3.1937 \(\int \frac{x^4}{(a+\frac{b}{x^2})^{3/2}} \, dx\)

Optimal. Leaf size=88 \[ \frac{16 b^2 x \sqrt{a+\frac{b}{x^2}}}{5 a^4}-\frac{8 b^2 x}{5 a^3 \sqrt{a+\frac{b}{x^2}}}-\frac{2 b x^3}{5 a^2 \sqrt{a+\frac{b}{x^2}}}+\frac{x^5}{5 a \sqrt{a+\frac{b}{x^2}}} \]

[Out]

(-8*b^2*x)/(5*a^3*Sqrt[a + b/x^2]) + (16*b^2*Sqrt[a + b/x^2]*x)/(5*a^4) - (2*b*x^3)/(5*a^2*Sqrt[a + b/x^2]) +
x^5/(5*a*Sqrt[a + b/x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0268412, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {271, 192, 191} \[ \frac{16 b^2 x \sqrt{a+\frac{b}{x^2}}}{5 a^4}-\frac{8 b^2 x}{5 a^3 \sqrt{a+\frac{b}{x^2}}}-\frac{2 b x^3}{5 a^2 \sqrt{a+\frac{b}{x^2}}}+\frac{x^5}{5 a \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b/x^2)^(3/2),x]

[Out]

(-8*b^2*x)/(5*a^3*Sqrt[a + b/x^2]) + (16*b^2*Sqrt[a + b/x^2]*x)/(5*a^4) - (2*b*x^3)/(5*a^2*Sqrt[a + b/x^2]) +
x^5/(5*a*Sqrt[a + b/x^2])

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{x^4}{\left (a+\frac{b}{x^2}\right )^{3/2}} \, dx &=\frac{x^5}{5 a \sqrt{a+\frac{b}{x^2}}}-\frac{(6 b) \int \frac{x^2}{\left (a+\frac{b}{x^2}\right )^{3/2}} \, dx}{5 a}\\ &=-\frac{2 b x^3}{5 a^2 \sqrt{a+\frac{b}{x^2}}}+\frac{x^5}{5 a \sqrt{a+\frac{b}{x^2}}}+\frac{\left (8 b^2\right ) \int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2}} \, dx}{5 a^2}\\ &=-\frac{8 b^2 x}{5 a^3 \sqrt{a+\frac{b}{x^2}}}-\frac{2 b x^3}{5 a^2 \sqrt{a+\frac{b}{x^2}}}+\frac{x^5}{5 a \sqrt{a+\frac{b}{x^2}}}+\frac{\left (16 b^2\right ) \int \frac{1}{\sqrt{a+\frac{b}{x^2}}} \, dx}{5 a^3}\\ &=-\frac{8 b^2 x}{5 a^3 \sqrt{a+\frac{b}{x^2}}}+\frac{16 b^2 \sqrt{a+\frac{b}{x^2}} x}{5 a^4}-\frac{2 b x^3}{5 a^2 \sqrt{a+\frac{b}{x^2}}}+\frac{x^5}{5 a \sqrt{a+\frac{b}{x^2}}}\\ \end{align*}

Mathematica [A]  time = 0.0242082, size = 52, normalized size = 0.59 \[ \frac{-2 a^2 b x^4+a^3 x^6+8 a b^2 x^2+16 b^3}{5 a^4 x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b/x^2)^(3/2),x]

[Out]

(16*b^3 + 8*a*b^2*x^2 - 2*a^2*b*x^4 + a^3*x^6)/(5*a^4*Sqrt[a + b/x^2]*x)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 60, normalized size = 0.7 \begin{align*}{\frac{ \left ( a{x}^{2}+b \right ) \left ({a}^{3}{x}^{6}-2\,{a}^{2}b{x}^{4}+8\,a{b}^{2}{x}^{2}+16\,{b}^{3} \right ) }{5\,{a}^{4}{x}^{3}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a+1/x^2*b)^(3/2),x)

[Out]

1/5*(a*x^2+b)*(a^3*x^6-2*a^2*b*x^4+8*a*b^2*x^2+16*b^3)/a^4/x^3/((a*x^2+b)/x^2)^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 1.00198, size = 93, normalized size = 1.06 \begin{align*} \frac{b^{3}}{\sqrt{a + \frac{b}{x^{2}}} a^{4} x} + \frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x^{5} - 5 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} b x^{3} + 15 \, \sqrt{a + \frac{b}{x^{2}}} b^{2} x}{5 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3/(sqrt(a + b/x^2)*a^4*x) + 1/5*((a + b/x^2)^(5/2)*x^5 - 5*(a + b/x^2)^(3/2)*b*x^3 + 15*sqrt(a + b/x^2)*b^2*
x)/a^4

________________________________________________________________________________________

Fricas [A]  time = 1.54923, size = 127, normalized size = 1.44 \begin{align*} \frac{{\left (a^{3} x^{7} - 2 \, a^{2} b x^{5} + 8 \, a b^{2} x^{3} + 16 \, b^{3} x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{5 \,{\left (a^{5} x^{2} + a^{4} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(a^3*x^7 - 2*a^2*b*x^5 + 8*a*b^2*x^3 + 16*b^3*x)*sqrt((a*x^2 + b)/x^2)/(a^5*x^2 + a^4*b)

________________________________________________________________________________________

Sympy [B]  time = 1.43174, size = 337, normalized size = 3.83 \begin{align*} \frac{a^{5} b^{\frac{19}{2}} x^{10} \sqrt{\frac{a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} + \frac{5 a^{3} b^{\frac{23}{2}} x^{6} \sqrt{\frac{a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} + \frac{30 a^{2} b^{\frac{25}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} + \frac{40 a b^{\frac{27}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} + \frac{16 b^{\frac{29}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*b**(19/2)*x**10*sqrt(a*x**2/b + 1)/(5*a**7*b**9*x**6 + 15*a**6*b**10*x**4 + 15*a**5*b**11*x**2 + 5*a**4*b
**12) + 5*a**3*b**(23/2)*x**6*sqrt(a*x**2/b + 1)/(5*a**7*b**9*x**6 + 15*a**6*b**10*x**4 + 15*a**5*b**11*x**2 +
5*a**4*b**12) + 30*a**2*b**(25/2)*x**4*sqrt(a*x**2/b + 1)/(5*a**7*b**9*x**6 + 15*a**6*b**10*x**4 + 15*a**5*b*
*11*x**2 + 5*a**4*b**12) + 40*a*b**(27/2)*x**2*sqrt(a*x**2/b + 1)/(5*a**7*b**9*x**6 + 15*a**6*b**10*x**4 + 15*
a**5*b**11*x**2 + 5*a**4*b**12) + 16*b**(29/2)*sqrt(a*x**2/b + 1)/(5*a**7*b**9*x**6 + 15*a**6*b**10*x**4 + 15*
a**5*b**11*x**2 + 5*a**4*b**12)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/(a + b/x^2)^(3/2), x)