3.43 \(\int x^3 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b n x^4 \]

[Out]

-1/16*b*n*x^4+1/4*x^4*(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} \[ \frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b n x^4 \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*Log[c*x^n]),x]

[Out]

-(b*n*x^4)/16 + (x^4*(a + b*Log[c*x^n]))/4

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^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {1}{16} b n x^4+\frac {1}{4} x^4 \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 \[ \frac {a x^4}{4}+\frac {1}{4} b x^4 \log \left (c x^n\right )-\frac {1}{16} b n x^4 \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*Log[c*x^n]),x]

[Out]

(a*x^4)/4 - (b*n*x^4)/16 + (b*x^4*Log[c*x^n])/4

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fricas [A]  time = 0.41, size = 30, normalized size = 1.11 \[ \frac {1}{4} \, b n x^{4} \log \relax (x) + \frac {1}{4} \, b x^{4} \log \relax (c) - \frac {1}{16} \, {\left (b n - 4 \, a\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

1/4*b*n*x^4*log(x) + 1/4*b*x^4*log(c) - 1/16*(b*n - 4*a)*x^4

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giac [A]  time = 0.23, size = 31, normalized size = 1.15 \[ \frac {1}{4} \, b n x^{4} \log \relax (x) - \frac {1}{16} \, b n x^{4} + \frac {1}{4} \, b x^{4} \log \relax (c) + \frac {1}{4} \, a x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/4*b*n*x^4*log(x) - 1/16*b*n*x^4 + 1/4*b*x^4*log(c) + 1/4*a*x^4

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maple [C]  time = 0.22, size = 112, normalized size = 4.15 \[ \frac {b \,x^{4} \ln \left (x^{n}\right )}{4}+\frac {\left (-2 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+2 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-b n +4 b \ln \relax (c )+4 a \right ) x^{4}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*ln(c*x^n)),x)

[Out]

1/4*b*x^4*ln(x^n)+1/16*x^4*(2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-
2*I*b*Pi*csgn(I*c*x^n)^3+2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+4*b*ln(c)-b*n+4*a)

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maxima [A]  time = 0.50, size = 26, normalized size = 0.96 \[ -\frac {1}{16} \, b n x^{4} + \frac {1}{4} \, b x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/16*b*n*x^4 + 1/4*b*x^4*log(c*x^n) + 1/4*a*x^4

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mupad [B]  time = 3.59, size = 25, normalized size = 0.93 \[ x^4\,\left (\frac {a}{4}-\frac {b\,n}{16}\right )+\frac {b\,x^4\,\ln \left (c\,x^n\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*log(c*x^n)),x)

[Out]

x^4*(a/4 - (b*n)/16) + (b*x^4*log(c*x^n))/4

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sympy [A]  time = 1.36, size = 36, normalized size = 1.33 \[ \frac {a x^{4}}{4} + \frac {b n x^{4} \log {\relax (x )}}{4} - \frac {b n x^{4}}{16} + \frac {b x^{4} \log {\relax (c )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*ln(c*x**n)),x)

[Out]

a*x**4/4 + b*n*x**4*log(x)/4 - b*n*x**4/16 + b*x**4*log(c)/4

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