3.582 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^{12}} \, dx\)

Optimal. Leaf size=167 \[ -\frac{a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^9 \left (a+b x^2\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )} \]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*x^11*(a + b*x^2)) - (a^2*b*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])/(3*x^9*(a + b*x^2)) - (3*a*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2
*x^4])/(7*x^7*(a + b*x^2)) - (b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*x^5*(a + b
*x^2))

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Rubi [A]  time = 0.125694, 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 \[ -\frac{a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^9 \left (a+b x^2\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^12,x]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*x^11*(a + b*x^2)) - (a^2*b*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])/(3*x^9*(a + b*x^2)) - (3*a*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2
*x^4])/(7*x^7*(a + b*x^2)) - (b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*x^5*(a + b
*x^2))

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Rubi in Sympy [A]  time = 16.7292, size = 138, normalized size = 0.83 \[ \frac{16 a b^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{1155 x^{7} \left (a + b x^{2}\right )} + \frac{2 a \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{33 x^{11}} - \frac{8 b^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{165 x^{7}} - \frac{5 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{33 x^{11}} \]

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**12,x)

[Out]

16*a*b**2*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(1155*x**7*(a + b*x**2)) + 2*a*(a
+ b*x**2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(33*x**11) - 8*b**2*sqrt(a**2 + 2*
a*b*x**2 + b**2*x**4)/(165*x**7) - 5*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(33*
x**11)

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Mathematica [A]  time = 0.0287521, size = 61, normalized size = 0.37 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (105 a^3+385 a^2 b x^2+495 a b^2 x^4+231 b^3 x^6\right )}{1155 x^{11} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^12,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(105*a^3 + 385*a^2*b*x^2 + 495*a*b^2*x^4 + 231*b^3*x^6))/(
1155*x^11*(a + b*x^2))

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Maple [A]  time = 0.01, size = 58, normalized size = 0.4 \[ -{\frac{231\,{b}^{3}{x}^{6}+495\,a{x}^{4}{b}^{2}+385\,{a}^{2}b{x}^{2}+105\,{a}^{3}}{1155\,{x}^{11} \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^12,x)

[Out]

-1/1155*(231*b^3*x^6+495*a*b^2*x^4+385*a^2*b*x^2+105*a^3)*((b*x^2+a)^2)^(3/2)/x^
11/(b*x^2+a)^3

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Maxima [A]  time = 0.692885, size = 50, normalized size = 0.3 \[ -\frac{231 \, b^{3} x^{6} + 495 \, a b^{2} x^{4} + 385 \, a^{2} b x^{2} + 105 \, a^{3}}{1155 \, x^{11}} \]

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^12,x, algorithm="maxima")

[Out]

-1/1155*(231*b^3*x^6 + 495*a*b^2*x^4 + 385*a^2*b*x^2 + 105*a^3)/x^11

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Fricas [A]  time = 0.260481, size = 50, normalized size = 0.3 \[ -\frac{231 \, b^{3} x^{6} + 495 \, a b^{2} x^{4} + 385 \, a^{2} b x^{2} + 105 \, a^{3}}{1155 \, x^{11}} \]

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^12,x, algorithm="fricas")

[Out]

-1/1155*(231*b^3*x^6 + 495*a*b^2*x^4 + 385*a^2*b*x^2 + 105*a^3)/x^11

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}{x^{12}}\, dx \]

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**12,x)

[Out]

Integral(((a + b*x**2)**2)**(3/2)/x**12, x)

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GIAC/XCAS [A]  time = 0.270948, size = 93, normalized size = 0.56 \[ -\frac{231 \, b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + 495 \, a b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 385 \, a^{2} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 105 \, a^{3}{\rm sign}\left (b x^{2} + a\right )}{1155 \, x^{11}} \]

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^12,x, algorithm="giac")

[Out]

-1/1155*(231*b^3*x^6*sign(b*x^2 + a) + 495*a*b^2*x^4*sign(b*x^2 + a) + 385*a^2*b
*x^2*sign(b*x^2 + a) + 105*a^3*sign(b*x^2 + a))/x^11