3.61 \(\int (a+b \log (c (d+e x)^n))^4 \, dx\)

Optimal. Leaf size=131 \[ \frac{12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-24 a b^3 n^3 x-\frac{4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}-\frac{24 b^4 n^3 (d+e x) \log \left (c (d+e x)^n\right )}{e}+24 b^4 n^4 x \]

[Out]

-24*a*b^3*n^3*x + 24*b^4*n^4*x - (24*b^4*n^3*(d + e*x)*Log[c*(d + e*x)^n])/e + (12*b^2*n^2*(d + e*x)*(a + b*Lo
g[c*(d + e*x)^n])^2)/e - (4*b*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e + ((d + e*x)*(a + b*Log[c*(d + e*x)^
n])^4)/e

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Rubi [A]  time = 0.0710391, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2389, 2296, 2295} \[ \frac{12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-24 a b^3 n^3 x-\frac{4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}-\frac{24 b^4 n^3 (d+e x) \log \left (c (d+e x)^n\right )}{e}+24 b^4 n^4 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x)^n])^4,x]

[Out]

-24*a*b^3*n^3*x + 24*b^4*n^4*x - (24*b^4*n^3*(d + e*x)*Log[c*(d + e*x)^n])/e + (12*b^2*n^2*(d + e*x)*(a + b*Lo
g[c*(d + e*x)^n])^2)/e - (4*b*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e + ((d + e*x)*(a + b*Log[c*(d + e*x)^
n])^4)/e

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin{align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^4 \, dx,x,d+e x\right )}{e}\\ &=\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}-\frac{(4 b n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e}\\ &=-\frac{4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}+\frac{\left (12 b^2 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}\\ &=\frac{12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}-\frac{\left (24 b^3 n^3\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}\\ &=-24 a b^3 n^3 x+\frac{12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}-\frac{\left (24 b^4 n^3\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}\\ &=-24 a b^3 n^3 x+24 b^4 n^4 x-\frac{24 b^4 n^3 (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}\\ \end{align*}

Mathematica [A]  time = 0.0304475, size = 112, normalized size = 0.85 \[ \frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4-4 b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3-3 b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b n \left (e x (a-b n)+b (d+e x) \log \left (c (d+e x)^n\right )\right )\right )\right )}{e} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])^4,x]

[Out]

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

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Maple [C]  time = 0.452, size = 15871, normalized size = 121.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(e*x+d)^n))^4,x)

[Out]

result too large to display

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Maxima [B]  time = 1.35097, size = 675, normalized size = 5.15 \begin{align*} b^{4} x \log \left ({\left (e x + d\right )}^{n} c\right )^{4} + 4 \, a b^{3} x \log \left ({\left (e x + d\right )}^{n} c\right )^{3} - 4 \, a^{3} b e n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} + 6 \, a^{2} b^{2} x \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 4 \, a^{3} b x \log \left ({\left (e x + d\right )}^{n} c\right ) - 6 \,{\left (2 \, e n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac{{\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n^{2}}{e}\right )} a^{2} b^{2} - 4 \,{\left (3 \, e n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} - e n{\left (\frac{{\left (d \log \left (e x + d\right )^{3} + 3 \, d \log \left (e x + d\right )^{2} - 6 \, e x + 6 \, d \log \left (e x + d\right )\right )} n^{2}}{e^{2}} - \frac{3 \,{\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n \log \left ({\left (e x + d\right )}^{n} c\right )}{e^{2}}\right )}\right )} a b^{3} -{\left (4 \, e n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{3} +{\left (e n{\left (\frac{{\left (d \log \left (e x + d\right )^{4} + 4 \, d \log \left (e x + d\right )^{3} + 12 \, d \log \left (e x + d\right )^{2} - 24 \, e x + 24 \, d \log \left (e x + d\right )\right )} n^{2}}{e^{3}} - \frac{4 \,{\left (d \log \left (e x + d\right )^{3} + 3 \, d \log \left (e x + d\right )^{2} - 6 \, e x + 6 \, d \log \left (e x + d\right )\right )} n \log \left ({\left (e x + d\right )}^{n} c\right )}{e^{3}}\right )} + \frac{6 \,{\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n \log \left ({\left (e x + d\right )}^{n} c\right )^{2}}{e^{2}}\right )} e n\right )} b^{4} + a^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^4,x, algorithm="maxima")

[Out]

b^4*x*log((e*x + d)^n*c)^4 + 4*a*b^3*x*log((e*x + d)^n*c)^3 - 4*a^3*b*e*n*(x/e - d*log(e*x + d)/e^2) + 6*a^2*b
^2*x*log((e*x + d)^n*c)^2 + 4*a^3*b*x*log((e*x + d)^n*c) - 6*(2*e*n*(x/e - d*log(e*x + d)/e^2)*log((e*x + d)^n
*c) + (d*log(e*x + d)^2 - 2*e*x + 2*d*log(e*x + d))*n^2/e)*a^2*b^2 - 4*(3*e*n*(x/e - d*log(e*x + d)/e^2)*log((
e*x + d)^n*c)^2 - e*n*((d*log(e*x + d)^3 + 3*d*log(e*x + d)^2 - 6*e*x + 6*d*log(e*x + d))*n^2/e^2 - 3*(d*log(e
*x + d)^2 - 2*e*x + 2*d*log(e*x + d))*n*log((e*x + d)^n*c)/e^2))*a*b^3 - (4*e*n*(x/e - d*log(e*x + d)/e^2)*log
((e*x + d)^n*c)^3 + (e*n*((d*log(e*x + d)^4 + 4*d*log(e*x + d)^3 + 12*d*log(e*x + d)^2 - 24*e*x + 24*d*log(e*x
+ d))*n^2/e^3 - 4*(d*log(e*x + d)^3 + 3*d*log(e*x + d)^2 - 6*e*x + 6*d*log(e*x + d))*n*log((e*x + d)^n*c)/e^3
) + 6*(d*log(e*x + d)^2 - 2*e*x + 2*d*log(e*x + d))*n*log((e*x + d)^n*c)^2/e^2)*e*n)*b^4 + a^4*x

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Fricas [B]  time = 2.10305, size = 1312, normalized size = 10.02 \begin{align*} \frac{b^{4} e x \log \left (c\right )^{4} +{\left (b^{4} e n^{4} x + b^{4} d n^{4}\right )} \log \left (e x + d\right )^{4} - 4 \,{\left (b^{4} e n - a b^{3} e\right )} x \log \left (c\right )^{3} - 4 \,{\left (b^{4} d n^{4} - a b^{3} d n^{3} +{\left (b^{4} e n^{4} - a b^{3} e n^{3}\right )} x -{\left (b^{4} e n^{3} x + b^{4} d n^{3}\right )} \log \left (c\right )\right )} \log \left (e x + d\right )^{3} + 6 \,{\left (2 \, b^{4} e n^{2} - 2 \, a b^{3} e n + a^{2} b^{2} e\right )} x \log \left (c\right )^{2} + 6 \,{\left (2 \, b^{4} d n^{4} - 2 \, a b^{3} d n^{3} + a^{2} b^{2} d n^{2} +{\left (b^{4} e n^{2} x + b^{4} d n^{2}\right )} \log \left (c\right )^{2} +{\left (2 \, b^{4} e n^{4} - 2 \, a b^{3} e n^{3} + a^{2} b^{2} e n^{2}\right )} x - 2 \,{\left (b^{4} d n^{3} - a b^{3} d n^{2} +{\left (b^{4} e n^{3} - a b^{3} e n^{2}\right )} x\right )} \log \left (c\right )\right )} \log \left (e x + d\right )^{2} - 4 \,{\left (6 \, b^{4} e n^{3} - 6 \, a b^{3} e n^{2} + 3 \, a^{2} b^{2} e n - a^{3} b e\right )} x \log \left (c\right ) +{\left (24 \, b^{4} e n^{4} - 24 \, a b^{3} e n^{3} + 12 \, a^{2} b^{2} e n^{2} - 4 \, a^{3} b e n + a^{4} e\right )} x - 4 \,{\left (6 \, b^{4} d n^{4} - 6 \, a b^{3} d n^{3} + 3 \, a^{2} b^{2} d n^{2} - a^{3} b d n -{\left (b^{4} e n x + b^{4} d n\right )} \log \left (c\right )^{3} + 3 \,{\left (b^{4} d n^{2} - a b^{3} d n +{\left (b^{4} e n^{2} - a b^{3} e n\right )} x\right )} \log \left (c\right )^{2} +{\left (6 \, b^{4} e n^{4} - 6 \, a b^{3} e n^{3} + 3 \, a^{2} b^{2} e n^{2} - a^{3} b e n\right )} x - 3 \,{\left (2 \, b^{4} d n^{3} - 2 \, a b^{3} d n^{2} + a^{2} b^{2} d n +{\left (2 \, b^{4} e n^{3} - 2 \, a b^{3} e n^{2} + a^{2} b^{2} e n\right )} x\right )} \log \left (c\right )\right )} \log \left (e x + d\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^4,x, algorithm="fricas")

[Out]

(b^4*e*x*log(c)^4 + (b^4*e*n^4*x + b^4*d*n^4)*log(e*x + d)^4 - 4*(b^4*e*n - a*b^3*e)*x*log(c)^3 - 4*(b^4*d*n^4
- a*b^3*d*n^3 + (b^4*e*n^4 - a*b^3*e*n^3)*x - (b^4*e*n^3*x + b^4*d*n^3)*log(c))*log(e*x + d)^3 + 6*(2*b^4*e*n
^2 - 2*a*b^3*e*n + a^2*b^2*e)*x*log(c)^2 + 6*(2*b^4*d*n^4 - 2*a*b^3*d*n^3 + a^2*b^2*d*n^2 + (b^4*e*n^2*x + b^4
*d*n^2)*log(c)^2 + (2*b^4*e*n^4 - 2*a*b^3*e*n^3 + a^2*b^2*e*n^2)*x - 2*(b^4*d*n^3 - a*b^3*d*n^2 + (b^4*e*n^3 -
a*b^3*e*n^2)*x)*log(c))*log(e*x + d)^2 - 4*(6*b^4*e*n^3 - 6*a*b^3*e*n^2 + 3*a^2*b^2*e*n - a^3*b*e)*x*log(c) +
(24*b^4*e*n^4 - 24*a*b^3*e*n^3 + 12*a^2*b^2*e*n^2 - 4*a^3*b*e*n + a^4*e)*x - 4*(6*b^4*d*n^4 - 6*a*b^3*d*n^3 +
3*a^2*b^2*d*n^2 - a^3*b*d*n - (b^4*e*n*x + b^4*d*n)*log(c)^3 + 3*(b^4*d*n^2 - a*b^3*d*n + (b^4*e*n^2 - a*b^3*
e*n)*x)*log(c)^2 + (6*b^4*e*n^4 - 6*a*b^3*e*n^3 + 3*a^2*b^2*e*n^2 - a^3*b*e*n)*x - 3*(2*b^4*d*n^3 - 2*a*b^3*d*
n^2 + a^2*b^2*d*n + (2*b^4*e*n^3 - 2*a*b^3*e*n^2 + a^2*b^2*e*n)*x)*log(c))*log(e*x + d))/e

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Sympy [A]  time = 6.32086, size = 1059, normalized size = 8.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(e*x+d)**n))**4,x)

[Out]

Piecewise((a**4*x + 4*a**3*b*d*n*log(d + e*x)/e + 4*a**3*b*n*x*log(d + e*x) - 4*a**3*b*n*x + 4*a**3*b*x*log(c)
+ 6*a**2*b**2*d*n**2*log(d + e*x)**2/e - 12*a**2*b**2*d*n**2*log(d + e*x)/e + 12*a**2*b**2*d*n*log(c)*log(d +
e*x)/e + 6*a**2*b**2*n**2*x*log(d + e*x)**2 - 12*a**2*b**2*n**2*x*log(d + e*x) + 12*a**2*b**2*n**2*x + 12*a**
2*b**2*n*x*log(c)*log(d + e*x) - 12*a**2*b**2*n*x*log(c) + 6*a**2*b**2*x*log(c)**2 + 4*a*b**3*d*n**3*log(d + e
*x)**3/e - 12*a*b**3*d*n**3*log(d + e*x)**2/e + 24*a*b**3*d*n**3*log(d + e*x)/e + 12*a*b**3*d*n**2*log(c)*log(
d + e*x)**2/e - 24*a*b**3*d*n**2*log(c)*log(d + e*x)/e + 12*a*b**3*d*n*log(c)**2*log(d + e*x)/e + 4*a*b**3*n**
3*x*log(d + e*x)**3 - 12*a*b**3*n**3*x*log(d + e*x)**2 + 24*a*b**3*n**3*x*log(d + e*x) - 24*a*b**3*n**3*x + 12
*a*b**3*n**2*x*log(c)*log(d + e*x)**2 - 24*a*b**3*n**2*x*log(c)*log(d + e*x) + 24*a*b**3*n**2*x*log(c) + 12*a*
b**3*n*x*log(c)**2*log(d + e*x) - 12*a*b**3*n*x*log(c)**2 + 4*a*b**3*x*log(c)**3 + b**4*d*n**4*log(d + e*x)**4
/e - 4*b**4*d*n**4*log(d + e*x)**3/e + 12*b**4*d*n**4*log(d + e*x)**2/e - 24*b**4*d*n**4*log(d + e*x)/e + 4*b*
*4*d*n**3*log(c)*log(d + e*x)**3/e - 12*b**4*d*n**3*log(c)*log(d + e*x)**2/e + 24*b**4*d*n**3*log(c)*log(d + e
*x)/e + 6*b**4*d*n**2*log(c)**2*log(d + e*x)**2/e - 12*b**4*d*n**2*log(c)**2*log(d + e*x)/e + 4*b**4*d*n*log(c
)**3*log(d + e*x)/e + b**4*n**4*x*log(d + e*x)**4 - 4*b**4*n**4*x*log(d + e*x)**3 + 12*b**4*n**4*x*log(d + e*x
)**2 - 24*b**4*n**4*x*log(d + e*x) + 24*b**4*n**4*x + 4*b**4*n**3*x*log(c)*log(d + e*x)**3 - 12*b**4*n**3*x*lo
g(c)*log(d + e*x)**2 + 24*b**4*n**3*x*log(c)*log(d + e*x) - 24*b**4*n**3*x*log(c) + 6*b**4*n**2*x*log(c)**2*lo
g(d + e*x)**2 - 12*b**4*n**2*x*log(c)**2*log(d + e*x) + 12*b**4*n**2*x*log(c)**2 + 4*b**4*n*x*log(c)**3*log(d
+ e*x) - 4*b**4*n*x*log(c)**3 + b**4*x*log(c)**4, Ne(e, 0)), (x*(a + b*log(c*d**n))**4, True))

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Giac [B]  time = 1.29681, size = 1050, normalized size = 8.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^4,x, algorithm="giac")

[Out]

(x*e + d)*b^4*n^4*e^(-1)*log(x*e + d)^4 - 4*(x*e + d)*b^4*n^4*e^(-1)*log(x*e + d)^3 + 4*(x*e + d)*b^4*n^3*e^(-
1)*log(x*e + d)^3*log(c) + 12*(x*e + d)*b^4*n^4*e^(-1)*log(x*e + d)^2 + 4*(x*e + d)*a*b^3*n^3*e^(-1)*log(x*e +
d)^3 - 12*(x*e + d)*b^4*n^3*e^(-1)*log(x*e + d)^2*log(c) + 6*(x*e + d)*b^4*n^2*e^(-1)*log(x*e + d)^2*log(c)^2
- 24*(x*e + d)*b^4*n^4*e^(-1)*log(x*e + d) - 12*(x*e + d)*a*b^3*n^3*e^(-1)*log(x*e + d)^2 + 24*(x*e + d)*b^4*
n^3*e^(-1)*log(x*e + d)*log(c) + 12*(x*e + d)*a*b^3*n^2*e^(-1)*log(x*e + d)^2*log(c) - 12*(x*e + d)*b^4*n^2*e^
(-1)*log(x*e + d)*log(c)^2 + 4*(x*e + d)*b^4*n*e^(-1)*log(x*e + d)*log(c)^3 + 24*(x*e + d)*b^4*n^4*e^(-1) + 24
*(x*e + d)*a*b^3*n^3*e^(-1)*log(x*e + d) + 6*(x*e + d)*a^2*b^2*n^2*e^(-1)*log(x*e + d)^2 - 24*(x*e + d)*b^4*n^
3*e^(-1)*log(c) - 24*(x*e + d)*a*b^3*n^2*e^(-1)*log(x*e + d)*log(c) + 12*(x*e + d)*b^4*n^2*e^(-1)*log(c)^2 + 1
2*(x*e + d)*a*b^3*n*e^(-1)*log(x*e + d)*log(c)^2 - 4*(x*e + d)*b^4*n*e^(-1)*log(c)^3 + (x*e + d)*b^4*e^(-1)*lo
g(c)^4 - 24*(x*e + d)*a*b^3*n^3*e^(-1) - 12*(x*e + d)*a^2*b^2*n^2*e^(-1)*log(x*e + d) + 24*(x*e + d)*a*b^3*n^2
*e^(-1)*log(c) + 12*(x*e + d)*a^2*b^2*n*e^(-1)*log(x*e + d)*log(c) - 12*(x*e + d)*a*b^3*n*e^(-1)*log(c)^2 + 4*
(x*e + d)*a*b^3*e^(-1)*log(c)^3 + 12*(x*e + d)*a^2*b^2*n^2*e^(-1) + 4*(x*e + d)*a^3*b*n*e^(-1)*log(x*e + d) -
12*(x*e + d)*a^2*b^2*n*e^(-1)*log(c) + 6*(x*e + d)*a^2*b^2*e^(-1)*log(c)^2 - 4*(x*e + d)*a^3*b*n*e^(-1) + 4*(x
*e + d)*a^3*b*e^(-1)*log(c) + (x*e + d)*a^4*e^(-1)