3.854 \(\int \frac{x^9}{\left (a+b x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 b^{5/2}}+\frac{3 x^2 \sqrt{a+b x^4}}{4 b^2}-\frac{x^6}{2 b \sqrt{a+b x^4}} \]

[Out]

-x^6/(2*b*Sqrt[a + b*x^4]) + (3*x^2*Sqrt[a + b*x^4])/(4*b^2) - (3*a*ArcTanh[(Sqr
t[b]*x^2)/Sqrt[a + b*x^4]])/(4*b^(5/2))

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Rubi [A]  time = 0.110273, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 b^{5/2}}+\frac{3 x^2 \sqrt{a+b x^4}}{4 b^2}-\frac{x^6}{2 b \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(a + b*x^4)^(3/2),x]

[Out]

-x^6/(2*b*Sqrt[a + b*x^4]) + (3*x^2*Sqrt[a + b*x^4])/(4*b^2) - (3*a*ArcTanh[(Sqr
t[b]*x^2)/Sqrt[a + b*x^4]])/(4*b^(5/2))

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Rubi in Sympy [A]  time = 11.7722, size = 66, normalized size = 0.89 \[ - \frac{3 a \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a + b x^{4}}} \right )}}{4 b^{\frac{5}{2}}} - \frac{x^{6}}{2 b \sqrt{a + b x^{4}}} + \frac{3 x^{2} \sqrt{a + b x^{4}}}{4 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(b*x**4+a)**(3/2),x)

[Out]

-3*a*atanh(sqrt(b)*x**2/sqrt(a + b*x**4))/(4*b**(5/2)) - x**6/(2*b*sqrt(a + b*x*
*4)) + 3*x**2*sqrt(a + b*x**4)/(4*b**2)

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Mathematica [A]  time = 0.108848, size = 65, normalized size = 0.88 \[ \frac{3 a x^2+b x^6}{4 b^2 \sqrt{a+b x^4}}-\frac{3 a \log \left (\sqrt{b} \sqrt{a+b x^4}+b x^2\right )}{4 b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(a + b*x^4)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.017, size = 61, normalized size = 0.8 \[{\frac{{x}^{6}}{4\,b}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{3\,a{x}^{2}}{4\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{3\,a}{4}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(b*x^4+a)^(3/2),x)

[Out]

1/4*x^6/b/(b*x^4+a)^(1/2)+3/4*a/b^2*x^2/(b*x^4+a)^(1/2)-3/4*a/b^(5/2)*ln(b^(1/2)
*x^2+(b*x^4+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^9/(b*x^4 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.327221, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (b x^{6} + 3 \, a x^{2}\right )} \sqrt{b x^{4} + a} \sqrt{b} + 3 \,{\left (a b x^{4} + a^{2}\right )} \log \left (2 \, \sqrt{b x^{4} + a} b x^{2} -{\left (2 \, b x^{4} + a\right )} \sqrt{b}\right )}{8 \,{\left (b^{3} x^{4} + a b^{2}\right )} \sqrt{b}}, \frac{{\left (b x^{6} + 3 \, a x^{2}\right )} \sqrt{b x^{4} + a} \sqrt{-b} - 3 \,{\left (a b x^{4} + a^{2}\right )} \arctan \left (\frac{\sqrt{-b} x^{2}}{\sqrt{b x^{4} + a}}\right )}{4 \,{\left (b^{3} x^{4} + a b^{2}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^4 + a)^(3/2),x, algorithm="fricas")

[Out]

[1/8*(2*(b*x^6 + 3*a*x^2)*sqrt(b*x^4 + a)*sqrt(b) + 3*(a*b*x^4 + a^2)*log(2*sqrt
(b*x^4 + a)*b*x^2 - (2*b*x^4 + a)*sqrt(b)))/((b^3*x^4 + a*b^2)*sqrt(b)), 1/4*((b
*x^6 + 3*a*x^2)*sqrt(b*x^4 + a)*sqrt(-b) - 3*(a*b*x^4 + a^2)*arctan(sqrt(-b)*x^2
/sqrt(b*x^4 + a)))/((b^3*x^4 + a*b^2)*sqrt(-b))]

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Sympy [A]  time = 12.393, size = 75, normalized size = 1.01 \[ \frac{3 \sqrt{a} x^{2}}{4 b^{2} \sqrt{1 + \frac{b x^{4}}{a}}} - \frac{3 a \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 b^{\frac{5}{2}}} + \frac{x^{6}}{4 \sqrt{a} b \sqrt{1 + \frac{b x^{4}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(b*x**4+a)**(3/2),x)

[Out]

3*sqrt(a)*x**2/(4*b**2*sqrt(1 + b*x**4/a)) - 3*a*asinh(sqrt(b)*x**2/sqrt(a))/(4*
b**(5/2)) + x**6/(4*sqrt(a)*b*sqrt(1 + b*x**4/a))

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GIAC/XCAS [A]  time = 0.233652, size = 74, normalized size = 1. \[ \frac{{\left (\frac{x^{4}}{b} + \frac{3 \, a}{b^{2}}\right )} x^{2}}{4 \, \sqrt{b x^{4} + a}} + \frac{3 \, a{\rm ln}\left ({\left | -\sqrt{b} x^{2} + \sqrt{b x^{4} + a} \right |}\right )}{4 \, b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^4 + a)^(3/2),x, algorithm="giac")

[Out]

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