3.73 \(\int x^4 \log (a^2+x^2) \, dx\)

Optimal. Leaf size=54 \[ \frac{2 a^2 x^3}{15}+\frac{1}{5} x^5 \log \left (a^2+x^2\right )-\frac{2 a^4 x}{5}+\frac{2}{5} a^5 \tan ^{-1}\left (\frac{x}{a}\right )-\frac{2 x^5}{25} \]

[Out]

(-2*a^4*x)/5 + (2*a^2*x^3)/15 - (2*x^5)/25 + (2*a^5*ArcTan[x/a])/5 + (x^5*Log[a^2 + x^2])/5

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Rubi [A]  time = 0.02527, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2455, 302, 203} \[ \frac{2 a^2 x^3}{15}+\frac{1}{5} x^5 \log \left (a^2+x^2\right )-\frac{2 a^4 x}{5}+\frac{2}{5} a^5 \tan ^{-1}\left (\frac{x}{a}\right )-\frac{2 x^5}{25} \]

Antiderivative was successfully verified.

[In]

Int[x^4*Log[a^2 + x^2],x]

[Out]

(-2*a^4*x)/5 + (2*a^2*x^3)/15 - (2*x^5)/25 + (2*a^5*ArcTan[x/a])/5 + (x^5*Log[a^2 + x^2])/5

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int x^4 \log \left (a^2+x^2\right ) \, dx &=\frac{1}{5} x^5 \log \left (a^2+x^2\right )-\frac{2}{5} \int \frac{x^6}{a^2+x^2} \, dx\\ &=\frac{1}{5} x^5 \log \left (a^2+x^2\right )-\frac{2}{5} \int \left (a^4-a^2 x^2+x^4-\frac{a^6}{a^2+x^2}\right ) \, dx\\ &=-\frac{2 a^4 x}{5}+\frac{2 a^2 x^3}{15}-\frac{2 x^5}{25}+\frac{1}{5} x^5 \log \left (a^2+x^2\right )+\frac{1}{5} \left (2 a^6\right ) \int \frac{1}{a^2+x^2} \, dx\\ &=-\frac{2 a^4 x}{5}+\frac{2 a^2 x^3}{15}-\frac{2 x^5}{25}+\frac{2}{5} a^5 \tan ^{-1}\left (\frac{x}{a}\right )+\frac{1}{5} x^5 \log \left (a^2+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0026944, size = 54, normalized size = 1. \[ \frac{2 a^2 x^3}{15}+\frac{1}{5} x^5 \log \left (a^2+x^2\right )-\frac{2 a^4 x}{5}+\frac{2}{5} a^5 \tan ^{-1}\left (\frac{x}{a}\right )-\frac{2 x^5}{25} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*Log[a^2 + x^2],x]

[Out]

(-2*a^4*x)/5 + (2*a^2*x^3)/15 - (2*x^5)/25 + (2*a^5*ArcTan[x/a])/5 + (x^5*Log[a^2 + x^2])/5

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Maple [A]  time = 0.003, size = 45, normalized size = 0.8 \begin{align*} -{\frac{2\,{a}^{4}x}{5}}+{\frac{2\,{a}^{2}{x}^{3}}{15}}-{\frac{2\,{x}^{5}}{25}}+{\frac{2\,{a}^{5}}{5}\arctan \left ({\frac{x}{a}} \right ) }+{\frac{{x}^{5}\ln \left ({a}^{2}+{x}^{2} \right ) }{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*ln(a^2+x^2),x)

[Out]

-2/5*a^4*x+2/15*a^2*x^3-2/25*x^5+2/5*a^5*arctan(x/a)+1/5*x^5*ln(a^2+x^2)

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Maxima [A]  time = 1.42888, size = 59, normalized size = 1.09 \begin{align*} \frac{2}{5} \, a^{5} \arctan \left (\frac{x}{a}\right ) + \frac{1}{5} \, x^{5} \log \left (a^{2} + x^{2}\right ) - \frac{2}{5} \, a^{4} x + \frac{2}{15} \, a^{2} x^{3} - \frac{2}{25} \, x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*log(a^2+x^2),x, algorithm="maxima")

[Out]

2/5*a^5*arctan(x/a) + 1/5*x^5*log(a^2 + x^2) - 2/5*a^4*x + 2/15*a^2*x^3 - 2/25*x^5

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Fricas [A]  time = 2.04898, size = 113, normalized size = 2.09 \begin{align*} \frac{2}{5} \, a^{5} \arctan \left (\frac{x}{a}\right ) + \frac{1}{5} \, x^{5} \log \left (a^{2} + x^{2}\right ) - \frac{2}{5} \, a^{4} x + \frac{2}{15} \, a^{2} x^{3} - \frac{2}{25} \, x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*log(a^2+x^2),x, algorithm="fricas")

[Out]

2/5*a^5*arctan(x/a) + 1/5*x^5*log(a^2 + x^2) - 2/5*a^4*x + 2/15*a^2*x^3 - 2/25*x^5

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Sympy [C]  time = 0.348756, size = 63, normalized size = 1.17 \begin{align*} - 2 a^{5} \left (\frac{i \log{\left (- i a + x \right )}}{10} - \frac{i \log{\left (i a + x \right )}}{10}\right ) - \frac{2 a^{4} x}{5} + \frac{2 a^{2} x^{3}}{15} + \frac{x^{5} \log{\left (a^{2} + x^{2} \right )}}{5} - \frac{2 x^{5}}{25} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*ln(a**2+x**2),x)

[Out]

-2*a**5*(I*log(-I*a + x)/10 - I*log(I*a + x)/10) - 2*a**4*x/5 + 2*a**2*x**3/15 + x**5*log(a**2 + x**2)/5 - 2*x
**5/25

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Giac [A]  time = 1.08867, size = 59, normalized size = 1.09 \begin{align*} \frac{2}{5} \, a^{5} \arctan \left (\frac{x}{a}\right ) + \frac{1}{5} \, x^{5} \log \left (a^{2} + x^{2}\right ) - \frac{2}{5} \, a^{4} x + \frac{2}{15} \, a^{2} x^{3} - \frac{2}{25} \, x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*log(a^2+x^2),x, algorithm="giac")

[Out]

2/5*a^5*arctan(x/a) + 1/5*x^5*log(a^2 + x^2) - 2/5*a^4*x + 2/15*a^2*x^3 - 2/25*x^5