3.442 \(\int (f x)^m (d+e x^r) (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=97 \[ \frac{d (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac{e x^{r+1} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{m+r+1}-\frac{b d n (f x)^{m+1}}{f (m+1)^2}-\frac{b e n x^{r+1} (f x)^m}{(m+r+1)^2} \]

[Out]

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

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Rubi [A]  time = 0.102707, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {14, 20, 30, 2350} \[ \frac{d (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac{e x^{r+1} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{m+r+1}-\frac{b d n (f x)^{m+1}}{f (m+1)^2}-\frac{b e n x^{r+1} (f x)^m}{(m+r+1)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2350

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

Mathematica [A]  time = 0.110588, size = 70, normalized size = 0.72 \[ x (f x)^m \left (\frac{d \left (a+b \log \left (c x^n\right )\right )}{m+1}+\frac{e x^r \left (a+b \log \left (c x^n\right )\right )}{m+r+1}-\frac{b d n}{(m+1)^2}-\frac{b e n x^r}{(m+r+1)^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.362, size = 2152, normalized size = 22.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x*(m*e*x^r+m*d+d*r+e*x^r+d)/(1+m)/(1+m+r)*exp(1/2*m*(-I*Pi*csgn(I*f*x)^3+I*Pi*csgn(I*f*x)^2*csgn(I*f)+I*Pi*c
sgn(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*x*(-2*a*d-2*a*d*m^3+
2*I*Pi*b*e*m*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-6*x^r*a*e*m-2*x^r*a*e*r+2*x^r*b*e*n+4*b*d*n*r+2*b*d*n+3
*I*Pi*b*e*m*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-I*Pi*b*e*m^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-6*a*d*m-2*x
^r*a*e-4*ln(c)*b*d*r-2*ln(c)*b*d*r^2-2*ln(c)*b*e*x^r*r-8*ln(c)*b*d*m*r-4*ln(c)*b*d*m^2*r-2*ln(c)*b*d*m*r^2-6*l
n(c)*b*d*m^2-6*ln(c)*b*d*m-2*ln(c)*b*d*m^3+4*I*Pi*b*d*m*r*csgn(I*c*x^n)^3-3*I*Pi*b*d*m*csgn(I*x^n)*csgn(I*c*x^
n)^2-3*I*Pi*b*d*m*csgn(I*c*x^n)^2*csgn(I*c)+I*Pi*b*d*m^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-2*ln(c)*b*e*x^r+2
*b*d*m^2*n-2*a*d*r^2-2*ln(c)*b*d-4*a*d*r+4*b*d*m*n-2*a*e*m^3*x^r-6*a*e*m^2*x^r+2*b*d*n*r^2-I*Pi*b*e*csgn(I*x^n
)*csgn(I*c*x^n)^2*x^r*r+I*Pi*b*e*csgn(I*c*x^n)^3*x^r+2*I*Pi*b*d*csgn(I*c*x^n)^3*r-I*Pi*b*d*csgn(I*x^n)*csgn(I*
c*x^n)^2+I*Pi*b*d*r^2*csgn(I*c*x^n)^3+I*Pi*b*d*m^3*csgn(I*c*x^n)^3-4*a*d*m^2*r-2*a*d*m*r^2+I*Pi*b*e*csgn(I*c*x
^n)^3*x^r*r-I*Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r*r-3*I*Pi*b*e*m^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-3*I*Pi*b*e
*m^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r+2*I*Pi*b*e*m*r*csgn(I*c*x^n)^3*x^r+I*Pi*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)*c
sgn(I*c)+4*b*d*m*n*r-8*a*d*m*r-I*Pi*b*e*m^2*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r+4*I*Pi*b*d*m*r*csgn(I*x^n)*csgn(I*
c*x^n)*csgn(I*c)+2*I*Pi*b*d*m^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r
-2*I*Pi*b*d*r*csgn(I*x^n)*csgn(I*c*x^n)^2-6*a*d*m^2-2*I*Pi*b*e*m*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r+3*I*Pi*b*e*m^
2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-2*I*Pi*b*e*m*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-6*ln(c)*b*e*m^2*x^r-6
*ln(c)*b*e*m*x^r-2*ln(c)*b*e*m^3*x^r-2*a*e*m^2*r*x^r+2*b*e*m^2*n*x^r-4*a*e*m*r*x^r+4*b*e*m*n*x^r+I*Pi*b*d*csgn
(I*c*x^n)^3+I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r*r-4*I*Pi*b*d*m*r*csgn(I*c*x^n)^2*csgn(I*c)-I*Pi*b
*d*csgn(I*c*x^n)^2*csgn(I*c)+I*Pi*b*e*m^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-I*Pi*b*d*r^2*csgn(I*c*x^n)
^2*csgn(I*c)+2*I*Pi*b*d*m^2*r*csgn(I*c*x^n)^3+3*I*Pi*b*e*m^2*csgn(I*c*x^n)^3*x^r+3*I*Pi*b*e*m*csgn(I*c*x^n)^3*
x^r-3*I*Pi*b*d*m^2*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*Pi*b*d*m^2*csgn(I*c*x^n)^2*csgn(I*c)+I*Pi*b*e*csgn(I*x^n)*c
sgn(I*c*x^n)*csgn(I*c)*x^r+2*I*Pi*b*d*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+3*I*Pi*b*d*m*csgn(I*x^n)*csgn(I*c*
x^n)*csgn(I*c)+3*I*Pi*b*d*m^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*Pi*b*d*m*r*csgn(I*x^n)*csgn(I*c*x^n)^2-4
*ln(c)*b*e*m*r*x^r-2*ln(c)*b*e*m^2*r*x^r+3*I*Pi*b*d*m^2*csgn(I*c*x^n)^3+I*Pi*b*d*m*r^2*csgn(I*c*x^n)^3+I*Pi*b*
e*m^3*csgn(I*c*x^n)^3*x^r-2*I*Pi*b*d*m^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2-2*I*Pi*b*d*m^2*r*csgn(I*c*x^n)^2*csgn(I
*c)-I*Pi*b*d*m*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*d*m*r^2*csgn(I*c*x^n)^2*csgn(I*c)+3*I*Pi*b*d*m*csgn(I*c*
x^n)^3+I*Pi*b*e*m^2*r*csgn(I*c*x^n)^3*x^r-3*I*Pi*b*e*m*csgn(I*c*x^n)^2*csgn(I*c)*x^r-I*Pi*b*e*m^3*csgn(I*x^n)*
csgn(I*c*x^n)^2*x^r-I*Pi*b*e*m^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r-3*I*Pi*b*e*m*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-I*
Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r-2*I*Pi*b*d*r*csgn(I*c*x^n)^2*csgn(I*c)+I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)-I*Pi*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+I*Pi*b*e*m^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+I*Pi*
b*d*m*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b*d*m^3*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*d*m^3*csgn(I*c*x
^n)^2*csgn(I*c))/(1+m)^2/(1+m+r)^2*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*(d+e*x^r)*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.39571, size = 1035, normalized size = 10.67 \begin{align*} \frac{{\left ({\left (b e m^{3} + 3 \, b e m^{2} + 3 \, b e m + b e +{\left (b e m^{2} + 2 \, b e m + b e\right )} r\right )} x \log \left (c\right ) +{\left ({\left (b e m^{2} + 2 \, b e m + b e\right )} n r +{\left (b e m^{3} + 3 \, b e m^{2} + 3 \, b e m + b e\right )} n\right )} x \log \left (x\right ) +{\left (a e m^{3} + 3 \, a e m^{2} + 3 \, a e m + a e -{\left (b e m^{2} + 2 \, b e m + b e\right )} n +{\left (a e m^{2} + 2 \, a e m + a e\right )} r\right )} x\right )} x^{r} e^{\left (m \log \left (f\right ) + m \log \left (x\right )\right )} +{\left ({\left (b d m^{3} + 3 \, b d m^{2} + 3 \, b d m +{\left (b d m + b d\right )} r^{2} + b d + 2 \,{\left (b d m^{2} + 2 \, b d m + b d\right )} r\right )} x \log \left (c\right ) +{\left ({\left (b d m + b d\right )} n r^{2} + 2 \,{\left (b d m^{2} + 2 \, b d m + b d\right )} n r +{\left (b d m^{3} + 3 \, b d m^{2} + 3 \, b d m + b d\right )} n\right )} x \log \left (x\right ) +{\left (a d m^{3} + 3 \, a d m^{2} + 3 \, a d m +{\left (a d m - b d n + a d\right )} r^{2} + a d -{\left (b d m^{2} + 2 \, b d m + b d\right )} n + 2 \,{\left (a d m^{2} + 2 \, a d m + a d -{\left (b d m + b d\right )} n\right )} r\right )} x\right )} e^{\left (m \log \left (f\right ) + m \log \left (x\right )\right )}}{m^{4} + 4 \, m^{3} +{\left (m^{2} + 2 \, m + 1\right )} r^{2} + 6 \, m^{2} + 2 \,{\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} r + 4 \, m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((b*e*m^3 + 3*b*e*m^2 + 3*b*e*m + b*e + (b*e*m^2 + 2*b*e*m + b*e)*r)*x*log(c) + ((b*e*m^2 + 2*b*e*m + b*e)*n*
r + (b*e*m^3 + 3*b*e*m^2 + 3*b*e*m + b*e)*n)*x*log(x) + (a*e*m^3 + 3*a*e*m^2 + 3*a*e*m + a*e - (b*e*m^2 + 2*b*
e*m + b*e)*n + (a*e*m^2 + 2*a*e*m + a*e)*r)*x)*x^r*e^(m*log(f) + m*log(x)) + ((b*d*m^3 + 3*b*d*m^2 + 3*b*d*m +
(b*d*m + b*d)*r^2 + b*d + 2*(b*d*m^2 + 2*b*d*m + b*d)*r)*x*log(c) + ((b*d*m + b*d)*n*r^2 + 2*(b*d*m^2 + 2*b*d
*m + b*d)*n*r + (b*d*m^3 + 3*b*d*m^2 + 3*b*d*m + b*d)*n)*x*log(x) + (a*d*m^3 + 3*a*d*m^2 + 3*a*d*m + (a*d*m -
b*d*n + a*d)*r^2 + a*d - (b*d*m^2 + 2*b*d*m + b*d)*n + 2*(a*d*m^2 + 2*a*d*m + a*d - (b*d*m + b*d)*n)*r)*x)*e^(
m*log(f) + m*log(x)))/(m^4 + 4*m^3 + (m^2 + 2*m + 1)*r^2 + 6*m^2 + 2*(m^3 + 3*m^2 + 3*m + 1)*r + 4*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*(d+e*x**r)*(a+b*ln(c*x**n)),x)

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.32141, size = 393, normalized size = 4.05 \begin{align*} \frac{b f^{m} m n x x^{m} x^{r} e \log \left (x\right )}{m^{2} + 2 \, m r + r^{2} + 2 \, m + 2 \, r + 1} + \frac{b f^{m} n r x x^{m} x^{r} e \log \left (x\right )}{m^{2} + 2 \, m r + r^{2} + 2 \, m + 2 \, r + 1} + \frac{b d f^{m} m n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac{b f^{m} n x x^{m} x^{r} e \log \left (x\right )}{m^{2} + 2 \, m r + r^{2} + 2 \, m + 2 \, r + 1} - \frac{b f^{m} n x x^{m} x^{r} e}{m^{2} + 2 \, m r + r^{2} + 2 \, m + 2 \, r + 1} + \frac{b f^{m} x x^{m} x^{r} e \log \left (c\right )}{m + r + 1} + \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 f^{m} x x^{m} x^{r} e}{m + r + 1} + \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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