3.387 \(\int (d+e x^r)^2 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=113 \[ d^2 x \left (a+b \log \left (c x^n\right )\right )+\frac{2 d e x^{r+1} \left (a+b \log \left (c x^n\right )\right )}{r+1}+\frac{e^2 x^{2 r+1} \left (a+b \log \left (c x^n\right )\right )}{2 r+1}-b d^2 n x-\frac{2 b d e n x^{r+1}}{(r+1)^2}-\frac{b e^2 n x^{2 r+1}}{(2 r+1)^2} \]

[Out]

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

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Rubi [A]  time = 0.0762582, antiderivative size = 95, normalized size of antiderivative = 0.84, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {244, 2313} \[ \left (d^2 x+\frac{2 d e x^{r+1}}{r+1}+\frac{e^2 x^{2 r+1}}{2 r+1}\right ) \left (a+b \log \left (c x^n\right )\right )-b d^2 n x-\frac{2 b d e n x^{r+1}}{(r+1)^2}-\frac{b e^2 n x^{2 r+1}}{(2 r+1)^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^r)^2*(a + b*Log[c*x^n]),x]

[Out]

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

Rule 244

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n},
x] && IGtQ[p, 0]

Rule 2313

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(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]

Rubi steps

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

Mathematica [A]  time = 0.15896, size = 107, normalized size = 0.95 \[ x \left (\frac{2 d e x^r \left (a+b \log \left (c x^n\right )\right )}{r+1}+\frac{e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r+1}+a d^2+b d^2 \log \left (c x^n\right )-b d^2 n-\frac{2 b d e n x^r}{(r+1)^2}-\frac{b e^2 n x^{2 r}}{(2 r+1)^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^r)^2*(a + b*Log[c*x^n]),x]

[Out]

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

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Maple [C]  time = 0.288, size = 1921, normalized size = 17. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+e*x^r)^2*(a+b*ln(c*x^n)),x)

[Out]

b*x*(e^2*(x^r)^2*r+2*d^2*r^2+4*d*e*x^r*r+e^2*(x^r)^2+3*d^2*r+2*d*e*x^r+d^2)/(1+2*r)/(1+r)*ln(x^n)-1/2*x*(8*b*d
^2*n*r^4+24*b*d^2*n*r^3-2*ln(c)*b*d^2+10*I*Pi*b*d*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+26*b*d^2*n*r^2+1
2*b*d^2*n*r+8*I*Pi*b*d*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+13*I*Pi*b*d^2*r^2*csgn(I*x^n)*csgn(I*c*x^
n)*csgn(I*c)-4*a*e^2*r^3*(x^r)^2-10*a*e^2*r^2*(x^r)^2-2*a*d^2+4*I*Pi*b*d^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(
I*c)-4*a*d*e*x^r+16*I*Pi*b*d*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+I*Pi*b*e^2*csgn(I*x^n)*csgn(I*c*x^n
)*csgn(I*c)*(x^r)^2-4*I*Pi*b*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2+
12*I*Pi*b*d^2*r^3*csgn(I*c*x^n)^3-4*I*Pi*b*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+10*I*Pi*b*d*e*r*csgn(I*c*x^
n)^3*x^r+16*I*Pi*b*d*e*r^2*csgn(I*c*x^n)^3*x^r+12*I*Pi*b*d^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+6*I*Pi*b*
d^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-8*a*e^2*r*(x^r)^2-26*a*d^2*r^2-12*a*d^2*r-8*a*d^2*r^4-24*a*d^2*r^3+2
*b*d^2*n+I*Pi*b*d^2*csgn(I*c*x^n)^3+4*b*d*e*n*x^r-4*ln(c)*b*e^2*r^3*(x^r)^2-4*ln(c)*b*d*e*x^r-10*ln(c)*b*e^2*r
^2*(x^r)^2-8*ln(c)*b*e^2*r*(x^r)^2-2*ln(c)*b*e^2*(x^r)^2+2*b*e^2*n*(x^r)^2-26*ln(c)*b*d^2*r^2-12*ln(c)*b*d^2*r
-2*a*e^2*(x^r)^2-8*ln(c)*b*d^2*r^4-24*ln(c)*b*d^2*r^3+2*b*e^2*n*r^2*(x^r)^2-16*a*d*e*r^3*x^r-32*a*d*e*r^2*x^r-
20*a*d*e*r*x^r+4*b*e^2*n*r*(x^r)^2-5*I*Pi*b*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-5*I*Pi*b*e^2*r^2*csgn(
I*c*x^n)^2*csgn(I*c)*(x^r)^2-2*I*Pi*b*d*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r-2*I*Pi*b*e^2*r^3*csgn(I*c*x^n)^2*csgn(
I*c)*(x^r)^2+8*I*Pi*b*d*e*r^3*csgn(I*c*x^n)^3*x^r-2*I*Pi*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-2*I*Pi*b*e^2*r^
3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+I*Pi*b*e^2*csgn(I*c*x^n)^3*(x^r)^2+6*I*Pi*b*d^2*r*csgn(I*c*x^n)^3-I*Pi*b
*d^2*csgn(I*c*x^n)^2*csgn(I*c)+4*I*Pi*b*d^2*r^4*csgn(I*c*x^n)^3-16*I*Pi*b*d*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*
x^r-16*I*Pi*b*d*e*r^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r+5*I*Pi*b*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)
^2-8*I*Pi*b*d*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-10*I*Pi*b*d*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-10*I*Pi*b*
d*e*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r-8*I*Pi*b*d*e*r^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r+2*I*Pi*b*e^2*r^3*csgn(I*x^n
)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+2*I*Pi*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+4*I*Pi*b*e^2*r*csgn(I*x
^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-4*I*Pi*b*d^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2-4*I*Pi*b*d^2*r^4*csgn(I*c*x^n
)^2*csgn(I*c)-13*I*Pi*b*d^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2-32*ln(c)*b*d*e*r^2*x^r-20*ln(c)*b*d*e*r*x^r-16*ln(
c)*b*d*e*r^3*x^r-I*Pi*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-6*I*Pi*b*d^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2-6*I
*Pi*b*d^2*r*csgn(I*c*x^n)^2*csgn(I*c)-13*I*Pi*b*d^2*r^2*csgn(I*c*x^n)^2*csgn(I*c)-12*I*Pi*b*d^2*r^3*csgn(I*x^n
)*csgn(I*c*x^n)^2-12*I*Pi*b*d^2*r^3*csgn(I*c*x^n)^2*csgn(I*c)+4*I*Pi*b*e^2*r*csgn(I*c*x^n)^3*(x^r)^2+16*b*d*e*
n*r*x^r+16*b*d*e*n*r^2*x^r+I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+5*I*Pi*b*e^2*r^2*csgn(I*c*x^n)^3*(x^
r)^2-I*Pi*b*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+2*I*Pi*b*d*e*csgn(I*c*x^n)^3*x^r+2*I*Pi*b*e^2*r^3*csgn(I*c*x
^n)^3*(x^r)^2+13*I*Pi*b*d^2*r^2*csgn(I*c*x^n)^3)/(1+2*r)^2/(1+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((d+e*x^r)^2*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.34927, size = 1041, normalized size = 9.21 \begin{align*} \frac{{\left (4 \, b d^{2} r^{4} + 12 \, b d^{2} r^{3} + 13 \, b d^{2} r^{2} + 6 \, b d^{2} r + b d^{2}\right )} x \log \left (c\right ) +{\left (4 \, b d^{2} n r^{4} + 12 \, b d^{2} n r^{3} + 13 \, b d^{2} n r^{2} + 6 \, b d^{2} n r + b d^{2} n\right )} x \log \left (x\right ) -{\left (4 \,{\left (b d^{2} n - a d^{2}\right )} r^{4} + b d^{2} n + 12 \,{\left (b d^{2} n - a d^{2}\right )} r^{3} - a d^{2} + 13 \,{\left (b d^{2} n - a d^{2}\right )} r^{2} + 6 \,{\left (b d^{2} n - a d^{2}\right )} r\right )} x +{\left ({\left (2 \, b e^{2} r^{3} + 5 \, b e^{2} r^{2} + 4 \, b e^{2} r + b e^{2}\right )} x \log \left (c\right ) +{\left (2 \, b e^{2} n r^{3} + 5 \, b e^{2} n r^{2} + 4 \, b e^{2} n r + b e^{2} n\right )} x \log \left (x\right ) +{\left (2 \, a e^{2} r^{3} - b e^{2} n + a e^{2} -{\left (b e^{2} n - 5 \, a e^{2}\right )} r^{2} - 2 \,{\left (b e^{2} n - 2 \, a e^{2}\right )} r\right )} x\right )} x^{2 \, r} + 2 \,{\left ({\left (4 \, b d e r^{3} + 8 \, b d e r^{2} + 5 \, b d e r + b d e\right )} x \log \left (c\right ) +{\left (4 \, b d e n r^{3} + 8 \, b d e n r^{2} + 5 \, b d e n r + b d e n\right )} x \log \left (x\right ) +{\left (4 \, a d e r^{3} - b d e n + a d e - 4 \,{\left (b d e n - 2 \, a d e\right )} r^{2} -{\left (4 \, b d e n - 5 \, a d e\right )} r\right )} x\right )} x^{r}}{4 \, r^{4} + 12 \, r^{3} + 13 \, r^{2} + 6 \, r + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)^2*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

((4*b*d^2*r^4 + 12*b*d^2*r^3 + 13*b*d^2*r^2 + 6*b*d^2*r + b*d^2)*x*log(c) + (4*b*d^2*n*r^4 + 12*b*d^2*n*r^3 +
13*b*d^2*n*r^2 + 6*b*d^2*n*r + b*d^2*n)*x*log(x) - (4*(b*d^2*n - a*d^2)*r^4 + b*d^2*n + 12*(b*d^2*n - a*d^2)*r
^3 - a*d^2 + 13*(b*d^2*n - a*d^2)*r^2 + 6*(b*d^2*n - a*d^2)*r)*x + ((2*b*e^2*r^3 + 5*b*e^2*r^2 + 4*b*e^2*r + b
*e^2)*x*log(c) + (2*b*e^2*n*r^3 + 5*b*e^2*n*r^2 + 4*b*e^2*n*r + b*e^2*n)*x*log(x) + (2*a*e^2*r^3 - b*e^2*n + a
*e^2 - (b*e^2*n - 5*a*e^2)*r^2 - 2*(b*e^2*n - 2*a*e^2)*r)*x)*x^(2*r) + 2*((4*b*d*e*r^3 + 8*b*d*e*r^2 + 5*b*d*e
*r + b*d*e)*x*log(c) + (4*b*d*e*n*r^3 + 8*b*d*e*n*r^2 + 5*b*d*e*n*r + b*d*e*n)*x*log(x) + (4*a*d*e*r^3 - b*d*e
*n + a*d*e - 4*(b*d*e*n - 2*a*d*e)*r^2 - (4*b*d*e*n - 5*a*d*e)*r)*x)*x^r)/(4*r^4 + 12*r^3 + 13*r^2 + 6*r + 1)

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Sympy [A]  time = 13.4626, size = 211, normalized size = 1.87 \begin{align*} a d^{2} x + 2 a d e \left (\begin{cases} \frac{x^{r + 1}}{r + 1} & \text{for}\: r \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right ) + a e^{2} \left (\begin{cases} \frac{x^{2 r + 1}}{2 r + 1} & \text{for}\: 2 r \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right ) - b d^{2} n x + b d^{2} x \log{\left (c x^{n} \right )} - 2 b d e n \left (\begin{cases} \frac{\begin{cases} \frac{x x^{r}}{r + 1} & \text{for}\: r \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases}}{r + 1} & \text{for}\: r > -\infty \wedge r < \infty \wedge r \neq -1 \\\frac{\log{\left (x \right )}^{2}}{2} & \text{otherwise} \end{cases}\right ) + 2 b d e \left (\begin{cases} \frac{x^{r + 1}}{r + 1} & \text{for}\: r \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )} - b e^{2} n \left (\begin{cases} \frac{\begin{cases} \frac{x x^{2 r}}{2 r + 1} & \text{for}\: r \neq - \frac{1}{2} \\\log{\left (x \right )} & \text{otherwise} \end{cases}}{2 r + 1} & \text{for}\: r > -\infty \wedge r < \infty \wedge r \neq - \frac{1}{2} \\\frac{\log{\left (x \right )}^{2}}{2} & \text{otherwise} \end{cases}\right ) + b e^{2} \left (\begin{cases} \frac{x^{2 r + 1}}{2 r + 1} & \text{for}\: 2 r \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x**r)**2*(a+b*ln(c*x**n)),x)

[Out]

a*d**2*x + 2*a*d*e*Piecewise((x**(r + 1)/(r + 1), Ne(r, -1)), (log(x), True)) + a*e**2*Piecewise((x**(2*r + 1)
/(2*r + 1), Ne(2*r, -1)), (log(x), True)) - b*d**2*n*x + b*d**2*x*log(c*x**n) - 2*b*d*e*n*Piecewise((Piecewise
((x*x**r/(r + 1), Ne(r, -1)), (log(x), True))/(r + 1), (r > -oo) & (r < oo) & Ne(r, -1)), (log(x)**2/2, True))
+ 2*b*d*e*Piecewise((x**(r + 1)/(r + 1), Ne(r, -1)), (log(x), True))*log(c*x**n) - b*e**2*n*Piecewise((Piecew
ise((x*x**(2*r)/(2*r + 1), Ne(r, -1/2)), (log(x), True))/(2*r + 1), (r > -oo) & (r < oo) & Ne(r, -1/2)), (log(
x)**2/2, True)) + b*e**2*Piecewise((x**(2*r + 1)/(2*r + 1), Ne(2*r, -1)), (log(x), True))*log(c*x**n)

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Giac [B]  time = 1.32161, size = 329, normalized size = 2.91 \begin{align*} \frac{2 \, b d n r x x^{r} e \log \left (x\right )}{r^{2} + 2 \, r + 1} + b d^{2} n x \log \left (x\right ) + \frac{2 \, b n r x x^{2 \, r} e^{2} \log \left (x\right )}{4 \, r^{2} + 4 \, r + 1} + \frac{2 \, b d n x x^{r} e \log \left (x\right )}{r^{2} + 2 \, r + 1} - b d^{2} n x - \frac{2 \, b d n x x^{r} e}{r^{2} + 2 \, r + 1} + b d^{2} x \log \left (c\right ) + \frac{2 \, b d x x^{r} e \log \left (c\right )}{r + 1} + \frac{b n x x^{2 \, r} e^{2} \log \left (x\right )}{4 \, r^{2} + 4 \, r + 1} + a d^{2} x - \frac{b n x x^{2 \, r} e^{2}}{4 \, r^{2} + 4 \, r + 1} + \frac{2 \, a d x x^{r} e}{r + 1} + \frac{b x x^{2 \, r} e^{2} \log \left (c\right )}{2 \, r + 1} + \frac{a x x^{2 \, r} e^{2}}{2 \, r + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)^2*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

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