3.128 \(\int \frac{x^7}{a+b x^2} \, dx\)

Optimal. Leaf size=53 \[ -\frac{a^3 \log \left (a+b x^2\right )}{2 b^4}+\frac{a^2 x^2}{2 b^3}-\frac{a x^4}{4 b^2}+\frac{x^6}{6 b} \]

[Out]

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

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Rubi [A]  time = 0.0849097, antiderivative size = 53, 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 \[ -\frac{a^3 \log \left (a+b x^2\right )}{2 b^4}+\frac{a^2 x^2}{2 b^3}-\frac{a x^4}{4 b^2}+\frac{x^6}{6 b} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + b*x^2),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(b*x**2+a),x)

[Out]

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

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Mathematica [A]  time = 0.0090984, size = 53, normalized size = 1. \[ -\frac{a^3 \log \left (a+b x^2\right )}{2 b^4}+\frac{a^2 x^2}{2 b^3}-\frac{a x^4}{4 b^2}+\frac{x^6}{6 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a + b*x^2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 46, normalized size = 0.9 \[{\frac{{a}^{2}{x}^{2}}{2\,{b}^{3}}}-{\frac{a{x}^{4}}{4\,{b}^{2}}}+{\frac{{x}^{6}}{6\,b}}-{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.34441, size = 62, normalized size = 1.17 \[ -\frac{a^{3} \log \left (b x^{2} + a\right )}{2 \, b^{4}} + \frac{2 \, b^{2} x^{6} - 3 \, a b x^{4} + 6 \, a^{2} x^{2}}{12 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^2 + a),x, algorithm="maxima")

[Out]

-1/2*a^3*log(b*x^2 + a)/b^4 + 1/12*(2*b^2*x^6 - 3*a*b*x^4 + 6*a^2*x^2)/b^3

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Fricas [A]  time = 0.19787, size = 61, normalized size = 1.15 \[ \frac{2 \, b^{3} x^{6} - 3 \, a b^{2} x^{4} + 6 \, a^{2} b x^{2} - 6 \, a^{3} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^2 + a),x, algorithm="fricas")

[Out]

1/12*(2*b^3*x^6 - 3*a*b^2*x^4 + 6*a^2*b*x^2 - 6*a^3*log(b*x^2 + a))/b^4

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Sympy [A]  time = 1.28111, size = 44, normalized size = 0.83 \[ - \frac{a^{3} \log{\left (a + b x^{2} \right )}}{2 b^{4}} + \frac{a^{2} x^{2}}{2 b^{3}} - \frac{a x^{4}}{4 b^{2}} + \frac{x^{6}}{6 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(b*x**2+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.209284, size = 63, normalized size = 1.19 \[ -\frac{a^{3}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} + \frac{2 \, b^{2} x^{6} - 3 \, a b x^{4} + 6 \, a^{2} x^{2}}{12 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^2 + a),x, algorithm="giac")

[Out]

-1/2*a^3*ln(abs(b*x^2 + a))/b^4 + 1/12*(2*b^2*x^6 - 3*a*b*x^4 + 6*a^2*x^2)/b^3