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

Optimal. Leaf size=340 \[ \frac{12 b^2 n^2 (d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}-\frac{3 b^3 g n^3 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac{3 b^2 g n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{24 a b^3 n^3 x (e f-d g)}{e}-\frac{4 b n (d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}+\frac{(d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e^2}-\frac{b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{2 e^2}-\frac{24 b^4 n^3 (d+e x) (e f-d g) \log \left (c (d+e x)^n\right )}{e^2}+\frac{3 b^4 g n^4 (d+e x)^2}{4 e^2}+\frac{24 b^4 n^4 x (e f-d g)}{e} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.280602, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \frac{12 b^2 n^2 (d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}-\frac{3 b^3 g n^3 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac{3 b^2 g n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{24 a b^3 n^3 x (e f-d g)}{e}-\frac{4 b n (d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}+\frac{(d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e^2}-\frac{b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{2 e^2}-\frac{24 b^4 n^3 (d+e x) (e f-d g) \log \left (c (d+e x)^n\right )}{e^2}+\frac{3 b^4 g n^4 (d+e x)^2}{4 e^2}+\frac{24 b^4 n^4 x (e f-d g)}{e} \]

Antiderivative was successfully verified.

[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 2401

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]

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]

Rule 2390

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 2305

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 2304

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
]

Rubi steps

\begin{align*} \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx &=\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 \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 0.217932, size = 258, normalized size = 0.76 \[ \frac{4 (d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^4-16 b n (e f-d g) \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3-3 b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b n \left (e x (a-b n)+b (d+e x) \log \left (c (d+e x)^n\right )\right )\right )\right )+2 g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^4-b g n \left (4 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3-3 b n \left (2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+b n \left (b e n x (2 d+e x)-2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )\right )\right )}{4 e^2} \]

Antiderivative was successfully verified.

[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 [C]  time = 2.669, size = 37938, normalized size = 111.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [B]  time = 1.36211, size = 1570, normalized size = 4.62 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 2.55082, size = 3646, normalized size = 10.72 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [A]  time = 19.2123, size = 2885, normalized size = 8.49 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Giac [B]  time = 1.31625, size = 3440, normalized size = 10.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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