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

Optimal. Leaf size=118 \[ -\frac{a^6}{10 b^7 \left (a+b x^2\right )^5}+\frac{3 a^5}{4 b^7 \left (a+b x^2\right )^4}-\frac{5 a^4}{2 b^7 \left (a+b x^2\right )^3}+\frac{5 a^3}{b^7 \left (a+b x^2\right )^2}-\frac{15 a^2}{2 b^7 \left (a+b x^2\right )}-\frac{3 a \log \left (a+b x^2\right )}{b^7}+\frac{x^2}{2 b^6} \]

[Out]

x^2/(2*b^6) - a^6/(10*b^7*(a + b*x^2)^5) + (3*a^5)/(4*b^7*(a + b*x^2)^4) - (5*a^
4)/(2*b^7*(a + b*x^2)^3) + (5*a^3)/(b^7*(a + b*x^2)^2) - (15*a^2)/(2*b^7*(a + b*
x^2)) - (3*a*Log[a + b*x^2])/b^7

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Rubi [A]  time = 0.255735, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{a^6}{10 b^7 \left (a+b x^2\right )^5}+\frac{3 a^5}{4 b^7 \left (a+b x^2\right )^4}-\frac{5 a^4}{2 b^7 \left (a+b x^2\right )^3}+\frac{5 a^3}{b^7 \left (a+b x^2\right )^2}-\frac{15 a^2}{2 b^7 \left (a+b x^2\right )}-\frac{3 a \log \left (a+b x^2\right )}{b^7}+\frac{x^2}{2 b^6} \]

Antiderivative was successfully verified.

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

[Out]

x^2/(2*b^6) - a^6/(10*b^7*(a + b*x^2)^5) + (3*a^5)/(4*b^7*(a + b*x^2)^4) - (5*a^
4)/(2*b^7*(a + b*x^2)^3) + (5*a^3)/(b^7*(a + b*x^2)^2) - (15*a^2)/(2*b^7*(a + b*
x^2)) - (3*a*Log[a + b*x^2])/b^7

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{6}}{10 b^{7} \left (a + b x^{2}\right )^{5}} + \frac{3 a^{5}}{4 b^{7} \left (a + b x^{2}\right )^{4}} - \frac{5 a^{4}}{2 b^{7} \left (a + b x^{2}\right )^{3}} + \frac{5 a^{3}}{b^{7} \left (a + b x^{2}\right )^{2}} - \frac{15 a^{2}}{2 b^{7} \left (a + b x^{2}\right )} - \frac{3 a \log{\left (a + b x^{2} \right )}}{b^{7}} + \frac{b^{6} \int ^{x^{2}} \frac{1}{b^{12}}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**6/(10*b**7*(a + b*x**2)**5) + 3*a**5/(4*b**7*(a + b*x**2)**4) - 5*a**4/(2*b*
*7*(a + b*x**2)**3) + 5*a**3/(b**7*(a + b*x**2)**2) - 15*a**2/(2*b**7*(a + b*x**
2)) - 3*a*log(a + b*x**2)/b**7 + b**6*Integral(b**(-12), (x, x**2))/2

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Mathematica [A]  time = 0.0507, size = 101, normalized size = 0.86 \[ -\frac{87 a^6+375 a^5 b x^2+600 a^4 b^2 x^4+400 a^3 b^3 x^6+50 a^2 b^4 x^8-50 a b^5 x^{10}+60 a \left (a+b x^2\right )^5 \log \left (a+b x^2\right )-10 b^6 x^{12}}{20 b^7 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

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

[Out]

-(87*a^6 + 375*a^5*b*x^2 + 600*a^4*b^2*x^4 + 400*a^3*b^3*x^6 + 50*a^2*b^4*x^8 -
50*a*b^5*x^10 - 10*b^6*x^12 + 60*a*(a + b*x^2)^5*Log[a + b*x^2])/(20*b^7*(a + b*
x^2)^5)

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Maple [A]  time = 0.02, size = 109, normalized size = 0.9 \[{\frac{{x}^{2}}{2\,{b}^{6}}}-{\frac{{a}^{6}}{10\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{3\,{a}^{5}}{4\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{4}}}-{\frac{5\,{a}^{4}}{2\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{3}}}+5\,{\frac{{a}^{3}}{{b}^{7} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,{a}^{2}}{2\,{b}^{7} \left ( b{x}^{2}+a \right ) }}-3\,{\frac{a\ln \left ( b{x}^{2}+a \right ) }{{b}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2/b^6-1/10*a^6/b^7/(b*x^2+a)^5+3/4*a^5/b^7/(b*x^2+a)^4-5/2*a^4/b^7/(b*x^2+
a)^3+5*a^3/b^7/(b*x^2+a)^2-15/2*a^2/b^7/(b*x^2+a)-3*a*ln(b*x^2+a)/b^7

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Maxima [A]  time = 0.699634, size = 178, normalized size = 1.51 \[ -\frac{150 \, a^{2} b^{4} x^{8} + 500 \, a^{3} b^{3} x^{6} + 650 \, a^{4} b^{2} x^{4} + 385 \, a^{5} b x^{2} + 87 \, a^{6}}{20 \,{\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}} + \frac{x^{2}}{2 \, b^{6}} - \frac{3 \, a \log \left (b x^{2} + a\right )}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(150*a^2*b^4*x^8 + 500*a^3*b^3*x^6 + 650*a^4*b^2*x^4 + 385*a^5*b*x^2 + 87*
a^6)/(b^12*x^10 + 5*a*b^11*x^8 + 10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b^8*x^
2 + a^5*b^7) + 1/2*x^2/b^6 - 3*a*log(b*x^2 + a)/b^7

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Fricas [A]  time = 0.26371, size = 257, normalized size = 2.18 \[ \frac{10 \, b^{6} x^{12} + 50 \, a b^{5} x^{10} - 50 \, a^{2} b^{4} x^{8} - 400 \, a^{3} b^{3} x^{6} - 600 \, a^{4} b^{2} x^{4} - 375 \, a^{5} b x^{2} - 87 \, a^{6} - 60 \,{\left (a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 10 \, a^{4} b^{2} x^{4} + 5 \, a^{5} b x^{2} + a^{6}\right )} \log \left (b x^{2} + a\right )}{20 \,{\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(10*b^6*x^12 + 50*a*b^5*x^10 - 50*a^2*b^4*x^8 - 400*a^3*b^3*x^6 - 600*a^4*b
^2*x^4 - 375*a^5*b*x^2 - 87*a^6 - 60*(a*b^5*x^10 + 5*a^2*b^4*x^8 + 10*a^3*b^3*x^
6 + 10*a^4*b^2*x^4 + 5*a^5*b*x^2 + a^6)*log(b*x^2 + a))/(b^12*x^10 + 5*a*b^11*x^
8 + 10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b^8*x^2 + a^5*b^7)

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Sympy [A]  time = 5.14548, size = 136, normalized size = 1.15 \[ - \frac{3 a \log{\left (a + b x^{2} \right )}}{b^{7}} - \frac{87 a^{6} + 385 a^{5} b x^{2} + 650 a^{4} b^{2} x^{4} + 500 a^{3} b^{3} x^{6} + 150 a^{2} b^{4} x^{8}}{20 a^{5} b^{7} + 100 a^{4} b^{8} x^{2} + 200 a^{3} b^{9} x^{4} + 200 a^{2} b^{10} x^{6} + 100 a b^{11} x^{8} + 20 b^{12} x^{10}} + \frac{x^{2}}{2 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a*log(a + b*x**2)/b**7 - (87*a**6 + 385*a**5*b*x**2 + 650*a**4*b**2*x**4 + 50
0*a**3*b**3*x**6 + 150*a**2*b**4*x**8)/(20*a**5*b**7 + 100*a**4*b**8*x**2 + 200*
a**3*b**9*x**4 + 200*a**2*b**10*x**6 + 100*a*b**11*x**8 + 20*b**12*x**10) + x**2
/(2*b**6)

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GIAC/XCAS [A]  time = 0.271691, size = 128, normalized size = 1.08 \[ \frac{x^{2}}{2 \, b^{6}} - \frac{3 \, a{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{b^{7}} + \frac{137 \, a b^{5} x^{10} + 535 \, a^{2} b^{4} x^{8} + 870 \, a^{3} b^{3} x^{6} + 720 \, a^{4} b^{2} x^{4} + 300 \, a^{5} b x^{2} + 50 \, a^{6}}{20 \,{\left (b x^{2} + a\right )}^{5} b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2/b^6 - 3*a*ln(abs(b*x^2 + a))/b^7 + 1/20*(137*a*b^5*x^10 + 535*a^2*b^4*x^
8 + 870*a^3*b^3*x^6 + 720*a^4*b^2*x^4 + 300*a^5*b*x^2 + 50*a^6)/((b*x^2 + a)^5*b
^7)