3.1937 \(\int \frac {x^4}{(a+\frac {b}{x^2})^{3/2}} \, dx\)

Optimal. Leaf size=88 \[ \frac {16 b^2 x \sqrt {a+\frac {b}{x^2}}}{5 a^4}-\frac {8 b^2 x}{5 a^3 \sqrt {a+\frac {b}{x^2}}}-\frac {2 b x^3}{5 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {x^5}{5 a \sqrt {a+\frac {b}{x^2}}} \]

[Out]

-8/5*b^2*x/a^3/(a+b/x^2)^(1/2)-2/5*b*x^3/a^2/(a+b/x^2)^(1/2)+1/5*x^5/a/(a+b/x^2)^(1/2)+16/5*b^2*x*(a+b/x^2)^(1
/2)/a^4

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Rubi [A]  time = 0.03, antiderivative size = 88, 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} \[ \frac {16 b^2 x \sqrt {a+\frac {b}{x^2}}}{5 a^4}-\frac {8 b^2 x}{5 a^3 \sqrt {a+\frac {b}{x^2}}}-\frac {2 b x^3}{5 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {x^5}{5 a \sqrt {a+\frac {b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b/x^2)^(3/2),x]

[Out]

(-8*b^2*x)/(5*a^3*Sqrt[a + b/x^2]) + (16*b^2*Sqrt[a + b/x^2]*x)/(5*a^4) - (2*b*x^3)/(5*a^2*Sqrt[a + b/x^2]) +
x^5/(5*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^4}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx &=\frac {x^5}{5 a \sqrt {a+\frac {b}{x^2}}}-\frac {(6 b) \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx}{5 a}\\ &=-\frac {2 b x^3}{5 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {x^5}{5 a \sqrt {a+\frac {b}{x^2}}}+\frac {\left (8 b^2\right ) \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx}{5 a^2}\\ &=-\frac {8 b^2 x}{5 a^3 \sqrt {a+\frac {b}{x^2}}}-\frac {2 b x^3}{5 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {x^5}{5 a \sqrt {a+\frac {b}{x^2}}}+\frac {\left (16 b^2\right ) \int \frac {1}{\sqrt {a+\frac {b}{x^2}}} \, dx}{5 a^3}\\ &=-\frac {8 b^2 x}{5 a^3 \sqrt {a+\frac {b}{x^2}}}+\frac {16 b^2 \sqrt {a+\frac {b}{x^2}} x}{5 a^4}-\frac {2 b x^3}{5 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {x^5}{5 a \sqrt {a+\frac {b}{x^2}}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 52, normalized size = 0.59 \[ \frac {a^3 x^6-2 a^2 b x^4+8 a b^2 x^2+16 b^3}{5 a^4 x \sqrt {a+\frac {b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b/x^2)^(3/2),x]

[Out]

(16*b^3 + 8*a*b^2*x^2 - 2*a^2*b*x^4 + a^3*x^6)/(5*a^4*Sqrt[a + b/x^2]*x)

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fricas [A]  time = 0.85, size = 62, normalized size = 0.70 \[ \frac {{\left (a^{3} x^{7} - 2 \, a^{2} b x^{5} + 8 \, a b^{2} x^{3} + 16 \, b^{3} x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{5 \, {\left (a^{5} x^{2} + a^{4} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b/x^2)^(3/2),x, algorithm="fricas")

[Out]

1/5*(a^3*x^7 - 2*a^2*b*x^5 + 8*a*b^2*x^3 + 16*b^3*x)*sqrt((a*x^2 + b)/x^2)/(a^5*x^2 + a^4*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^4/(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 [17,97.031638477,-16]Warning, choosing
root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-30,21.978851114
4,56]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters v
alues [66,51.1688777674,-43]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,
4]%%%}] at parameters values [28,28.9556208979,28]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,
1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-71,67.2689954746,93]Warning, choosing root of [1,0,%%%{-2
,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-13,34.9002507734,-2]Warning, choos
ing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-32,42.20937
15044,-53]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at paramet
ers values [-68,89.6047224309,-49]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1
,[2,0,4]%%%}] at parameters values [-67,31.1777770203,-77]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%
%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [98,37.8441393101,-21]b^3/a^4/(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)))+(1/5*a^26*(-sqrt(b)/a*sig
n(x)+sqrt(a*x^2+b)/a/sign(x))^5+2*a^24*b*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))^3+2*a^23*sqrt(b)*b*(-sqr
t(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^22*b^2*(-sqrt(b)/
a*sign(x)+sqrt(a*x^2+b)/a/sign(x))-a^24*b*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))^3-3*a^22*b^2*(-sqrt(b)/
a*sign(x)+sqrt(a*x^2+b)/a/sign(x))+3*a^22*b^2*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))-3*a^23*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^25*sqrt(b)*(-s
qrt(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^25-b^2*sqrt(b)
/a^4*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 = 0.68 \[ \frac {\left (a \,x^{2}+b \right ) \left (a^{3} x^{6}-2 a^{2} b \,x^{4}+8 a \,b^{2} x^{2}+16 b^{3}\right )}{5 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a+b/x^2)^(3/2),x)

[Out]

1/5*(a*x^2+b)*(a^3*x^6-2*a^2*b*x^4+8*a*b^2*x^2+16*b^3)/a^4/x^3/((a*x^2+b)/x^2)^(3/2)

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maxima [A]  time = 0.87, size = 69, normalized size = 0.78 \[ \frac {b^{3}}{\sqrt {a + \frac {b}{x^{2}}} a^{4} x} + \frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}} x^{5} - 5 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} b x^{3} + 15 \, \sqrt {a + \frac {b}{x^{2}}} b^{2} x}{5 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b/x^2)^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.72, size = 48, normalized size = 0.55 \[ \frac {a^3\,x^6-2\,a^2\,b\,x^4+8\,a\,b^2\,x^2+16\,b^3}{5\,a^4\,x\,\sqrt {a+\frac {b}{x^2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a + b/x^2)^(3/2),x)

[Out]

(16*b^3 + a^3*x^6 + 8*a*b^2*x^2 - 2*a^2*b*x^4)/(5*a^4*x*(a + b/x^2)^(1/2))

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sympy [B]  time = 1.64, size = 337, normalized size = 3.83 \[ \frac {a^{5} b^{\frac {19}{2}} x^{10} \sqrt {\frac {a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} + \frac {5 a^{3} b^{\frac {23}{2}} x^{6} \sqrt {\frac {a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} + \frac {30 a^{2} b^{\frac {25}{2}} x^{4} \sqrt {\frac {a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} + \frac {40 a b^{\frac {27}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} + \frac {16 b^{\frac {29}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(a+b/x**2)**(3/2),x)

[Out]

a**5*b**(19/2)*x**10*sqrt(a*x**2/b + 1)/(5*a**7*b**9*x**6 + 15*a**6*b**10*x**4 + 15*a**5*b**11*x**2 + 5*a**4*b
**12) + 5*a**3*b**(23/2)*x**6*sqrt(a*x**2/b + 1)/(5*a**7*b**9*x**6 + 15*a**6*b**10*x**4 + 15*a**5*b**11*x**2 +
5*a**4*b**12) + 30*a**2*b**(25/2)*x**4*sqrt(a*x**2/b + 1)/(5*a**7*b**9*x**6 + 15*a**6*b**10*x**4 + 15*a**5*b*
*11*x**2 + 5*a**4*b**12) + 40*a*b**(27/2)*x**2*sqrt(a*x**2/b + 1)/(5*a**7*b**9*x**6 + 15*a**6*b**10*x**4 + 15*
a**5*b**11*x**2 + 5*a**4*b**12) + 16*b**(29/2)*sqrt(a*x**2/b + 1)/(5*a**7*b**9*x**6 + 15*a**6*b**10*x**4 + 15*
a**5*b**11*x**2 + 5*a**4*b**12)

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