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

Optimal. Leaf size=68 \[ -\frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{5/2}}+\frac{3 x \sqrt{a+b x^2}}{2 b^2}-\frac{x^3}{b \sqrt{a+b x^2}} \]

[Out]

-(x^3/(b*Sqrt[a + b*x^2])) + (3*x*Sqrt[a + b*x^2])/(2*b^2) - (3*a*ArcTanh[(Sqrt[
b]*x)/Sqrt[a + b*x^2]])/(2*b^(5/2))

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Rubi [A]  time = 0.0717978, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{5/2}}+\frac{3 x \sqrt{a+b x^2}}{2 b^2}-\frac{x^3}{b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^2)^(3/2),x]

[Out]

-(x^3/(b*Sqrt[a + b*x^2])) + (3*x*Sqrt[a + b*x^2])/(2*b^2) - (3*a*ArcTanh[(Sqrt[
b]*x)/Sqrt[a + b*x^2]])/(2*b^(5/2))

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Rubi in Sympy [A]  time = 9.42943, size = 61, normalized size = 0.9 \[ - \frac{3 a \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2 b^{\frac{5}{2}}} - \frac{x^{3}}{b \sqrt{a + b x^{2}}} + \frac{3 x \sqrt{a + b x^{2}}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(2*b**(5/2)) - x**3/(b*sqrt(a + b*x**2))
+ 3*x*sqrt(a + b*x**2)/(2*b**2)

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Mathematica [A]  time = 0.0936453, size = 61, normalized size = 0.9 \[ \frac{3 a x+b x^3}{2 b^2 \sqrt{a+b x^2}}-\frac{3 a \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{2 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(3*a*x + b*x^3)/(2*b^2*Sqrt[a + b*x^2]) - (3*a*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]
])/(2*b^(5/2))

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Maple [A]  time = 0.009, size = 57, normalized size = 0.8 \[{\frac{{x}^{3}}{2\,b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{3\,ax}{2\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{3\,a}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2+a)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246054, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (b x^{3} + 3 \, a x\right )} \sqrt{b x^{2} + a} \sqrt{b} + 3 \,{\left (a b x^{2} + a^{2}\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{4 \,{\left (b^{3} x^{2} + a b^{2}\right )} \sqrt{b}}, \frac{{\left (b x^{3} + 3 \, a x\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 3 \,{\left (a b x^{2} + a^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.6197, size = 71, normalized size = 1.04 \[ \frac{3 \sqrt{a} x}{2 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{5}{2}}} + \frac{x^{3}}{2 \sqrt{a} b \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(a)*x/(2*b**2*sqrt(1 + b*x**2/a)) - 3*a*asinh(sqrt(b)*x/sqrt(a))/(2*b**(5/
2)) + x**3/(2*sqrt(a)*b*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.216809, size = 69, normalized size = 1.01 \[ \frac{x{\left (\frac{x^{2}}{b} + \frac{3 \, a}{b^{2}}\right )}}{2 \, \sqrt{b x^{2} + a}} + \frac{3 \, a{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{2 \, b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x*(x^2/b + 3*a/b^2)/sqrt(b*x^2 + a) + 3/2*a*ln(abs(-sqrt(b)*x + sqrt(b*x^2 +
a)))/b^(5/2)