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

Optimal. Leaf size=247 \[ \frac{b^5 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac{5 a b^4 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{2 a^2 b^3 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}+\frac{5 a^4 b x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{10 a^3 b^2 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]

[Out]

-((a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(x*(a + b*x^2))) + (5*a^4*b*x*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])/(a + b*x^2) + (10*a^3*b^2*x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*
x^4])/(3*(a + b*x^2)) + (2*a^2*b^3*x^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(a + b*x
^2) + (5*a*b^4*x^7*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*(a + b*x^2)) + (b^5*x^9*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(9*(a + b*x^2))

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Rubi [A]  time = 0.17594, antiderivative size = 247, 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 \[ \frac{b^5 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac{5 a b^4 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{2 a^2 b^3 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}+\frac{5 a^4 b x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{10 a^3 b^2 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(x*(a + b*x^2))) + (5*a^4*b*x*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])/(a + b*x^2) + (10*a^3*b^2*x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*
x^4])/(3*(a + b*x^2)) + (2*a^2*b^3*x^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(a + b*x
^2) + (5*a*b^4*x^7*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*(a + b*x^2)) + (b^5*x^9*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(9*(a + b*x^2))

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Rubi in Sympy [A]  time = 26.1457, size = 196, normalized size = 0.79 \[ - \frac{256 a^{5} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{63 x \left (a + b x^{2}\right )} + \frac{128 a^{4} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{63 x} + \frac{32 a^{3} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{63 x} + \frac{16 a^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{63 x} + \frac{10 a \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{63 x} + \frac{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{9 x} \]

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**2,x)

[Out]

-256*a**5*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(63*x*(a + b*x**2)) + 128*a**4*sqr
t(a**2 + 2*a*b*x**2 + b**2*x**4)/(63*x) + 32*a**3*(a + b*x**2)*sqrt(a**2 + 2*a*b
*x**2 + b**2*x**4)/(63*x) + 16*a**2*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(63*x
) + 10*a*(a + b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(63*x) + (a**2 + 2*
a*b*x**2 + b**2*x**4)**(5/2)/(9*x)

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Mathematica [A]  time = 0.0390482, size = 83, normalized size = 0.34 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (-63 a^5+315 a^4 b x^2+210 a^3 b^2 x^4+126 a^2 b^3 x^6+45 a b^4 x^8+7 b^5 x^{10}\right )}{63 x \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^2)^2]*(-63*a^5 + 315*a^4*b*x^2 + 210*a^3*b^2*x^4 + 126*a^2*b^3*x^
6 + 45*a*b^4*x^8 + 7*b^5*x^10))/(63*x*(a + b*x^2))

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Maple [A]  time = 0.009, size = 80, normalized size = 0.3 \[ -{\frac{-7\,{b}^{5}{x}^{10}-45\,a{b}^{4}{x}^{8}-126\,{a}^{2}{b}^{3}{x}^{6}-210\,{a}^{3}{b}^{2}{x}^{4}-315\,{a}^{4}b{x}^{2}+63\,{a}^{5}}{63\,x \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^2,x)

[Out]

-1/63*(-7*b^5*x^10-45*a*b^4*x^8-126*a^2*b^3*x^6-210*a^3*b^2*x^4-315*a^4*b*x^2+63
*a^5)*((b*x^2+a)^2)^(5/2)/x/(b*x^2+a)^5

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Maxima [A]  time = 0.705841, size = 80, normalized size = 0.32 \[ \frac{7 \, b^{5} x^{10} + 45 \, a b^{4} x^{8} + 126 \, a^{2} b^{3} x^{6} + 210 \, a^{3} b^{2} x^{4} + 315 \, a^{4} b x^{2} - 63 \, a^{5}}{63 \, x} \]

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^2,x, algorithm="maxima")

[Out]

1/63*(7*b^5*x^10 + 45*a*b^4*x^8 + 126*a^2*b^3*x^6 + 210*a^3*b^2*x^4 + 315*a^4*b*
x^2 - 63*a^5)/x

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Fricas [A]  time = 0.259916, size = 80, normalized size = 0.32 \[ \frac{7 \, b^{5} x^{10} + 45 \, a b^{4} x^{8} + 126 \, a^{2} b^{3} x^{6} + 210 \, a^{3} b^{2} x^{4} + 315 \, a^{4} b x^{2} - 63 \, a^{5}}{63 \, x} \]

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^2,x, algorithm="fricas")

[Out]

1/63*(7*b^5*x^10 + 45*a*b^4*x^8 + 126*a^2*b^3*x^6 + 210*a^3*b^2*x^4 + 315*a^4*b*
x^2 - 63*a^5)/x

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

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**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.271463, size = 139, normalized size = 0.56 \[ \frac{1}{9} \, b^{5} x^{9}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{7} \, a b^{4} x^{7}{\rm sign}\left (b x^{2} + a\right ) + 2 \, a^{2} b^{3} x^{5}{\rm sign}\left (b x^{2} + a\right ) + \frac{10}{3} \, a^{3} b^{2} x^{3}{\rm sign}\left (b x^{2} + a\right ) + 5 \, a^{4} b x{\rm sign}\left (b x^{2} + a\right ) - \frac{a^{5}{\rm sign}\left (b x^{2} + a\right )}{x} \]

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^2,x, algorithm="giac")

[Out]

1/9*b^5*x^9*sign(b*x^2 + a) + 5/7*a*b^4*x^7*sign(b*x^2 + a) + 2*a^2*b^3*x^5*sign
(b*x^2 + a) + 10/3*a^3*b^2*x^3*sign(b*x^2 + a) + 5*a^4*b*x*sign(b*x^2 + a) - a^5
*sign(b*x^2 + a)/x