\(\int (d x)^m (a+\frac {a (1+m) \log (c x^n)}{n}) \, dx\) [149]

Optimal result

Integrand size = 22, antiderivative size = 21 \[ \int (d x)^m \left (a+\frac {a (1+m) \log \left (c x^n\right )}{n}\right ) \, dx=\frac {a (d x)^{1+m} \log \left (c x^n\right )}{d n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2340} \[ \int (d x)^m \left (a+\frac {a (1+m) \log \left (c x^n\right )}{n}\right ) \, dx=\frac {a (d x)^{m+1} \log \left (c x^n\right )}{d n} \]

[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 2340

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*} \text {integral}& = \frac {a (d x)^{1+m} \log \left (c x^n\right )}{d n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int (d x)^m \left (a+\frac {a (1+m) \log \left (c x^n\right )}{n}\right ) \, dx=\frac {a x (d x)^m \log \left (c x^n\right )}{n} \]

[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

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86

method	result	size
parallelrisch	\(\frac {x \left (d x \right )^{m} \ln \left (c \,x^{n}\right ) a}{n}\)	\(18\)
risch	\(\frac {a x \,x^{m} d^{m} {\mathrm e}^{\frac {i \operatorname {csgn}\left (i d x \right ) \pi m \left (\operatorname {csgn}\left (i d x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i d x \right )+\operatorname {csgn}\left (i d \right )\right )}{2}} \ln \left (x^{n}\right )}{n}+\frac {a \left (-i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 \ln \left (c \right )\right ) x \,x^{m} d^{m} {\mathrm e}^{\frac {i \operatorname {csgn}\left (i d x \right ) \pi m \left (\operatorname {csgn}\left (i d x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i d x \right )+\operatorname {csgn}\left (i d \right )\right )}{2}}}{2 n}\)	\(194\)

[In]

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

[Out]

x*(d*x)^m*ln(c*x^n)*a/n

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int (d x)^m \left (a+\frac {a (1+m) \log \left (c x^n\right )}{n}\right ) \, dx=\frac {{\left (a n x \log \left (x\right ) + a x \log \left (c\right )\right )} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )}}{n} \]

[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

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int (d x)^m \left (a+\frac {a (1+m) \log \left (c x^n\right )}{n}\right ) \, dx=\frac {a x \left (d x\right )^{m} \log {\left (c x^{n} \right )}}{n} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (21) = 42\).

Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.86 \[ \int (d x)^m \left (a+\frac {a (1+m) \log \left (c x^n\right )}{n}\right ) \, dx=-\frac {a d^{m} m x x^{m}}{{\left (m + 1\right )}^{2}} - \frac {a d^{m} x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {\left (d x\right )^{m + 1} a m \log \left (c x^{n}\right )}{d {\left (m + 1\right )} n} + \frac {\left (d x\right )^{m + 1} a}{d {\left (m + 1\right )}} + \frac {\left (d x\right )^{m + 1} a \log \left (c x^{n}\right )}{d {\left (m + 1\right )} n} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (21) = 42\).

Time = 0.41 (sec) , antiderivative size = 175, normalized size of antiderivative = 8.33 \[ \int (d x)^m \left (a+\frac {a (1+m) \log \left (c x^n\right )}{n}\right ) \, dx=\frac {a d^{m} m^{2} x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac {2 \, a d^{m} m x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac {a d^{m} m x x^{m}}{m^{2} + 2 \, m + 1} + \frac {a d^{m + 1} m x x^{m} \log \left (c\right )}{{\left (d m + d\right )} n} + \frac {a d^{m} x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac {a d^{m + 1} x x^{m}}{d m + d} - \frac {a d^{m} x x^{m}}{m^{2} + 2 \, m + 1} + \frac {a d^{m + 1} x x^{m} \log \left (c\right )}{{\left (d m + d\right )} n} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (d x)^m \left (a+\frac {a (1+m) \log \left (c x^n\right )}{n}\right ) \, dx=\int {\left (d\,x\right )}^m\,\left (a+\frac {a\,\ln \left (c\,x^n\right )\,\left (m+1\right )}{n}\right ) \,d x \]

[In]

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

[Out]

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