3.165 \(\int (f x)^m (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=46 \[ \frac{(f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}-\frac{b n (f x)^{m+1}}{f (m+1)^2} \]

[Out]

-((b*n*(f*x)^(1 + m))/(f*(1 + m)^2)) + ((f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m))

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Rubi [A]  time = 0.0165148, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {2304} \[ \frac{(f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}-\frac{b n (f x)^{m+1}}{f (m+1)^2} \]

Antiderivative was successfully verified.

[In]

Int[(f*x)^m*(a + b*Log[c*x^n]),x]

[Out]

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

Mathematica [A]  time = 0.0130727, size = 32, normalized size = 0.7 \[ \frac{x (f x)^m \left (a m+a+b (m+1) \log \left (c x^n\right )-b n\right )}{(m+1)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(f*x)^m*(a + a*m - b*n + b*(1 + m)*Log[c*x^n]))/(1 + m)^2

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Maple [C]  time = 0.109, size = 371, normalized size = 8.1 \begin{align*}{\frac{bx\ln \left ({x}^{n} \right ) }{1+m}{{\rm e}^{{\frac{m \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-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -2\,b\ln \left ( c \right ) m-2\,b\ln \left ( c \right ) -2\,am+2\,bn-2\,a \right ) x}{2\, \left ( 1+m \right ) ^{2}}{{\rm e}^{{\frac{m \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)^m*(a+b*ln(c*x^n)),x)

[Out]

b/(1+m)*x*exp(1/2*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)))*ln(x^n)-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-I*b*Pi*csgn(I*x^n)*c
sgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+I*b*Pi*csgn(I*c*x^n)^3-I*b*Pi*csgn(I*c*x^n)^2*csgn(I
*c)-2*b*ln(c)*m-2*b*ln(c)-2*a*m+2*b*n-2*a)/(1+m)^2*x*exp(1/2*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)^m*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.28986, size = 142, normalized size = 3.09 \begin{align*} \frac{{\left ({\left (b m + b\right )} n x \log \left (x\right ) +{\left (b m + b\right )} x \log \left (c\right ) +{\left (a m - b n + a\right )} x\right )} e^{\left (m \log \left (f\right ) + m \log \left (x\right )\right )}}{m^{2} + 2 \, m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)**m*(a+b*ln(c*x**n)),x)

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.32657, size = 128, normalized size = 2.78 \begin{align*} \frac{b f^{m} m n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac{b f^{m} n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac{b f^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac{\left (f x\right )^{m} b x \log \left (c\right )}{m + 1} + \frac{\left (f x\right )^{m} a x}{m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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