3.601 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{19}} \, dx\)

Optimal. Leaf size=255 \[ -\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac{a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^{10} \left (a+b x^2\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^{12} \left (a+b x^2\right )}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{18 x^{18} \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^{14} \left (a+b x^2\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(18*x^18*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(16*x^16*(a + b*x^2)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^2
+ b^2*x^4])/(7*x^14*(a + b*x^2)) - (5*a^2*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/
(6*x^12*(a + b*x^2)) - (a*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(2*x^10*(a + b*x^
2)) - (b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*x^8*(a + b*x^2))

_______________________________________________________________________________________

Rubi [A]  time = 0.360323, 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 \[ -\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac{a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^{10} \left (a+b x^2\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^{12} \left (a+b x^2\right )}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{18 x^{18} \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^{14} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^19,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(18*x^18*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(16*x^16*(a + b*x^2)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^2
+ b^2*x^4])/(7*x^14*(a + b*x^2)) - (5*a^2*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/
(6*x^12*(a + b*x^2)) - (a*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(2*x^10*(a + b*x^
2)) - (b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*x^8*(a + b*x^2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.2613, size = 158, normalized size = 0.62 \[ - \frac{\left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{36 a x^{18}} + \frac{b \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{96 a^{2} x^{16}} - \frac{b^{2} \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{288 a^{3} x^{14}} + \frac{b^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{7}{2}}}{1008 a^{4} x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**19,x)

[Out]

-(2*a + 2*b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(36*a*x**18) + b*(2*a +
2*b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(96*a**2*x**16) - b**2*(2*a +
2*b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(288*a**3*x**14) + b**2*(a**2 +
2*a*b*x**2 + b**2*x**4)**(7/2)/(1008*a**4*x**14)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0308624, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (56 a^5+315 a^4 b x^2+720 a^3 b^2 x^4+840 a^2 b^3 x^6+504 a b^4 x^8+126 b^5 x^{10}\right )}{1008 x^{18} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^19,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(56*a^5 + 315*a^4*b*x^2 + 720*a^3*b^2*x^4 + 840*a^2*b^3*x^
6 + 504*a*b^4*x^8 + 126*b^5*x^10))/(1008*x^18*(a + b*x^2))

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 80, normalized size = 0.3 \[ -{\frac{126\,{b}^{5}{x}^{10}+504\,a{b}^{4}{x}^{8}+840\,{a}^{2}{b}^{3}{x}^{6}+720\,{a}^{3}{b}^{2}{x}^{4}+315\,{a}^{4}b{x}^{2}+56\,{a}^{5}}{1008\,{x}^{18} \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^19,x)

[Out]

-1/1008*(126*b^5*x^10+504*a*b^4*x^8+840*a^2*b^3*x^6+720*a^3*b^2*x^4+315*a^4*b*x^
2+56*a^5)*((b*x^2+a)^2)^(5/2)/x^18/(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^19,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.26662, size = 80, normalized size = 0.31 \[ -\frac{126 \, b^{5} x^{10} + 504 \, a b^{4} x^{8} + 840 \, a^{2} b^{3} x^{6} + 720 \, a^{3} b^{2} x^{4} + 315 \, a^{4} b x^{2} + 56 \, a^{5}}{1008 \, x^{18}} \]

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^19,x, algorithm="fricas")

[Out]

-1/1008*(126*b^5*x^10 + 504*a*b^4*x^8 + 840*a^2*b^3*x^6 + 720*a^3*b^2*x^4 + 315*
a^4*b*x^2 + 56*a^5)/x^18

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}{x^{19}}\, dx \]

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**19,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**19, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.271549, size = 144, normalized size = 0.56 \[ -\frac{126 \, b^{5} x^{10}{\rm sign}\left (b x^{2} + a\right ) + 504 \, a b^{4} x^{8}{\rm sign}\left (b x^{2} + a\right ) + 840 \, a^{2} b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + 720 \, a^{3} b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 315 \, a^{4} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 56 \, a^{5}{\rm sign}\left (b x^{2} + a\right )}{1008 \, x^{18}} \]

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^19,x, algorithm="giac")

[Out]

-1/1008*(126*b^5*x^10*sign(b*x^2 + a) + 504*a*b^4*x^8*sign(b*x^2 + a) + 840*a^2*
b^3*x^6*sign(b*x^2 + a) + 720*a^3*b^2*x^4*sign(b*x^2 + a) + 315*a^4*b*x^2*sign(b
*x^2 + a) + 56*a^5*sign(b*x^2 + a))/x^18