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

Optimal. Leaf size=64 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{a} b^{5/2}}-\frac{3 x}{8 b^2 \left (a+b x^2\right )}-\frac{x^3}{4 b \left (a+b x^2\right )^2} \]

[Out]

-x^3/(4*b*(a + b*x^2)^2) - (3*x)/(8*b^2*(a + b*x^2)) + (3*ArcTan[(Sqrt[b]*x)/Sqr
t[a]])/(8*Sqrt[a]*b^(5/2))

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Rubi [A]  time = 0.0625032, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{a} b^{5/2}}-\frac{3 x}{8 b^2 \left (a+b x^2\right )}-\frac{x^3}{4 b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^2)^3,x]

[Out]

-x^3/(4*b*(a + b*x^2)^2) - (3*x)/(8*b^2*(a + b*x^2)) + (3*ArcTan[(Sqrt[b]*x)/Sqr
t[a]])/(8*Sqrt[a]*b^(5/2))

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Rubi in Sympy [A]  time = 10.3164, size = 56, normalized size = 0.88 \[ - \frac{x^{3}}{4 b \left (a + b x^{2}\right )^{2}} - \frac{3 x}{8 b^{2} \left (a + b x^{2}\right )} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 \sqrt{a} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.076753, size = 55, normalized size = 0.86 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{a} b^{5/2}}-\frac{3 a x+5 b x^3}{8 b^2 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x^2)^3,x]

[Out]

-(3*a*x + 5*b*x^3)/(8*b^2*(a + b*x^2)^2) + (3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*Sq
rt[a]*b^(5/2))

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Maple [A]  time = 0.012, size = 47, normalized size = 0.7 \[{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( -{\frac{5\,{x}^{3}}{8\,b}}-{\frac{3\,ax}{8\,{b}^{2}}} \right ) }+{\frac{3}{8\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2+a)^3,x)

[Out]

(-5/8*x^3/b-3/8*a*x/b^2)/(b*x^2+a)^2+3/8/b^2/(a*b)^(1/2)*arctan(x*b/(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/(b*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.211152, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (5 \, b x^{3} + 3 \, a x\right )} \sqrt{-a b}}{16 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \sqrt{-a b}}, \frac{3 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (5 \, b x^{3} + 3 \, a x\right )} \sqrt{a b}}{8 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(3*(b^2*x^4 + 2*a*b*x^2 + a^2)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x
^2 + a)) - 2*(5*b*x^3 + 3*a*x)*sqrt(-a*b))/((b^4*x^4 + 2*a*b^3*x^2 + a^2*b^2)*sq
rt(-a*b)), 1/8*(3*(b^2*x^4 + 2*a*b*x^2 + a^2)*arctan(sqrt(a*b)*x/a) - (5*b*x^3 +
3*a*x)*sqrt(a*b))/((b^4*x^4 + 2*a*b^3*x^2 + a^2*b^2)*sqrt(a*b))]

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Sympy [A]  time = 1.84902, size = 109, normalized size = 1.7 \[ - \frac{3 \sqrt{- \frac{1}{a b^{5}}} \log{\left (- a b^{2} \sqrt{- \frac{1}{a b^{5}}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a b^{5}}} \log{\left (a b^{2} \sqrt{- \frac{1}{a b^{5}}} + x \right )}}{16} - \frac{3 a x + 5 b x^{3}}{8 a^{2} b^{2} + 16 a b^{3} x^{2} + 8 b^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**2+a)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.211024, size = 61, normalized size = 0.95 \[ \frac{3 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{2}} - \frac{5 \, b x^{3} + 3 \, a x}{8 \,{\left (b x^{2} + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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