3.1012 \(\int \frac{x^4}{\sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx\)

Optimal. Leaf size=73 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{5/2}}-\frac{3 a x \sqrt{a+b x^2}}{8 b^2}+\frac{x^3 \sqrt{a+b x^2}}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.0685916, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{5/2}}-\frac{3 a x \sqrt{a+b x^2}}{8 b^2}+\frac{x^3 \sqrt{a+b x^2}}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 8.48802, size = 66, normalized size = 0.9 \[ \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8 b^{\frac{5}{2}}} - \frac{3 a x \sqrt{a + b x^{2}}}{8 b^{2}} + \frac{x^{3} \sqrt{a + b x^{2}}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**2+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0551104, size = 67, normalized size = 0.92 \[ \frac{3 a^2 \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{8 b^{5/2}}+\sqrt{a+b x^2} \left (\frac{x^3}{4 b}-\frac{3 a x}{8 b^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4],x]

[Out]

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

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Maple [A]  time = 0.01, size = 59, normalized size = 0.8 \[{\frac{{x}^{3}}{4\,b}\sqrt{b{x}^{2}+a}}-{\frac{3\,ax}{8\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{a}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2+a)^(1/2),x)

[Out]

1/4*x^3*(b*x^2+a)^(1/2)/b-3/8*a*x*(b*x^2+a)^(1/2)/b^2+3/8*a^2/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/sqrt(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.288658, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (2 \, b x^{3} - 3 \, a x\right )} \sqrt{b x^{2} + a} \sqrt{b}}{16 \, b^{\frac{5}{2}}}, \frac{3 \, a^{2} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (2 \, b x^{3} - 3 \, a x\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{8 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.429, size = 95, normalized size = 1.3 \[ - \frac{3 a^{\frac{3}{2}} x}{8 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} x^{3}}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} + \frac{x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**2+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.283285, size = 73, normalized size = 1. \[ \frac{1}{8} \, \sqrt{b x^{2} + a} x{\left (\frac{2 \, x^{2}}{b} - \frac{3 \, a}{b^{2}}\right )} - \frac{3 \, a^{2}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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