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

Optimal. Leaf size=118 \[ -\frac{3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+3 d e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+e^3 x \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n}{x}-\frac{b d^3 n}{4 x^2}-\frac{3}{2} b d e^2 n \log ^2(x)-b e^3 n x \]

[Out]

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

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Rubi [A]  time = 0.0909902, antiderivative size = 91, normalized size of antiderivative = 0.77, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {43, 2334, 2301} \[ -\frac{1}{2} \left (\frac{6 d^2 e}{x}+\frac{d^3}{x^2}-6 d e^2 \log (x)-2 e^3 x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n}{x}-\frac{b d^3 n}{4 x^2}-\frac{3}{2} b d e^2 n \log ^2(x)-b e^3 n x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0778604, size = 115, normalized size = 0.97 \[ -\frac{3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac{3 d e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+a e^3 x+b e^3 x \log \left (c x^n\right )-\frac{3 b d^2 e n}{x}-\frac{b d^3 n}{4 x^2}-b e^3 n x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.263, size = 586, normalized size = 5. \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^3,x)

[Out]

-1/2*b*(-6*d*e^2*ln(x)*x^2-2*e^3*x^3+6*d^2*e*x+d^3)/x^2*ln(x^n)-1/4*(2*a*d^3+6*I*ln(x)*Pi*b*d*e^2*csgn(I*x^n)*
csgn(I*c*x^n)*csgn(I*c)*x^2-6*I*Pi*b*d^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-6*I*ln(x)*Pi*b*d*e^2*csgn(I*x
^n)*csgn(I*c*x^n)^2*x^2-6*I*ln(x)*Pi*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*x^2+I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c
)+2*I*Pi*b*e^3*x^3*csgn(I*c*x^n)^3+I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2-4*ln(c)*b*e^3*x^3+12*a*d^2*e*x+2*ln(
c)*b*d^3-12*ln(x)*ln(c)*b*d*e^2*x^2+6*b*d*e^2*n*ln(x)^2*x^2+6*I*ln(x)*Pi*b*d*e^2*csgn(I*c*x^n)^3*x^2+12*ln(c)*
b*d^2*e*x+6*I*Pi*b*d^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2+2*I*Pi*b*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+6*
I*Pi*b*d^2*e*x*csgn(I*c*x^n)^2*csgn(I*c)-4*a*e^3*x^3+b*d^3*n-2*I*Pi*b*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2-2*I*
Pi*b*e^3*x^3*csgn(I*c*x^n)^2*csgn(I*c)-I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b*d^3*csgn(I*c*x^n)
^3-6*I*Pi*b*d^2*e*x*csgn(I*c*x^n)^3-12*ln(x)*a*d*e^2*x^2+4*b*e^3*n*x^3+12*b*d^2*e*n*x)/x^2

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Maxima [A]  time = 1.09208, size = 169, normalized size = 1.43 \begin{align*} -b e^{3} n x + b e^{3} x \log \left (c x^{n}\right ) + a e^{3} x + \frac{3 \, b d e^{2} \log \left (c x^{n}\right )^{2}}{2 \, n} + 3 \, a d e^{2} \log \left (x\right ) - \frac{3 \, b d^{2} e n}{x} - \frac{3 \, b d^{2} e \log \left (c x^{n}\right )}{x} - \frac{b d^{3} n}{4 \, x^{2}} - \frac{3 \, a d^{2} e}{x} - \frac{b d^{3} \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac{a d^{3}}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.01498, size = 338, normalized size = 2.86 \begin{align*} \frac{6 \, b d e^{2} n x^{2} \log \left (x\right )^{2} - b d^{3} n - 2 \, a d^{3} - 4 \,{\left (b e^{3} n - a e^{3}\right )} x^{3} - 12 \,{\left (b d^{2} e n + a d^{2} e\right )} x + 2 \,{\left (2 \, b e^{3} x^{3} - 6 \, b d^{2} e x - b d^{3}\right )} \log \left (c\right ) + 2 \,{\left (2 \, b e^{3} n x^{3} + 6 \, b d e^{2} x^{2} \log \left (c\right ) - 6 \, b d^{2} e n x + 6 \, a d e^{2} x^{2} - b d^{3} n\right )} \log \left (x\right )}{4 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.94532, size = 182, normalized size = 1.54 \begin{align*} - \frac{a d^{3}}{2 x^{2}} - \frac{3 a d^{2} e}{x} + 3 a d e^{2} \log{\left (x \right )} + a e^{3} x - \frac{b d^{3} n \log{\left (x \right )}}{2 x^{2}} - \frac{b d^{3} n}{4 x^{2}} - \frac{b d^{3} \log{\left (c \right )}}{2 x^{2}} - \frac{3 b d^{2} e n \log{\left (x \right )}}{x} - \frac{3 b d^{2} e n}{x} - \frac{3 b d^{2} e \log{\left (c \right )}}{x} + \frac{3 b d e^{2} n \log{\left (x \right )}^{2}}{2} + 3 b d e^{2} \log{\left (c \right )} \log{\left (x \right )} + b e^{3} n x \log{\left (x \right )} - b e^{3} n x + b e^{3} x \log{\left (c \right )} \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**3,x)

[Out]

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

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Giac [A]  time = 1.33339, size = 208, normalized size = 1.76 \begin{align*} \frac{6 \, b d n x^{2} e^{2} \log \left (x\right )^{2} + 4 \, b n x^{3} e^{3} \log \left (x\right ) - 12 \, b d^{2} n x e \log \left (x\right ) + 12 \, b d x^{2} e^{2} \log \left (c\right ) \log \left (x\right ) - 4 \, b n x^{3} e^{3} - 12 \, b d^{2} n x e + 4 \, b x^{3} e^{3} \log \left (c\right ) - 12 \, b d^{2} x e \log \left (c\right ) - 2 \, b d^{3} n \log \left (x\right ) + 12 \, a d x^{2} e^{2} \log \left (x\right ) - b d^{3} n + 4 \, a x^{3} e^{3} - 12 \, a d^{2} x e - 2 \, b d^{3} \log \left (c\right ) - 2 \, a d^{3}}{4 \, x^{2}} \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^3,x, algorithm="giac")

[Out]

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