3.614 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=249 \[ \frac{b^5 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac{5 a b^4 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{10 a^2 b^3 x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*x^5*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])/(3*x^3*(a + b*x^2)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(x*(a + b*x^2)) + (10*a^2*b^3*x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(a +
b*x^2) + (5*a*b^4*x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*(a + b*x^2)) + (b^5*x^
5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*(a + b*x^2))

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Rubi [A]  time = 0.177201, antiderivative size = 249, 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 \[ \frac{b^5 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac{5 a b^4 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{10 a^2 b^3 x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^6,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*x^5*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])/(3*x^3*(a + b*x^2)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(x*(a + b*x^2)) + (10*a^2*b^3*x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(a +
b*x^2) + (5*a*b^4*x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*(a + b*x^2)) + (b^5*x^
5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*(a + b*x^2))

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Rubi in Sympy [A]  time = 26.7095, size = 209, normalized size = 0.84 \[ - \frac{256 a^{3} b^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{15 x \left (a + b x^{2}\right )} + \frac{128 a^{2} b^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{15 x} + \frac{32 a b^{2} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{15 x} + \frac{2 a \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{3 x^{5}} + \frac{16 b^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{15 x} - \frac{13 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{15 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**6,x)

[Out]

-256*a**3*b**2*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(15*x*(a + b*x**2)) + 128*a**
2*b**2*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(15*x) + 32*a*b**2*(a + b*x**2)*sqrt(
a**2 + 2*a*b*x**2 + b**2*x**4)/(15*x) + 2*a*(a + b*x**2)*(a**2 + 2*a*b*x**2 + b*
*2*x**4)**(3/2)/(3*x**5) + 16*b**2*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(15*x)
- 13*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(15*x**5)

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Mathematica [A]  time = 0.0449608, size = 83, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (-3 a^5-25 a^4 b x^2-150 a^3 b^2 x^4+150 a^2 b^3 x^6+25 a b^4 x^8+3 b^5 x^{10}\right )}{15 x^5 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^6,x]

[Out]

(Sqrt[(a + b*x^2)^2]*(-3*a^5 - 25*a^4*b*x^2 - 150*a^3*b^2*x^4 + 150*a^2*b^3*x^6
+ 25*a*b^4*x^8 + 3*b^5*x^10))/(15*x^5*(a + b*x^2))

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Maple [A]  time = 0.009, size = 80, normalized size = 0.3 \[ -{\frac{-3\,{b}^{5}{x}^{10}-25\,a{b}^{4}{x}^{8}-150\,{a}^{2}{b}^{3}{x}^{6}+150\,{a}^{3}{b}^{2}{x}^{4}+25\,{a}^{4}b{x}^{2}+3\,{a}^{5}}{15\,{x}^{5} \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^6,x)

[Out]

-1/15*(-3*b^5*x^10-25*a*b^4*x^8-150*a^2*b^3*x^6+150*a^3*b^2*x^4+25*a^4*b*x^2+3*a
^5)*((b*x^2+a)^2)^(5/2)/x^5/(b*x^2+a)^5

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Maxima [A]  time = 0.69996, size = 80, normalized size = 0.32 \[ \frac{3 \, b^{5} x^{10} + 25 \, a b^{4} x^{8} + 150 \, a^{2} b^{3} x^{6} - 150 \, a^{3} b^{2} x^{4} - 25 \, a^{4} b x^{2} - 3 \, a^{5}}{15 \, x^{5}} \]

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^6,x, algorithm="maxima")

[Out]

1/15*(3*b^5*x^10 + 25*a*b^4*x^8 + 150*a^2*b^3*x^6 - 150*a^3*b^2*x^4 - 25*a^4*b*x
^2 - 3*a^5)/x^5

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Fricas [A]  time = 0.262312, size = 80, normalized size = 0.32 \[ \frac{3 \, b^{5} x^{10} + 25 \, a b^{4} x^{8} + 150 \, a^{2} b^{3} x^{6} - 150 \, a^{3} b^{2} x^{4} - 25 \, a^{4} b x^{2} - 3 \, a^{5}}{15 \, x^{5}} \]

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^6,x, algorithm="fricas")

[Out]

1/15*(3*b^5*x^10 + 25*a*b^4*x^8 + 150*a^2*b^3*x^6 - 150*a^3*b^2*x^4 - 25*a^4*b*x
^2 - 3*a^5)/x^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}{x^{6}}\, dx \]

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**6,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**6, x)

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GIAC/XCAS [A]  time = 0.272048, size = 143, normalized size = 0.57 \[ \frac{1}{5} \, b^{5} x^{5}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{3} \, a b^{4} x^{3}{\rm sign}\left (b x^{2} + a\right ) + 10 \, a^{2} b^{3} x{\rm sign}\left (b x^{2} + a\right ) - \frac{150 \, a^{3} b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 25 \, a^{4} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 3 \, a^{5}{\rm sign}\left (b x^{2} + a\right )}{15 \, x^{5}} \]

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^6,x, algorithm="giac")

[Out]

1/5*b^5*x^5*sign(b*x^2 + a) + 5/3*a*b^4*x^3*sign(b*x^2 + a) + 10*a^2*b^3*x*sign(
b*x^2 + a) - 1/15*(150*a^3*b^2*x^4*sign(b*x^2 + a) + 25*a^4*b*x^2*sign(b*x^2 + a
) + 3*a^5*sign(b*x^2 + a))/x^5