3.570 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^{15}} \, dx\)

Optimal. Leaf size=167 \[ -\frac{a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 x^{12} \left (a+b x^2\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )} \]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(14*x^14*(a + b*x^2)) - (a^2*b*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])/(4*x^12*(a + b*x^2)) - (3*a*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^
2*x^4])/(10*x^10*(a + b*x^2)) - (b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*x^8*(a
+ b*x^2))

_______________________________________________________________________________________

Rubi [A]  time = 0.256836, 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 \[ -\frac{a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 x^{12} \left (a+b x^2\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^15,x]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(14*x^14*(a + b*x^2)) - (a^2*b*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])/(4*x^12*(a + b*x^2)) - (3*a*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^
2*x^4])/(10*x^10*(a + b*x^2)) - (b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*x^8*(a
+ b*x^2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.8263, size = 133, normalized size = 0.8 \[ \frac{a b^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{280 x^{10} \left (a + b x^{2}\right )} + \frac{a \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{28 x^{14}} - \frac{b^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{56 x^{10}} - \frac{3 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{28 x^{14}} \]

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**15,x)

[Out]

a*b**2*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(280*x**10*(a + b*x**2)) + a*(a + b*x
**2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(28*x**14) - b**2*sqrt(a**2 + 2*a*b*x**
2 + b**2*x**4)/(56*x**10) - 3*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(28*x**14)

_______________________________________________________________________________________

Mathematica [A]  time = 0.028362, size = 61, normalized size = 0.37 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (20 a^3+70 a^2 b x^2+84 a b^2 x^4+35 b^3 x^6\right )}{280 x^{14} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^15,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(20*a^3 + 70*a^2*b*x^2 + 84*a*b^2*x^4 + 35*b^3*x^6))/(280*
x^14*(a + b*x^2))

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 58, normalized size = 0.4 \[ -{\frac{35\,{b}^{3}{x}^{6}+84\,a{x}^{4}{b}^{2}+70\,{a}^{2}b{x}^{2}+20\,{a}^{3}}{280\,{x}^{14} \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^15,x)

[Out]

-1/280*(35*b^3*x^6+84*a*b^2*x^4+70*a^2*b*x^2+20*a^3)*((b*x^2+a)^2)^(3/2)/x^14/(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^15,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.261451, size = 50, normalized size = 0.3 \[ -\frac{35 \, b^{3} x^{6} + 84 \, a b^{2} x^{4} + 70 \, a^{2} b x^{2} + 20 \, a^{3}}{280 \, x^{14}} \]

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^15,x, algorithm="fricas")

[Out]

-1/280*(35*b^3*x^6 + 84*a*b^2*x^4 + 70*a^2*b*x^2 + 20*a^3)/x^14

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}{x^{15}}\, dx \]

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**15,x)

[Out]

Integral(((a + b*x**2)**2)**(3/2)/x**15, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270106, size = 93, normalized size = 0.56 \[ -\frac{35 \, b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + 84 \, a b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 70 \, a^{2} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 20 \, a^{3}{\rm sign}\left (b x^{2} + a\right )}{280 \, x^{14}} \]

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^15,x, algorithm="giac")

[Out]

-1/280*(35*b^3*x^6*sign(b*x^2 + a) + 84*a*b^2*x^4*sign(b*x^2 + a) + 70*a^2*b*x^2
*sign(b*x^2 + a) + 20*a^3*sign(b*x^2 + a))/x^14