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]

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

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Rubi [A]  time = 0.0125367, antiderivative size = 27, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0020941, 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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Maple [C]  time = 0.21, size = 112, normalized size = 4.2 \begin{align*}{\frac{b{x}^{4}\ln \left ({x}^{n} \right ) }{4}}+{\frac{{x}^{4} \left ( 2\,ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-2\,ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -2\,ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +4\,b\ln \left ( c \right ) -bn+4\,a \right ) }{16}} \end{align*}

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 = 1.10864, size = 35, normalized size = 1.3 \begin{align*} -\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} \end{align*}

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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Fricas [A]  time = 0.787481, size = 84, normalized size = 3.11 \begin{align*} \frac{1}{4} \, b n x^{4} \log \left (x\right ) + \frac{1}{4} \, b x^{4} \log \left (c\right ) - \frac{1}{16} \,{\left (b n - 4 \, a\right )} x^{4} \end{align*}

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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Sympy [A]  time = 1.39695, size = 36, normalized size = 1.33 \begin{align*} \frac{a x^{4}}{4} + \frac{b n x^{4} \log{\left (x \right )}}{4} - \frac{b n x^{4}}{16} + \frac{b x^{4} \log{\left (c \right )}}{4} \end{align*}

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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Giac [A]  time = 1.11166, size = 42, normalized size = 1.56 \begin{align*} \frac{1}{4} \, b n x^{4} \log \left (x\right ) - \frac{1}{16} \, b n x^{4} + \frac{1}{4} \, b x^{4} \log \left (c\right ) + \frac{1}{4} \, a x^{4} \end{align*}

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