3.569 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^{13}} \, dx\)

Optimal. Leaf size=167 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left (a+b x^2\right )}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{12 x^{12} \left (a+b x^2\right )} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.253587, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left (a+b x^2\right )}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{12 x^{12} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^13,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.2065, size = 112, normalized size = 0.67 \[ - \frac{\left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{24 a x^{12}} + \frac{b \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{48 a^{2} x^{10}} - \frac{b \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{120 a^{3} x^{10}} \]

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**13,x)

[Out]

-(2*a + 2*b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(24*a*x**12) + b*(2*a +
2*b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(48*a**2*x**10) - b*(a**2 + 2*
a*b*x**2 + b**2*x**4)**(5/2)/(120*a**3*x**10)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0236637, size = 61, normalized size = 0.37 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (10 a^3+36 a^2 b x^2+45 a b^2 x^4+20 b^3 x^6\right )}{120 x^{12} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^13,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(10*a^3 + 36*a^2*b*x^2 + 45*a*b^2*x^4 + 20*b^3*x^6))/(120*
x^12*(a + b*x^2))

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 58, normalized size = 0.4 \[ -{\frac{20\,{b}^{3}{x}^{6}+45\,a{x}^{4}{b}^{2}+36\,{a}^{2}b{x}^{2}+10\,{a}^{3}}{120\,{x}^{12} \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^13,x)

[Out]

-1/120*(20*b^3*x^6+45*a*b^2*x^4+36*a^2*b*x^2+10*a^3)*((b*x^2+a)^2)^(3/2)/x^12/(b
*x^2+a)^3

_______________________________________________________________________________________

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)^(3/2)/x^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.264136, size = 50, normalized size = 0.3 \[ -\frac{20 \, b^{3} x^{6} + 45 \, a b^{2} x^{4} + 36 \, a^{2} b x^{2} + 10 \, a^{3}}{120 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/x^13,x, algorithm="fricas")

[Out]

-1/120*(20*b^3*x^6 + 45*a*b^2*x^4 + 36*a^2*b*x^2 + 10*a^3)/x^12

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}{x^{13}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)**(3/2)/x**13,x)

[Out]

Integral(((a + b*x**2)**2)**(3/2)/x**13, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.271521, size = 93, normalized size = 0.56 \[ -\frac{20 \, b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + 45 \, a b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 36 \, a^{2} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 10 \, a^{3}{\rm sign}\left (b x^{2} + a\right )}{120 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/x^13,x, algorithm="giac")

[Out]

-1/120*(20*b^3*x^6*sign(b*x^2 + a) + 45*a*b^2*x^4*sign(b*x^2 + a) + 36*a^2*b*x^2
*sign(b*x^2 + a) + 10*a^3*sign(b*x^2 + a))/x^12