3.149 \(\int (d x)^m (a+\frac{a (1+m) \log (c x^n)}{n}) \, dx\)

Optimal. Leaf size=21 \[ \frac{a (d x)^{m+1} \log \left (c x^n\right )}{d n} \]

[Out]

(a*(d*x)^(1 + m)*Log[c*x^n])/(d*n)

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Rubi [A]  time = 0.0204304, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {2303} \[ \frac{a (d x)^{m+1} \log \left (c x^n\right )}{d n} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + (a*(1 + m)*Log[c*x^n])/n),x]

[Out]

(a*(d*x)^(1 + m)*Log[c*x^n])/(d*n)

Rule 2303

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*(d*x)^(m + 1)*Log[c*x^n])/(
d*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && EqQ[a*(m + 1) - b*n, 0]

Rubi steps

\begin{align*} \int (d x)^m \left (a+\frac{a (1+m) \log \left (c x^n\right )}{n}\right ) \, dx &=\frac{a (d x)^{1+m} \log \left (c x^n\right )}{d n}\\ \end{align*}

Mathematica [A]  time = 0.0113577, size = 17, normalized size = 0.81 \[ \frac{a x (d x)^m \log \left (c x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a + (a*(1 + m)*Log[c*x^n])/n),x]

[Out]

(a*x*(d*x)^m*Log[c*x^n])/n

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Maple [C]  time = 0.117, size = 260, normalized size = 12.4 \begin{align*}{\frac{ax\ln \left ({x}^{n} \right ) }{n}{{\rm e}^{{\frac{m \left ( -i \left ({\it csgn} \left ( idx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( id \right ) \pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i\pi \,{\it csgn} \left ( idx \right ){\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ) +2\,\ln \left ( x \right ) +2\,\ln \left ( d \right ) \right ) }{2}}}}}+{\frac{a \left ( i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,\ln \left ( c \right ) \right ) x}{2\,n}{{\rm e}^{{\frac{m \left ( -i \left ({\it csgn} \left ( idx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( id \right ) \pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i\pi \,{\it csgn} \left ( idx \right ){\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ) +2\,\ln \left ( x \right ) +2\,\ln \left ( d \right ) \right ) }{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a+a*(1+m)*ln(c*x^n)/n),x)

[Out]

a/n*x*exp(1/2*m*(-I*csgn(I*d*x)^3*Pi+I*csgn(I*d*x)^2*csgn(I*d)*Pi+I*csgn(I*d*x)^2*csgn(I*x)*Pi-I*csgn(I*d*x)*c
sgn(I*d)*csgn(I*x)*Pi+2*ln(x)+2*ln(d)))*ln(x^n)+1/2*a*(I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*csgn(I*x^n)*csgn(
I*c*x^n)*csgn(I*c)-I*Pi*csgn(I*c*x^n)^3+I*Pi*csgn(I*c*x^n)^2*csgn(I*c)+2*ln(c))*x/n*exp(1/2*m*(-I*csgn(I*d*x)^
3*Pi+I*csgn(I*d*x)^2*csgn(I*d)*Pi+I*csgn(I*d*x)^2*csgn(I*x)*Pi-I*csgn(I*d*x)*csgn(I*d)*csgn(I*x)*Pi+2*ln(x)+2*
ln(d)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+a*(1+m)*log(c*x^n)/n),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.00632, size = 74, normalized size = 3.52 \begin{align*} \frac{{\left (a n x \log \left (x\right ) + a x \log \left (c\right )\right )} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )}}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+a*(1+m)*log(c*x^n)/n),x, algorithm="fricas")

[Out]

(a*n*x*log(x) + a*x*log(c))*e^(m*log(d) + m*log(x))/n

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Sympy [A]  time = 1.35617, size = 27, normalized size = 1.29 \begin{align*} a d^{m} x x^{m} \log{\left (x \right )} + \frac{a d^{m} x x^{m} \log{\left (c \right )}}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(a+a*(1+m)*ln(c*x**n)/n),x)

[Out]

a*d**m*x*x**m*log(x) + a*d**m*x*x**m*log(c)/n

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Giac [B]  time = 1.3511, size = 289, normalized size = 13.76 \begin{align*} \frac{a d^{2} \frac{1}{d}^{m} m x x^{m}{\left | d \right |}^{2 \, m} \log \left (c\right )}{{\left (d^{2} m + d^{2}\right )} n} + \frac{a d^{2} \frac{1}{d}^{m} x x^{m}{\left | d \right |}^{2 \, m}}{d^{2} m + d^{2}} + \frac{a d^{2} \frac{1}{d}^{m} x x^{m}{\left | d \right |}^{2 \, m} \log \left (c\right )}{{\left (d^{2} m + d^{2}\right )} n} + \frac{a d^{m} m^{2} x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac{2 \, a d^{m} m x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac{a d^{m} m x x^{m}}{m^{2} + 2 \, m + 1} + \frac{a d^{m} x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac{a d^{m} x x^{m}}{m^{2} + 2 \, m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+a*(1+m)*log(c*x^n)/n),x, algorithm="giac")

[Out]

a*d^2*(1/d)^m*m*x*x^m*abs(d)^(2*m)*log(c)/((d^2*m + d^2)*n) + a*d^2*(1/d)^m*x*x^m*abs(d)^(2*m)/(d^2*m + d^2) +
a*d^2*(1/d)^m*x*x^m*abs(d)^(2*m)*log(c)/((d^2*m + d^2)*n) + a*d^m*m^2*x*x^m*log(x)/(m^2 + 2*m + 1) + 2*a*d^m*
m*x*x^m*log(x)/(m^2 + 2*m + 1) - a*d^m*m*x*x^m/(m^2 + 2*m + 1) + a*d^m*x*x^m*log(x)/(m^2 + 2*m + 1) - a*d^m*x*
x^m/(m^2 + 2*m + 1)