3.23 \(\int \frac{(d+e x)^3 (a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=122 \[ 3 d^2 e x \left (a+b \log \left (c x^n\right )\right )+d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac{3}{2} d e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{3} e^3 x^3 \left (a+b \log \left (c x^n\right )\right )-3 b d^2 e n x-\frac{1}{2} b d^3 n \log ^2(x)-\frac{3}{4} b d e^2 n x^2-\frac{1}{9} b e^3 n x^3 \]

[Out]

-3*b*d^2*e*n*x - (3*b*d*e^2*n*x^2)/4 - (b*e^3*n*x^3)/9 - (b*d^3*n*Log[x]^2)/2 + 3*d^2*e*x*(a + b*Log[c*x^n]) +
(3*d*e^2*x^2*(a + b*Log[c*x^n]))/2 + (e^3*x^3*(a + b*Log[c*x^n]))/3 + d^3*Log[x]*(a + b*Log[c*x^n])

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Rubi [A]  time = 0.0882038, antiderivative size = 94, normalized size of antiderivative = 0.77, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {43, 2334, 2301} \[ \frac{1}{6} \left (18 d^2 e x+6 d^3 \log (x)+9 d e^2 x^2+2 e^3 x^3\right ) \left (a+b \log \left (c x^n\right )\right )-3 b d^2 e n x-\frac{1}{2} b d^3 n \log ^2(x)-\frac{3}{4} b d e^2 n x^2-\frac{1}{9} b e^3 n x^3 \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^3*(a + b*Log[c*x^n]))/x,x]

[Out]

-3*b*d^2*e*n*x - (3*b*d*e^2*n*x^2)/4 - (b*e^3*n*x^3)/9 - (b*d^3*n*Log[x]^2)/2 + ((18*d^2*e*x + 9*d*e^2*x^2 + 2
*e^3*x^3 + 6*d^3*Log[x])*(a + b*Log[c*x^n]))/6

Rule 43

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 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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin{align*} \int \frac{(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{1}{6} \left (18 d^2 e x+9 d e^2 x^2+2 e^3 x^3+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac{1}{6} e \left (18 d^2+9 d e x+2 e^2 x^2\right )+\frac{d^3 \log (x)}{x}\right ) \, dx\\ &=\frac{1}{6} \left (18 d^2 e x+9 d e^2 x^2+2 e^3 x^3+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\left (b d^3 n\right ) \int \frac{\log (x)}{x} \, dx-\frac{1}{6} (b e n) \int \left (18 d^2+9 d e x+2 e^2 x^2\right ) \, dx\\ &=-3 b d^2 e n x-\frac{3}{4} b d e^2 n x^2-\frac{1}{9} b e^3 n x^3-\frac{1}{2} b d^3 n \log ^2(x)+\frac{1}{6} \left (18 d^2 e x+9 d e^2 x^2+2 e^3 x^3+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0612481, size = 123, normalized size = 1.01 \[ \frac{d^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+\frac{3}{2} d e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{3} e^3 x^3 \left (a+b \log \left (c x^n\right )\right )+3 a d^2 e x+3 b d^2 e x \log \left (c x^n\right )-3 b d^2 e n x-\frac{3}{4} b d e^2 n x^2-\frac{1}{9} b e^3 n x^3 \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^3*(a + b*Log[c*x^n]))/x,x]

[Out]

3*a*d^2*e*x - 3*b*d^2*e*n*x - (3*b*d*e^2*n*x^2)/4 - (b*e^3*n*x^3)/9 + 3*b*d^2*e*x*Log[c*x^n] + (3*d*e^2*x^2*(a
+ b*Log[c*x^n]))/2 + (e^3*x^3*(a + b*Log[c*x^n]))/3 + (d^3*(a + b*Log[c*x^n])^2)/(2*b*n)

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Maple [C]  time = 0.246, size = 579, normalized size = 4.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(a+b*ln(c*x^n))/x,x)

[Out]

(1/3*b*e^3*x^3+3/2*b*d*e^2*x^2+3*b*d^2*e*x+b*d^3*ln(x))*ln(x^n)+1/3*ln(c)*b*e^3*x^3+3/2*a*d*e^2*x^2+3*a*d^2*e*
x-1/6*I*Pi*b*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/6*I*Pi*b*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2+1/6*I*
Pi*b*e^3*x^3*csgn(I*c*x^n)^2*csgn(I*c)+3/2*I*Pi*b*d^2*e*x*csgn(I*c*x^n)^2*csgn(I*c)+3/4*I*Pi*b*d*e^2*x^2*csgn(
I*c*x^n)^2*csgn(I*c)+3/2*I*Pi*b*d^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2-3/4*I*Pi*b*d*e^2*x^2*csgn(I*x^n)*csgn(I*c*
x^n)*csgn(I*c)-3/2*I*Pi*b*d^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+ln(x)*ln(c)*b*d^3-1/2*I*ln(x)*Pi*b*d^3*c
sgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+3/4*I*Pi*b*d*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-1/6*I*Pi*b*e^3*x^3*csgn(I*
c*x^n)^3-1/2*I*ln(x)*Pi*b*d^3*csgn(I*c*x^n)^3+3/2*ln(c)*b*d*e^2*x^2+3*ln(c)*b*d^2*e*x+1/3*a*e^3*x^3+ln(x)*a*d^
3-3/4*I*Pi*b*d*e^2*x^2*csgn(I*c*x^n)^3-3/2*I*Pi*b*d^2*e*x*csgn(I*c*x^n)^3+1/2*I*ln(x)*Pi*b*d^3*csgn(I*x^n)*csg
n(I*c*x^n)^2+1/2*I*ln(x)*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)-1/9*b*e^3*n*x^3-3*b*d^2*e*n*x-3/4*b*d*e^2*n*x^2-1/
2*b*d^3*n*ln(x)^2

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Maxima [A]  time = 1.23623, size = 171, normalized size = 1.4 \begin{align*} -\frac{1}{9} \, b e^{3} n x^{3} + \frac{1}{3} \, b e^{3} x^{3} \log \left (c x^{n}\right ) - \frac{3}{4} \, b d e^{2} n x^{2} + \frac{1}{3} \, a e^{3} x^{3} + \frac{3}{2} \, b d e^{2} x^{2} \log \left (c x^{n}\right ) - 3 \, b d^{2} e n x + \frac{3}{2} \, a d e^{2} x^{2} + 3 \, b d^{2} e x \log \left (c x^{n}\right ) + 3 \, a d^{2} e x + \frac{b d^{3} \log \left (c x^{n}\right )^{2}}{2 \, n} + a d^{3} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*log(c*x^n))/x,x, algorithm="maxima")

[Out]

-1/9*b*e^3*n*x^3 + 1/3*b*e^3*x^3*log(c*x^n) - 3/4*b*d*e^2*n*x^2 + 1/3*a*e^3*x^3 + 3/2*b*d*e^2*x^2*log(c*x^n) -
3*b*d^2*e*n*x + 3/2*a*d*e^2*x^2 + 3*b*d^2*e*x*log(c*x^n) + 3*a*d^2*e*x + 1/2*b*d^3*log(c*x^n)^2/n + a*d^3*log
(x)

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Fricas [A]  time = 1.03711, size = 355, normalized size = 2.91 \begin{align*} \frac{1}{2} \, b d^{3} n \log \left (x\right )^{2} - \frac{1}{9} \,{\left (b e^{3} n - 3 \, a e^{3}\right )} x^{3} - \frac{3}{4} \,{\left (b d e^{2} n - 2 \, a d e^{2}\right )} x^{2} - 3 \,{\left (b d^{2} e n - a d^{2} e\right )} x + \frac{1}{6} \,{\left (2 \, b e^{3} x^{3} + 9 \, b d e^{2} x^{2} + 18 \, b d^{2} e x\right )} \log \left (c\right ) + \frac{1}{6} \,{\left (2 \, b e^{3} n x^{3} + 9 \, b d e^{2} n x^{2} + 18 \, b d^{2} e n x + 6 \, b d^{3} \log \left (c\right ) + 6 \, a d^{3}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*log(c*x^n))/x,x, algorithm="fricas")

[Out]

1/2*b*d^3*n*log(x)^2 - 1/9*(b*e^3*n - 3*a*e^3)*x^3 - 3/4*(b*d*e^2*n - 2*a*d*e^2)*x^2 - 3*(b*d^2*e*n - a*d^2*e)
*x + 1/6*(2*b*e^3*x^3 + 9*b*d*e^2*x^2 + 18*b*d^2*e*x)*log(c) + 1/6*(2*b*e^3*n*x^3 + 9*b*d*e^2*n*x^2 + 18*b*d^2
*e*n*x + 6*b*d^3*log(c) + 6*a*d^3)*log(x)

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Sympy [A]  time = 2.57997, size = 199, normalized size = 1.63 \begin{align*} a d^{3} \log{\left (x \right )} + 3 a d^{2} e x + \frac{3 a d e^{2} x^{2}}{2} + \frac{a e^{3} x^{3}}{3} + \frac{b d^{3} n \log{\left (x \right )}^{2}}{2} + b d^{3} \log{\left (c \right )} \log{\left (x \right )} + 3 b d^{2} e n x \log{\left (x \right )} - 3 b d^{2} e n x + 3 b d^{2} e x \log{\left (c \right )} + \frac{3 b d e^{2} n x^{2} \log{\left (x \right )}}{2} - \frac{3 b d e^{2} n x^{2}}{4} + \frac{3 b d e^{2} x^{2} \log{\left (c \right )}}{2} + \frac{b e^{3} n x^{3} \log{\left (x \right )}}{3} - \frac{b e^{3} n x^{3}}{9} + \frac{b e^{3} x^{3} \log{\left (c \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(a+b*ln(c*x**n))/x,x)

[Out]

a*d**3*log(x) + 3*a*d**2*e*x + 3*a*d*e**2*x**2/2 + a*e**3*x**3/3 + b*d**3*n*log(x)**2/2 + b*d**3*log(c)*log(x)
+ 3*b*d**2*e*n*x*log(x) - 3*b*d**2*e*n*x + 3*b*d**2*e*x*log(c) + 3*b*d*e**2*n*x**2*log(x)/2 - 3*b*d*e**2*n*x*
*2/4 + 3*b*d*e**2*x**2*log(c)/2 + b*e**3*n*x**3*log(x)/3 - b*e**3*n*x**3/9 + b*e**3*x**3*log(c)/3

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Giac [A]  time = 1.23394, size = 203, normalized size = 1.66 \begin{align*} \frac{1}{3} \, b n x^{3} e^{3} \log \left (x\right ) + \frac{3}{2} \, b d n x^{2} e^{2} \log \left (x\right ) + 3 \, b d^{2} n x e \log \left (x\right ) + \frac{1}{2} \, b d^{3} n \log \left (x\right )^{2} - \frac{1}{9} \, b n x^{3} e^{3} - \frac{3}{4} \, b d n x^{2} e^{2} - 3 \, b d^{2} n x e + \frac{1}{3} \, b x^{3} e^{3} \log \left (c\right ) + \frac{3}{2} \, b d x^{2} e^{2} \log \left (c\right ) + 3 \, b d^{2} x e \log \left (c\right ) + b d^{3} \log \left (c\right ) \log \left (x\right ) + \frac{1}{3} \, a x^{3} e^{3} + \frac{3}{2} \, a d x^{2} e^{2} + 3 \, a d^{2} x e + a d^{3} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*log(c*x^n))/x,x, algorithm="giac")

[Out]

1/3*b*n*x^3*e^3*log(x) + 3/2*b*d*n*x^2*e^2*log(x) + 3*b*d^2*n*x*e*log(x) + 1/2*b*d^3*n*log(x)^2 - 1/9*b*n*x^3*
e^3 - 3/4*b*d*n*x^2*e^2 - 3*b*d^2*n*x*e + 1/3*b*x^3*e^3*log(c) + 3/2*b*d*x^2*e^2*log(c) + 3*b*d^2*x*e*log(c) +
b*d^3*log(c)*log(x) + 1/3*a*x^3*e^3 + 3/2*a*d*x^2*e^2 + 3*a*d^2*x*e + a*d^3*log(x)