3.44 \(\int (f+g x)^3 (a+b \log (c (d+e x)^n))^2 \, dx\)

Optimal. Leaf size=365 \[ -\frac{2 b g^2 n (d+e x)^3 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^4}-\frac{b n (e f-d g)^4 \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^4 g}-\frac{2 b n (d+e x) (e f-d g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^4}-\frac{3 b g n (d+e x)^2 (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^4}-\frac{b g^3 n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{8 e^4}+\frac{(f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac{2 b^2 g^2 n^2 (d+e x)^3 (e f-d g)}{9 e^4}+\frac{2 b^2 n^2 x (e f-d g)^3}{e^3}+\frac{3 b^2 g n^2 (d+e x)^2 (e f-d g)^2}{4 e^4}+\frac{b^2 n^2 (e f-d g)^4 \log ^2(d+e x)}{4 e^4 g}+\frac{b^2 g^3 n^2 (d+e x)^4}{32 e^4} \]

[Out]

(2*b^2*(e*f - d*g)^3*n^2*x)/e^3 + (3*b^2*g*(e*f - d*g)^2*n^2*(d + e*x)^2)/(4*e^4) + (2*b^2*g^2*(e*f - d*g)*n^2
*(d + e*x)^3)/(9*e^4) + (b^2*g^3*n^2*(d + e*x)^4)/(32*e^4) + (b^2*(e*f - d*g)^4*n^2*Log[d + e*x]^2)/(4*e^4*g)
- (2*b*(e*f - d*g)^3*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/e^4 - (3*b*g*(e*f - d*g)^2*n*(d + e*x)^2*(a + b*L
og[c*(d + e*x)^n]))/(2*e^4) - (2*b*g^2*(e*f - d*g)*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(3*e^4) - (b*g^3*
n*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n]))/(8*e^4) - (b*(e*f - d*g)^4*n*Log[d + e*x]*(a + b*Log[c*(d + e*x)^n])
)/(2*e^4*g) + ((f + g*x)^4*(a + b*Log[c*(d + e*x)^n])^2)/(4*g)

________________________________________________________________________________________

Rubi [A]  time = 0.535377, antiderivative size = 301, normalized size of antiderivative = 0.82, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2398, 2411, 43, 2334, 12, 2301} \[ -\frac{b n \left (\frac{36 g^2 (d+e x)^2 (e f-d g)^2}{e^4}+\frac{16 g^3 (d+e x)^3 (e f-d g)}{e^4}+\frac{48 g (d+e x) (e f-d g)^3}{e^4}+\frac{12 (e f-d g)^4 \log (d+e x)}{e^4}+\frac{3 g^4 (d+e x)^4}{e^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{24 g}+\frac{(f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac{2 b^2 g^2 n^2 (d+e x)^3 (e f-d g)}{9 e^4}+\frac{2 b^2 n^2 x (e f-d g)^3}{e^3}+\frac{3 b^2 g n^2 (d+e x)^2 (e f-d g)^2}{4 e^4}+\frac{b^2 n^2 (e f-d g)^4 \log ^2(d+e x)}{4 e^4 g}+\frac{b^2 g^3 n^2 (d+e x)^4}{32 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^3*(a + b*Log[c*(d + e*x)^n])^2,x]

[Out]

(2*b^2*(e*f - d*g)^3*n^2*x)/e^3 + (3*b^2*g*(e*f - d*g)^2*n^2*(d + e*x)^2)/(4*e^4) + (2*b^2*g^2*(e*f - d*g)*n^2
*(d + e*x)^3)/(9*e^4) + (b^2*g^3*n^2*(d + e*x)^4)/(32*e^4) + (b^2*(e*f - d*g)^4*n^2*Log[d + e*x]^2)/(4*e^4*g)
- (b*n*((48*g*(e*f - d*g)^3*(d + e*x))/e^4 + (36*g^2*(e*f - d*g)^2*(d + e*x)^2)/e^4 + (16*g^3*(e*f - d*g)*(d +
e*x)^3)/e^4 + (3*g^4*(d + e*x)^4)/e^4 + (12*(e*f - d*g)^4*Log[d + e*x])/e^4)*(a + b*Log[c*(d + e*x)^n]))/(24*
g) + ((f + g*x)^4*(a + b*Log[c*(d + e*x)^n])^2)/(4*g)

Rule 2398

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q
+ 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 (f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx &=\frac{(f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}-\frac{(b e n) \int \frac{(f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx}{2 g}\\ &=\frac{(f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}-\frac{(b n) \operatorname{Subst}\left (\int \frac{\left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^4 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )}{2 g}\\ &=-\frac{b n \left (\frac{48 g (e f-d g)^3 (d+e x)}{e^4}+\frac{36 g^2 (e f-d g)^2 (d+e x)^2}{e^4}+\frac{16 g^3 (e f-d g) (d+e x)^3}{e^4}+\frac{3 g^4 (d+e x)^4}{e^4}+\frac{12 (e f-d g)^4 \log (d+e x)}{e^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{24 g}+\frac{(f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{48 g (e f-d g)^3+36 g^2 (e f-d g)^2 x+16 g^3 (e f-d g) x^2+3 g^4 x^3+\frac{12 (e f-d g)^4 \log (x)}{x}}{12 e^4} \, dx,x,d+e x\right )}{2 g}\\ &=-\frac{b n \left (\frac{48 g (e f-d g)^3 (d+e x)}{e^4}+\frac{36 g^2 (e f-d g)^2 (d+e x)^2}{e^4}+\frac{16 g^3 (e f-d g) (d+e x)^3}{e^4}+\frac{3 g^4 (d+e x)^4}{e^4}+\frac{12 (e f-d g)^4 \log (d+e x)}{e^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{24 g}+\frac{(f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \left (48 g (e f-d g)^3+36 g^2 (e f-d g)^2 x+16 g^3 (e f-d g) x^2+3 g^4 x^3+\frac{12 (e f-d g)^4 \log (x)}{x}\right ) \, dx,x,d+e x\right )}{24 e^4 g}\\ &=\frac{2 b^2 (e f-d g)^3 n^2 x}{e^3}+\frac{3 b^2 g (e f-d g)^2 n^2 (d+e x)^2}{4 e^4}+\frac{2 b^2 g^2 (e f-d g) n^2 (d+e x)^3}{9 e^4}+\frac{b^2 g^3 n^2 (d+e x)^4}{32 e^4}-\frac{b n \left (\frac{48 g (e f-d g)^3 (d+e x)}{e^4}+\frac{36 g^2 (e f-d g)^2 (d+e x)^2}{e^4}+\frac{16 g^3 (e f-d g) (d+e x)^3}{e^4}+\frac{3 g^4 (d+e x)^4}{e^4}+\frac{12 (e f-d g)^4 \log (d+e x)}{e^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{24 g}+\frac{(f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac{\left (b^2 (e f-d g)^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,d+e x\right )}{2 e^4 g}\\ &=\frac{2 b^2 (e f-d g)^3 n^2 x}{e^3}+\frac{3 b^2 g (e f-d g)^2 n^2 (d+e x)^2}{4 e^4}+\frac{2 b^2 g^2 (e f-d g) n^2 (d+e x)^3}{9 e^4}+\frac{b^2 g^3 n^2 (d+e x)^4}{32 e^4}+\frac{b^2 (e f-d g)^4 n^2 \log ^2(d+e x)}{4 e^4 g}-\frac{b n \left (\frac{48 g (e f-d g)^3 (d+e x)}{e^4}+\frac{36 g^2 (e f-d g)^2 (d+e x)^2}{e^4}+\frac{16 g^3 (e f-d g) (d+e x)^3}{e^4}+\frac{3 g^4 (d+e x)^4}{e^4}+\frac{12 (e f-d g)^4 \log (d+e x)}{e^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{24 g}+\frac{(f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}\\ \end{align*}

Mathematica [A]  time = 0.250792, size = 360, normalized size = 0.99 \[ \frac{64 b g^2 n (e f-d g) \left (b e n x \left (3 d^2+3 d e x+e^2 x^2\right )-3 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )+9 b g^3 n \left (b e n x \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )-4 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )+288 g^2 (d+e x)^3 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+432 g (d+e x)^2 (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+288 (d+e x) (e f-d g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2-576 b n (e f-d g)^3 \left (e x (a-b n)+b (d+e x) \log \left (c (d+e x)^n\right )\right )+216 b g n (e f-d g)^2 \left (b e n x (2 d+e x)-2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )+72 g^3 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{288 e^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^3*(a + b*Log[c*(d + e*x)^n])^2,x]

[Out]

(288*(e*f - d*g)^3*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2 + 432*g*(e*f - d*g)^2*(d + e*x)^2*(a + b*Log[c*(d +
e*x)^n])^2 + 288*g^2*(e*f - d*g)*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^2 + 72*g^3*(d + e*x)^4*(a + b*Log[c*(d
+ e*x)^n])^2 - 576*b*(e*f - d*g)^3*n*(e*(a - b*n)*x + b*(d + e*x)*Log[c*(d + e*x)^n]) + 216*b*g*(e*f - d*g)^2
*n*(b*e*n*x*(2*d + e*x) - 2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])) + 64*b*g^2*(e*f - d*g)*n*(b*e*n*x*(3*d^2 +
3*d*e*x + e^2*x^2) - 3*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])) + 9*b*g^3*n*(b*e*n*x*(4*d^3 + 6*d^2*e*x + 4*d*
e^2*x^2 + e^3*x^3) - 4*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n])))/(288*e^4)

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Maple [C]  time = 0.888, size = 6770, normalized size = 18.6 \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)^3*(a+b*ln(c*(e*x+d)^n))^2,x)

[Out]

result too large to display

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Maxima [B]  time = 1.30261, size = 1116, normalized size = 3.06 \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)^3*(a+b*log(c*(e*x+d)^n))^2,x, algorithm="maxima")

[Out]

1/4*b^2*g^3*x^4*log((e*x + d)^n*c)^2 + 1/2*a*b*g^3*x^4*log((e*x + d)^n*c) + b^2*f*g^2*x^3*log((e*x + d)^n*c)^2
+ 1/4*a^2*g^3*x^4 + 2*a*b*f*g^2*x^3*log((e*x + d)^n*c) + 3/2*b^2*f^2*g*x^2*log((e*x + d)^n*c)^2 + a^2*f*g^2*x
^3 - 2*a*b*e*f^3*n*(x/e - d*log(e*x + d)/e^2) - 1/24*a*b*e*g^3*n*(12*d^4*log(e*x + d)/e^5 + (3*e^3*x^4 - 4*d*e
^2*x^3 + 6*d^2*e*x^2 - 12*d^3*x)/e^4) + 1/3*a*b*e*f*g^2*n*(6*d^3*log(e*x + d)/e^4 - (2*e^2*x^3 - 3*d*e*x^2 + 6
*d^2*x)/e^3) - 3/2*a*b*e*f^2*g*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2) + 3*a*b*f^2*g*x^2*log((e*x + d
)^n*c) + b^2*f^3*x*log((e*x + d)^n*c)^2 + 3/2*a^2*f^2*g*x^2 + 2*a*b*f^3*x*log((e*x + d)^n*c) - (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)*b^2*f^3 - 3/4*(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)*b^2*f^2*g + 1/18*(6*e*n*(6*d^3*log(e*x + d)/e^4 - (2*e^2*x^3 - 3*d*e*x^2 +
6*d^2*x)/e^3)*log((e*x + d)^n*c) + (4*e^3*x^3 - 15*d*e^2*x^2 - 18*d^3*log(e*x + d)^2 + 66*d^2*e*x - 66*d^3*log
(e*x + d))*n^2/e^3)*b^2*f*g^2 - 1/288*(12*e*n*(12*d^4*log(e*x + d)/e^5 + (3*e^3*x^4 - 4*d*e^2*x^3 + 6*d^2*e*x^
2 - 12*d^3*x)/e^4)*log((e*x + d)^n*c) - (9*e^4*x^4 - 28*d*e^3*x^3 + 78*d^2*e^2*x^2 + 72*d^4*log(e*x + d)^2 - 3
00*d^3*e*x + 300*d^4*log(e*x + d))*n^2/e^4)*b^2*g^3 + a^2*f^3*x

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Fricas [B]  time = 2.52976, size = 2429, normalized size = 6.65 \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)^3*(a+b*log(c*(e*x+d)^n))^2,x, algorithm="fricas")

[Out]

1/288*(9*(b^2*e^4*g^3*n^2 - 4*a*b*e^4*g^3*n + 8*a^2*e^4*g^3)*x^4 + 4*(72*a^2*e^4*f*g^2 + (16*b^2*e^4*f*g^2 - 7
*b^2*d*e^3*g^3)*n^2 - 12*(4*a*b*e^4*f*g^2 - a*b*d*e^3*g^3)*n)*x^3 + 6*(72*a^2*e^4*f^2*g + (36*b^2*e^4*f^2*g -
40*b^2*d*e^3*f*g^2 + 13*b^2*d^2*e^2*g^3)*n^2 - 12*(6*a*b*e^4*f^2*g - 4*a*b*d*e^3*f*g^2 + a*b*d^2*e^2*g^3)*n)*x
^2 + 72*(b^2*e^4*g^3*n^2*x^4 + 4*b^2*e^4*f*g^2*n^2*x^3 + 6*b^2*e^4*f^2*g*n^2*x^2 + 4*b^2*e^4*f^3*n^2*x + (4*b^
2*d*e^3*f^3 - 6*b^2*d^2*e^2*f^2*g + 4*b^2*d^3*e*f*g^2 - b^2*d^4*g^3)*n^2)*log(e*x + d)^2 + 72*(b^2*e^4*g^3*x^4
+ 4*b^2*e^4*f*g^2*x^3 + 6*b^2*e^4*f^2*g*x^2 + 4*b^2*e^4*f^3*x)*log(c)^2 + 12*(24*a^2*e^4*f^3 + (48*b^2*e^4*f^
3 - 108*b^2*d*e^3*f^2*g + 88*b^2*d^2*e^2*f*g^2 - 25*b^2*d^3*e*g^3)*n^2 - 12*(4*a*b*e^4*f^3 - 6*a*b*d*e^3*f^2*g
+ 4*a*b*d^2*e^2*f*g^2 - a*b*d^3*e*g^3)*n)*x - 12*(3*(b^2*e^4*g^3*n^2 - 4*a*b*e^4*g^3*n)*x^4 - 4*(12*a*b*e^4*f
*g^2*n - (4*b^2*e^4*f*g^2 - b^2*d*e^3*g^3)*n^2)*x^3 + (48*b^2*d*e^3*f^3 - 108*b^2*d^2*e^2*f^2*g + 88*b^2*d^3*e
*f*g^2 - 25*b^2*d^4*g^3)*n^2 - 6*(12*a*b*e^4*f^2*g*n - (6*b^2*e^4*f^2*g - 4*b^2*d*e^3*f*g^2 + b^2*d^2*e^2*g^3)
*n^2)*x^2 - 12*(4*a*b*d*e^3*f^3 - 6*a*b*d^2*e^2*f^2*g + 4*a*b*d^3*e*f*g^2 - a*b*d^4*g^3)*n - 12*(4*a*b*e^4*f^3
*n - (4*b^2*e^4*f^3 - 6*b^2*d*e^3*f^2*g + 4*b^2*d^2*e^2*f*g^2 - b^2*d^3*e*g^3)*n^2)*x - 12*(b^2*e^4*g^3*n*x^4
+ 4*b^2*e^4*f*g^2*n*x^3 + 6*b^2*e^4*f^2*g*n*x^2 + 4*b^2*e^4*f^3*n*x + (4*b^2*d*e^3*f^3 - 6*b^2*d^2*e^2*f^2*g +
4*b^2*d^3*e*f*g^2 - b^2*d^4*g^3)*n)*log(c))*log(e*x + d) - 12*(3*(b^2*e^4*g^3*n - 4*a*b*e^4*g^3)*x^4 - 4*(12*
a*b*e^4*f*g^2 - (4*b^2*e^4*f*g^2 - b^2*d*e^3*g^3)*n)*x^3 - 6*(12*a*b*e^4*f^2*g - (6*b^2*e^4*f^2*g - 4*b^2*d*e^
3*f*g^2 + b^2*d^2*e^2*g^3)*n)*x^2 - 12*(4*a*b*e^4*f^3 - (4*b^2*e^4*f^3 - 6*b^2*d*e^3*f^2*g + 4*b^2*d^2*e^2*f*g
^2 - b^2*d^3*e*g^3)*n)*x)*log(c))/e^4

________________________________________________________________________________________

Sympy [A]  time = 20.5847, size = 1744, normalized size = 4.78 \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)**3*(a+b*ln(c*(e*x+d)**n))**2,x)

[Out]

Piecewise((a**2*f**3*x + 3*a**2*f**2*g*x**2/2 + a**2*f*g**2*x**3 + a**2*g**3*x**4/4 - a*b*d**4*g**3*n*log(d +
e*x)/(2*e**4) + 2*a*b*d**3*f*g**2*n*log(d + e*x)/e**3 + a*b*d**3*g**3*n*x/(2*e**3) - 3*a*b*d**2*f**2*g*n*log(d
+ e*x)/e**2 - 2*a*b*d**2*f*g**2*n*x/e**2 - a*b*d**2*g**3*n*x**2/(4*e**2) + 2*a*b*d*f**3*n*log(d + e*x)/e + 3*
a*b*d*f**2*g*n*x/e + a*b*d*f*g**2*n*x**2/e + a*b*d*g**3*n*x**3/(6*e) + 2*a*b*f**3*n*x*log(d + e*x) - 2*a*b*f**
3*n*x + 2*a*b*f**3*x*log(c) + 3*a*b*f**2*g*n*x**2*log(d + e*x) - 3*a*b*f**2*g*n*x**2/2 + 3*a*b*f**2*g*x**2*log
(c) + 2*a*b*f*g**2*n*x**3*log(d + e*x) - 2*a*b*f*g**2*n*x**3/3 + 2*a*b*f*g**2*x**3*log(c) + a*b*g**3*n*x**4*lo
g(d + e*x)/2 - a*b*g**3*n*x**4/8 + a*b*g**3*x**4*log(c)/2 - b**2*d**4*g**3*n**2*log(d + e*x)**2/(4*e**4) + 25*
b**2*d**4*g**3*n**2*log(d + e*x)/(24*e**4) - b**2*d**4*g**3*n*log(c)*log(d + e*x)/(2*e**4) + b**2*d**3*f*g**2*
n**2*log(d + e*x)**2/e**3 - 11*b**2*d**3*f*g**2*n**2*log(d + e*x)/(3*e**3) + 2*b**2*d**3*f*g**2*n*log(c)*log(d
+ e*x)/e**3 + b**2*d**3*g**3*n**2*x*log(d + e*x)/(2*e**3) - 25*b**2*d**3*g**3*n**2*x/(24*e**3) + b**2*d**3*g*
*3*n*x*log(c)/(2*e**3) - 3*b**2*d**2*f**2*g*n**2*log(d + e*x)**2/(2*e**2) + 9*b**2*d**2*f**2*g*n**2*log(d + e*
x)/(2*e**2) - 3*b**2*d**2*f**2*g*n*log(c)*log(d + e*x)/e**2 - 2*b**2*d**2*f*g**2*n**2*x*log(d + e*x)/e**2 + 11
*b**2*d**2*f*g**2*n**2*x/(3*e**2) - 2*b**2*d**2*f*g**2*n*x*log(c)/e**2 - b**2*d**2*g**3*n**2*x**2*log(d + e*x)
/(4*e**2) + 13*b**2*d**2*g**3*n**2*x**2/(48*e**2) - b**2*d**2*g**3*n*x**2*log(c)/(4*e**2) + b**2*d*f**3*n**2*l
og(d + e*x)**2/e - 2*b**2*d*f**3*n**2*log(d + e*x)/e + 2*b**2*d*f**3*n*log(c)*log(d + e*x)/e + 3*b**2*d*f**2*g
*n**2*x*log(d + e*x)/e - 9*b**2*d*f**2*g*n**2*x/(2*e) + 3*b**2*d*f**2*g*n*x*log(c)/e + b**2*d*f*g**2*n**2*x**2
*log(d + e*x)/e - 5*b**2*d*f*g**2*n**2*x**2/(6*e) + b**2*d*f*g**2*n*x**2*log(c)/e + b**2*d*g**3*n**2*x**3*log(
d + e*x)/(6*e) - 7*b**2*d*g**3*n**2*x**3/(72*e) + b**2*d*g**3*n*x**3*log(c)/(6*e) + b**2*f**3*n**2*x*log(d + e
*x)**2 - 2*b**2*f**3*n**2*x*log(d + e*x) + 2*b**2*f**3*n**2*x + 2*b**2*f**3*n*x*log(c)*log(d + e*x) - 2*b**2*f
**3*n*x*log(c) + b**2*f**3*x*log(c)**2 + 3*b**2*f**2*g*n**2*x**2*log(d + e*x)**2/2 - 3*b**2*f**2*g*n**2*x**2*l
og(d + e*x)/2 + 3*b**2*f**2*g*n**2*x**2/4 + 3*b**2*f**2*g*n*x**2*log(c)*log(d + e*x) - 3*b**2*f**2*g*n*x**2*lo
g(c)/2 + 3*b**2*f**2*g*x**2*log(c)**2/2 + b**2*f*g**2*n**2*x**3*log(d + e*x)**2 - 2*b**2*f*g**2*n**2*x**3*log(
d + e*x)/3 + 2*b**2*f*g**2*n**2*x**3/9 + 2*b**2*f*g**2*n*x**3*log(c)*log(d + e*x) - 2*b**2*f*g**2*n*x**3*log(c
)/3 + b**2*f*g**2*x**3*log(c)**2 + b**2*g**3*n**2*x**4*log(d + e*x)**2/4 - b**2*g**3*n**2*x**4*log(d + e*x)/8
+ b**2*g**3*n**2*x**4/32 + b**2*g**3*n*x**4*log(c)*log(d + e*x)/2 - b**2*g**3*n*x**4*log(c)/8 + b**2*g**3*x**4
*log(c)**2/4, Ne(e, 0)), ((a + b*log(c*d**n))**2*(f**3*x + 3*f**2*g*x**2/2 + f*g**2*x**3 + g**3*x**4/4), True)
)

________________________________________________________________________________________

Giac [B]  time = 1.33397, size = 3220, normalized size = 8.82 \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)^3*(a+b*log(c*(e*x+d)^n))^2,x, algorithm="giac")

[Out]

1/4*(x*e + d)^4*b^2*g^3*n^2*e^(-4)*log(x*e + d)^2 - (x*e + d)^3*b^2*d*g^3*n^2*e^(-4)*log(x*e + d)^2 + 3/2*(x*e
+ d)^2*b^2*d^2*g^3*n^2*e^(-4)*log(x*e + d)^2 - (x*e + d)*b^2*d^3*g^3*n^2*e^(-4)*log(x*e + d)^2 - 1/8*(x*e + d
)^4*b^2*g^3*n^2*e^(-4)*log(x*e + d) + 2/3*(x*e + d)^3*b^2*d*g^3*n^2*e^(-4)*log(x*e + d) - 3/2*(x*e + d)^2*b^2*
d^2*g^3*n^2*e^(-4)*log(x*e + d) + 2*(x*e + d)*b^2*d^3*g^3*n^2*e^(-4)*log(x*e + d) + (x*e + d)^3*b^2*f*g^2*n^2*
e^(-3)*log(x*e + d)^2 - 3*(x*e + d)^2*b^2*d*f*g^2*n^2*e^(-3)*log(x*e + d)^2 + 3*(x*e + d)*b^2*d^2*f*g^2*n^2*e^
(-3)*log(x*e + d)^2 + 1/2*(x*e + d)^4*b^2*g^3*n*e^(-4)*log(x*e + d)*log(c) - 2*(x*e + d)^3*b^2*d*g^3*n*e^(-4)*
log(x*e + d)*log(c) + 3*(x*e + d)^2*b^2*d^2*g^3*n*e^(-4)*log(x*e + d)*log(c) - 2*(x*e + d)*b^2*d^3*g^3*n*e^(-4
)*log(x*e + d)*log(c) + 1/32*(x*e + d)^4*b^2*g^3*n^2*e^(-4) - 2/9*(x*e + d)^3*b^2*d*g^3*n^2*e^(-4) + 3/4*(x*e
+ d)^2*b^2*d^2*g^3*n^2*e^(-4) - 2*(x*e + d)*b^2*d^3*g^3*n^2*e^(-4) - 2/3*(x*e + d)^3*b^2*f*g^2*n^2*e^(-3)*log(
x*e + d) + 3*(x*e + d)^2*b^2*d*f*g^2*n^2*e^(-3)*log(x*e + d) - 6*(x*e + d)*b^2*d^2*f*g^2*n^2*e^(-3)*log(x*e +
d) + 1/2*(x*e + d)^4*a*b*g^3*n*e^(-4)*log(x*e + d) - 2*(x*e + d)^3*a*b*d*g^3*n*e^(-4)*log(x*e + d) + 3*(x*e +
d)^2*a*b*d^2*g^3*n*e^(-4)*log(x*e + d) - 2*(x*e + d)*a*b*d^3*g^3*n*e^(-4)*log(x*e + d) + 3/2*(x*e + d)^2*b^2*f
^2*g*n^2*e^(-2)*log(x*e + d)^2 - 3*(x*e + d)*b^2*d*f^2*g*n^2*e^(-2)*log(x*e + d)^2 - 1/8*(x*e + d)^4*b^2*g^3*n
*e^(-4)*log(c) + 2/3*(x*e + d)^3*b^2*d*g^3*n*e^(-4)*log(c) - 3/2*(x*e + d)^2*b^2*d^2*g^3*n*e^(-4)*log(c) + 2*(
x*e + d)*b^2*d^3*g^3*n*e^(-4)*log(c) + 2*(x*e + d)^3*b^2*f*g^2*n*e^(-3)*log(x*e + d)*log(c) - 6*(x*e + d)^2*b^
2*d*f*g^2*n*e^(-3)*log(x*e + d)*log(c) + 6*(x*e + d)*b^2*d^2*f*g^2*n*e^(-3)*log(x*e + d)*log(c) + 1/4*(x*e + d
)^4*b^2*g^3*e^(-4)*log(c)^2 - (x*e + d)^3*b^2*d*g^3*e^(-4)*log(c)^2 + 3/2*(x*e + d)^2*b^2*d^2*g^3*e^(-4)*log(c
)^2 - (x*e + d)*b^2*d^3*g^3*e^(-4)*log(c)^2 + 2/9*(x*e + d)^3*b^2*f*g^2*n^2*e^(-3) - 3/2*(x*e + d)^2*b^2*d*f*g
^2*n^2*e^(-3) + 6*(x*e + d)*b^2*d^2*f*g^2*n^2*e^(-3) - 1/8*(x*e + d)^4*a*b*g^3*n*e^(-4) + 2/3*(x*e + d)^3*a*b*
d*g^3*n*e^(-4) - 3/2*(x*e + d)^2*a*b*d^2*g^3*n*e^(-4) + 2*(x*e + d)*a*b*d^3*g^3*n*e^(-4) - 3/2*(x*e + d)^2*b^2
*f^2*g*n^2*e^(-2)*log(x*e + d) + 6*(x*e + d)*b^2*d*f^2*g*n^2*e^(-2)*log(x*e + d) + 2*(x*e + d)^3*a*b*f*g^2*n*e
^(-3)*log(x*e + d) - 6*(x*e + d)^2*a*b*d*f*g^2*n*e^(-3)*log(x*e + d) + 6*(x*e + d)*a*b*d^2*f*g^2*n*e^(-3)*log(
x*e + d) + (x*e + d)*b^2*f^3*n^2*e^(-1)*log(x*e + d)^2 - 2/3*(x*e + d)^3*b^2*f*g^2*n*e^(-3)*log(c) + 3*(x*e +
d)^2*b^2*d*f*g^2*n*e^(-3)*log(c) - 6*(x*e + d)*b^2*d^2*f*g^2*n*e^(-3)*log(c) + 1/2*(x*e + d)^4*a*b*g^3*e^(-4)*
log(c) - 2*(x*e + d)^3*a*b*d*g^3*e^(-4)*log(c) + 3*(x*e + d)^2*a*b*d^2*g^3*e^(-4)*log(c) - 2*(x*e + d)*a*b*d^3
*g^3*e^(-4)*log(c) + 3*(x*e + d)^2*b^2*f^2*g*n*e^(-2)*log(x*e + d)*log(c) - 6*(x*e + d)*b^2*d*f^2*g*n*e^(-2)*l
og(x*e + d)*log(c) + (x*e + d)^3*b^2*f*g^2*e^(-3)*log(c)^2 - 3*(x*e + d)^2*b^2*d*f*g^2*e^(-3)*log(c)^2 + 3*(x*
e + d)*b^2*d^2*f*g^2*e^(-3)*log(c)^2 + 3/4*(x*e + d)^2*b^2*f^2*g*n^2*e^(-2) - 6*(x*e + d)*b^2*d*f^2*g*n^2*e^(-
2) - 2/3*(x*e + d)^3*a*b*f*g^2*n*e^(-3) + 3*(x*e + d)^2*a*b*d*f*g^2*n*e^(-3) - 6*(x*e + d)*a*b*d^2*f*g^2*n*e^(
-3) + 1/4*(x*e + d)^4*a^2*g^3*e^(-4) - (x*e + d)^3*a^2*d*g^3*e^(-4) + 3/2*(x*e + d)^2*a^2*d^2*g^3*e^(-4) - (x*
e + d)*a^2*d^3*g^3*e^(-4) - 2*(x*e + d)*b^2*f^3*n^2*e^(-1)*log(x*e + d) + 3*(x*e + d)^2*a*b*f^2*g*n*e^(-2)*log
(x*e + d) - 6*(x*e + d)*a*b*d*f^2*g*n*e^(-2)*log(x*e + d) - 3/2*(x*e + d)^2*b^2*f^2*g*n*e^(-2)*log(c) + 6*(x*e
+ d)*b^2*d*f^2*g*n*e^(-2)*log(c) + 2*(x*e + d)^3*a*b*f*g^2*e^(-3)*log(c) - 6*(x*e + d)^2*a*b*d*f*g^2*e^(-3)*l
og(c) + 6*(x*e + d)*a*b*d^2*f*g^2*e^(-3)*log(c) + 2*(x*e + d)*b^2*f^3*n*e^(-1)*log(x*e + d)*log(c) + 3/2*(x*e
+ d)^2*b^2*f^2*g*e^(-2)*log(c)^2 - 3*(x*e + d)*b^2*d*f^2*g*e^(-2)*log(c)^2 + 2*(x*e + d)*b^2*f^3*n^2*e^(-1) -
3/2*(x*e + d)^2*a*b*f^2*g*n*e^(-2) + 6*(x*e + d)*a*b*d*f^2*g*n*e^(-2) + (x*e + d)^3*a^2*f*g^2*e^(-3) - 3*(x*e
+ d)^2*a^2*d*f*g^2*e^(-3) + 3*(x*e + d)*a^2*d^2*f*g^2*e^(-3) + 2*(x*e + d)*a*b*f^3*n*e^(-1)*log(x*e + d) - 2*(
x*e + d)*b^2*f^3*n*e^(-1)*log(c) + 3*(x*e + d)^2*a*b*f^2*g*e^(-2)*log(c) - 6*(x*e + d)*a*b*d*f^2*g*e^(-2)*log(
c) + (x*e + d)*b^2*f^3*e^(-1)*log(c)^2 - 2*(x*e + d)*a*b*f^3*n*e^(-1) + 3/2*(x*e + d)^2*a^2*f^2*g*e^(-2) - 3*(
x*e + d)*a^2*d*f^2*g*e^(-2) + 2*(x*e + d)*a*b*f^3*e^(-1)*log(c) + (x*e + d)*a^2*f^3*e^(-1)