3.617 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{12}} \, dx\)

Optimal. Leaf size=251 \[ -\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac{2 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^5 \left (a+b x^2\right )}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*x^11*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(9*x^9*(a + b*x^2)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^2
+ b^2*x^4])/(7*x^7*(a + b*x^2)) - (2*a^2*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(x
^5*(a + b*x^2)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*x^3*(a + b*x^2))
- (b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(x*(a + b*x^2))

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Rubi [A]  time = 0.17755, antiderivative size = 251, 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 \[ -\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac{2 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^5 \left (a+b x^2\right )}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^12,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*x^11*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(9*x^9*(a + b*x^2)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^2
+ b^2*x^4])/(7*x^7*(a + b*x^2)) - (2*a^2*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(x
^5*(a + b*x^2)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*x^3*(a + b*x^2))
- (b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(x*(a + b*x^2))

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Rubi in Sympy [A]  time = 26.7293, size = 211, normalized size = 0.84 \[ \frac{256 a b^{4} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{693 x^{3} \left (a + b x^{2}\right )} + \frac{32 a b^{2} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{231 x^{7}} + \frac{10 a \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{99 x^{11}} - \frac{128 b^{4} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{231 x^{3}} - \frac{16 b^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{63 x^{7}} - \frac{19 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{99 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**12,x)

[Out]

256*a*b**4*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(693*x**3*(a + b*x**2)) + 32*a*b*
*2*(a + b*x**2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(231*x**7) + 10*a*(a + b*x**
2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(99*x**11) - 128*b**4*sqrt(a**2 + 2*a*
b*x**2 + b**2*x**4)/(231*x**3) - 16*b**2*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/
(63*x**7) - 19*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(99*x**11)

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Mathematica [A]  time = 0.0337643, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (63 a^5+385 a^4 b x^2+990 a^3 b^2 x^4+1386 a^2 b^3 x^6+1155 a b^4 x^8+693 b^5 x^{10}\right )}{693 x^{11} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^12,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(63*a^5 + 385*a^4*b*x^2 + 990*a^3*b^2*x^4 + 1386*a^2*b^3*x
^6 + 1155*a*b^4*x^8 + 693*b^5*x^10))/(693*x^11*(a + b*x^2))

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Maple [A]  time = 0.01, size = 80, normalized size = 0.3 \[ -{\frac{693\,{b}^{5}{x}^{10}+1155\,a{b}^{4}{x}^{8}+1386\,{a}^{2}{b}^{3}{x}^{6}+990\,{a}^{3}{b}^{2}{x}^{4}+385\,{a}^{4}b{x}^{2}+63\,{a}^{5}}{693\,{x}^{11} \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^12,x)

[Out]

-1/693*(693*b^5*x^10+1155*a*b^4*x^8+1386*a^2*b^3*x^6+990*a^3*b^2*x^4+385*a^4*b*x
^2+63*a^5)*((b*x^2+a)^2)^(5/2)/x^11/(b*x^2+a)^5

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Maxima [A]  time = 0.700499, size = 80, normalized size = 0.32 \[ -\frac{693 \, b^{5} x^{10} + 1155 \, a b^{4} x^{8} + 1386 \, a^{2} b^{3} x^{6} + 990 \, a^{3} b^{2} x^{4} + 385 \, a^{4} b x^{2} + 63 \, a^{5}}{693 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)/x^12,x, algorithm="maxima")

[Out]

-1/693*(693*b^5*x^10 + 1155*a*b^4*x^8 + 1386*a^2*b^3*x^6 + 990*a^3*b^2*x^4 + 385
*a^4*b*x^2 + 63*a^5)/x^11

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Fricas [A]  time = 0.260375, size = 80, normalized size = 0.32 \[ -\frac{693 \, b^{5} x^{10} + 1155 \, a b^{4} x^{8} + 1386 \, a^{2} b^{3} x^{6} + 990 \, a^{3} b^{2} x^{4} + 385 \, a^{4} b x^{2} + 63 \, a^{5}}{693 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)/x^12,x, algorithm="fricas")

[Out]

-1/693*(693*b^5*x^10 + 1155*a*b^4*x^8 + 1386*a^2*b^3*x^6 + 990*a^3*b^2*x^4 + 385
*a^4*b*x^2 + 63*a^5)/x^11

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}{x^{12}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**12,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**12, x)

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GIAC/XCAS [A]  time = 0.2716, size = 144, normalized size = 0.57 \[ -\frac{693 \, b^{5} x^{10}{\rm sign}\left (b x^{2} + a\right ) + 1155 \, a b^{4} x^{8}{\rm sign}\left (b x^{2} + a\right ) + 1386 \, a^{2} b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + 990 \, a^{3} b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 385 \, a^{4} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 63 \, a^{5}{\rm sign}\left (b x^{2} + a\right )}{693 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)/x^12,x, algorithm="giac")

[Out]

-1/693*(693*b^5*x^10*sign(b*x^2 + a) + 1155*a*b^4*x^8*sign(b*x^2 + a) + 1386*a^2
*b^3*x^6*sign(b*x^2 + a) + 990*a^3*b^2*x^4*sign(b*x^2 + a) + 385*a^4*b*x^2*sign(
b*x^2 + a) + 63*a^5*sign(b*x^2 + a))/x^11