\(\int (f x)^m (d+e x) (a+b \log (c x^n)) \, dx\) [164]

Optimal result

Integrand size = 21, antiderivative size = 95 \[ \int (f x)^m (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {b d n (f x)^{1+m}}{f (1+m)^2}-\frac {b e n (f x)^{2+m}}{f^2 (2+m)^2}+\frac {d (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac {e (f x)^{2+m} \left (a+b \log \left (c x^n\right )\right )}{f^2 (2+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {45, 2392} \[ \int (f x)^m (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {d (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {e (f x)^{m+2} \left (a+b \log \left (c x^n\right )\right )}{f^2 (m+2)}-\frac {b d n (f x)^{m+1}}{f (m+1)^2}-\frac {b e n (f x)^{m+2}}{f^2 (m+2)^2} \]

[In]

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

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2392

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x,
x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {d (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac {e (f x)^{2+m} \left (a+b \log \left (c x^n\right )\right )}{f^2 (2+m)}-(b n) \int (f x)^m \left (\frac {d}{1+m}+\frac {e x}{2+m}\right ) \, dx \\ & = \frac {d (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac {e (f x)^{2+m} \left (a+b \log \left (c x^n\right )\right )}{f^2 (2+m)}-(b n) \int \left (\frac {d (f x)^m}{1+m}+\frac {e (f x)^{1+m}}{f (2+m)}\right ) \, dx \\ & = -\frac {b d n (f x)^{1+m}}{f (1+m)^2}-\frac {b e n (f x)^{2+m}}{f^2 (2+m)^2}+\frac {d (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac {e (f x)^{2+m} \left (a+b \log \left (c x^n\right )\right )}{f^2 (2+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.67 \[ \int (f x)^m (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx=x (f x)^m \left (-\frac {b d n}{(1+m)^2}-\frac {b e n x}{(2+m)^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{1+m}+\frac {e x \left (a+b \log \left (c x^n\right )\right )}{2+m}\right ) \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(352\) vs. \(2(95)=190\).

Time = 0.22 (sec) , antiderivative size = 353, normalized size of antiderivative = 3.72

method	result	size
parallelrisch	\(-\frac {-2 x^{2} \left (f x \right )^{m} a e -4 x \left (f x \right )^{m} a d -x^{2} \left (f x \right )^{m} a e \,m^{3}-4 x^{2} \left (f x \right )^{m} a e \,m^{2}-x \left (f x \right )^{m} a d \,m^{3}-5 x^{2} \left (f x \right )^{m} a e m +x^{2} \left (f x \right )^{m} b e n -5 x \left (f x \right )^{m} a d \,m^{2}-8 x \left (f x \right )^{m} a d m +4 x \left (f x \right )^{m} b d n -4 x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d -2 x^{2} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b e -x^{2} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b e \,m^{3}-4 x^{2} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b e \,m^{2}+x^{2} \left (f x \right )^{m} b e \,m^{2} n -x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d \,m^{3}-5 x^{2} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b e m +2 x^{2} \left (f x \right )^{m} b e m n -5 x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d \,m^{2}+x \left (f x \right )^{m} b d \,m^{2} n -8 x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d m +4 x \left (f x \right )^{m} b d m n}{\left (m^{2}+4 m +4\right ) \left (m^{2}+2 m +1\right )}\)	\(353\)
risch	\(\text {Expression too large to display}\)	\(1056\)

[In]

int((f*x)^m*(e*x+d)*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)

[Out]

-(-2*x^2*(f*x)^m*a*e-4*x*(f*x)^m*a*d-x^2*(f*x)^m*a*e*m^3-4*x^2*(f*x)^m*a*e*m^2-x*(f*x)^m*a*d*m^3-5*x^2*(f*x)^m
*a*e*m+x^2*(f*x)^m*b*e*n-5*x*(f*x)^m*a*d*m^2-8*x*(f*x)^m*a*d*m+4*x*(f*x)^m*b*d*n-4*x*(f*x)^m*ln(c*x^n)*b*d-2*x
^2*(f*x)^m*ln(c*x^n)*b*e-x^2*(f*x)^m*ln(c*x^n)*b*e*m^3-4*x^2*(f*x)^m*ln(c*x^n)*b*e*m^2+x^2*(f*x)^m*b*e*m^2*n-x
*(f*x)^m*ln(c*x^n)*b*d*m^3-5*x^2*(f*x)^m*ln(c*x^n)*b*e*m+2*x^2*(f*x)^m*b*e*m*n-5*x*(f*x)^m*ln(c*x^n)*b*d*m^2+x
*(f*x)^m*b*d*m^2*n-8*x*(f*x)^m*ln(c*x^n)*b*d*m+4*x*(f*x)^m*b*d*m*n)/(m^2+4*m+4)/(m^2+2*m+1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (95) = 190\).

Time = 0.33 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.47 \[ \int (f x)^m (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {{\left ({\left (a e m^{3} + 4 \, a e m^{2} + 5 \, a e m + 2 \, a e - {\left (b e m^{2} + 2 \, b e m + b e\right )} n\right )} x^{2} + {\left (a d m^{3} + 5 \, a d m^{2} + 8 \, a d m + 4 \, a d - {\left (b d m^{2} + 4 \, b d m + 4 \, b d\right )} n\right )} x + {\left ({\left (b e m^{3} + 4 \, b e m^{2} + 5 \, b e m + 2 \, b e\right )} x^{2} + {\left (b d m^{3} + 5 \, b d m^{2} + 8 \, b d m + 4 \, b d\right )} x\right )} \log \left (c\right ) + {\left ({\left (b e m^{3} + 4 \, b e m^{2} + 5 \, b e m + 2 \, b e\right )} n x^{2} + {\left (b d m^{3} + 5 \, b d m^{2} + 8 \, b d m + 4 \, b d\right )} n x\right )} \log \left (x\right )\right )} e^{\left (m \log \left (f\right ) + m \log \left (x\right )\right )}}{m^{4} + 6 \, m^{3} + 13 \, m^{2} + 12 \, m + 4} \]

[In]

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

[Out]

((a*e*m^3 + 4*a*e*m^2 + 5*a*e*m + 2*a*e - (b*e*m^2 + 2*b*e*m + b*e)*n)*x^2 + (a*d*m^3 + 5*a*d*m^2 + 8*a*d*m +
4*a*d - (b*d*m^2 + 4*b*d*m + 4*b*d)*n)*x + ((b*e*m^3 + 4*b*e*m^2 + 5*b*e*m + 2*b*e)*x^2 + (b*d*m^3 + 5*b*d*m^2
+ 8*b*d*m + 4*b*d)*x)*log(c) + ((b*e*m^3 + 4*b*e*m^2 + 5*b*e*m + 2*b*e)*n*x^2 + (b*d*m^3 + 5*b*d*m^2 + 8*b*d*
m + 4*b*d)*n*x)*log(x))*e^(m*log(f) + m*log(x))/(m^4 + 6*m^3 + 13*m^2 + 12*m + 4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 899 vs. \(2 (87) = 174\).

Time = 2.39 (sec) , antiderivative size = 899, normalized size of antiderivative = 9.46 \[ \int (f x)^m (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \frac {- \frac {a d}{x} + a e \log {\left (x \right )} + b d \left (- \frac {n}{x} - \frac {\log {\left (c x^{n} \right )}}{x}\right ) - b e \left (\begin {cases} - \log {\left (c \right )} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases}\right )}{f^{2}} & \text {for}\: m = -2 \\\frac {\frac {a d \log {\left (c x^{n} \right )}}{n} + a e x + \frac {b d \log {\left (c x^{n} \right )}^{2}}{2 n} - b e n x + b e x \log {\left (c x^{n} \right )}}{f} & \text {for}\: m = -1 \\\frac {a d m^{3} x \left (f x\right )^{m}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} + \frac {5 a d m^{2} x \left (f x\right )^{m}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} + \frac {8 a d m x \left (f x\right )^{m}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} + \frac {4 a d x \left (f x\right )^{m}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} + \frac {a e m^{3} x^{2} \left (f x\right )^{m}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} + \frac {4 a e m^{2} x^{2} \left (f x\right )^{m}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} + \frac {5 a e m x^{2} \left (f x\right )^{m}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} + \frac {2 a e x^{2} \left (f x\right )^{m}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} + \frac {b d m^{3} x \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} - \frac {b d m^{2} n x \left (f x\right )^{m}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} + \frac {5 b d m^{2} x \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} - \frac {4 b d m n x \left (f x\right )^{m}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} + \frac {8 b d m x \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} - \frac {4 b d n x \left (f x\right )^{m}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} + \frac {4 b d x \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} + \frac {b e m^{3} x^{2} \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} - \frac {b e m^{2} n x^{2} \left (f x\right )^{m}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} + \frac {4 b e m^{2} x^{2} \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} - \frac {2 b e m n x^{2} \left (f x\right )^{m}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} + \frac {5 b e m x^{2} \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} - \frac {b e n x^{2} \left (f x\right )^{m}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} + \frac {2 b e x^{2} \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 6 m^{3} + 13 m^{2} + 12 m + 4} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise(((-a*d/x + a*e*log(x) + b*d*(-n/x - log(c*x**n)/x) - b*e*Piecewise((-log(c)*log(x), Eq(n, 0)), (-log
(c*x**n)**2/(2*n), True)))/f**2, Eq(m, -2)), ((a*d*log(c*x**n)/n + a*e*x + b*d*log(c*x**n)**2/(2*n) - b*e*n*x
+ b*e*x*log(c*x**n))/f, Eq(m, -1)), (a*d*m**3*x*(f*x)**m/(m**4 + 6*m**3 + 13*m**2 + 12*m + 4) + 5*a*d*m**2*x*(
f*x)**m/(m**4 + 6*m**3 + 13*m**2 + 12*m + 4) + 8*a*d*m*x*(f*x)**m/(m**4 + 6*m**3 + 13*m**2 + 12*m + 4) + 4*a*d
*x*(f*x)**m/(m**4 + 6*m**3 + 13*m**2 + 12*m + 4) + a*e*m**3*x**2*(f*x)**m/(m**4 + 6*m**3 + 13*m**2 + 12*m + 4)
+ 4*a*e*m**2*x**2*(f*x)**m/(m**4 + 6*m**3 + 13*m**2 + 12*m + 4) + 5*a*e*m*x**2*(f*x)**m/(m**4 + 6*m**3 + 13*m
**2 + 12*m + 4) + 2*a*e*x**2*(f*x)**m/(m**4 + 6*m**3 + 13*m**2 + 12*m + 4) + b*d*m**3*x*(f*x)**m*log(c*x**n)/(
m**4 + 6*m**3 + 13*m**2 + 12*m + 4) - b*d*m**2*n*x*(f*x)**m/(m**4 + 6*m**3 + 13*m**2 + 12*m + 4) + 5*b*d*m**2*
x*(f*x)**m*log(c*x**n)/(m**4 + 6*m**3 + 13*m**2 + 12*m + 4) - 4*b*d*m*n*x*(f*x)**m/(m**4 + 6*m**3 + 13*m**2 +
12*m + 4) + 8*b*d*m*x*(f*x)**m*log(c*x**n)/(m**4 + 6*m**3 + 13*m**2 + 12*m + 4) - 4*b*d*n*x*(f*x)**m/(m**4 + 6
*m**3 + 13*m**2 + 12*m + 4) + 4*b*d*x*(f*x)**m*log(c*x**n)/(m**4 + 6*m**3 + 13*m**2 + 12*m + 4) + b*e*m**3*x**
2*(f*x)**m*log(c*x**n)/(m**4 + 6*m**3 + 13*m**2 + 12*m + 4) - b*e*m**2*n*x**2*(f*x)**m/(m**4 + 6*m**3 + 13*m**
2 + 12*m + 4) + 4*b*e*m**2*x**2*(f*x)**m*log(c*x**n)/(m**4 + 6*m**3 + 13*m**2 + 12*m + 4) - 2*b*e*m*n*x**2*(f*
x)**m/(m**4 + 6*m**3 + 13*m**2 + 12*m + 4) + 5*b*e*m*x**2*(f*x)**m*log(c*x**n)/(m**4 + 6*m**3 + 13*m**2 + 12*m
+ 4) - b*e*n*x**2*(f*x)**m/(m**4 + 6*m**3 + 13*m**2 + 12*m + 4) + 2*b*e*x**2*(f*x)**m*log(c*x**n)/(m**4 + 6*m
**3 + 13*m**2 + 12*m + 4), True))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.25 \[ \int (f x)^m (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e f^{m} x^{2} x^{m} \log \left (c x^{n}\right )}{m + 2} + \frac {a e f^{m} x^{2} x^{m}}{m + 2} - \frac {b e f^{m} n x^{2} x^{m}}{{\left (m + 2\right )}^{2}} - \frac {b d f^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {\left (f x\right )^{m + 1} b d \log \left (c x^{n}\right )}{f {\left (m + 1\right )}} + \frac {\left (f x\right )^{m + 1} a d}{f {\left (m + 1\right )}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (95) = 190\).

Time = 0.33 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.23 \[ \int (f x)^m (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e f^{m} m n x^{2} x^{m} \log \left (x\right )}{m^{2} + 4 \, m + 4} + \frac {b d f^{m} m n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac {2 \, b e f^{m} n x^{2} x^{m} \log \left (x\right )}{m^{2} + 4 \, m + 4} - \frac {b e f^{m} n x^{2} x^{m}}{m^{2} + 4 \, m + 4} + \frac {b e f^{m} x^{2} x^{m} \log \left (c\right )}{m + 2} + \frac {b d f^{m} n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac {b d f^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac {a e f^{m} x^{2} x^{m}}{m + 2} + \frac {\left (f x\right )^{m} b d x \log \left (c\right )}{m + 1} + \frac {\left (f x\right )^{m} a d x}{m + 1} \]

[In]

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

[Out]

b*e*f^m*m*n*x^2*x^m*log(x)/(m^2 + 4*m + 4) + b*d*f^m*m*n*x*x^m*log(x)/(m^2 + 2*m + 1) + 2*b*e*f^m*n*x^2*x^m*lo
g(x)/(m^2 + 4*m + 4) - b*e*f^m*n*x^2*x^m/(m^2 + 4*m + 4) + b*e*f^m*x^2*x^m*log(c)/(m + 2) + b*d*f^m*n*x*x^m*lo
g(x)/(m^2 + 2*m + 1) - b*d*f^m*n*x*x^m/(m^2 + 2*m + 1) + a*e*f^m*x^2*x^m/(m + 2) + (f*x)^m*b*d*x*log(c)/(m + 1
) + (f*x)^m*a*d*x/(m + 1)

Mupad [F(-1)]

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

[In]

int((f*x)^m*(a + b*log(c*x^n))*(d + e*x),x)

[Out]

int((f*x)^m*(a + b*log(c*x^n))*(d + e*x), x)