\(\int (f+g x) (a+b \log (c (d+e x)^n))^4 \, dx\) [60]

Optimal result

Integrand size = 22, antiderivative size = 340 \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx=-\frac {24 a b^3 (e f-d g) n^3 x}{e}+\frac {24 b^4 (e f-d g) n^4 x}{e}+\frac {3 b^4 g n^4 (d+e x)^2}{4 e^2}-\frac {24 b^4 (e f-d g) n^3 (d+e x) \log \left (c (d+e x)^n\right )}{e^2}-\frac {3 b^3 g n^3 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {12 b^2 (e f-d g) n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {3 b^2 g n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {4 b (e f-d g) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}-\frac {b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}+\frac {(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e^2}+\frac {g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{2 e^2} \]

[Out]

-24*a*b^3*(-d*g+e*f)*n^3*x/e+24*b^4*(-d*g+e*f)*n^4*x/e+3/4*b^4*g*n^4*(e*x+d)^2/e^2-24*b^4*(-d*g+e*f)*n^3*(e*x+
d)*ln(c*(e*x+d)^n)/e^2-3/2*b^3*g*n^3*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))/e^2+12*b^2*(-d*g+e*f)*n^2*(e*x+d)*(a+b*ln
(c*(e*x+d)^n))^2/e^2+3/2*b^2*g*n^2*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))^2/e^2-4*b*(-d*g+e*f)*n*(e*x+d)*(a+b*ln(c*(e
*x+d)^n))^3/e^2-b*g*n*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))^3/e^2+(-d*g+e*f)*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^4/e^2+1/2
*g*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))^4/e^2

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx=-\frac {3 b^3 g n^3 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {24 a b^3 n^3 x (e f-d g)}{e}+\frac {12 b^2 n^2 (d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {3 b^2 g n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {4 b n (d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}+\frac {(d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e^2}-\frac {b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}+\frac {g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{2 e^2}-\frac {24 b^4 n^3 (d+e x) (e f-d g) \log \left (c (d+e x)^n\right )}{e^2}+\frac {3 b^4 g n^4 (d+e x)^2}{4 e^2}+\frac {24 b^4 n^4 x (e f-d g)}{e} \]

[In]

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

[Out]

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

Rule 2332

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

Rule 2333

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 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2436

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 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2448

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.76 \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx=\frac {4 (e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4+2 g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^4-16 b (e f-d g) n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3-3 b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b n \left (e (a-b n) x+b (d+e x) \log \left (c (d+e x)^n\right )\right )\right )\right )-b g n \left (4 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3-3 b n \left (2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+b n \left (b e n x (2 d+e x)-2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )\right )\right )}{4 e^2} \]

[In]

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

[Out]

(4*(e*f - d*g)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^4 + 2*g*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^4 - 16*b*(e
*f - d*g)*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]))) - b*g*n*(4*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^3 - 3*b*n*(
2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2 + b*n*(b*e*n*x*(2*d + e*x) - 2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n
])))))/(4*e^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1492\) vs. \(2(332)=664\).

Time = 3.52 (sec) , antiderivative size = 1493, normalized size of antiderivative = 4.39

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1493\)
risch	\(\text {Expression too large to display}\)	\(37938\)

[In]

int((g*x+f)*(a+b*ln(c*(e*x+d)^n))^4,x,method=_RETURNVERBOSE)

[Out]

-1/4*(-16*a^3*b*d*e*f*n-90*b^4*n^4*d^2*g+4*a^4*d*e*f-2*a^4*e^2*g*x^2+8*a^3*b*d^2*g*n-4*a^4*e^2*f*x-36*b^2*n^2*
a^2*d^2*g+84*b^3*n^3*a*d^2*g-3*b^4*n^4*e^2*g*x^2-96*b^4*n^4*e^2*f*x+6*b^3*n^3*a*e^2*g*x^2+90*b^4*n^4*d*e*g*x-6
*b^2*n^2*a^2*e^2*g*x^2+96*b^3*n^3*a*e^2*f*x-48*b^2*n^2*a^2*e^2*f*x+4*b*n*a^3*e^2*g*x^2+16*b*n*a^3*e^2*f*x-174*
ln(e*x+d)*b^4*d^2*g*n^4-96*b^3*n^3*a*d*e*f+48*b^2*n^2*a^2*d*e*f+96*b^4*n^4*d*e*f+192*ln(e*x+d)*b^4*d*e*f*n^4+1
56*ln(e*x+d)*a*b^3*d^2*g*n^3-60*ln(e*x+d)*a^2*b^2*d^2*g*n^2+8*ln(e*x+d)*a^3*b*d^2*g*n-2*x^2*ln(c*(e*x+d)^n)^4*
b^4*e^2*g-4*x*ln(c*(e*x+d)^n)^4*b^4*e^2*f-4*ln(c*(e*x+d)^n)^4*b^4*d*e*f-12*ln(c*(e*x+d)^n)^3*b^4*d^2*g*n+42*ln
(c*(e*x+d)^n)^2*b^4*d^2*g*n^2+84*ln(c*(e*x+d)^n)*b^4*d^2*g*n^3+8*ln(c*(e*x+d)^n)^3*a*b^3*d^2*g+12*ln(c*(e*x+d)
^n)^2*a^2*b^2*d^2*g+2*ln(c*(e*x+d)^n)^4*b^4*d^2*g+4*x^2*ln(c*(e*x+d)^n)^3*b^4*e^2*g*n-6*x^2*ln(c*(e*x+d)^n)^2*
b^4*e^2*g*n^2+6*x^2*ln(c*(e*x+d)^n)*b^4*e^2*g*n^3-8*x^2*ln(c*(e*x+d)^n)^3*a*b^3*e^2*g+16*x*ln(c*(e*x+d)^n)^3*b
^4*e^2*f*n-48*x*ln(c*(e*x+d)^n)^2*b^4*e^2*f*n^2+96*x*ln(c*(e*x+d)^n)*b^4*e^2*f*n^3-12*x^2*ln(c*(e*x+d)^n)^2*a^
2*b^2*e^2*g-16*x*ln(c*(e*x+d)^n)^3*a*b^3*e^2*f+16*ln(c*(e*x+d)^n)^3*b^4*d*e*f*n-84*b^3*n^3*a*d*e*g*x+36*b^2*n^
2*a^2*d*e*g*x-192*ln(e*x+d)*a*b^3*d*e*f*n^3+96*ln(e*x+d)*a^2*b^2*d*e*f*n^2-32*ln(e*x+d)*a^3*b*d*e*f*n-24*x*ln(
c*(e*x+d)^n)^2*a*b^3*d*e*g*n-48*ln(c*(e*x+d)^n)^2*b^4*d*e*f*n^2-96*ln(c*(e*x+d)^n)*b^4*d*e*f*n^3-8*x^2*ln(c*(e
*x+d)^n)*a^3*b*e^2*g-24*x*ln(c*(e*x+d)^n)^2*a^2*b^2*e^2*f-16*ln(c*(e*x+d)^n)^3*a*b^3*d*e*f-36*ln(c*(e*x+d)^n)^
2*a*b^3*d^2*g*n-72*ln(c*(e*x+d)^n)*a*b^3*d^2*g*n^2-16*x*ln(c*(e*x+d)^n)*a^3*b*e^2*f-24*ln(c*(e*x+d)^n)^2*a^2*b
^2*d*e*f+24*ln(c*(e*x+d)^n)*a^2*b^2*d^2*g*n+16*ln(c*(e*x+d)^n)*a^3*b*d*e*f+12*x^2*ln(c*(e*x+d)^n)^2*a*b^3*e^2*
g*n-12*x^2*ln(c*(e*x+d)^n)*a*b^3*e^2*g*n^2-8*x*ln(c*(e*x+d)^n)^3*b^4*d*e*g*n+36*x*ln(c*(e*x+d)^n)^2*b^4*d*e*g*
n^2-84*x*ln(c*(e*x+d)^n)*b^4*d*e*g*n^3+12*x^2*ln(c*(e*x+d)^n)*a^2*b^2*e^2*g*n+48*x*ln(c*(e*x+d)^n)^2*a*b^3*e^2
*f*n-96*x*ln(c*(e*x+d)^n)*a*b^3*e^2*f*n^2+48*x*ln(c*(e*x+d)^n)*a^2*b^2*e^2*f*n+48*ln(c*(e*x+d)^n)^2*a*b^3*d*e*
f*n+96*ln(c*(e*x+d)^n)*a*b^3*d*e*f*n^2-48*ln(c*(e*x+d)^n)*a^2*b^2*d*e*f*n+72*x*ln(c*(e*x+d)^n)*a*b^3*d*e*g*n^2
-24*x*ln(c*(e*x+d)^n)*a^2*b^2*d*e*g*n-8*a^3*b*d*e*g*n*x)/e^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1756 vs. \(2 (332) = 664\).

Time = 0.31 (sec) , antiderivative size = 1756, normalized size of antiderivative = 5.16 \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(2*(b^4*e^2*g*n^4*x^2 + 2*b^4*e^2*f*n^4*x + (2*b^4*d*e*f - b^4*d^2*g)*n^4)*log(e*x + d)^4 + 2*(b^4*e^2*g*x
^2 + 2*b^4*e^2*f*x)*log(c)^4 - 4*((4*b^4*d*e*f - 3*b^4*d^2*g)*n^4 - 2*(2*a*b^3*d*e*f - a*b^3*d^2*g)*n^3 + (b^4
*e^2*g*n^4 - 2*a*b^3*e^2*g*n^3)*x^2 - 2*(2*a*b^3*e^2*f*n^3 - (2*b^4*e^2*f - b^4*d*e*g)*n^4)*x - 2*(b^4*e^2*g*n
^3*x^2 + 2*b^4*e^2*f*n^3*x + (2*b^4*d*e*f - b^4*d^2*g)*n^3)*log(c))*log(e*x + d)^3 - 4*((b^4*e^2*g*n - 2*a*b^3
*e^2*g)*x^2 - 2*(2*a*b^3*e^2*f - (2*b^4*e^2*f - b^4*d*e*g)*n)*x)*log(c)^3 + (3*b^4*e^2*g*n^4 - 6*a*b^3*e^2*g*n
^3 + 6*a^2*b^2*e^2*g*n^2 - 4*a^3*b*e^2*g*n + 2*a^4*e^2*g)*x^2 + 6*((8*b^4*d*e*f - 7*b^4*d^2*g)*n^4 - 2*(4*a*b^
3*d*e*f - 3*a*b^3*d^2*g)*n^3 + 2*(2*a^2*b^2*d*e*f - a^2*b^2*d^2*g)*n^2 + (b^4*e^2*g*n^4 - 2*a*b^3*e^2*g*n^3 +
2*a^2*b^2*e^2*g*n^2)*x^2 + 2*(b^4*e^2*g*n^2*x^2 + 2*b^4*e^2*f*n^2*x + (2*b^4*d*e*f - b^4*d^2*g)*n^2)*log(c)^2
+ 2*(2*a^2*b^2*e^2*f*n^2 + (4*b^4*e^2*f - 3*b^4*d*e*g)*n^4 - 2*(2*a*b^3*e^2*f - a*b^3*d*e*g)*n^3)*x - 2*((4*b^
4*d*e*f - 3*b^4*d^2*g)*n^3 - 2*(2*a*b^3*d*e*f - a*b^3*d^2*g)*n^2 + (b^4*e^2*g*n^3 - 2*a*b^3*e^2*g*n^2)*x^2 - 2
*(2*a*b^3*e^2*f*n^2 - (2*b^4*e^2*f - b^4*d*e*g)*n^3)*x)*log(c))*log(e*x + d)^2 + 6*((b^4*e^2*g*n^2 - 2*a*b^3*e
^2*g*n + 2*a^2*b^2*e^2*g)*x^2 + 2*(2*a^2*b^2*e^2*f + (4*b^4*e^2*f - 3*b^4*d*e*g)*n^2 - 2*(2*a*b^3*e^2*f - a*b^
3*d*e*g)*n)*x)*log(c)^2 + 2*(2*a^4*e^2*f + 3*(16*b^4*e^2*f - 15*b^4*d*e*g)*n^4 - 6*(8*a*b^3*e^2*f - 7*a*b^3*d*
e*g)*n^3 + 6*(4*a^2*b^2*e^2*f - 3*a^2*b^2*d*e*g)*n^2 - 4*(2*a^3*b*e^2*f - a^3*b*d*e*g)*n)*x - 2*(3*(16*b^4*d*e
*f - 15*b^4*d^2*g)*n^4 - 6*(8*a*b^3*d*e*f - 7*a*b^3*d^2*g)*n^3 - 4*(b^4*e^2*g*n*x^2 + 2*b^4*e^2*f*n*x + (2*b^4
*d*e*f - b^4*d^2*g)*n)*log(c)^3 + 6*(4*a^2*b^2*d*e*f - 3*a^2*b^2*d^2*g)*n^2 + (3*b^4*e^2*g*n^4 - 6*a*b^3*e^2*g
*n^3 + 6*a^2*b^2*e^2*g*n^2 - 4*a^3*b*e^2*g*n)*x^2 + 6*((4*b^4*d*e*f - 3*b^4*d^2*g)*n^2 + (b^4*e^2*g*n^2 - 2*a*
b^3*e^2*g*n)*x^2 - 2*(2*a*b^3*d*e*f - a*b^3*d^2*g)*n - 2*(2*a*b^3*e^2*f*n - (2*b^4*e^2*f - b^4*d*e*g)*n^2)*x)*
log(c)^2 - 4*(2*a^3*b*d*e*f - a^3*b*d^2*g)*n - 2*(4*a^3*b*e^2*f*n - 3*(8*b^4*e^2*f - 7*b^4*d*e*g)*n^4 + 6*(4*a
*b^3*e^2*f - 3*a*b^3*d*e*g)*n^3 - 6*(2*a^2*b^2*e^2*f - a^2*b^2*d*e*g)*n^2)*x - 6*((8*b^4*d*e*f - 7*b^4*d^2*g)*
n^3 - 2*(4*a*b^3*d*e*f - 3*a*b^3*d^2*g)*n^2 + (b^4*e^2*g*n^3 - 2*a*b^3*e^2*g*n^2 + 2*a^2*b^2*e^2*g*n)*x^2 + 2*
(2*a^2*b^2*d*e*f - a^2*b^2*d^2*g)*n + 2*(2*a^2*b^2*e^2*f*n + (4*b^4*e^2*f - 3*b^4*d*e*g)*n^3 - 2*(2*a*b^3*e^2*
f - a*b^3*d*e*g)*n^2)*x)*log(c))*log(e*x + d) - 2*((3*b^4*e^2*g*n^3 - 6*a*b^3*e^2*g*n^2 + 6*a^2*b^2*e^2*g*n -
4*a^3*b*e^2*g)*x^2 - 2*(4*a^3*b*e^2*f - 3*(8*b^4*e^2*f - 7*b^4*d*e*g)*n^3 + 6*(4*a*b^3*e^2*f - 3*a*b^3*d*e*g)*
n^2 - 6*(2*a^2*b^2*e^2*f - a^2*b^2*d*e*g)*n)*x)*log(c))/e^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1372 vs. \(2 (332) = 664\).

Time = 2.31 (sec) , antiderivative size = 1372, normalized size of antiderivative = 4.04 \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((a**4*f*x + a**4*g*x**2/2 - 2*a**3*b*d**2*g*log(c*(d + e*x)**n)/e**2 + 4*a**3*b*d*f*log(c*(d + e*x)*
*n)/e + 2*a**3*b*d*g*n*x/e - 4*a**3*b*f*n*x + 4*a**3*b*f*x*log(c*(d + e*x)**n) - a**3*b*g*n*x**2 + 2*a**3*b*g*
x**2*log(c*(d + e*x)**n) + 9*a**2*b**2*d**2*g*n*log(c*(d + e*x)**n)/e**2 - 3*a**2*b**2*d**2*g*log(c*(d + e*x)*
*n)**2/e**2 - 12*a**2*b**2*d*f*n*log(c*(d + e*x)**n)/e + 6*a**2*b**2*d*f*log(c*(d + e*x)**n)**2/e - 9*a**2*b**
2*d*g*n**2*x/e + 6*a**2*b**2*d*g*n*x*log(c*(d + e*x)**n)/e + 12*a**2*b**2*f*n**2*x - 12*a**2*b**2*f*n*x*log(c*
(d + e*x)**n) + 6*a**2*b**2*f*x*log(c*(d + e*x)**n)**2 + 3*a**2*b**2*g*n**2*x**2/2 - 3*a**2*b**2*g*n*x**2*log(
c*(d + e*x)**n) + 3*a**2*b**2*g*x**2*log(c*(d + e*x)**n)**2 - 21*a*b**3*d**2*g*n**2*log(c*(d + e*x)**n)/e**2 +
9*a*b**3*d**2*g*n*log(c*(d + e*x)**n)**2/e**2 - 2*a*b**3*d**2*g*log(c*(d + e*x)**n)**3/e**2 + 24*a*b**3*d*f*n
**2*log(c*(d + e*x)**n)/e - 12*a*b**3*d*f*n*log(c*(d + e*x)**n)**2/e + 4*a*b**3*d*f*log(c*(d + e*x)**n)**3/e +
21*a*b**3*d*g*n**3*x/e - 18*a*b**3*d*g*n**2*x*log(c*(d + e*x)**n)/e + 6*a*b**3*d*g*n*x*log(c*(d + e*x)**n)**2
/e - 24*a*b**3*f*n**3*x + 24*a*b**3*f*n**2*x*log(c*(d + e*x)**n) - 12*a*b**3*f*n*x*log(c*(d + e*x)**n)**2 + 4*
a*b**3*f*x*log(c*(d + e*x)**n)**3 - 3*a*b**3*g*n**3*x**2/2 + 3*a*b**3*g*n**2*x**2*log(c*(d + e*x)**n) - 3*a*b*
*3*g*n*x**2*log(c*(d + e*x)**n)**2 + 2*a*b**3*g*x**2*log(c*(d + e*x)**n)**3 + 45*b**4*d**2*g*n**3*log(c*(d + e
*x)**n)/(2*e**2) - 21*b**4*d**2*g*n**2*log(c*(d + e*x)**n)**2/(2*e**2) + 3*b**4*d**2*g*n*log(c*(d + e*x)**n)**
3/e**2 - b**4*d**2*g*log(c*(d + e*x)**n)**4/(2*e**2) - 24*b**4*d*f*n**3*log(c*(d + e*x)**n)/e + 12*b**4*d*f*n*
*2*log(c*(d + e*x)**n)**2/e - 4*b**4*d*f*n*log(c*(d + e*x)**n)**3/e + b**4*d*f*log(c*(d + e*x)**n)**4/e - 45*b
**4*d*g*n**4*x/(2*e) + 21*b**4*d*g*n**3*x*log(c*(d + e*x)**n)/e - 9*b**4*d*g*n**2*x*log(c*(d + e*x)**n)**2/e +
2*b**4*d*g*n*x*log(c*(d + e*x)**n)**3/e + 24*b**4*f*n**4*x - 24*b**4*f*n**3*x*log(c*(d + e*x)**n) + 12*b**4*f
*n**2*x*log(c*(d + e*x)**n)**2 - 4*b**4*f*n*x*log(c*(d + e*x)**n)**3 + b**4*f*x*log(c*(d + e*x)**n)**4 + 3*b**
4*g*n**4*x**2/4 - 3*b**4*g*n**3*x**2*log(c*(d + e*x)**n)/2 + 3*b**4*g*n**2*x**2*log(c*(d + e*x)**n)**2/2 - b**
4*g*n*x**2*log(c*(d + e*x)**n)**3 + b**4*g*x**2*log(c*(d + e*x)**n)**4/2, Ne(e, 0)), ((a + b*log(c*d**n))**4*(
f*x + g*x**2/2), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1163 vs. \(2 (332) = 664\).

Time = 0.24 (sec) , antiderivative size = 1163, normalized size of antiderivative = 3.42 \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*b^4*g*x^2*log((e*x + d)^n*c)^4 + 2*a*b^3*g*x^2*log((e*x + d)^n*c)^3 + b^4*f*x*log((e*x + d)^n*c)^4 + 3*a^2
*b^2*g*x^2*log((e*x + d)^n*c)^2 + 4*a*b^3*f*x*log((e*x + d)^n*c)^3 - 4*a^3*b*e*f*n*(x/e - d*log(e*x + d)/e^2)
- a^3*b*e*g*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2) + 2*a^3*b*g*x^2*log((e*x + d)^n*c) + 6*a^2*b^2*f*
x*log((e*x + d)^n*c)^2 + 1/2*a^4*g*x^2 + 4*a^3*b*f*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*f - 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*f - (4*e*n*(x/e - d*lo
g(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*f
- 3/2*(2*e*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2)*log((e*x + d)^n*c) - (e^2*x^2 + 2*d^2*log(e*x + d)
^2 - 6*d*e*x + 6*d^2*log(e*x + d))*n^2/e^2)*a^2*b^2*g - 1/2*(6*e*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e
^2)*log((e*x + d)^n*c)^2 + e*n*((4*d^2*log(e*x + d)^3 + 3*e^2*x^2 + 18*d^2*log(e*x + d)^2 - 42*d*e*x + 42*d^2*
log(e*x + d))*n^2/e^3 - 6*(e^2*x^2 + 2*d^2*log(e*x + d)^2 - 6*d*e*x + 6*d^2*log(e*x + d))*n*log((e*x + d)^n*c)
/e^3))*a*b^3*g - 1/4*(4*e*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2)*log((e*x + d)^n*c)^3 - (e*n*((2*d^2
*log(e*x + d)^4 + 12*d^2*log(e*x + d)^3 + 3*e^2*x^2 + 42*d^2*log(e*x + d)^2 - 90*d*e*x + 90*d^2*log(e*x + d))*
n^2/e^4 - 2*(4*d^2*log(e*x + d)^3 + 3*e^2*x^2 + 18*d^2*log(e*x + d)^2 - 42*d*e*x + 42*d^2*log(e*x + d))*n*log(
(e*x + d)^n*c)/e^4) + 6*(e^2*x^2 + 2*d^2*log(e*x + d)^2 - 6*d*e*x + 6*d^2*log(e*x + d))*n*log((e*x + d)^n*c)^2
/e^3)*e*n)*b^4*g + a^4*f*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2488 vs. \(2 (332) = 664\).

Time = 0.35 (sec) , antiderivative size = 2488, normalized size of antiderivative = 7.32 \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.82 (sec) , antiderivative size = 823, normalized size of antiderivative = 2.42 \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx=x\,\left (\frac {2\,a^4\,d\,g+2\,a^4\,e\,f-42\,b^4\,d\,g\,n^4+48\,b^4\,e\,f\,n^4+36\,a\,b^3\,d\,g\,n^3-48\,a\,b^3\,e\,f\,n^3-12\,a^2\,b^2\,d\,g\,n^2+24\,a^2\,b^2\,e\,f\,n^2-8\,a^3\,b\,e\,f\,n}{2\,e}-\frac {d\,g\,\left (2\,a^4-4\,a^3\,b\,n+6\,a^2\,b^2\,n^2-6\,a\,b^3\,n^3+3\,b^4\,n^4\right )}{2\,e}\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^4\,\left (\frac {b^4\,g\,x^2}{2}-\frac {d\,\left (b^4\,d\,g-2\,b^4\,e\,f\right )}{2\,e^2}+b^4\,f\,x\right )+\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (\frac {b\,g\,\left (4\,a^3-6\,a^2\,b\,n+6\,a\,b^2\,n^2-3\,b^3\,n^3\right )\,x^2}{2}+\left (\frac {4\,a^3\,b\,d\,g+4\,a^3\,b\,e\,f+18\,b^4\,d\,g\,n^3-24\,b^4\,e\,f\,n^3-12\,a^2\,b^2\,e\,f\,n-12\,a\,b^3\,d\,g\,n^2+24\,a\,b^3\,e\,f\,n^2}{e}-\frac {b\,d\,g\,\left (4\,a^3-6\,a^2\,b\,n+6\,a\,b^2\,n^2-3\,b^3\,n^3\right )}{e}\right )\,x\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^3\,\left (x\,\left (\frac {4\,b^3\,\left (a\,d\,g+a\,e\,f-b\,e\,f\,n\right )}{e}-\frac {2\,b^3\,d\,g\,\left (2\,a-b\,n\right )}{e}\right )-\frac {d\,\left (2\,a\,b^3\,d\,g-4\,a\,b^3\,e\,f-3\,b^4\,d\,g\,n+4\,b^4\,e\,f\,n\right )}{e^2}+b^3\,g\,x^2\,\left (2\,a-b\,n\right )\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (x\,\left (\frac {6\,a^2\,b^2\,d\,g+6\,a^2\,b^2\,e\,f-6\,b^4\,d\,g\,n^2+12\,b^4\,e\,f\,n^2-12\,a\,b^3\,e\,f\,n}{e}-\frac {3\,b^2\,d\,g\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{e}\right )-\frac {3\,d\,\left (2\,a^2\,b^2\,d\,g-4\,a^2\,b^2\,e\,f+7\,b^4\,d\,g\,n^2-8\,b^4\,e\,f\,n^2-6\,a\,b^3\,d\,g\,n+8\,a\,b^3\,e\,f\,n\right )}{2\,e^2}+\frac {3\,b^2\,g\,x^2\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-4\,g\,a^3\,b\,d^2\,n+8\,e\,f\,a^3\,b\,d\,n+18\,g\,a^2\,b^2\,d^2\,n^2-24\,e\,f\,a^2\,b^2\,d\,n^2-42\,g\,a\,b^3\,d^2\,n^3+48\,e\,f\,a\,b^3\,d\,n^3+45\,g\,b^4\,d^2\,n^4-48\,e\,f\,b^4\,d\,n^4\right )}{2\,e^2}+\frac {g\,x^2\,\left (2\,a^4-4\,a^3\,b\,n+6\,a^2\,b^2\,n^2-6\,a\,b^3\,n^3+3\,b^4\,n^4\right )}{4} \]

[In]

int((f + g*x)*(a + b*log(c*(d + e*x)^n))^4,x)

[Out]

x*((2*a^4*d*g + 2*a^4*e*f - 42*b^4*d*g*n^4 + 48*b^4*e*f*n^4 + 36*a*b^3*d*g*n^3 - 48*a*b^3*e*f*n^3 - 12*a^2*b^2
*d*g*n^2 + 24*a^2*b^2*e*f*n^2 - 8*a^3*b*e*f*n)/(2*e) - (d*g*(2*a^4 + 3*b^4*n^4 - 6*a*b^3*n^3 + 6*a^2*b^2*n^2 -
4*a^3*b*n))/(2*e)) + log(c*(d + e*x)^n)^4*((b^4*g*x^2)/2 - (d*(b^4*d*g - 2*b^4*e*f))/(2*e^2) + b^4*f*x) + log
(c*(d + e*x)^n)*(x*((4*a^3*b*d*g + 4*a^3*b*e*f + 18*b^4*d*g*n^3 - 24*b^4*e*f*n^3 - 12*a^2*b^2*e*f*n - 12*a*b^3
*d*g*n^2 + 24*a*b^3*e*f*n^2)/e - (b*d*g*(4*a^3 - 3*b^3*n^3 + 6*a*b^2*n^2 - 6*a^2*b*n))/e) + (b*g*x^2*(4*a^3 -
3*b^3*n^3 + 6*a*b^2*n^2 - 6*a^2*b*n))/2) + log(c*(d + e*x)^n)^3*(x*((4*b^3*(a*d*g + a*e*f - b*e*f*n))/e - (2*b
^3*d*g*(2*a - b*n))/e) - (d*(2*a*b^3*d*g - 4*a*b^3*e*f - 3*b^4*d*g*n + 4*b^4*e*f*n))/e^2 + b^3*g*x^2*(2*a - b*
n)) + log(c*(d + e*x)^n)^2*(x*((6*a^2*b^2*d*g + 6*a^2*b^2*e*f - 6*b^4*d*g*n^2 + 12*b^4*e*f*n^2 - 12*a*b^3*e*f*
n)/e - (3*b^2*d*g*(2*a^2 + b^2*n^2 - 2*a*b*n))/e) - (3*d*(2*a^2*b^2*d*g - 4*a^2*b^2*e*f + 7*b^4*d*g*n^2 - 8*b^
4*e*f*n^2 - 6*a*b^3*d*g*n + 8*a*b^3*e*f*n))/(2*e^2) + (3*b^2*g*x^2*(2*a^2 + b^2*n^2 - 2*a*b*n))/2) + (log(d +
e*x)*(45*b^4*d^2*g*n^4 - 4*a^3*b*d^2*g*n - 48*b^4*d*e*f*n^4 - 42*a*b^3*d^2*g*n^3 + 18*a^2*b^2*d^2*g*n^2 + 8*a^
3*b*d*e*f*n + 48*a*b^3*d*e*f*n^3 - 24*a^2*b^2*d*e*f*n^2))/(2*e^2) + (g*x^2*(2*a^4 + 3*b^4*n^4 - 6*a*b^3*n^3 +
6*a^2*b^2*n^2 - 4*a^3*b*n))/4