3.589 \(\int x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx\)

Optimal. Leaf size=119 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^7}{16 b^3}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^6}{7 b^3}+\frac{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^3} \]

[Out]

(a^2*(a + b*x^2)^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(12*b^3) - (a*(a + b*x^2)^6*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*b^3) + ((a + b*x^2)^7*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(16*b^3)

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Rubi [A]  time = 0.238568, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^7}{16 b^3}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^6}{7 b^3}+\frac{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(a^2*(a + b*x^2)^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(12*b^3) - (a*(a + b*x^2)^6*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*b^3) + ((a + b*x^2)^7*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(16*b^3)

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Rubi in Sympy [A]  time = 20.591, size = 107, normalized size = 0.9 \[ \frac{a^{2} \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{96 b^{3}} - \frac{a \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{7}{2}}}{56 b^{3}} + \frac{x^{4} \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{32 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

a**2*(2*a + 2*b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(96*b**3) - a*(a**2
+ 2*a*b*x**2 + b**2*x**4)**(7/2)/(56*b**3) + x**4*(2*a + 2*b*x**2)*(a**2 + 2*a*
b*x**2 + b**2*x**4)**(5/2)/(32*b)

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Mathematica [A]  time = 0.0370608, size = 83, normalized size = 0.7 \[ \frac{x^6 \sqrt{\left (a+b x^2\right )^2} \left (56 a^5+210 a^4 b x^2+336 a^3 b^2 x^4+280 a^2 b^3 x^6+120 a b^4 x^8+21 b^5 x^{10}\right )}{336 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(x^6*Sqrt[(a + b*x^2)^2]*(56*a^5 + 210*a^4*b*x^2 + 336*a^3*b^2*x^4 + 280*a^2*b^3
*x^6 + 120*a*b^4*x^8 + 21*b^5*x^10))/(336*(a + b*x^2))

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Maple [A]  time = 0.009, size = 80, normalized size = 0.7 \[{\frac{{x}^{6} \left ( 21\,{b}^{5}{x}^{10}+120\,a{b}^{4}{x}^{8}+280\,{a}^{2}{b}^{3}{x}^{6}+336\,{a}^{3}{b}^{2}{x}^{4}+210\,{a}^{4}b{x}^{2}+56\,{a}^{5} \right ) }{336\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/336*x^6*(21*b^5*x^10+120*a*b^4*x^8+280*a^2*b^3*x^6+336*a^3*b^2*x^4+210*a^4*b*x
^2+56*a^5)*((b*x^2+a)^2)^(5/2)/(b*x^2+a)^5

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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((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.256256, size = 76, normalized size = 0.64 \[ \frac{1}{16} \, b^{5} x^{16} + \frac{5}{14} \, a b^{4} x^{14} + \frac{5}{6} \, a^{2} b^{3} x^{12} + a^{3} b^{2} x^{10} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{6} \, a^{5} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*x^5,x, algorithm="fricas")

[Out]

1/16*b^5*x^16 + 5/14*a*b^4*x^14 + 5/6*a^2*b^3*x^12 + a^3*b^2*x^10 + 5/8*a^4*b*x^
8 + 1/6*a^5*x^6

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{5} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral(x**5*((a + b*x**2)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.271189, size = 140, normalized size = 1.18 \[ \frac{1}{16} \, b^{5} x^{16}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{14} \, a b^{4} x^{14}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{6} \, a^{2} b^{3} x^{12}{\rm sign}\left (b x^{2} + a\right ) + a^{3} b^{2} x^{10}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{8} \, a^{4} b x^{8}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{6} \, a^{5} x^{6}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*x^5,x, algorithm="giac")

[Out]

1/16*b^5*x^16*sign(b*x^2 + a) + 5/14*a*b^4*x^14*sign(b*x^2 + a) + 5/6*a^2*b^3*x^
12*sign(b*x^2 + a) + a^3*b^2*x^10*sign(b*x^2 + a) + 5/8*a^4*b*x^8*sign(b*x^2 + a
) + 1/6*a^5*x^6*sign(b*x^2 + a)