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

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 = 0.01, size = 32, normalized size = 0.70 \[ \frac {x (f x)^m \left (a m+a+b (m+1) \log \left (c x^n\right )-b n\right )}{(m+1)^2} \]

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

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 52, normalized size = 1.13 \[ \frac {{\left ({\left (b m + b\right )} n x \log \relax (x) + {\left (b m + b\right )} x \log \relax (c) + {\left (a m - b n + a\right )} x\right )} e^{\left (m \log \relax (f) + m \log \relax (x)\right )}}{m^{2} + 2 \, m + 1} \]

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)

________________________________________________________________________________________

giac [B]  time = 0.36, size = 95, normalized size = 2.07 \[ \frac {b f^{m} m n x x^{m} \log \relax (x)}{m^{2} + 2 \, m + 1} + \frac {b f^{m} n x x^{m} \log \relax (x)}{m^{2} + 2 \, m + 1} - \frac {b f^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac {\left (f x\right )^{m} b x \log \relax (c)}{m + 1} + \frac {\left (f x\right )^{m} a x}{m + 1} \]

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)

________________________________________________________________________________________

maple [C]  time = 0.07, size = 371, normalized size = 8.07 \[ \frac {b x \,{\mathrm e}^{\frac {\left (-i \pi \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i f x \right )+i \pi \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i f x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i f x \right )^{2}-i \pi \mathrm {csgn}\left (i f x \right )^{3}+2 \ln \relax (f )+2 \ln \relax (x )\right ) m}{2}} \ln \left (x^{n}\right )}{m +1}-\frac {\left (i \pi b m \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-i \pi b m \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b m \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b m \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-2 b m \ln \relax (c )-2 a m +2 b n -2 b \ln \relax (c )-2 a \right ) x \,{\mathrm e}^{\frac {\left (-i \pi \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i f x \right )+i \pi \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i f x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i f x \right )^{2}-i \pi \mathrm {csgn}\left (i f x \right )^{3}+2 \ln \relax (f )+2 \ln \relax (x )\right ) m}{2}}}{2 \left (m +1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.02, size = 57, normalized size = 1.24 \[ -\frac {b f^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {\left (f x\right )^{m + 1} b \log \left (c x^{n}\right )}{f {\left (m + 1\right )}} + \frac {\left (f x\right )^{m + 1} a}{f {\left (m + 1\right )}} \]

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]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (f\,x\right )}^m\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*(a + b*log(c*x^n)),x)

[Out]

int((f*x)^m*(a + b*log(c*x^n)), x)

________________________________________________________________________________________

sympy [A]  time = 9.89, size = 192, normalized size = 4.17 \[ \begin {cases} \frac {a f^{m} m x x^{m}}{m^{2} + 2 m + 1} + \frac {a f^{m} x x^{m}}{m^{2} + 2 m + 1} + \frac {b f^{m} m n x x^{m} \log {\relax (x )}}{m^{2} + 2 m + 1} + \frac {b f^{m} m x x^{m} \log {\relax (c )}}{m^{2} + 2 m + 1} + \frac {b f^{m} n x x^{m} \log {\relax (x )}}{m^{2} + 2 m + 1} - \frac {b f^{m} n x x^{m}}{m^{2} + 2 m + 1} + \frac {b f^{m} x x^{m} \log {\relax (c )}}{m^{2} + 2 m + 1} & \text {for}\: m \neq -1 \\\frac {\begin {cases} a \log {\relax (x )} & \text {for}\: b = 0 \\- \left (- a - b \log {\relax (c )}\right ) \log {\relax (x )} & \text {for}\: n = 0 \\\frac {\left (- a - b \log {\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text {otherwise} \end {cases}}{f} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*f**m*m*x*x**m/(m**2 + 2*m + 1) + a*f**m*x*x**m/(m**2 + 2*m + 1) + b*f**m*m*n*x*x**m*log(x)/(m**2
+ 2*m + 1) + b*f**m*m*x*x**m*log(c)/(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) + b*f**m*x*x**m*log(c)/(m**2 + 2*m + 1), Ne(m, -1)), (Piecewise((a*log(x), Eq(b, 0)), (-(-
a - b*log(c))*log(x), Eq(n, 0)), ((-a - b*log(c*x**n))**2/(2*b*n), True))/f, True))

________________________________________________________________________________________