3.550 \(\int x^4 \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx\)

Optimal. Leaf size=79 \[ \frac{b x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{a x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )} \]

[Out]

(a*x^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*(a + b*x^2)) + (b*x^7*Sqrt[a^2 + 2*a*
b*x^2 + b^2*x^4])/(7*(a + b*x^2))

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Rubi [A]  time = 0.0727897, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{b x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{a x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

(a*x^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*(a + b*x^2)) + (b*x^7*Sqrt[a^2 + 2*a*
b*x^2 + b^2*x^4])/(7*(a + b*x^2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{4} \sqrt{\left (a + b x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**4*sqrt((a + b*x**2)**2), x)

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Mathematica [A]  time = 0.0118893, size = 39, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (7 a x^5+5 b x^7\right )}{35 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

(Sqrt[(a + b*x^2)^2]*(7*a*x^5 + 5*b*x^7))/(35*(a + b*x^2))

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Maple [A]  time = 0.005, size = 36, normalized size = 0.5 \[{\frac{{x}^{5} \left ( 5\,b{x}^{2}+7\,a \right ) }{35\,b{x}^{2}+35\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*((b*x^2+a)^2)^(1/2),x)

[Out]

1/35*x^5*(5*b*x^2+7*a)*((b*x^2+a)^2)^(1/2)/(b*x^2+a)

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Maxima [A]  time = 0.695197, size = 18, normalized size = 0.23 \[ \frac{1}{7} \, b x^{7} + \frac{1}{5} \, a x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b*x^7 + 1/5*a*x^5

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Fricas [A]  time = 0.255577, size = 18, normalized size = 0.23 \[ \frac{1}{7} \, b x^{7} + \frac{1}{5} \, a x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b*x^7 + 1/5*a*x^5

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Sympy [A]  time = 0.165378, size = 12, normalized size = 0.15 \[ \frac{a x^{5}}{5} + \frac{b x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**5/5 + b*x**7/7

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GIAC/XCAS [A]  time = 0.269305, size = 39, normalized size = 0.49 \[ \frac{1}{7} \, b x^{7}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{5} \, a x^{5}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b*x^7*sign(b*x^2 + a) + 1/5*a*x^5*sign(b*x^2 + a)