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

Optimal. Leaf size=59 \[ \frac{1}{2} \left (d x^2+\frac{2 e x^{r+2}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} b d n x^2-\frac{b e n x^{r+2}}{(r+2)^2} \]

[Out]

-(b*d*n*x^2)/4 - (b*e*n*x^(2 + r))/(2 + r)^2 + ((d*x^2 + (2*e*x^(2 + r))/(2 + r))*(a + b*Log[c*x^n]))/2

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Rubi [A]  time = 0.0634667, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {14, 2334, 12} \[ \frac{1}{2} \left (d x^2+\frac{2 e x^{r+2}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} b d n x^2-\frac{b e n x^{r+2}}{(r+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d*n*x^2)/4 - (b*e*n*x^(2 + r))/(2 + r)^2 + ((d*x^2 + (2*e*x^(2 + r))/(2 + r))*(a + b*Log[c*x^n]))/2

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 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin{align*} \int x \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{2} \left (d x^2+\frac{2 e x^{2+r}}{2+r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{2} x \left (d+\frac{2 e x^r}{2+r}\right ) \, dx\\ &=\frac{1}{2} \left (d x^2+\frac{2 e x^{2+r}}{2+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} (b n) \int x \left (d+\frac{2 e x^r}{2+r}\right ) \, dx\\ &=\frac{1}{2} \left (d x^2+\frac{2 e x^{2+r}}{2+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} (b n) \int \left (d x+\frac{2 e x^{1+r}}{2+r}\right ) \, dx\\ &=-\frac{1}{4} b d n x^2-\frac{b e n x^{2+r}}{(2+r)^2}+\frac{1}{2} \left (d x^2+\frac{2 e x^{2+r}}{2+r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0895245, size = 73, normalized size = 1.24 \[ \frac{x^2 \left (2 a (r+2) \left (d (r+2)+2 e x^r\right )+2 b (r+2) \log \left (c x^n\right ) \left (d (r+2)+2 e x^r\right )-b n \left (d (r+2)^2+4 e x^r\right )\right )}{4 (r+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(2*a*(2 + r)*(d*(2 + r) + 2*e*x^r) - b*n*(d*(2 + r)^2 + 4*e*x^r) + 2*b*(2 + r)*(d*(2 + r) + 2*e*x^r)*Log[
c*x^n]))/(4*(2 + r)^2)

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Maple [C]  time = 0.234, size = 613, normalized size = 10.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2*(d*r+2*e*x^r+2*d)/(2+r)*ln(x^n)-1/4*x^2*(-8*a*d-4*x^r*a*e*r+4*x^r*b*e*n+2*I*Pi*b*e*csgn(I*x^n)*csgn(
I*c*x^n)*csgn(I*c)*x^r*r+4*b*d*n*r+4*I*Pi*b*d*csgn(I*c*x^n)^3+4*b*d*n-8*x^r*a*e-8*ln(c)*b*d*r-2*ln(c)*b*d*r^2-
4*ln(c)*b*e*x^r*r+I*Pi*b*d*r^2*csgn(I*c*x^n)^3+4*I*Pi*b*d*csgn(I*c*x^n)^3*r+4*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^
n)*csgn(I*c)*x^r-2*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r*r-8*ln(c)*b*e*x^r-2*a*d*r^2+4*I*Pi*b*d*csgn(I*x^n)
*csgn(I*c*x^n)*csgn(I*c)*r-2*I*Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r*r-8*ln(c)*b*d-8*a*d*r+I*Pi*b*d*r^2*csgn(I*
x^n)*csgn(I*c*x^n)*csgn(I*c)+b*d*n*r^2-4*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2-4*I*Pi*b*d*csgn(I*c*x^n)^2*csgn(
I*c)+4*I*Pi*b*e*csgn(I*c*x^n)^3*x^r-I*Pi*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*d*r^2*csgn(I*c*x^n)^2*csgn
(I*c)-4*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2*r-4*I*Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r-4*I*Pi*b*e*csgn(I*x^n)
*csgn(I*c*x^n)^2*x^r+2*I*Pi*b*e*csgn(I*c*x^n)^3*x^r*r-4*I*Pi*b*d*csgn(I*c*x^n)^2*csgn(I*c)*r+4*I*Pi*b*d*csgn(I
*x^n)*csgn(I*c*x^n)*csgn(I*c))/(2+r)^2

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.34566, size = 379, normalized size = 6.42 \begin{align*} \frac{2 \,{\left (b d r^{2} + 4 \, b d r + 4 \, b d\right )} x^{2} \log \left (c\right ) + 2 \,{\left (b d n r^{2} + 4 \, b d n r + 4 \, b d n\right )} x^{2} \log \left (x\right ) -{\left (4 \, b d n +{\left (b d n - 2 \, a d\right )} r^{2} - 8 \, a d + 4 \,{\left (b d n - 2 \, a d\right )} r\right )} x^{2} + 4 \,{\left ({\left (b e r + 2 \, b e\right )} x^{2} \log \left (c\right ) +{\left (b e n r + 2 \, b e n\right )} x^{2} \log \left (x\right ) -{\left (b e n - a e r - 2 \, a e\right )} x^{2}\right )} x^{r}}{4 \,{\left (r^{2} + 4 \, r + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*(b*d*r^2 + 4*b*d*r + 4*b*d)*x^2*log(c) + 2*(b*d*n*r^2 + 4*b*d*n*r + 4*b*d*n)*x^2*log(x) - (4*b*d*n + (b
*d*n - 2*a*d)*r^2 - 8*a*d + 4*(b*d*n - 2*a*d)*r)*x^2 + 4*((b*e*r + 2*b*e)*x^2*log(c) + (b*e*n*r + 2*b*e*n)*x^2
*log(x) - (b*e*n - a*e*r - 2*a*e)*x^2)*x^r)/(r^2 + 4*r + 4)

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Sympy [A]  time = 7.41702, size = 525, normalized size = 8.9 \begin{align*} \begin{cases} \frac{2 a d r^{2} x^{2}}{4 r^{2} + 16 r + 16} + \frac{8 a d r x^{2}}{4 r^{2} + 16 r + 16} + \frac{8 a d x^{2}}{4 r^{2} + 16 r + 16} + \frac{4 a e r x^{2} x^{r}}{4 r^{2} + 16 r + 16} + \frac{8 a e x^{2} x^{r}}{4 r^{2} + 16 r + 16} + \frac{2 b d n r^{2} x^{2} \log{\left (x \right )}}{4 r^{2} + 16 r + 16} - \frac{b d n r^{2} x^{2}}{4 r^{2} + 16 r + 16} + \frac{8 b d n r x^{2} \log{\left (x \right )}}{4 r^{2} + 16 r + 16} - \frac{4 b d n r x^{2}}{4 r^{2} + 16 r + 16} + \frac{8 b d n x^{2} \log{\left (x \right )}}{4 r^{2} + 16 r + 16} - \frac{4 b d n x^{2}}{4 r^{2} + 16 r + 16} + \frac{2 b d r^{2} x^{2} \log{\left (c \right )}}{4 r^{2} + 16 r + 16} + \frac{8 b d r x^{2} \log{\left (c \right )}}{4 r^{2} + 16 r + 16} + \frac{8 b d x^{2} \log{\left (c \right )}}{4 r^{2} + 16 r + 16} + \frac{4 b e n r x^{2} x^{r} \log{\left (x \right )}}{4 r^{2} + 16 r + 16} + \frac{8 b e n x^{2} x^{r} \log{\left (x \right )}}{4 r^{2} + 16 r + 16} - \frac{4 b e n x^{2} x^{r}}{4 r^{2} + 16 r + 16} + \frac{4 b e r x^{2} x^{r} \log{\left (c \right )}}{4 r^{2} + 16 r + 16} + \frac{8 b e x^{2} x^{r} \log{\left (c \right )}}{4 r^{2} + 16 r + 16} & \text{for}\: r \neq -2 \\\frac{a d x^{2}}{2} + a e \log{\left (x \right )} + \frac{b d n x^{2} \log{\left (x \right )}}{2} - \frac{b d n x^{2}}{4} + \frac{b d x^{2} \log{\left (c \right )}}{2} + \frac{b e n \log{\left (x \right )}^{2}}{2} + b e \log{\left (c \right )} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*a*d*r**2*x**2/(4*r**2 + 16*r + 16) + 8*a*d*r*x**2/(4*r**2 + 16*r + 16) + 8*a*d*x**2/(4*r**2 + 16*
r + 16) + 4*a*e*r*x**2*x**r/(4*r**2 + 16*r + 16) + 8*a*e*x**2*x**r/(4*r**2 + 16*r + 16) + 2*b*d*n*r**2*x**2*lo
g(x)/(4*r**2 + 16*r + 16) - b*d*n*r**2*x**2/(4*r**2 + 16*r + 16) + 8*b*d*n*r*x**2*log(x)/(4*r**2 + 16*r + 16)
- 4*b*d*n*r*x**2/(4*r**2 + 16*r + 16) + 8*b*d*n*x**2*log(x)/(4*r**2 + 16*r + 16) - 4*b*d*n*x**2/(4*r**2 + 16*r
+ 16) + 2*b*d*r**2*x**2*log(c)/(4*r**2 + 16*r + 16) + 8*b*d*r*x**2*log(c)/(4*r**2 + 16*r + 16) + 8*b*d*x**2*l
og(c)/(4*r**2 + 16*r + 16) + 4*b*e*n*r*x**2*x**r*log(x)/(4*r**2 + 16*r + 16) + 8*b*e*n*x**2*x**r*log(x)/(4*r**
2 + 16*r + 16) - 4*b*e*n*x**2*x**r/(4*r**2 + 16*r + 16) + 4*b*e*r*x**2*x**r*log(c)/(4*r**2 + 16*r + 16) + 8*b*
e*x**2*x**r*log(c)/(4*r**2 + 16*r + 16), Ne(r, -2)), (a*d*x**2/2 + a*e*log(x) + b*d*n*x**2*log(x)/2 - b*d*n*x*
*2/4 + b*d*x**2*log(c)/2 + b*e*n*log(x)**2/2 + b*e*log(c)*log(x), True))

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Giac [B]  time = 1.32026, size = 185, normalized size = 3.14 \begin{align*} \frac{b n r x^{2} x^{r} e \log \left (x\right )}{r^{2} + 4 \, r + 4} + \frac{1}{2} \, b d n x^{2} \log \left (x\right ) + \frac{2 \, b n x^{2} x^{r} e \log \left (x\right )}{r^{2} + 4 \, r + 4} - \frac{1}{4} \, b d n x^{2} - \frac{b n x^{2} x^{r} e}{r^{2} + 4 \, r + 4} + \frac{1}{2} \, b d x^{2} \log \left (c\right ) + \frac{b x^{2} x^{r} e \log \left (c\right )}{r + 2} + \frac{1}{2} \, a d x^{2} + \frac{a x^{2} x^{r} e}{r + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*n*r*x^2*x^r*e*log(x)/(r^2 + 4*r + 4) + 1/2*b*d*n*x^2*log(x) + 2*b*n*x^2*x^r*e*log(x)/(r^2 + 4*r + 4) - 1/4*b
*d*n*x^2 - b*n*x^2*x^r*e/(r^2 + 4*r + 4) + 1/2*b*d*x^2*log(c) + b*x^2*x^r*e*log(c)/(r + 2) + 1/2*a*d*x^2 + a*x
^2*x^r*e/(r + 2)