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]

8/15*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]  time = 0.02, antiderivative size = 66, 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 = {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 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]

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]

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 = 0.03, 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)

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 40, normalized size = 0.61 \[ \frac {{\left (3 \, a^{2} x^{5} - 4 \, a b x^{3} + 8 \, b^{2} x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{15 \, a^{3}} \]

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

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, choosing root of [1,0,%%%{-2,[1
,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-13,6.59772427287,-13]Warning, choosin
g root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [88,60.96805573
16,-81]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters
values [-43,80.6998703005,-84]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2
,0,4]%%%}] at parameters values [26,27.1915548402,44]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,
[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [56,99.5462650466,-1]Warning, choosing root of [1,0,%%%{
-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [67,64.8330923682,-48](1/5*a^4*(-s
qrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))^5+2*a^2*b*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))^3+2*a*sqrt(b)
*b*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))^2*sign(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))+b^2*(-sqrt(
b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))-2/3*a^2*b*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))^3-2*b^2*(-sqrt(b)
/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))-2*a*sqrt(b)*b*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))^2*sign(-sqrt(b)
/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))+a^3*sqrt(b)*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))^4*sign(-sqrt(b)/a
*sign(x)+sqrt(a*x^2+b)/a/sign(x)))/a^2+b^2/a^2*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 50, normalized size = 0.76 \[ \frac {\left (a \,x^{2}+b \right ) \left (3 a^{2} x^{4}-4 a b \,x^{2}+8 b^{2}\right )}{15 \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a+b/x^2)^(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)

________________________________________________________________________________________

maxima [A]  time = 0.79, size = 50, normalized size = 0.76 \[ \frac {3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}} x^{5} - 10 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} b x^{3} + 15 \, \sqrt {a + \frac {b}{x^{2}}} b^{2} x}{15 \, a^{3}} \]

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

________________________________________________________________________________________

mupad [B]  time = 1.56, size = 38, normalized size = 0.58 \[ \frac {x^5\,\sqrt {a+\frac {b}{x^2}}\,\left (3\,a^2+\frac {8\,b^2}{x^4}-\frac {4\,a\,b}{x^2}\right )}{15\,a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [B]  time = 1.25, size = 279, normalized size = 4.23 \[ \frac {3 a^{4} b^{\frac {9}{2}} x^{8} \sqrt {\frac {a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} + \frac {2 a^{3} b^{\frac {11}{2}} x^{6} \sqrt {\frac {a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} + \frac {3 a^{2} b^{\frac {13}{2}} x^{4} \sqrt {\frac {a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} + \frac {12 a b^{\frac {15}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} + \frac {8 b^{\frac {17}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} \]

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)

________________________________________________________________________________________