3.603 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{23}} \, dx\)

Optimal. Leaf size=255 \[ -\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{12 x^{12} \left (a+b x^2\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^{16} \left (a+b x^2\right )}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{22 x^{22} \left (a+b x^2\right )}-\frac{a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 x^{20} \left (a+b x^2\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^{18} \left (a+b x^2\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(22*x^22*(a + b*x^2)) - (a^4*b*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])/(4*x^20*(a + b*x^2)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(9*x^18*(a + b*x^2)) - (5*a^2*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*
x^16*(a + b*x^2)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(14*x^14*(a + b*x^
2)) - (b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(12*x^12*(a + b*x^2))

_______________________________________________________________________________________

Rubi [A]  time = 0.365483, antiderivative size = 255, 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 \[ -\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{12 x^{12} \left (a+b x^2\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^{16} \left (a+b x^2\right )}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{22 x^{22} \left (a+b x^2\right )}-\frac{a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 x^{20} \left (a+b x^2\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^{18} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^23,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(22*x^22*(a + b*x^2)) - (a^4*b*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])/(4*x^20*(a + b*x^2)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(9*x^18*(a + b*x^2)) - (5*a^2*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*
x^16*(a + b*x^2)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(14*x^14*(a + b*x^
2)) - (b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(12*x^12*(a + b*x^2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 26.5308, size = 202, normalized size = 0.79 \[ \frac{a b^{4} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{5544 x^{14} \left (a + b x^{2}\right )} + \frac{a b^{2} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{264 x^{18}} + \frac{a \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{44 x^{22}} - \frac{b^{4} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{792 x^{14}} - \frac{b^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{72 x^{18}} - \frac{3 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{44 x^{22}} \]

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**23,x)

[Out]

a*b**4*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(5544*x**14*(a + b*x**2)) + a*b**2*(a
+ b*x**2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(264*x**18) + a*(a + b*x**2)*(a**
2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(44*x**22) - b**4*sqrt(a**2 + 2*a*b*x**2 + b*
*2*x**4)/(792*x**14) - b**2*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(72*x**18) -
3*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(44*x**22)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0351357, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (252 a^5+1386 a^4 b x^2+3080 a^3 b^2 x^4+3465 a^2 b^3 x^6+1980 a b^4 x^8+462 b^5 x^{10}\right )}{5544 x^{22} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^23,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(252*a^5 + 1386*a^4*b*x^2 + 3080*a^3*b^2*x^4 + 3465*a^2*b^
3*x^6 + 1980*a*b^4*x^8 + 462*b^5*x^10))/(5544*x^22*(a + b*x^2))

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 80, normalized size = 0.3 \[ -{\frac{462\,{b}^{5}{x}^{10}+1980\,a{b}^{4}{x}^{8}+3465\,{a}^{2}{b}^{3}{x}^{6}+3080\,{a}^{3}{b}^{2}{x}^{4}+1386\,{a}^{4}b{x}^{2}+252\,{a}^{5}}{5544\,{x}^{22} \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^23,x)

[Out]

-1/5544*(462*b^5*x^10+1980*a*b^4*x^8+3465*a^2*b^3*x^6+3080*a^3*b^2*x^4+1386*a^4*
b*x^2+252*a^5)*((b*x^2+a)^2)^(5/2)/x^22/(b*x^2+a)^5

_______________________________________________________________________________________

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)^(5/2)/x^23,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.267657, size = 80, normalized size = 0.31 \[ -\frac{462 \, b^{5} x^{10} + 1980 \, a b^{4} x^{8} + 3465 \, a^{2} b^{3} x^{6} + 3080 \, a^{3} b^{2} x^{4} + 1386 \, a^{4} b x^{2} + 252 \, a^{5}}{5544 \, x^{22}} \]

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^23,x, algorithm="fricas")

[Out]

-1/5544*(462*b^5*x^10 + 1980*a*b^4*x^8 + 3465*a^2*b^3*x^6 + 3080*a^3*b^2*x^4 + 1
386*a^4*b*x^2 + 252*a^5)/x^22

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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**23,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.273884, size = 144, normalized size = 0.56 \[ -\frac{462 \, b^{5} x^{10}{\rm sign}\left (b x^{2} + a\right ) + 1980 \, a b^{4} x^{8}{\rm sign}\left (b x^{2} + a\right ) + 3465 \, a^{2} b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + 3080 \, a^{3} b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 1386 \, a^{4} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 252 \, a^{5}{\rm sign}\left (b x^{2} + a\right )}{5544 \, x^{22}} \]

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^23,x, algorithm="giac")

[Out]

-1/5544*(462*b^5*x^10*sign(b*x^2 + a) + 1980*a*b^4*x^8*sign(b*x^2 + a) + 3465*a^
2*b^3*x^6*sign(b*x^2 + a) + 3080*a^3*b^2*x^4*sign(b*x^2 + a) + 1386*a^4*b*x^2*si
gn(b*x^2 + a) + 252*a^5*sign(b*x^2 + a))/x^22