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

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Rubi [A]  time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.47, size = 26, normalized size = 1.24 \[ \frac {{\left (a n x \log \relax (x) + a x \log \relax (c)\right )} e^{\left (m \log \relax (d) + m \log \relax (x)\right )}}{n} \]

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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giac [B]  time = 0.46, size = 214, normalized size = 10.19 \[ \frac {a d^{2} \frac {1}{d}^{m} m x x^{m} {\left | d \right |}^{2 \, m} \log \relax (c)}{{\left (d^{2} m + d^{2}\right )} n} + \frac {a d^{2} \frac {1}{d}^{m} x x^{m} {\left | d \right |}^{2 \, m}}{d^{2} m + d^{2}} + \frac {a d^{2} \frac {1}{d}^{m} x x^{m} {\left | d \right |}^{2 \, m} \log \relax (c)}{{\left (d^{2} m + d^{2}\right )} n} + \frac {a d^{m} m^{2} x x^{m} \log \relax (x)}{m^{2} + 2 \, m + 1} + \frac {2 \, a d^{m} m x x^{m} \log \relax (x)}{m^{2} + 2 \, m + 1} - \frac {a d^{m} m x x^{m}}{m^{2} + 2 \, m + 1} + \frac {a d^{m} x x^{m} \log \relax (x)}{m^{2} + 2 \, m + 1} - \frac {a d^{m} x x^{m}}{m^{2} + 2 \, m + 1} \]

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)

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maple [C]  time = 0.17, size = 260, normalized size = 12.38 \[ \frac {a x \,{\mathrm e}^{\frac {\left (-i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )+i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2}-i \pi \mathrm {csgn}\left (i d x \right )^{3}+2 \ln \relax (d )+2 \ln \relax (x )\right ) m}{2}} \ln \left (x^{n}\right )}{n}+\frac {\left (-i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 \ln \relax (c )\right ) a x \,{\mathrm e}^{\frac {\left (-i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )+i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2}-i \pi \mathrm {csgn}\left (i d x \right )^{3}+2 \ln \relax (d )+2 \ln \relax (x )\right ) m}{2}}}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.61, size = 102, normalized size = 4.86 \[ -\frac {a d^{m} m x x^{m}}{{\left (m + 1\right )}^{2}} - \frac {a d^{m} x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {\left (d x\right )^{m + 1} a m \log \left (c x^{n}\right )}{d {\left (m + 1\right )} n} + \frac {\left (d x\right )^{m + 1} a}{d {\left (m + 1\right )}} + \frac {\left (d x\right )^{m + 1} a \log \left (c x^{n}\right )}{d {\left (m + 1\right )} n} \]

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]

-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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ \int {\left (d\,x\right )}^m\,\left (a+\frac {a\,\ln \left (c\,x^n\right )\,\left (m+1\right )}{n}\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.98, size = 27, normalized size = 1.29 \[ a d^{m} x x^{m} \log {\relax (x )} + \frac {a d^{m} x x^{m} \log {\relax (c )}}{n} \]

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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