3.1922 \(\int \frac{x^4}{\sqrt{a+\frac{b}{x^2}}} \, dx\)

Optimal. Leaf size=66 \[ \frac{8 b^2 x \sqrt{a+\frac{b}{x^2}}}{15 a^3}-\frac{4 b x^3 \sqrt{a+\frac{b}{x^2}}}{15 a^2}+\frac{x^5 \sqrt{a+\frac{b}{x^2}}}{5 a} \]

[Out]

(8*b^2*Sqrt[a + b/x^2]*x)/(15*a^3) - (4*b*Sqrt[a + b/x^2]*x^3)/(15*a^2) + (Sqrt[a + b/x^2]*x^5)/(5*a)

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Rubi [A]  time = 0.0189321, antiderivative size = 66, 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 = {271, 191} \[ \frac{8 b^2 x \sqrt{a+\frac{b}{x^2}}}{15 a^3}-\frac{4 b x^3 \sqrt{a+\frac{b}{x^2}}}{15 a^2}+\frac{x^5 \sqrt{a+\frac{b}{x^2}}}{5 a} \]

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[a + b/x^2],x]

[Out]

(8*b^2*Sqrt[a + b/x^2]*x)/(15*a^3) - (4*b*Sqrt[a + b/x^2]*x^3)/(15*a^2) + (Sqrt[a + b/x^2]*x^5)/(5*a)

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 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}{\sqrt{a+\frac{b}{x^2}}} \, dx &=\frac{\sqrt{a+\frac{b}{x^2}} x^5}{5 a}-\frac{(4 b) \int \frac{x^2}{\sqrt{a+\frac{b}{x^2}}} \, dx}{5 a}\\ &=-\frac{4 b \sqrt{a+\frac{b}{x^2}} x^3}{15 a^2}+\frac{\sqrt{a+\frac{b}{x^2}} x^5}{5 a}+\frac{\left (8 b^2\right ) \int \frac{1}{\sqrt{a+\frac{b}{x^2}}} \, dx}{15 a^2}\\ &=\frac{8 b^2 \sqrt{a+\frac{b}{x^2}} x}{15 a^3}-\frac{4 b \sqrt{a+\frac{b}{x^2}} x^3}{15 a^2}+\frac{\sqrt{a+\frac{b}{x^2}} x^5}{5 a}\\ \end{align*}

Mathematica [A]  time = 0.0241804, size = 40, normalized size = 0.61 \[ \frac{x \sqrt{a+\frac{b}{x^2}} \left (3 a^2 x^4-4 a b x^2+8 b^2\right )}{15 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/Sqrt[a + b/x^2],x]

[Out]

(Sqrt[a + b/x^2]*x*(8*b^2 - 4*a*b*x^2 + 3*a^2*x^4))/(15*a^3)

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Maple [A]  time = 0.004, size = 50, normalized size = 0.8 \begin{align*}{\frac{ \left ( a{x}^{2}+b \right ) \left ( 3\,{a}^{2}{x}^{4}-4\,ab{x}^{2}+8\,{b}^{2} \right ) }{15\,{a}^{3}x}{\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(a*x^2+b)*(3*a^2*x^4-4*a*b*x^2+8*b^2)/a^3/x/((a*x^2+b)/x^2)^(1/2)

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Maxima [A]  time = 1.01776, size = 68, normalized size = 1.03 \begin{align*} \frac{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x^{5} - 10 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} b x^{3} + 15 \, \sqrt{a + \frac{b}{x^{2}}} b^{2} x}{15 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b/x^2)^(1/2),x, algorithm="maxima")

[Out]

1/15*(3*(a + b/x^2)^(5/2)*x^5 - 10*(a + b/x^2)^(3/2)*b*x^3 + 15*sqrt(a + b/x^2)*b^2*x)/a^3

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Fricas [A]  time = 1.546, size = 89, normalized size = 1.35 \begin{align*} \frac{{\left (3 \, a^{2} x^{5} - 4 \, a b x^{3} + 8 \, b^{2} x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{15 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b/x^2)^(1/2),x, algorithm="fricas")

[Out]

1/15*(3*a^2*x^5 - 4*a*b*x^3 + 8*b^2*x)*sqrt((a*x^2 + b)/x^2)/a^3

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Sympy [B]  time = 1.53797, size = 279, normalized size = 4.23 \begin{align*} \frac{3 a^{4} b^{\frac{9}{2}} x^{8} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} + \frac{2 a^{3} b^{\frac{11}{2}} x^{6} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} + \frac{3 a^{2} b^{\frac{13}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} + \frac{12 a b^{\frac{15}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} + \frac{8 b^{\frac{17}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(a+b/x**2)**(1/2),x)

[Out]

3*a**4*b**(9/2)*x**8*sqrt(a*x**2/b + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**2 + 15*a**3*b**6) + 2*a**3*b**(11
/2)*x**6*sqrt(a*x**2/b + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**2 + 15*a**3*b**6) + 3*a**2*b**(13/2)*x**4*sqr
t(a*x**2/b + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**2 + 15*a**3*b**6) + 12*a*b**(15/2)*x**2*sqrt(a*x**2/b + 1
)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**2 + 15*a**3*b**6) + 8*b**(17/2)*sqrt(a*x**2/b + 1)/(15*a**5*b**4*x**4 +
30*a**4*b**5*x**2 + 15*a**3*b**6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b/x^2)^(1/2),x, algorithm="giac")

[Out]

integrate(x^4/sqrt(a + b/x^2), x)