3.1938 \(\int \frac {x^2}{(a+\frac {b}{x^2})^{3/2}} \, dx\)

Optimal. Leaf size=62 \[ -\frac {8 b x \sqrt {a+\frac {b}{x^2}}}{3 a^3}+\frac {4 b x}{3 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {x^3}{3 a \sqrt {a+\frac {b}{x^2}}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {271, 192, 191} \[ -\frac {8 b x \sqrt {a+\frac {b}{x^2}}}{3 a^3}+\frac {4 b x}{3 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {x^3}{3 a \sqrt {a+\frac {b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b/x^2)^(3/2),x]

[Out]

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

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

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Mathematica [A]  time = 0.02, size = 41, normalized size = 0.66 \[ \frac {a^2 x^4-4 a b x^2-8 b^2}{3 a^3 x \sqrt {a+\frac {b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b/x^2)^(3/2),x]

[Out]

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

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fricas [A]  time = 0.78, size = 51, normalized size = 0.82 \[ \frac {{\left (a^{2} x^{5} - 4 \, a b x^{3} - 8 \, b^{2} x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, {\left (a^{4} x^{2} + a^{3} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^2/(a+b/x^2)^(3/2),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 [87,11.8623251389,-77]Warning, choosing
root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [42,66.639742349
7,-22]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters
values [-79,4.84063049592,-53]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,
0,4]%%%}] at parameters values [-8,67.6494641076,-79]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,
[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-5,8.73413453996,60]Warning, choosing root of [1,0,%%%{
-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [81,31.1594383004,58]1/a*(-b^2/a^2
/(a*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))+sqrt(b)*sign(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x)))+(1/
3*a^10*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))^3+a^8*b*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))-2*a^8
*b*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))+a^9*sqrt(b)*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))^2*sig
n(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x)))/a^9+b*sqrt(b)/a^2*sign(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(
x)))

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maple [A]  time = 0.01, size = 49, normalized size = 0.79 \[ \frac {\left (a \,x^{2}+b \right ) \left (a^{2} x^{4}-4 a b \,x^{2}-8 b^{2}\right )}{3 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.82, size = 53, normalized size = 0.85 \[ \frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} x^{3} - 6 \, \sqrt {a + \frac {b}{x^{2}}} b x}{3 \, a^{3}} - \frac {b^{2}}{\sqrt {a + \frac {b}{x^{2}}} a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((a + b/x^2)^(3/2)*x^3 - 6*sqrt(a + b/x^2)*b*x)/a^3 - b^2/(sqrt(a + b/x^2)*a^3*x)

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mupad [B]  time = 1.59, size = 38, normalized size = 0.61 \[ -\frac {-a^2\,x^4+4\,a\,b\,x^2+8\,b^2}{3\,a^3\,x\,\sqrt {a+\frac {b}{x^2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b/x^2)^(3/2),x)

[Out]

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

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sympy [B]  time = 1.28, size = 219, normalized size = 3.53 \[ \frac {a^{3} b^{\frac {9}{2}} x^{6} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} - \frac {3 a^{2} b^{\frac {11}{2}} x^{4} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} - \frac {12 a b^{\frac {13}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} - \frac {8 b^{\frac {15}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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