3.44 \(\int x^2 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \]

[Out]

-1/9*b*n*x^3+1/3*x^3*(a+b*ln(c*x^n))

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2304} \[ \frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*Log[c*x^n]),x]

[Out]

-(b*n*x^3)/9 + (x^3*(a + b*Log[c*x^n]))/3

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {align*} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {1}{9} b n x^3+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 32, normalized size = 1.19 \[ \frac {a x^3}{3}+\frac {1}{3} b x^3 \log \left (c x^n\right )-\frac {1}{9} b n x^3 \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*Log[c*x^n]),x]

[Out]

(a*x^3)/3 - (b*n*x^3)/9 + (b*x^3*Log[c*x^n])/3

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fricas [A]  time = 0.42, size = 30, normalized size = 1.11 \[ \frac {1}{3} \, b n x^{3} \log \relax (x) + \frac {1}{3} \, b x^{3} \log \relax (c) - \frac {1}{9} \, {\left (b n - 3 \, a\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

1/3*b*n*x^3*log(x) + 1/3*b*x^3*log(c) - 1/9*(b*n - 3*a)*x^3

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giac [A]  time = 0.29, size = 31, normalized size = 1.15 \[ \frac {1}{3} \, b n x^{3} \log \relax (x) - \frac {1}{9} \, b n x^{3} + \frac {1}{3} \, b x^{3} \log \relax (c) + \frac {1}{3} \, a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/3*b*n*x^3*log(x) - 1/9*b*n*x^3 + 1/3*b*x^3*log(c) + 1/3*a*x^3

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maple [C]  time = 0.18, size = 112, normalized size = 4.15 \[ \frac {b \,x^{3} \ln \left (x^{n}\right )}{3}+\frac {\left (-3 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+3 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-2 b n +6 b \ln \relax (c )+6 a \right ) x^{3}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*ln(c*x^n)),x)

[Out]

1/3*b*x^3*ln(x^n)+1/18*x^3*(3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-
3*I*b*Pi*csgn(I*c*x^n)^3+3*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+6*b*ln(c)-2*b*n+6*a)

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maxima [A]  time = 0.55, size = 26, normalized size = 0.96 \[ -\frac {1}{9} \, b n x^{3} + \frac {1}{3} \, b x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/9*b*n*x^3 + 1/3*b*x^3*log(c*x^n) + 1/3*a*x^3

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mupad [B]  time = 3.36, size = 25, normalized size = 0.93 \[ x^3\,\left (\frac {a}{3}-\frac {b\,n}{9}\right )+\frac {b\,x^3\,\ln \left (c\,x^n\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*log(c*x^n)),x)

[Out]

x^3*(a/3 - (b*n)/9) + (b*x^3*log(c*x^n))/3

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sympy [A]  time = 0.78, size = 36, normalized size = 1.33 \[ \frac {a x^{3}}{3} + \frac {b n x^{3} \log {\relax (x )}}{3} - \frac {b n x^{3}}{9} + \frac {b x^{3} \log {\relax (c )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*ln(c*x**n)),x)

[Out]

a*x**3/3 + b*n*x**3*log(x)/3 - b*n*x**3/9 + b*x**3*log(c)/3

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