3.124 \(\int \frac{x^{11}}{a+b x^2} \, dx\)

Optimal. Leaf size=79 \[ -\frac{a^5 \log \left (a+b x^2\right )}{2 b^6}+\frac{a^4 x^2}{2 b^5}-\frac{a^3 x^4}{4 b^4}+\frac{a^2 x^6}{6 b^3}-\frac{a x^8}{8 b^2}+\frac{x^{10}}{10 b} \]

[Out]

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

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Rubi [A]  time = 0.13527, antiderivative size = 79, 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 \[ -\frac{a^5 \log \left (a+b x^2\right )}{2 b^6}+\frac{a^4 x^2}{2 b^5}-\frac{a^3 x^4}{4 b^4}+\frac{a^2 x^6}{6 b^3}-\frac{a x^8}{8 b^2}+\frac{x^{10}}{10 b} \]

Antiderivative was successfully verified.

[In]  Int[x^11/(a + b*x^2),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00985356, size = 79, normalized size = 1. \[ -\frac{a^5 \log \left (a+b x^2\right )}{2 b^6}+\frac{a^4 x^2}{2 b^5}-\frac{a^3 x^4}{4 b^4}+\frac{a^2 x^6}{6 b^3}-\frac{a x^8}{8 b^2}+\frac{x^{10}}{10 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(a + b*x^2),x]

[Out]

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

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Maple [A]  time = 0.005, size = 68, normalized size = 0.9 \[{\frac{{a}^{4}{x}^{2}}{2\,{b}^{5}}}-{\frac{{a}^{3}{x}^{4}}{4\,{b}^{4}}}+{\frac{{a}^{2}{x}^{6}}{6\,{b}^{3}}}-{\frac{a{x}^{8}}{8\,{b}^{2}}}+{\frac{{x}^{10}}{10\,b}}-{\frac{{a}^{5}\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.34686, size = 92, normalized size = 1.16 \[ -\frac{a^{5} \log \left (b x^{2} + a\right )}{2 \, b^{6}} + \frac{12 \, b^{4} x^{10} - 15 \, a b^{3} x^{8} + 20 \, a^{2} b^{2} x^{6} - 30 \, a^{3} b x^{4} + 60 \, a^{4} x^{2}}{120 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*a^5*log(b*x^2 + a)/b^6 + 1/120*(12*b^4*x^10 - 15*a*b^3*x^8 + 20*a^2*b^2*x^6
- 30*a^3*b*x^4 + 60*a^4*x^2)/b^5

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Fricas [A]  time = 0.200843, size = 90, normalized size = 1.14 \[ \frac{12 \, b^{5} x^{10} - 15 \, a b^{4} x^{8} + 20 \, a^{2} b^{3} x^{6} - 30 \, a^{3} b^{2} x^{4} + 60 \, a^{4} b x^{2} - 60 \, a^{5} \log \left (b x^{2} + a\right )}{120 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/120*(12*b^5*x^10 - 15*a*b^4*x^8 + 20*a^2*b^3*x^6 - 30*a^3*b^2*x^4 + 60*a^4*b*x
^2 - 60*a^5*log(b*x^2 + a))/b^6

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Sympy [A]  time = 1.3307, size = 68, normalized size = 0.86 \[ - \frac{a^{5} \log{\left (a + b x^{2} \right )}}{2 b^{6}} + \frac{a^{4} x^{2}}{2 b^{5}} - \frac{a^{3} x^{4}}{4 b^{4}} + \frac{a^{2} x^{6}}{6 b^{3}} - \frac{a x^{8}}{8 b^{2}} + \frac{x^{10}}{10 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(b*x**2+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.214059, size = 93, normalized size = 1.18 \[ -\frac{a^{5}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} + \frac{12 \, b^{4} x^{10} - 15 \, a b^{3} x^{8} + 20 \, a^{2} b^{2} x^{6} - 30 \, a^{3} b x^{4} + 60 \, a^{4} x^{2}}{120 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*a^5*ln(abs(b*x^2 + a))/b^6 + 1/120*(12*b^4*x^10 - 15*a*b^3*x^8 + 20*a^2*b^2
*x^6 - 30*a^3*b*x^4 + 60*a^4*x^2)/b^5