3.170 \(\int \frac{x^{11}}{\left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=87 \[ \frac{a^5}{4 b^6 \left (a+b x^2\right )^2}-\frac{5 a^4}{2 b^6 \left (a+b x^2\right )}-\frac{5 a^3 \log \left (a+b x^2\right )}{b^6}+\frac{3 a^2 x^2}{b^5}-\frac{3 a x^4}{4 b^4}+\frac{x^6}{6 b^3} \]

[Out]

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

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Rubi [A]  time = 0.166276, antiderivative size = 87, 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 \[ \frac{a^5}{4 b^6 \left (a+b x^2\right )^2}-\frac{5 a^4}{2 b^6 \left (a+b x^2\right )}-\frac{5 a^3 \log \left (a+b x^2\right )}{b^6}+\frac{3 a^2 x^2}{b^5}-\frac{3 a x^4}{4 b^4}+\frac{x^6}{6 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^11/(a + b*x^2)^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(b*x**2+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0437977, size = 75, normalized size = 0.86 \[ \frac{\frac{3 a^5}{\left (a+b x^2\right )^2}-\frac{30 a^4}{a+b x^2}-60 a^3 \log \left (a+b x^2\right )+36 a^2 b x^2-9 a b^2 x^4+2 b^3 x^6}{12 b^6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(a + b*x^2)^3,x]

[Out]

(36*a^2*b*x^2 - 9*a*b^2*x^4 + 2*b^3*x^6 + (3*a^5)/(a + b*x^2)^2 - (30*a^4)/(a +
b*x^2) - 60*a^3*Log[a + b*x^2])/(12*b^6)

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Maple [A]  time = 0.015, size = 80, normalized size = 0.9 \[ 3\,{\frac{{a}^{2}{x}^{2}}{{b}^{5}}}-{\frac{3\,a{x}^{4}}{4\,{b}^{4}}}+{\frac{{x}^{6}}{6\,{b}^{3}}}+{\frac{{a}^{5}}{4\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,{a}^{4}}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) }}-5\,{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) }{{b}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(b*x^2+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.35859, size = 120, normalized size = 1.38 \[ -\frac{10 \, a^{4} b x^{2} + 9 \, a^{5}}{4 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} - \frac{5 \, a^{3} \log \left (b x^{2} + a\right )}{b^{6}} + \frac{2 \, b^{2} x^{6} - 9 \, a b x^{4} + 36 \, a^{2} x^{2}}{12 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b*x^2 + a)^3,x, algorithm="maxima")

[Out]

-1/4*(10*a^4*b*x^2 + 9*a^5)/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6) - 5*a^3*log(b*x^2
+ a)/b^6 + 1/12*(2*b^2*x^6 - 9*a*b*x^4 + 36*a^2*x^2)/b^5

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Fricas [A]  time = 0.206862, size = 155, normalized size = 1.78 \[ \frac{2 \, b^{5} x^{10} - 5 \, a b^{4} x^{8} + 20 \, a^{2} b^{3} x^{6} + 63 \, a^{3} b^{2} x^{4} + 6 \, a^{4} b x^{2} - 27 \, a^{5} - 60 \,{\left (a^{3} b^{2} x^{4} + 2 \, a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{12 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b*x^2 + a)^3,x, algorithm="fricas")

[Out]

1/12*(2*b^5*x^10 - 5*a*b^4*x^8 + 20*a^2*b^3*x^6 + 63*a^3*b^2*x^4 + 6*a^4*b*x^2 -
27*a^5 - 60*(a^3*b^2*x^4 + 2*a^4*b*x^2 + a^5)*log(b*x^2 + a))/(b^8*x^4 + 2*a*b^
7*x^2 + a^2*b^6)

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Sympy [A]  time = 2.24815, size = 90, normalized size = 1.03 \[ - \frac{5 a^{3} \log{\left (a + b x^{2} \right )}}{b^{6}} + \frac{3 a^{2} x^{2}}{b^{5}} - \frac{3 a x^{4}}{4 b^{4}} - \frac{9 a^{5} + 10 a^{4} b x^{2}}{4 a^{2} b^{6} + 8 a b^{7} x^{2} + 4 b^{8} x^{4}} + \frac{x^{6}}{6 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(b*x**2+a)**3,x)

[Out]

-5*a**3*log(a + b*x**2)/b**6 + 3*a**2*x**2/b**5 - 3*a*x**4/(4*b**4) - (9*a**5 +
10*a**4*b*x**2)/(4*a**2*b**6 + 8*a*b**7*x**2 + 4*b**8*x**4) + x**6/(6*b**3)

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GIAC/XCAS [A]  time = 0.213021, size = 124, normalized size = 1.43 \[ -\frac{5 \, a^{3}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{b^{6}} + \frac{30 \, a^{3} b^{2} x^{4} + 50 \, a^{4} b x^{2} + 21 \, a^{5}}{4 \,{\left (b x^{2} + a\right )}^{2} b^{6}} + \frac{2 \, b^{6} x^{6} - 9 \, a b^{5} x^{4} + 36 \, a^{2} b^{4} x^{2}}{12 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b*x^2 + a)^3,x, algorithm="giac")

[Out]

-5*a^3*ln(abs(b*x^2 + a))/b^6 + 1/4*(30*a^3*b^2*x^4 + 50*a^4*b*x^2 + 21*a^5)/((b
*x^2 + a)^2*b^6) + 1/12*(2*b^6*x^6 - 9*a*b^5*x^4 + 36*a^2*b^4*x^2)/b^9