\(\int \frac {x^4}{\sqrt {a+\frac {b}{x^2}}} \, dx\) [1922]

Optimal result

Integrand size = 15, antiderivative size = 66 \[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {8 b^2 \sqrt {a+\frac {b}{x^2}} x}{15 a^3}-\frac {4 b \sqrt {a+\frac {b}{x^2}} x^3}{15 a^2}+\frac {\sqrt {a+\frac {b}{x^2}} x^5}{5 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 197} \[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {8 b^2 x \sqrt {a+\frac {b}{x^2}}}{15 a^3}-\frac {4 b x^3 \sqrt {a+\frac {b}{x^2}}}{15 a^2}+\frac {x^5 \sqrt {a+\frac {b}{x^2}}}{5 a} \]

[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 197

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]

Rule 277

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.61 \[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {\sqrt {a+\frac {b}{x^2}} x \left (8 b^2-4 a b x^2+3 a^2 x^4\right )}{15 a^3} \]

[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)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.68

method	result	size
trager	\(\frac {\left (3 a^{2} x^{4}-4 a b \,x^{2}+8 b^{2}\right ) x \sqrt {-\frac {-a \,x^{2}-b}{x^{2}}}}{15 a^{3}}\)	\(45\)
gosper	\(\frac {\left (a \,x^{2}+b \right ) \left (3 a^{2} x^{4}-4 a b \,x^{2}+8 b^{2}\right )}{15 a^{3} x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\)	\(50\)
default	\(\frac {\left (a \,x^{2}+b \right ) \left (3 a^{2} x^{4}-4 a b \,x^{2}+8 b^{2}\right )}{15 a^{3} x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\)	\(50\)
risch	\(\frac {\left (a \,x^{2}+b \right ) \left (3 a^{2} x^{4}-4 a b \,x^{2}+8 b^{2}\right )}{15 a^{3} x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\)	\(50\)

[In]

int(x^4/(a+b/x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.61 \[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {{\left (3 \, a^{2} x^{5} - 4 \, a b x^{3} + 8 \, b^{2} x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{15 \, a^{3}} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (60) = 120\).

Time = 0.65 (sec) , antiderivative size = 279, normalized size of antiderivative = 4.23 \[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {3 a^{4} b^{\frac {9}{2}} x^{8} \sqrt {\frac {a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} + \frac {2 a^{3} b^{\frac {11}{2}} x^{6} \sqrt {\frac {a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} + \frac {3 a^{2} b^{\frac {13}{2}} x^{4} \sqrt {\frac {a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} + \frac {12 a b^{\frac {15}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} + \frac {8 b^{\frac {17}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.76 \[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}} x^{5} - 10 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} b x^{3} + 15 \, \sqrt {a + \frac {b}{x^{2}}} b^{2} x}{15 \, a^{3}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97 \[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^2}}} \, dx=-\frac {8 \, b^{\frac {5}{2}} \mathrm {sgn}\left (x\right )}{15 \, a^{3}} + \frac {\sqrt {a x^{2} + b} b^{2}}{a^{3} \mathrm {sgn}\left (x\right )} + \frac {3 \, {\left (a x^{2} + b\right )}^{\frac {5}{2}} - 10 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} b}{15 \, a^{3} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

-8/15*b^(5/2)*sgn(x)/a^3 + sqrt(a*x^2 + b)*b^2/(a^3*sgn(x)) + 1/15*(3*(a*x^2 + b)^(5/2) - 10*(a*x^2 + b)^(3/2)
*b)/(a^3*sgn(x))

Mupad [B] (verification not implemented)

Time = 6.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.58 \[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {x^5\,\sqrt {a+\frac {b}{x^2}}\,\left (3\,a^2+\frac {8\,b^2}{x^4}-\frac {4\,a\,b}{x^2}\right )}{15\,a^3} \]

[In]

int(x^4/(a + b/x^2)^(1/2),x)

[Out]

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