3.1951 \(\int \frac {x^2}{(a+\frac {b}{x^2})^{5/2}} \, dx\)

Optimal. Leaf size=82 \[ -\frac {16 b x \sqrt {a+\frac {b}{x^2}}}{3 a^4}+\frac {8 b x}{3 a^3 \sqrt {a+\frac {b}{x^2}}}+\frac {2 b x}{3 a^2 \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {x^3}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {271, 192, 191} \[ -\frac {16 b x \sqrt {a+\frac {b}{x^2}}}{3 a^4}+\frac {8 b x}{3 a^3 \sqrt {a+\frac {b}{x^2}}}+\frac {2 b x}{3 a^2 \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {x^3}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b/x^2)^(5/2),x]

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 61, normalized size = 0.74 \[ \frac {a^3 x^6-6 a^2 b x^4-24 a b^2 x^2-16 b^3}{3 a^4 x \sqrt {a+\frac {b}{x^2}} \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b/x^2)^(5/2),x]

[Out]

(-16*b^3 - 24*a*b^2*x^2 - 6*a^2*b*x^4 + a^3*x^6)/(3*a^4*Sqrt[a + b/x^2]*x*(b + a*x^2))

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fricas [A]  time = 1.04, size = 73, normalized size = 0.89 \[ \frac {{\left (a^{3} x^{7} - 6 \, a^{2} b x^{5} - 24 \, a b^{2} x^{3} - 16 \, b^{3} x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, {\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b/x^2)^(5/2),x, algorithm="fricas")

[Out]

1/3*(a^3*x^7 - 6*a^2*b*x^5 - 24*a*b^2*x^3 - 16*b^3*x)*sqrt((a*x^2 + b)/x^2)/(a^6*x^4 + 2*a^5*b*x^2 + a^4*b^2)

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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)^(5/2),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 [-37,79.8475059881,-6]Warning, choosing
root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [95,91.138003463
4,2]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters va
lues [70,77.1190297226,8]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%
%%}] at parameters values [-26,73.1070263633,63]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,
0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-51,40.7394986684,-52]Warning, choosing root of [1,0,%%%{-2,
[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [42,71.9334104825,-88]Warning, choosi
ng root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-98,19.147825
0967,-53]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at paramete
rs values [-65,12.0642261468,-46]1/a*((1/3*a^18*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))^3+a^16*b*(-sqrt(b
)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))-3*a^16*b*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))+a^17*sqrt(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^18+1/3*(-9*a^2*b^
2*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))^2-9*b^3+b^3-18*a*sqrt(b)*b^2*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/
a/sign(x))*sign(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x)))/a^3/(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)))^3+8*b*sqrt(b)/3/a^3*sign(-sqrt(b)/a*sign(x)+sqrt(
a*x^2+b)/a/sign(x)))

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maple [A]  time = 0.01, size = 60, normalized size = 0.73 \[ \frac {\left (a \,x^{2}+b \right ) \left (a^{3} x^{6}-6 a^{2} b \,x^{4}-24 a \,b^{2} x^{2}-16 b^{3}\right )}{3 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} a^{4} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+b/x^2)^(5/2),x)

[Out]

1/3*(a*x^2+b)*(a^3*x^6-6*a^2*b*x^4-24*a*b^2*x^2-16*b^3)/a^4/x^5/((a*x^2+b)/x^2)^(5/2)

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maxima [A]  time = 0.91, size = 71, normalized size = 0.87 \[ \frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} x^{3} - 9 \, \sqrt {a + \frac {b}{x^{2}}} b x}{3 \, a^{4}} - \frac {9 \, {\left (a + \frac {b}{x^{2}}\right )} b^{2} x^{2} - b^{3}}{3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b/x^2)^(5/2),x, algorithm="maxima")

[Out]

1/3*((a + b/x^2)^(3/2)*x^3 - 9*sqrt(a + b/x^2)*b*x)/a^4 - 1/3*(9*(a + b/x^2)*b^2*x^2 - b^3)/((a + b/x^2)^(3/2)
*a^4*x^3)

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mupad [B]  time = 1.73, size = 81, normalized size = 0.99 \[ \frac {6\,a^2\,\left (a+\frac {b}{x^2}\right )-24\,a\,{\left (a+\frac {b}{x^2}\right )}^2+16\,{\left (a+\frac {b}{x^2}\right )}^3+a^3}{\left (\frac {3\,a^5}{b\,x}-\frac {3\,a^4\,\left (a+\frac {b}{x^2}\right )}{b\,x}\right )\,{\left (a+\frac {b}{x^2}\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b/x^2)^(5/2),x)

[Out]

(6*a^2*(a + b/x^2) - 24*a*(a + b/x^2)^2 + 16*(a + b/x^2)^3 + a^3)/(((3*a^5)/(b*x) - (3*a^4*(a + b/x^2))/(b*x))
*(a + b/x^2)^(3/2))

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sympy [B]  time = 1.88, size = 337, normalized size = 4.11 \[ \frac {a^{4} b^{\frac {19}{2}} x^{8} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{7} b^{9} x^{6} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{2} + 3 a^{4} b^{12}} - \frac {5 a^{3} b^{\frac {21}{2}} x^{6} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{7} b^{9} x^{6} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{2} + 3 a^{4} b^{12}} - \frac {30 a^{2} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{7} b^{9} x^{6} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{2} + 3 a^{4} b^{12}} - \frac {40 a b^{\frac {25}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{7} b^{9} x^{6} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{2} + 3 a^{4} b^{12}} - \frac {16 b^{\frac {27}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{7} b^{9} x^{6} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{2} + 3 a^{4} b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a+b/x**2)**(5/2),x)

[Out]

a**4*b**(19/2)*x**8*sqrt(a*x**2/b + 1)/(3*a**7*b**9*x**6 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**2 + 3*a**4*b**1
2) - 5*a**3*b**(21/2)*x**6*sqrt(a*x**2/b + 1)/(3*a**7*b**9*x**6 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**2 + 3*a*
*4*b**12) - 30*a**2*b**(23/2)*x**4*sqrt(a*x**2/b + 1)/(3*a**7*b**9*x**6 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**
2 + 3*a**4*b**12) - 40*a*b**(25/2)*x**2*sqrt(a*x**2/b + 1)/(3*a**7*b**9*x**6 + 9*a**6*b**10*x**4 + 9*a**5*b**1
1*x**2 + 3*a**4*b**12) - 16*b**(27/2)*sqrt(a*x**2/b + 1)/(3*a**7*b**9*x**6 + 9*a**6*b**10*x**4 + 9*a**5*b**11*
x**2 + 3*a**4*b**12)

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