3.598 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{13}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{\left (a+b x^2\right )^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{12 a x^{12}} \]

[Out]

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

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Rubi [A]  time = 0.104701, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{\left (a+b x^2\right )^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{12 a x^{12}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^13,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.23687, size = 39, normalized size = 0.95 \[ - \frac{\left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{24 a x^{12}} \]

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**13,x)

[Out]

-(2*a + 2*b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(24*a*x**12)

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Mathematica [A]  time = 0.0302659, size = 81, normalized size = 1.98 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (a^5+6 a^4 b x^2+15 a^3 b^2 x^4+20 a^2 b^3 x^6+15 a b^4 x^8+6 b^5 x^{10}\right )}{12 x^{12} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^13,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(a^5 + 6*a^4*b*x^2 + 15*a^3*b^2*x^4 + 20*a^2*b^3*x^6 + 15*
a*b^4*x^8 + 6*b^5*x^10))/(12*x^12*(a + b*x^2))

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Maple [B]  time = 0.01, size = 78, normalized size = 1.9 \[ -{\frac{6\,{b}^{5}{x}^{10}+15\,a{b}^{4}{x}^{8}+20\,{a}^{2}{b}^{3}{x}^{6}+15\,{a}^{3}{b}^{2}{x}^{4}+6\,{a}^{4}b{x}^{2}+{a}^{5}}{12\,{x}^{12} \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^13,x)

[Out]

-1/12*(6*b^5*x^10+15*a*b^4*x^8+20*a^2*b^3*x^6+15*a^3*b^2*x^4+6*a^4*b*x^2+a^5)*((
b*x^2+a)^2)^(5/2)/x^12/(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^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267498, size = 77, normalized size = 1.88 \[ -\frac{6 \, b^{5} x^{10} + 15 \, a b^{4} x^{8} + 20 \, a^{2} b^{3} x^{6} + 15 \, a^{3} b^{2} x^{4} + 6 \, a^{4} b x^{2} + a^{5}}{12 \, x^{12}} \]

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^13,x, algorithm="fricas")

[Out]

-1/12*(6*b^5*x^10 + 15*a*b^4*x^8 + 20*a^2*b^3*x^6 + 15*a^3*b^2*x^4 + 6*a^4*b*x^2
+ a^5)/x^12

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}{x^{13}}\, dx \]

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**13,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**13, x)

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GIAC/XCAS [A]  time = 0.27369, size = 143, normalized size = 3.49 \[ -\frac{6 \, b^{5} x^{10}{\rm sign}\left (b x^{2} + a\right ) + 15 \, a b^{4} x^{8}{\rm sign}\left (b x^{2} + a\right ) + 20 \, a^{2} b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + 15 \, a^{3} b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 6 \, a^{4} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + a^{5}{\rm sign}\left (b x^{2} + a\right )}{12 \, x^{12}} \]

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^13,x, algorithm="giac")

[Out]

-1/12*(6*b^5*x^10*sign(b*x^2 + a) + 15*a*b^4*x^8*sign(b*x^2 + a) + 20*a^2*b^3*x^
6*sign(b*x^2 + a) + 15*a^3*b^2*x^4*sign(b*x^2 + a) + 6*a^4*b*x^2*sign(b*x^2 + a)
+ a^5*sign(b*x^2 + a))/x^12