*a**(25/3)*b**2*x**4
*(1 + b*x**2/a)**(1/3)/(28*a**8*b**3 + 84*a**7*b**4*x**2 + 84*a**6*b**5*x**4 + 28*a**5*b**6*x**6) - 81*a**(25/
3)*b**2*x**4/(28*a**8*b**3 + 84*a**7*b**4*x**2 + 84*a**6*b**5*x**4 + 28*a**5*b**6*x**6) + 18*a**(22/3)*b**3*x*
*6*(1 + b*x**2/a)**(1/3)/(28*a**8*b**3 + 84*a**7*b**4*x**2 + 84*a**6*b**5*x**4 + 28*a**5*b**6*x**6) - 27*a**(2
2/3)*b**3*x**6/(28*a**8*b**3 + 84*a**7*b**4*x**2 + 84*a**6*b**5*x**4 + 28*a**5*b**6*x**6) + 9*a**(19/3)*b**4*x
**8*(1 + b*x**2/a)**(1/3)/(28*a**8*b**3 + 84*a**7*b**4*x**2 + 84*a**6*b**5*x**4 + 28*a**5*b**6*x**6) + 6*a**(1
6/3)*b**5*x**10*(1 + b*x**2/a)**(1/3)/(28*a**8*b**3 + 84*a**7*b**4*x**2 + 84*a**6*b**5*x**4 + 28*a**5*b**6*x**
6)

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Giac [A]
time = 1.15, size = 47, normalized size = 0.80 \begin {gather*} \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{2}}{2 \, b^{3}} + \frac {3 \, {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {7}{3}} - 7 \, {\left (b x^{2} + a\right )}^{\frac {4}{3}} a\right )}}{28 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(b*x^2 + a)^(1/3)*a^2/b^3 + 3/28*(2*(b*x^2 + a)^(7/3) - 7*(b*x^2 + a)^(4/3)*a)/b^3

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Mupad [B]
time = 4.77, size = 36, normalized size = 0.61 \begin {gather*} {\left (b\,x^2+a\right )}^{1/3}\,\left (\frac {27\,a^2}{28\,b^3}+\frac {3\,x^4}{14\,b}-\frac {9\,a\,x^2}{28\,b^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a + b*x^2)^(2/3),x)

[Out]

(a + b*x^2)^(1/3)*((27*a^2)/(28*b^3) + (3*x^4)/(14*b) - (9*a*x^2)/(28*b^2))

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