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

Optimal. Leaf size=38 \[ \frac {3 \left (a+b x^2\right )^{4/3}}{8 b^2}-\frac {3 a \sqrt [3]{a+b x^2}}{2 b^2} \]

[Out]

-3/2*a*(b*x^2+a)^(1/3)/b^2+3/8*(b*x^2+a)^(4/3)/b^2

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Rubi [A]  time = 0.02, antiderivative size = 38, 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} \[ \frac {3 \left (a+b x^2\right )^{4/3}}{8 b^2}-\frac {3 a \sqrt [3]{a+b x^2}}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^2)^(2/3),x]

[Out]

(-3*a*(a + b*x^2)^(1/3))/(2*b^2) + (3*(a + b*x^2)^(4/3))/(8*b^2)

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

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.71 \[ \frac {3 \left (b x^2-3 a\right ) \sqrt [3]{a+b x^2}}{8 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^2)^(2/3),x]

[Out]

(3*(-3*a + b*x^2)*(a + b*x^2)^(1/3))/(8*b^2)

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fricas [A]  time = 0.51, size = 23, normalized size = 0.61 \[ \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} {\left (b x^{2} - 3 \, a\right )}}{8 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*(b*x^2 + a)^(1/3)*(b*x^2 - 3*a)/b^2

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giac [A]  time = 0.59, size = 30, normalized size = 0.79 \[ \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {4}{3}}}{8 \, b^{2}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*(b*x^2 + a)^(4/3)/b^2 - 3/2*(b*x^2 + a)^(1/3)*a/b^2

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maple [A]  time = 0.00, size = 25, normalized size = 0.66 \[ -\frac {3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} \left (-b \,x^{2}+3 a \right )}{8 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*(b*x^2+a)^(1/3)*(-b*x^2+3*a)/b^2

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maxima [A]  time = 1.34, size = 30, normalized size = 0.79 \[ \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {4}{3}}}{8 \, b^{2}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*(b*x^2 + a)^(4/3)/b^2 - 3/2*(b*x^2 + a)^(1/3)*a/b^2

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mupad [B]  time = 4.79, size = 24, normalized size = 0.63 \[ -\frac {3\,{\left (b\,x^2+a\right )}^{1/3}\,\left (3\,a-b\,x^2\right )}{8\,b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x^2)^(2/3),x)

[Out]

-(3*(a + b*x^2)^(1/3)*(3*a - b*x^2))/(8*b^2)

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sympy [B]  time = 1.14, size = 178, normalized size = 4.68 \[ - \frac {9 a^{\frac {10}{3}} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} + \frac {9 a^{\frac {10}{3}}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} - \frac {6 a^{\frac {7}{3}} b x^{2} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} + \frac {9 a^{\frac {7}{3}} b x^{2}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} + \frac {3 a^{\frac {4}{3}} b^{2} x^{4} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*a**(10/3)*(1 + b*x**2/a)**(1/3)/(8*a**2*b**2 + 8*a*b**3*x**2) + 9*a**(10/3)/(8*a**2*b**2 + 8*a*b**3*x**2) -
6*a**(7/3)*b*x**2*(1 + b*x**2/a)**(1/3)/(8*a**2*b**2 + 8*a*b**3*x**2) + 9*a**(7/3)*b*x**2/(8*a**2*b**2 + 8*a*
b**3*x**2) + 3*a**(4/3)*b**2*x**4*(1 + b*x**2/a)**(1/3)/(8*a**2*b**2 + 8*a*b**3*x**2)

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