3.583 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^{14}} \, dx\)

Optimal. Leaf size=167 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac{a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^9 \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left (a+b x^2\right )} \]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(13*x^13*(a + b*x^2)) - (3*a^2*b*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(11*x^11*(a + b*x^2)) - (a*b^2*Sqrt[a^2 + 2*a*b*x^2 + b
^2*x^4])/(3*x^9*(a + b*x^2)) - (b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*x^7*(a +
b*x^2))

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Rubi [A]  time = 0.128521, antiderivative size = 167, 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 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac{a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^9 \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^14,x]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(13*x^13*(a + b*x^2)) - (3*a^2*b*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(11*x^11*(a + b*x^2)) - (a*b^2*Sqrt[a^2 + 2*a*b*x^2 + b
^2*x^4])/(3*x^9*(a + b*x^2)) - (b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*x^7*(a +
b*x^2))

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Rubi in Sympy [A]  time = 16.7848, size = 138, normalized size = 0.83 \[ \frac{16 a b^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{3003 x^{9} \left (a + b x^{2}\right )} + \frac{6 a \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{143 x^{13}} - \frac{24 b^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{1001 x^{9}} - \frac{17 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{143 x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**(3/2)/x**14,x)

[Out]

16*a*b**2*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(3003*x**9*(a + b*x**2)) + 6*a*(a
+ b*x**2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(143*x**13) - 24*b**2*sqrt(a**2 +
2*a*b*x**2 + b**2*x**4)/(1001*x**9) - 17*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/
(143*x**13)

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Mathematica [A]  time = 0.0238637, size = 61, normalized size = 0.37 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (231 a^3+819 a^2 b x^2+1001 a b^2 x^4+429 b^3 x^6\right )}{3003 x^{13} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^14,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(231*a^3 + 819*a^2*b*x^2 + 1001*a*b^2*x^4 + 429*b^3*x^6))/
(3003*x^13*(a + b*x^2))

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Maple [A]  time = 0.01, size = 58, normalized size = 0.4 \[ -{\frac{429\,{b}^{3}{x}^{6}+1001\,a{x}^{4}{b}^{2}+819\,{a}^{2}b{x}^{2}+231\,{a}^{3}}{3003\,{x}^{13} \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^(3/2)/x^14,x)

[Out]

-1/3003*(429*b^3*x^6+1001*a*b^2*x^4+819*a^2*b*x^2+231*a^3)*((b*x^2+a)^2)^(3/2)/x
^13/(b*x^2+a)^3

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Maxima [A]  time = 0.691072, size = 50, normalized size = 0.3 \[ -\frac{429 \, b^{3} x^{6} + 1001 \, a b^{2} x^{4} + 819 \, a^{2} b x^{2} + 231 \, a^{3}}{3003 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/x^14,x, algorithm="maxima")

[Out]

-1/3003*(429*b^3*x^6 + 1001*a*b^2*x^4 + 819*a^2*b*x^2 + 231*a^3)/x^13

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Fricas [A]  time = 0.257893, size = 50, normalized size = 0.3 \[ -\frac{429 \, b^{3} x^{6} + 1001 \, a b^{2} x^{4} + 819 \, a^{2} b x^{2} + 231 \, a^{3}}{3003 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/x^14,x, algorithm="fricas")

[Out]

-1/3003*(429*b^3*x^6 + 1001*a*b^2*x^4 + 819*a^2*b*x^2 + 231*a^3)/x^13

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}{x^{14}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)**(3/2)/x**14,x)

[Out]

Integral(((a + b*x**2)**2)**(3/2)/x**14, x)

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GIAC/XCAS [A]  time = 0.26926, size = 93, normalized size = 0.56 \[ -\frac{429 \, b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + 1001 \, a b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 819 \, a^{2} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 231 \, a^{3}{\rm sign}\left (b x^{2} + a\right )}{3003 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/x^14,x, algorithm="giac")

[Out]

-1/3003*(429*b^3*x^6*sign(b*x^2 + a) + 1001*a*b^2*x^4*sign(b*x^2 + a) + 819*a^2*
b*x^2*sign(b*x^2 + a) + 231*a^3*sign(b*x^2 + a))/x^13