3.616 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{10}} \, dx\)

Optimal. Leaf size=246 \[ \frac{b^5 x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{2 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^5 \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.173745, antiderivative size = 246, 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 \[ \frac{b^5 x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{2 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^5 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^10,x]

[Out]

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

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Rubi in Sympy [A]  time = 26.4847, size = 207, normalized size = 0.84 \[ - \frac{256 a b^{4} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{63 x \left (a + b x^{2}\right )} + \frac{32 a b^{2} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{63 x^{5}} + \frac{10 a \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{63 x^{9}} + \frac{128 b^{4} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{63 x} - \frac{16 b^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{21 x^{5}} - \frac{17 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{63 x^{9}} \]

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**10,x)

[Out]

-256*a*b**4*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(63*x*(a + b*x**2)) + 32*a*b**2*
(a + b*x**2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(63*x**5) + 10*a*(a + b*x**2)*(
a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(63*x**9) + 128*b**4*sqrt(a**2 + 2*a*b*x**
2 + b**2*x**4)/(63*x) - 16*b**2*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(21*x**5)
- 17*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(63*x**9)

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Mathematica [A]  time = 0.0312822, size = 83, normalized size = 0.34 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (7 a^5+45 a^4 b x^2+126 a^3 b^2 x^4+210 a^2 b^3 x^6+315 a b^4 x^8-63 b^5 x^{10}\right )}{63 x^9 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^10,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(7*a^5 + 45*a^4*b*x^2 + 126*a^3*b^2*x^4 + 210*a^2*b^3*x^6
+ 315*a*b^4*x^8 - 63*b^5*x^10))/(63*x^9*(a + b*x^2))

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Maple [A]  time = 0.008, size = 80, normalized size = 0.3 \[ -{\frac{-63\,{b}^{5}{x}^{10}+315\,a{b}^{4}{x}^{8}+210\,{a}^{2}{b}^{3}{x}^{6}+126\,{a}^{3}{b}^{2}{x}^{4}+45\,{a}^{4}b{x}^{2}+7\,{a}^{5}}{63\,{x}^{9} \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^10,x)

[Out]

-1/63*(-63*b^5*x^10+315*a*b^4*x^8+210*a^2*b^3*x^6+126*a^3*b^2*x^4+45*a^4*b*x^2+7
*a^5)*((b*x^2+a)^2)^(5/2)/x^9/(b*x^2+a)^5

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Maxima [A]  time = 0.71963, size = 80, normalized size = 0.33 \[ \frac{63 \, b^{5} x^{10} - 315 \, a b^{4} x^{8} - 210 \, a^{2} b^{3} x^{6} - 126 \, a^{3} b^{2} x^{4} - 45 \, a^{4} b x^{2} - 7 \, a^{5}}{63 \, x^{9}} \]

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^10,x, algorithm="maxima")

[Out]

1/63*(63*b^5*x^10 - 315*a*b^4*x^8 - 210*a^2*b^3*x^6 - 126*a^3*b^2*x^4 - 45*a^4*b
*x^2 - 7*a^5)/x^9

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Fricas [A]  time = 0.263454, size = 80, normalized size = 0.33 \[ \frac{63 \, b^{5} x^{10} - 315 \, a b^{4} x^{8} - 210 \, a^{2} b^{3} x^{6} - 126 \, a^{3} b^{2} x^{4} - 45 \, a^{4} b x^{2} - 7 \, a^{5}}{63 \, x^{9}} \]

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^10,x, algorithm="fricas")

[Out]

1/63*(63*b^5*x^10 - 315*a*b^4*x^8 - 210*a^2*b^3*x^6 - 126*a^3*b^2*x^4 - 45*a^4*b
*x^2 - 7*a^5)/x^9

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}{x^{10}}\, dx \]

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**10,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**10, x)

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GIAC/XCAS [A]  time = 0.271795, size = 142, normalized size = 0.58 \[ b^{5} x{\rm sign}\left (b x^{2} + a\right ) - \frac{315 \, a b^{4} x^{8}{\rm sign}\left (b x^{2} + a\right ) + 210 \, a^{2} b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + 126 \, a^{3} b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 45 \, a^{4} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 7 \, a^{5}{\rm sign}\left (b x^{2} + a\right )}{63 \, x^{9}} \]

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^10,x, algorithm="giac")

[Out]

b^5*x*sign(b*x^2 + a) - 1/63*(315*a*b^4*x^8*sign(b*x^2 + a) + 210*a^2*b^3*x^6*si
gn(b*x^2 + a) + 126*a^3*b^2*x^4*sign(b*x^2 + a) + 45*a^4*b*x^2*sign(b*x^2 + a) +
7*a^5*sign(b*x^2 + a))/x^9