3.354 \(\int (f x)^{-1+m} (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=38 \[ \frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )}{f m}-\frac{b n (f x)^m}{f m^2} \]

[Out]

-((b*n*(f*x)^m)/(f*m^2)) + ((f*x)^m*(a + b*Log[c*x^n]))/(f*m)

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Rubi [A]  time = 0.0173125, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {2304} \[ \frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )}{f m}-\frac{b n (f x)^m}{f m^2} \]

Antiderivative was successfully verified.

[In]

Int[(f*x)^(-1 + m)*(a + b*Log[c*x^n]),x]

[Out]

-((b*n*(f*x)^m)/(f*m^2)) + ((f*x)^m*(a + b*Log[c*x^n]))/(f*m)

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 (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{b n (f x)^m}{f m^2}+\frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )}{f m}\\ \end{align*}

Mathematica [A]  time = 0.0094612, size = 29, normalized size = 0.76 \[ \frac{(f x)^m \left (a m+b m \log \left (c x^n\right )-b n\right )}{f m^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(f*x)^(-1 + m)*(a + b*Log[c*x^n]),x]

[Out]

((f*x)^m*(a*m - b*n + b*m*Log[c*x^n]))/(f*m^2)

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Maple [C]  time = 0.102, size = 281, normalized size = 7.4 \begin{align*}{\frac{bx\ln \left ({x}^{n} \right ) }{m}{{\rm e}^{{\frac{ \left ( -1+m \right ) \left ( -i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{3}+i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( if \right ) +i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) -i\pi \,{\it csgn} \left ( ifx \right ){\it csgn} \left ( if \right ){\it csgn} \left ( ix \right ) +2\,\ln \left ( f \right ) +2\,\ln \left ( x \right ) \right ) }{2}}}}}+{\frac{ \left ( i\pi \,b{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}m-i\pi \,b{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) m-i\pi \,b \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}m+i\pi \,b \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) m+2\,b\ln \left ( c \right ) m+2\,am-2\,bn \right ) x}{2\,{m}^{2}}{{\rm e}^{{\frac{ \left ( -1+m \right ) \left ( -i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{3}+i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( if \right ) +i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) -i\pi \,{\it csgn} \left ( ifx \right ){\it csgn} \left ( if \right ){\it csgn} \left ( ix \right ) +2\,\ln \left ( f \right ) +2\,\ln \left ( x \right ) \right ) }{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^(-1+m)*(a+b*ln(c*x^n)),x)

[Out]

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

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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((f*x)^(-1+m)*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.27176, size = 120, normalized size = 3.16 \begin{align*} \frac{{\left (b m n x \log \left (x\right ) + b m x \log \left (c\right ) +{\left (a m - b n\right )} x\right )} e^{\left ({\left (m - 1\right )} \log \left (f\right ) +{\left (m - 1\right )} \log \left (x\right )\right )}}{m^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^(-1+m)*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

(b*m*n*x*log(x) + b*m*x*log(c) + (a*m - b*n)*x)*e^((m - 1)*log(f) + (m - 1)*log(x))/m^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)**(-1+m)*(a+b*ln(c*x**n)),x)

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.3117, size = 86, normalized size = 2.26 \begin{align*} \frac{b f^{m} n x^{m} \log \left (x\right )}{f m} + \frac{b f^{m} x^{m} \log \left (c\right )}{f m} + \frac{a f^{m} x^{m}}{f m} - \frac{b f^{m} n x^{m}}{f m^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^(-1+m)*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

b*f^m*n*x^m*log(x)/(f*m) + b*f^m*x^m*log(c)/(f*m) + a*f^m*x^m/(f*m) - b*f^m*n*x^m/(f*m^2)