3.145 \(\int \frac{x^{13}}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=94 \[ -\frac{a^6}{2 b^7 \left (a+b x^2\right )}-\frac{3 a^5 \log \left (a+b x^2\right )}{b^7}+\frac{5 a^4 x^2}{2 b^6}-\frac{a^3 x^4}{b^5}+\frac{a^2 x^6}{2 b^4}-\frac{a x^8}{4 b^3}+\frac{x^{10}}{10 b^2} \]

[Out]

(5*a^4*x^2)/(2*b^6) - (a^3*x^4)/b^5 + (a^2*x^6)/(2*b^4) - (a*x^8)/(4*b^3) + x^10
/(10*b^2) - a^6/(2*b^7*(a + b*x^2)) - (3*a^5*Log[a + b*x^2])/b^7

_______________________________________________________________________________________

Rubi [A]  time = 0.181404, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{a^6}{2 b^7 \left (a+b x^2\right )}-\frac{3 a^5 \log \left (a+b x^2\right )}{b^7}+\frac{5 a^4 x^2}{2 b^6}-\frac{a^3 x^4}{b^5}+\frac{a^2 x^6}{2 b^4}-\frac{a x^8}{4 b^3}+\frac{x^{10}}{10 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^13/(a + b*x^2)^2,x]

[Out]

(5*a^4*x^2)/(2*b^6) - (a^3*x^4)/b^5 + (a^2*x^6)/(2*b^4) - (a*x^8)/(4*b^3) + x^10
/(10*b^2) - a^6/(2*b^7*(a + b*x^2)) - (3*a^5*Log[a + b*x^2])/b^7

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{6}}{2 b^{7} \left (a + b x^{2}\right )} - \frac{3 a^{5} \log{\left (a + b x^{2} \right )}}{b^{7}} + \frac{5 a^{4} x^{2}}{2 b^{6}} - \frac{2 a^{3} \int ^{x^{2}} x\, dx}{b^{5}} + \frac{a^{2} x^{6}}{2 b^{4}} - \frac{a x^{8}}{4 b^{3}} + \frac{x^{10}}{10 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**13/(b*x**2+a)**2,x)

[Out]

-a**6/(2*b**7*(a + b*x**2)) - 3*a**5*log(a + b*x**2)/b**7 + 5*a**4*x**2/(2*b**6)
- 2*a**3*Integral(x, (x, x**2))/b**5 + a**2*x**6/(2*b**4) - a*x**8/(4*b**3) + x
**10/(10*b**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0565087, size = 83, normalized size = 0.88 \[ \frac{-\frac{10 a^6}{a+b x^2}-60 a^5 \log \left (a+b x^2\right )+50 a^4 b x^2-20 a^3 b^2 x^4+10 a^2 b^3 x^6-5 a b^4 x^8+2 b^5 x^{10}}{20 b^7} \]

Antiderivative was successfully verified.

[In]  Integrate[x^13/(a + b*x^2)^2,x]

[Out]

(50*a^4*b*x^2 - 20*a^3*b^2*x^4 + 10*a^2*b^3*x^6 - 5*a*b^4*x^8 + 2*b^5*x^10 - (10
*a^6)/(a + b*x^2) - 60*a^5*Log[a + b*x^2])/(20*b^7)

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 85, normalized size = 0.9 \[{\frac{5\,{a}^{4}{x}^{2}}{2\,{b}^{6}}}-{\frac{{a}^{3}{x}^{4}}{{b}^{5}}}+{\frac{{a}^{2}{x}^{6}}{2\,{b}^{4}}}-{\frac{a{x}^{8}}{4\,{b}^{3}}}+{\frac{{x}^{10}}{10\,{b}^{2}}}-{\frac{{a}^{6}}{2\,{b}^{7} \left ( b{x}^{2}+a \right ) }}-3\,{\frac{{a}^{5}\ln \left ( b{x}^{2}+a \right ) }{{b}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^13/(b*x^2+a)^2,x)

[Out]

5/2*a^4*x^2/b^6-a^3*x^4/b^5+1/2*a^2*x^6/b^4-1/4*a*x^8/b^3+1/10*x^10/b^2-1/2*a^6/
b^7/(b*x^2+a)-3*a^5*ln(b*x^2+a)/b^7

_______________________________________________________________________________________

Maxima [A]  time = 1.34635, size = 119, normalized size = 1.27 \[ -\frac{a^{6}}{2 \,{\left (b^{8} x^{2} + a b^{7}\right )}} - \frac{3 \, a^{5} \log \left (b x^{2} + a\right )}{b^{7}} + \frac{2 \, b^{4} x^{10} - 5 \, a b^{3} x^{8} + 10 \, a^{2} b^{2} x^{6} - 20 \, a^{3} b x^{4} + 50 \, a^{4} x^{2}}{20 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*a^6/(b^8*x^2 + a*b^7) - 3*a^5*log(b*x^2 + a)/b^7 + 1/20*(2*b^4*x^10 - 5*a*b
^3*x^8 + 10*a^2*b^2*x^6 - 20*a^3*b*x^4 + 50*a^4*x^2)/b^6

_______________________________________________________________________________________

Fricas [A]  time = 0.197825, size = 140, normalized size = 1.49 \[ \frac{2 \, b^{6} x^{12} - 3 \, a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} - 10 \, a^{3} b^{3} x^{6} + 30 \, a^{4} b^{2} x^{4} + 50 \, a^{5} b x^{2} - 10 \, a^{6} - 60 \,{\left (a^{5} b x^{2} + a^{6}\right )} \log \left (b x^{2} + a\right )}{20 \,{\left (b^{8} x^{2} + a b^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(2*b^6*x^12 - 3*a*b^5*x^10 + 5*a^2*b^4*x^8 - 10*a^3*b^3*x^6 + 30*a^4*b^2*x^
4 + 50*a^5*b*x^2 - 10*a^6 - 60*(a^5*b*x^2 + a^6)*log(b*x^2 + a))/(b^8*x^2 + a*b^
7)

_______________________________________________________________________________________

Sympy [A]  time = 1.74405, size = 88, normalized size = 0.94 \[ - \frac{a^{6}}{2 a b^{7} + 2 b^{8} x^{2}} - \frac{3 a^{5} \log{\left (a + b x^{2} \right )}}{b^{7}} + \frac{5 a^{4} x^{2}}{2 b^{6}} - \frac{a^{3} x^{4}}{b^{5}} + \frac{a^{2} x^{6}}{2 b^{4}} - \frac{a x^{8}}{4 b^{3}} + \frac{x^{10}}{10 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**6/(2*a*b**7 + 2*b**8*x**2) - 3*a**5*log(a + b*x**2)/b**7 + 5*a**4*x**2/(2*b*
*6) - a**3*x**4/b**5 + a**2*x**6/(2*b**4) - a*x**8/(4*b**3) + x**10/(10*b**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211721, size = 139, normalized size = 1.48 \[ -\frac{3 \, a^{5}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{b^{7}} + \frac{6 \, a^{5} b x^{2} + 5 \, a^{6}}{2 \,{\left (b x^{2} + a\right )} b^{7}} + \frac{2 \, b^{8} x^{10} - 5 \, a b^{7} x^{8} + 10 \, a^{2} b^{6} x^{6} - 20 \, a^{3} b^{5} x^{4} + 50 \, a^{4} b^{4} x^{2}}{20 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a^5*ln(abs(b*x^2 + a))/b^7 + 1/2*(6*a^5*b*x^2 + 5*a^6)/((b*x^2 + a)*b^7) + 1/
20*(2*b^8*x^10 - 5*a*b^7*x^8 + 10*a^2*b^6*x^6 - 20*a^3*b^5*x^4 + 50*a^4*b^4*x^2)
/b^10