3.1873 \(\int \frac{x^4}{\left (a+\frac{b}{x^2}\right )^3} \, dx\)

Optimal. Leaf size=98 \[ -\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{11/2}}+\frac{63 b^2 x}{8 a^5}-\frac{21 b x^3}{8 a^4}+\frac{63 x^5}{40 a^3}-\frac{9 x^7}{8 a^2 \left (a x^2+b\right )}-\frac{x^9}{4 a \left (a x^2+b\right )^2} \]

[Out]

(63*b^2*x)/(8*a^5) - (21*b*x^3)/(8*a^4) + (63*x^5)/(40*a^3) - x^9/(4*a*(b + a*x^
2)^2) - (9*x^7)/(8*a^2*(b + a*x^2)) - (63*b^(5/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(
8*a^(11/2))

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Rubi [A]  time = 0.129064, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{11/2}}+\frac{63 b^2 x}{8 a^5}-\frac{21 b x^3}{8 a^4}+\frac{63 x^5}{40 a^3}-\frac{9 x^7}{8 a^2 \left (a x^2+b\right )}-\frac{x^9}{4 a \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b/x^2)^3,x]

[Out]

(63*b^2*x)/(8*a^5) - (21*b*x^3)/(8*a^4) + (63*x^5)/(40*a^3) - x^9/(4*a*(b + a*x^
2)^2) - (9*x^7)/(8*a^2*(b + a*x^2)) - (63*b^(5/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(
8*a^(11/2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{x^{9}}{4 a \left (a x^{2} + b\right )^{2}} + \frac{63 b^{2} \int \frac{1}{a^{3}}\, dx}{8 a^{2}} - \frac{9 x^{7}}{8 a^{2} \left (a x^{2} + b\right )} + \frac{63 x^{5}}{40 a^{3}} - \frac{21 b x^{3}}{8 a^{4}} - \frac{63 b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(a+b/x**2)**3,x)

[Out]

-x**9/(4*a*(a*x**2 + b)**2) + 63*b**2*Integral(a**(-3), x)/(8*a**2) - 9*x**7/(8*
a**2*(a*x**2 + b)) + 63*x**5/(40*a**3) - 21*b*x**3/(8*a**4) - 63*b**(5/2)*atan(s
qrt(a)*x/sqrt(b))/(8*a**(11/2))

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Mathematica [A]  time = 0.114053, size = 88, normalized size = 0.9 \[ \frac{8 a^4 x^9-24 a^3 b x^7+168 a^2 b^2 x^5+525 a b^3 x^3+315 b^4 x}{40 a^5 \left (a x^2+b\right )^2}-\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b/x^2)^3,x]

[Out]

(315*b^4*x + 525*a*b^3*x^3 + 168*a^2*b^2*x^5 - 24*a^3*b*x^7 + 8*a^4*x^9)/(40*a^5
*(b + a*x^2)^2) - (63*b^(5/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(8*a^(11/2))

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Maple [A]  time = 0.008, size = 88, normalized size = 0.9 \[{\frac{{x}^{5}}{5\,{a}^{3}}}-{\frac{b{x}^{3}}{{a}^{4}}}+6\,{\frac{{b}^{2}x}{{a}^{5}}}+{\frac{17\,{b}^{3}{x}^{3}}{8\,{a}^{4} \left ( a{x}^{2}+b \right ) ^{2}}}+{\frac{15\,{b}^{4}x}{8\,{a}^{5} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{63\,{b}^{3}}{8\,{a}^{5}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(a+b/x^2)^3,x)

[Out]

1/5*x^5/a^3-b*x^3/a^4+6*b^2*x/a^5+17/8/a^4*b^3/(a*x^2+b)^2*x^3+15/8/a^5*b^4/(a*x
^2+b)^2*x-63/8/a^5*b^3/(a*b)^(1/2)*arctan(a*x/(a*b)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23055, size = 1, normalized size = 0.01 \[ \left [\frac{16 \, a^{4} x^{9} - 48 \, a^{3} b x^{7} + 336 \, a^{2} b^{2} x^{5} + 1050 \, a b^{3} x^{3} + 630 \, b^{4} x + 315 \,{\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right )}{80 \,{\left (a^{7} x^{4} + 2 \, a^{6} b x^{2} + a^{5} b^{2}\right )}}, \frac{8 \, a^{4} x^{9} - 24 \, a^{3} b x^{7} + 168 \, a^{2} b^{2} x^{5} + 525 \, a b^{3} x^{3} + 315 \, b^{4} x - 315 \,{\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{x}{\sqrt{\frac{b}{a}}}\right )}{40 \,{\left (a^{7} x^{4} + 2 \, a^{6} b x^{2} + a^{5} b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/80*(16*a^4*x^9 - 48*a^3*b*x^7 + 336*a^2*b^2*x^5 + 1050*a*b^3*x^3 + 630*b^4*x
+ 315*(a^2*b^2*x^4 + 2*a*b^3*x^2 + b^4)*sqrt(-b/a)*log((a*x^2 - 2*a*x*sqrt(-b/a)
- b)/(a*x^2 + b)))/(a^7*x^4 + 2*a^6*b*x^2 + a^5*b^2), 1/40*(8*a^4*x^9 - 24*a^3*
b*x^7 + 168*a^2*b^2*x^5 + 525*a*b^3*x^3 + 315*b^4*x - 315*(a^2*b^2*x^4 + 2*a*b^3
*x^2 + b^4)*sqrt(b/a)*arctan(x/sqrt(b/a)))/(a^7*x^4 + 2*a^6*b*x^2 + a^5*b^2)]

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Sympy [A]  time = 2.35364, size = 144, normalized size = 1.47 \[ \frac{63 \sqrt{- \frac{b^{5}}{a^{11}}} \log{\left (- \frac{a^{5} \sqrt{- \frac{b^{5}}{a^{11}}}}{b^{2}} + x \right )}}{16} - \frac{63 \sqrt{- \frac{b^{5}}{a^{11}}} \log{\left (\frac{a^{5} \sqrt{- \frac{b^{5}}{a^{11}}}}{b^{2}} + x \right )}}{16} + \frac{17 a b^{3} x^{3} + 15 b^{4} x}{8 a^{7} x^{4} + 16 a^{6} b x^{2} + 8 a^{5} b^{2}} + \frac{x^{5}}{5 a^{3}} - \frac{b x^{3}}{a^{4}} + \frac{6 b^{2} x}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(a+b/x**2)**3,x)

[Out]

63*sqrt(-b**5/a**11)*log(-a**5*sqrt(-b**5/a**11)/b**2 + x)/16 - 63*sqrt(-b**5/a*
*11)*log(a**5*sqrt(-b**5/a**11)/b**2 + x)/16 + (17*a*b**3*x**3 + 15*b**4*x)/(8*a
**7*x**4 + 16*a**6*b*x**2 + 8*a**5*b**2) + x**5/(5*a**3) - b*x**3/a**4 + 6*b**2*
x/a**5

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GIAC/XCAS [A]  time = 0.23177, size = 113, normalized size = 1.15 \[ -\frac{63 \, b^{3} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{5}} + \frac{17 \, a b^{3} x^{3} + 15 \, b^{4} x}{8 \,{\left (a x^{2} + b\right )}^{2} a^{5}} + \frac{a^{12} x^{5} - 5 \, a^{11} b x^{3} + 30 \, a^{10} b^{2} x}{5 \, a^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(a + b/x^2)^3,x, algorithm="giac")

[Out]

-63/8*b^3*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*a^5) + 1/8*(17*a*b^3*x^3 + 15*b^4*x)/
((a*x^2 + b)^2*a^5) + 1/5*(a^12*x^5 - 5*a^11*b*x^3 + 30*a^10*b^2*x)/a^15