3.1.44 \(\int (f+g x)^3 (a+b \log (c (d+e x)^n))^2 \, dx\) [44]

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

[Out]

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

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Rubi [A]
time = 0.36, antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2445, 2458, 45, 2372, 12, 2338} \begin {gather*} -\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 {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}+\frac {2 b^2 n^2 x (e f-d g)^3}{e^3} \end {gather*}

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)
- (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)

Rule 12

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

Rule 45

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 2338

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]

Rule 2372

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]}, Dist[a + b*Log[c*x^n], u, 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 2445

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 2458

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]

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) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.38, size = 612, normalized size = 1.68 \begin {gather*} \frac {72 b^2 d \left (-4 e^3 f^3+6 d e^2 f^2 g-4 d^2 e f g^2+d^3 g^3\right ) n^2 \log ^2(d+e x)-12 b d n \log (d+e x) \left (-12 a \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right )+b \left (48 e^3 f^3-108 d e^2 f^2 g+88 d^2 e f g^2-25 d^3 g^3\right ) n-12 b \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right ) \log \left (c (d+e x)^n\right )\right )+e x \left (72 a^2 e^3 \left (4 f^3+6 f^2 g x+4 f g^2 x^2+g^3 x^3\right )-12 a b n \left (-12 d^3 g^3+6 d^2 e g^2 (8 f+g x)-4 d e^2 g \left (18 f^2+6 f g x+g^2 x^2\right )+e^3 \left (48 f^3+36 f^2 g x+16 f g^2 x^2+3 g^3 x^3\right )\right )+b^2 n^2 \left (-300 d^3 g^3+6 d^2 e g^2 (176 f+13 g x)-4 d e^2 g \left (324 f^2+60 f g x+7 g^2 x^2\right )+e^3 \left (576 f^3+216 f^2 g x+64 f g^2 x^2+9 g^3 x^3\right )\right )+12 b \left (12 a e^3 \left (4 f^3+6 f^2 g x+4 f g^2 x^2+g^3 x^3\right )-b n \left (-12 d^3 g^3+6 d^2 e g^2 (8 f+g x)-4 d e^2 g \left (18 f^2+6 f g x+g^2 x^2\right )+e^3 \left (48 f^3+36 f^2 g x+16 f g^2 x^2+3 g^3 x^3\right )\right )\right ) \log \left (c (d+e x)^n\right )+72 b^2 e^3 \left (4 f^3+6 f^2 g x+4 f g^2 x^2+g^3 x^3\right ) \log ^2\left (c (d+e x)^n\right )\right )}{288 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(72*b^2*d*(-4*e^3*f^3 + 6*d*e^2*f^2*g - 4*d^2*e*f*g^2 + d^3*g^3)*n^2*Log[d + e*x]^2 - 12*b*d*n*Log[d + e*x]*(-
12*a*(4*e^3*f^3 - 6*d*e^2*f^2*g + 4*d^2*e*f*g^2 - d^3*g^3) + b*(48*e^3*f^3 - 108*d*e^2*f^2*g + 88*d^2*e*f*g^2
- 25*d^3*g^3)*n - 12*b*(4*e^3*f^3 - 6*d*e^2*f^2*g + 4*d^2*e*f*g^2 - d^3*g^3)*Log[c*(d + e*x)^n]) + e*x*(72*a^2
*e^3*(4*f^3 + 6*f^2*g*x + 4*f*g^2*x^2 + g^3*x^3) - 12*a*b*n*(-12*d^3*g^3 + 6*d^2*e*g^2*(8*f + g*x) - 4*d*e^2*g
*(18*f^2 + 6*f*g*x + g^2*x^2) + e^3*(48*f^3 + 36*f^2*g*x + 16*f*g^2*x^2 + 3*g^3*x^3)) + b^2*n^2*(-300*d^3*g^3
+ 6*d^2*e*g^2*(176*f + 13*g*x) - 4*d*e^2*g*(324*f^2 + 60*f*g*x + 7*g^2*x^2) + e^3*(576*f^3 + 216*f^2*g*x + 64*
f*g^2*x^2 + 9*g^3*x^3)) + 12*b*(12*a*e^3*(4*f^3 + 6*f^2*g*x + 4*f*g^2*x^2 + g^3*x^3) - b*n*(-12*d^3*g^3 + 6*d^
2*e*g^2*(8*f + g*x) - 4*d*e^2*g*(18*f^2 + 6*f*g*x + g^2*x^2) + e^3*(48*f^3 + 36*f^2*g*x + 16*f*g^2*x^2 + 3*g^3
*x^3)))*Log[c*(d + e*x)^n] + 72*b^2*e^3*(4*f^3 + 6*f^2*g*x + 4*f*g^2*x^2 + g^3*x^3)*Log[c*(d + e*x)^n]^2))/(28
8*e^4)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.02, size = 6770, normalized size = 18.55

method	result	size
risch	\(\text {Expression too large to display}\)	\(6770\)

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,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 839 vs. \(2 (360) = 720\).
time = 0.30, size = 839, normalized size = 2.30 \begin {gather*} \frac {1}{4} \, b^{2} g^{3} x^{4} \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + \frac {1}{2} \, a b g^{3} x^{4} \log \left ({\left (x e + d\right )}^{n} c\right ) + b^{2} f g^{2} x^{3} \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + \frac {1}{4} \, a^{2} g^{3} x^{4} + 2 \, a b f g^{2} x^{3} \log \left ({\left (x e + d\right )}^{n} c\right ) + \frac {3}{2} \, b^{2} f^{2} g x^{2} \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + a^{2} f g^{2} x^{3} + 2 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} a b f^{3} n e - \frac {3}{2} \, {\left (2 \, d^{2} e^{\left (-3\right )} \log \left (x e + d\right ) + {\left (x^{2} e - 2 \, d x\right )} e^{\left (-2\right )}\right )} a b f^{2} g n e + \frac {1}{3} \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (x e + d\right ) - {\left (2 \, x^{3} e^{2} - 3 \, d x^{2} e + 6 \, d^{2} x\right )} e^{\left (-3\right )}\right )} a b f g^{2} n e - \frac {1}{24} \, {\left (12 \, d^{4} e^{\left (-5\right )} \log \left (x e + d\right ) + {\left (3 \, x^{4} e^{3} - 4 \, d x^{3} e^{2} + 6 \, d^{2} x^{2} e - 12 \, d^{3} x\right )} e^{\left (-4\right )}\right )} a b g^{3} n e + 3 \, a b f^{2} g x^{2} \log \left ({\left (x e + d\right )}^{n} c\right ) + b^{2} f^{3} x \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + \frac {3}{2} \, a^{2} f^{2} g x^{2} + 2 \, a b f^{3} x \log \left ({\left (x e + d\right )}^{n} c\right ) - {\left ({\left (d \log \left (x e + d\right )^{2} - 2 \, x e + 2 \, d \log \left (x e + d\right )\right )} n^{2} e^{\left (-1\right )} - 2 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )\right )} b^{2} f^{3} + \frac {3}{4} \, {\left ({\left (2 \, d^{2} \log \left (x e + d\right )^{2} + x^{2} e^{2} - 6 \, d x e + 6 \, d^{2} \log \left (x e + d\right )\right )} n^{2} e^{\left (-2\right )} - 2 \, {\left (2 \, d^{2} e^{\left (-3\right )} \log \left (x e + d\right ) + {\left (x^{2} e - 2 \, d x\right )} e^{\left (-2\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )\right )} b^{2} f^{2} g - \frac {1}{18} \, {\left ({\left (18 \, d^{3} \log \left (x e + d\right )^{2} - 4 \, x^{3} e^{3} + 15 \, d x^{2} e^{2} - 66 \, d^{2} x e + 66 \, d^{3} \log \left (x e + d\right )\right )} n^{2} e^{\left (-3\right )} - 6 \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (x e + d\right ) - {\left (2 \, x^{3} e^{2} - 3 \, d x^{2} e + 6 \, d^{2} x\right )} e^{\left (-3\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )\right )} b^{2} f g^{2} + \frac {1}{288} \, {\left ({\left (72 \, d^{4} \log \left (x e + d\right )^{2} + 9 \, x^{4} e^{4} - 28 \, d x^{3} e^{3} + 78 \, d^{2} x^{2} e^{2} - 300 \, d^{3} x e + 300 \, d^{4} \log \left (x e + d\right )\right )} n^{2} e^{\left (-4\right )} - 12 \, {\left (12 \, d^{4} e^{\left (-5\right )} \log \left (x e + d\right ) + {\left (3 \, x^{4} e^{3} - 4 \, d x^{3} e^{2} + 6 \, d^{2} x^{2} e - 12 \, d^{3} x\right )} e^{\left (-4\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )\right )} b^{2} g^{3} + a^{2} f^{3} x \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1127 vs. \(2 (360) = 720\).
time = 0.39, size = 1127, normalized size = 3.09 \begin {gather*} \frac {1}{288} \, {\left (72 \, {\left (b^{2} g^{3} x^{4} + 4 \, b^{2} f g^{2} x^{3} + 6 \, b^{2} f^{2} g x^{2} + 4 \, b^{2} f^{3} x\right )} e^{4} \log \left (c\right )^{2} - 12 \, {\left (25 \, b^{2} d^{3} g^{3} n^{2} - 12 \, a b d^{3} g^{3} n\right )} x e - 72 \, {\left (b^{2} d^{4} g^{3} n^{2} - 4 \, b^{2} d^{3} f g^{2} n^{2} e + 6 \, b^{2} d^{2} f^{2} g n^{2} e^{2} - 4 \, b^{2} d f^{3} n^{2} e^{3} - {\left (b^{2} g^{3} n^{2} x^{4} + 4 \, b^{2} f g^{2} n^{2} x^{3} + 6 \, b^{2} f^{2} g n^{2} x^{2} + 4 \, b^{2} f^{3} n^{2} x\right )} e^{4}\right )} \log \left (x e + d\right )^{2} + {\left (9 \, {\left (b^{2} g^{3} n^{2} - 4 \, a b g^{3} n + 8 \, a^{2} g^{3}\right )} x^{4} + 32 \, {\left (2 \, b^{2} f g^{2} n^{2} - 6 \, a b f g^{2} n + 9 \, a^{2} f g^{2}\right )} x^{3} + 216 \, {\left (b^{2} f^{2} g n^{2} - 2 \, a b f^{2} g n + 2 \, a^{2} f^{2} g\right )} x^{2} + 288 \, {\left (2 \, b^{2} f^{3} n^{2} - 2 \, a b f^{3} n + a^{2} f^{3}\right )} x\right )} e^{4} - 4 \, {\left ({\left (7 \, b^{2} d g^{3} n^{2} - 12 \, a b d g^{3} n\right )} x^{3} + 12 \, {\left (5 \, b^{2} d f g^{2} n^{2} - 6 \, a b d f g^{2} n\right )} x^{2} + 108 \, {\left (3 \, b^{2} d f^{2} g n^{2} - 2 \, a b d f^{2} g n\right )} x\right )} e^{3} + 6 \, {\left ({\left (13 \, b^{2} d^{2} g^{3} n^{2} - 12 \, a b d^{2} g^{3} n\right )} x^{2} + 16 \, {\left (11 \, b^{2} d^{2} f g^{2} n^{2} - 6 \, a b d^{2} f g^{2} n\right )} x\right )} e^{2} + 12 \, {\left (25 \, b^{2} d^{4} g^{3} n^{2} - 12 \, a b d^{4} g^{3} n - {\left (3 \, {\left (b^{2} g^{3} n^{2} - 4 \, a b g^{3} n\right )} x^{4} + 16 \, {\left (b^{2} f g^{2} n^{2} - 3 \, a b f g^{2} n\right )} x^{3} + 36 \, {\left (b^{2} f^{2} g n^{2} - 2 \, a b f^{2} g n\right )} x^{2} + 48 \, {\left (b^{2} f^{3} n^{2} - a b f^{3} n\right )} x\right )} e^{4} + 4 \, {\left (b^{2} d g^{3} n^{2} x^{3} + 6 \, b^{2} d f g^{2} n^{2} x^{2} + 18 \, b^{2} d f^{2} g n^{2} x - 12 \, b^{2} d f^{3} n^{2} + 12 \, a b d f^{3} n\right )} e^{3} - 6 \, {\left (b^{2} d^{2} g^{3} n^{2} x^{2} + 8 \, b^{2} d^{2} f g^{2} n^{2} x - 18 \, b^{2} d^{2} f^{2} g n^{2} + 12 \, a b d^{2} f^{2} g n\right )} e^{2} + 4 \, {\left (3 \, b^{2} d^{3} g^{3} n^{2} x - 22 \, b^{2} d^{3} f g^{2} n^{2} + 12 \, a b d^{3} f g^{2} n\right )} e - 12 \, {\left (b^{2} d^{4} g^{3} n - 4 \, b^{2} d^{3} f g^{2} n e + 6 \, b^{2} d^{2} f^{2} g n e^{2} - 4 \, b^{2} d f^{3} n e^{3} - {\left (b^{2} g^{3} n x^{4} + 4 \, b^{2} f g^{2} n x^{3} + 6 \, b^{2} f^{2} g n x^{2} + 4 \, b^{2} f^{3} n x\right )} e^{4}\right )} \log \left (c\right )\right )} \log \left (x e + d\right ) + 12 \, {\left (12 \, b^{2} d^{3} g^{3} n x e - {\left (3 \, {\left (b^{2} g^{3} n - 4 \, a b g^{3}\right )} x^{4} + 16 \, {\left (b^{2} f g^{2} n - 3 \, a b f g^{2}\right )} x^{3} + 36 \, {\left (b^{2} f^{2} g n - 2 \, a b f^{2} g\right )} x^{2} + 48 \, {\left (b^{2} f^{3} n - a b f^{3}\right )} x\right )} e^{4} + 4 \, {\left (b^{2} d g^{3} n x^{3} + 6 \, b^{2} d f g^{2} n x^{2} + 18 \, b^{2} d f^{2} g n x\right )} e^{3} - 6 \, {\left (b^{2} d^{2} g^{3} n x^{2} + 8 \, b^{2} d^{2} f g^{2} n x\right )} e^{2}\right )} \log \left (c\right )\right )} e^{\left (-4\right )} \end {gather*}

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*(72*(b^2*g^3*x^4 + 4*b^2*f*g^2*x^3 + 6*b^2*f^2*g*x^2 + 4*b^2*f^3*x)*e^4*log(c)^2 - 12*(25*b^2*d^3*g^3*n^
2 - 12*a*b*d^3*g^3*n)*x*e - 72*(b^2*d^4*g^3*n^2 - 4*b^2*d^3*f*g^2*n^2*e + 6*b^2*d^2*f^2*g*n^2*e^2 - 4*b^2*d*f^
3*n^2*e^3 - (b^2*g^3*n^2*x^4 + 4*b^2*f*g^2*n^2*x^3 + 6*b^2*f^2*g*n^2*x^2 + 4*b^2*f^3*n^2*x)*e^4)*log(x*e + d)^
2 + (9*(b^2*g^3*n^2 - 4*a*b*g^3*n + 8*a^2*g^3)*x^4 + 32*(2*b^2*f*g^2*n^2 - 6*a*b*f*g^2*n + 9*a^2*f*g^2)*x^3 +
216*(b^2*f^2*g*n^2 - 2*a*b*f^2*g*n + 2*a^2*f^2*g)*x^2 + 288*(2*b^2*f^3*n^2 - 2*a*b*f^3*n + a^2*f^3)*x)*e^4 - 4
*((7*b^2*d*g^3*n^2 - 12*a*b*d*g^3*n)*x^3 + 12*(5*b^2*d*f*g^2*n^2 - 6*a*b*d*f*g^2*n)*x^2 + 108*(3*b^2*d*f^2*g*n
^2 - 2*a*b*d*f^2*g*n)*x)*e^3 + 6*((13*b^2*d^2*g^3*n^2 - 12*a*b*d^2*g^3*n)*x^2 + 16*(11*b^2*d^2*f*g^2*n^2 - 6*a
*b*d^2*f*g^2*n)*x)*e^2 + 12*(25*b^2*d^4*g^3*n^2 - 12*a*b*d^4*g^3*n - (3*(b^2*g^3*n^2 - 4*a*b*g^3*n)*x^4 + 16*(
b^2*f*g^2*n^2 - 3*a*b*f*g^2*n)*x^3 + 36*(b^2*f^2*g*n^2 - 2*a*b*f^2*g*n)*x^2 + 48*(b^2*f^3*n^2 - a*b*f^3*n)*x)*
e^4 + 4*(b^2*d*g^3*n^2*x^3 + 6*b^2*d*f*g^2*n^2*x^2 + 18*b^2*d*f^2*g*n^2*x - 12*b^2*d*f^3*n^2 + 12*a*b*d*f^3*n)
*e^3 - 6*(b^2*d^2*g^3*n^2*x^2 + 8*b^2*d^2*f*g^2*n^2*x - 18*b^2*d^2*f^2*g*n^2 + 12*a*b*d^2*f^2*g*n)*e^2 + 4*(3*
b^2*d^3*g^3*n^2*x - 22*b^2*d^3*f*g^2*n^2 + 12*a*b*d^3*f*g^2*n)*e - 12*(b^2*d^4*g^3*n - 4*b^2*d^3*f*g^2*n*e + 6
*b^2*d^2*f^2*g*n*e^2 - 4*b^2*d*f^3*n*e^3 - (b^2*g^3*n*x^4 + 4*b^2*f*g^2*n*x^3 + 6*b^2*f^2*g*n*x^2 + 4*b^2*f^3*
n*x)*e^4)*log(c))*log(x*e + d) + 12*(12*b^2*d^3*g^3*n*x*e - (3*(b^2*g^3*n - 4*a*b*g^3)*x^4 + 16*(b^2*f*g^2*n -
3*a*b*f*g^2)*x^3 + 36*(b^2*f^2*g*n - 2*a*b*f^2*g)*x^2 + 48*(b^2*f^3*n - a*b*f^3)*x)*e^4 + 4*(b^2*d*g^3*n*x^3
+ 6*b^2*d*f*g^2*n*x^2 + 18*b^2*d*f^2*g*n*x)*e^3 - 6*(b^2*d^2*g^3*n*x^2 + 8*b^2*d^2*f*g^2*n*x)*e^2)*log(c))*e^(
-4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1241 vs. \(2 (348) = 696\).
time = 2.55, size = 1241, normalized size = 3.40 \begin {gather*} \begin {cases} a^{2} f^{3} x + \frac {3 a^{2} f^{2} g x^{2}}{2} + a^{2} f g^{2} x^{3} + \frac {a^{2} g^{3} x^{4}}{4} - \frac {a b d^{4} g^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{2 e^{4}} + \frac {2 a b d^{3} f g^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e^{3}} + \frac {a b d^{3} g^{3} n x}{2 e^{3}} - \frac {3 a b d^{2} f^{2} g \log {\left (c \left (d + e x\right )^{n} \right )}}{e^{2}} - \frac {2 a b d^{2} f g^{2} n x}{e^{2}} - \frac {a b d^{2} g^{3} n x^{2}}{4 e^{2}} + \frac {2 a b d f^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {3 a b d f^{2} g n x}{e} + \frac {a b d f g^{2} n x^{2}}{e} + \frac {a b d g^{3} n x^{3}}{6 e} - 2 a b f^{3} n x + 2 a b f^{3} x \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {3 a b f^{2} g n x^{2}}{2} + 3 a b f^{2} g x^{2} \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {2 a b f g^{2} n x^{3}}{3} + 2 a b f g^{2} x^{3} \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {a b g^{3} n x^{4}}{8} + \frac {a b g^{3} x^{4} \log {\left (c \left (d + e x\right )^{n} \right )}}{2} + \frac {25 b^{2} d^{4} g^{3} n \log {\left (c \left (d + e x\right )^{n} \right )}}{24 e^{4}} - \frac {b^{2} d^{4} g^{3} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{4 e^{4}} - \frac {11 b^{2} d^{3} f g^{2} n \log {\left (c \left (d + e x\right )^{n} \right )}}{3 e^{3}} + \frac {b^{2} d^{3} f g^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e^{3}} - \frac {25 b^{2} d^{3} g^{3} n^{2} x}{24 e^{3}} + \frac {b^{2} d^{3} g^{3} n x \log {\left (c \left (d + e x\right )^{n} \right )}}{2 e^{3}} + \frac {9 b^{2} d^{2} f^{2} g n \log {\left (c \left (d + e x\right )^{n} \right )}}{2 e^{2}} - \frac {3 b^{2} d^{2} f^{2} g \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{2 e^{2}} + \frac {11 b^{2} d^{2} f g^{2} n^{2} x}{3 e^{2}} - \frac {2 b^{2} d^{2} f g^{2} n x \log {\left (c \left (d + e x\right )^{n} \right )}}{e^{2}} + \frac {13 b^{2} d^{2} g^{3} n^{2} x^{2}}{48 e^{2}} - \frac {b^{2} d^{2} g^{3} n x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{4 e^{2}} - \frac {2 b^{2} d f^{3} n \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {b^{2} d f^{3} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} - \frac {9 b^{2} d f^{2} g n^{2} x}{2 e} + \frac {3 b^{2} d f^{2} g n x \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {5 b^{2} d f g^{2} n^{2} x^{2}}{6 e} + \frac {b^{2} d f g^{2} n x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {7 b^{2} d g^{3} n^{2} x^{3}}{72 e} + \frac {b^{2} d g^{3} n x^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{6 e} + 2 b^{2} f^{3} n^{2} x - 2 b^{2} f^{3} n x \log {\left (c \left (d + e x\right )^{n} \right )} + b^{2} f^{3} x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + \frac {3 b^{2} f^{2} g n^{2} x^{2}}{4} - \frac {3 b^{2} f^{2} g n x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{2} + \frac {3 b^{2} f^{2} g x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{2} + \frac {2 b^{2} f g^{2} n^{2} x^{3}}{9} - \frac {2 b^{2} f g^{2} n x^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{3} + b^{2} f g^{2} x^{3} \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + \frac {b^{2} g^{3} n^{2} x^{4}}{32} - \frac {b^{2} g^{3} n x^{4} \log {\left (c \left (d + e x\right )^{n} \right )}}{8} + \frac {b^{2} g^{3} x^{4} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{4} & \text {for}\: e \neq 0 \\\left (a + b \log {\left (c d^{n} \right )}\right )^{2} \left (f^{3} x + \frac {3 f^{2} g x^{2}}{2} + f g^{2} x^{3} + \frac {g^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}

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*log(c*(d +
e*x)**n)/(2*e**4) + 2*a*b*d**3*f*g**2*log(c*(d + e*x)**n)/e**3 + a*b*d**3*g**3*n*x/(2*e**3) - 3*a*b*d**2*f**2
*g*log(c*(d + e*x)**n)/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*log(c*
(d + e*x)**n)/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 + 2*
a*b*f**3*x*log(c*(d + e*x)**n) - 3*a*b*f**2*g*n*x**2/2 + 3*a*b*f**2*g*x**2*log(c*(d + e*x)**n) - 2*a*b*f*g**2*
n*x**3/3 + 2*a*b*f*g**2*x**3*log(c*(d + e*x)**n) - a*b*g**3*n*x**4/8 + a*b*g**3*x**4*log(c*(d + e*x)**n)/2 + 2
5*b**2*d**4*g**3*n*log(c*(d + e*x)**n)/(24*e**4) - b**2*d**4*g**3*log(c*(d + e*x)**n)**2/(4*e**4) - 11*b**2*d*
*3*f*g**2*n*log(c*(d + e*x)**n)/(3*e**3) + b**2*d**3*f*g**2*log(c*(d + e*x)**n)**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*(d + e*x)**n)/(2*e**3) + 9*b**2*d**2*f**2*g*n*log(c*(d + e*x)**n)/(2
*e**2) - 3*b**2*d**2*f**2*g*log(c*(d + e*x)**n)**2/(2*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*(d + e*x)**n)/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*(d
+ e*x)**n)/(4*e**2) - 2*b**2*d*f**3*n*log(c*(d + e*x)**n)/e + b**2*d*f**3*log(c*(d + e*x)**n)**2/e - 9*b**2*d
*f**2*g*n**2*x/(2*e) + 3*b**2*d*f**2*g*n*x*log(c*(d + e*x)**n)/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*(d + e*x)**n)/e - 7*b**2*d*g**3*n**2*x**3/(72*e) + b**2*d*g**3*n*x**3*log(c*(d + e*x)**n)/(6
*e) + 2*b**2*f**3*n**2*x - 2*b**2*f**3*n*x*log(c*(d + e*x)**n) + b**2*f**3*x*log(c*(d + e*x)**n)**2 + 3*b**2*f
**2*g*n**2*x**2/4 - 3*b**2*f**2*g*n*x**2*log(c*(d + e*x)**n)/2 + 3*b**2*f**2*g*x**2*log(c*(d + e*x)**n)**2/2 +
2*b**2*f*g**2*n**2*x**3/9 - 2*b**2*f*g**2*n*x**3*log(c*(d + e*x)**n)/3 + b**2*f*g**2*x**3*log(c*(d + e*x)**n)
**2 + b**2*g**3*n**2*x**4/32 - b**2*g**3*n*x**4*log(c*(d + e*x)**n)/8 + b**2*g**3*x**4*log(c*(d + e*x)**n)**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))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2385 vs. \(2 (360) = 720\).
time = 3.38, size = 2385, normalized size = 6.53 \begin {gather*} \text {Too large to display} \end {gather*}

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)

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Mupad [B]
time = 0.74, size = 1051, normalized size = 2.88 \begin {gather*} x\,\left (\frac {72\,a^2\,d\,e^2\,f^2\,g+24\,a^2\,e^3\,f^3-48\,a\,b\,e^3\,f^3\,n-12\,b^2\,d^3\,g^3\,n^2+48\,b^2\,d^2\,e\,f\,g^2\,n^2-72\,b^2\,d\,e^2\,f^2\,g\,n^2+48\,b^2\,e^3\,f^3\,n^2}{24\,e^3}+\frac {d\,\left (\frac {d\,\left (\frac {g^2\,\left (6\,a^2\,d\,g+18\,a^2\,e\,f-b^2\,d\,g\,n^2+4\,b^2\,e\,f\,n^2-12\,a\,b\,e\,f\,n\right )}{6\,e}-\frac {d\,g^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{8\,e}\right )}{e}-\frac {g\,\left (12\,a^2\,d\,e\,f\,g+12\,a^2\,e^2\,f^2-12\,a\,b\,e^2\,f^2\,n+b^2\,d^2\,g^2\,n^2-4\,b^2\,d\,e\,f\,g\,n^2+6\,b^2\,e^2\,f^2\,n^2\right )}{4\,e^2}\right )}{e}\right )-x^2\,\left (\frac {d\,\left (\frac {g^2\,\left (6\,a^2\,d\,g+18\,a^2\,e\,f-b^2\,d\,g\,n^2+4\,b^2\,e\,f\,n^2-12\,a\,b\,e\,f\,n\right )}{6\,e}-\frac {d\,g^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{8\,e}\right )}{2\,e}-\frac {g\,\left (12\,a^2\,d\,e\,f\,g+12\,a^2\,e^2\,f^2-12\,a\,b\,e^2\,f^2\,n+b^2\,d^2\,g^2\,n^2-4\,b^2\,d\,e\,f\,g\,n^2+6\,b^2\,e^2\,f^2\,n^2\right )}{8\,e^2}\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (b^2\,f^3\,x-\frac {d\,\left (b^2\,d^3\,g^3-4\,b^2\,d^2\,e\,f\,g^2+6\,b^2\,d\,e^2\,f^2\,g-4\,b^2\,e^3\,f^3\right )}{4\,e^4}+\frac {b^2\,g^3\,x^4}{4}+\frac {3\,b^2\,f^2\,g\,x^2}{2}+b^2\,f\,g^2\,x^3\right )+x^3\,\left (\frac {g^2\,\left (6\,a^2\,d\,g+18\,a^2\,e\,f-b^2\,d\,g\,n^2+4\,b^2\,e\,f\,n^2-12\,a\,b\,e\,f\,n\right )}{18\,e}-\frac {d\,g^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{24\,e}\right )+\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (\frac {x\,\left (\frac {d\,\left (\frac {d\,\left (\frac {8\,b\,g^2\,\left (a\,d\,g+3\,a\,e\,f-b\,e\,f\,n\right )}{e}-\frac {2\,b\,d\,g^3\,\left (4\,a-b\,n\right )}{e}\right )}{e}-\frac {12\,b\,f\,g\,\left (2\,a\,d\,g+2\,a\,e\,f-b\,e\,f\,n\right )}{e}\right )}{2\,e}+\frac {4\,b\,f^2\,\left (3\,a\,d\,g+a\,e\,f-b\,e\,f\,n\right )}{e}\right )}{2}+\frac {x^3\,\left (\frac {4\,b\,g^2\,\left (a\,d\,g+3\,a\,e\,f-b\,e\,f\,n\right )}{3\,e}-\frac {b\,d\,g^3\,\left (4\,a-b\,n\right )}{3\,e}\right )}{2}-\frac {x^2\,\left (\frac {d\,\left (\frac {8\,b\,g^2\,\left (a\,d\,g+3\,a\,e\,f-b\,e\,f\,n\right )}{e}-\frac {2\,b\,d\,g^3\,\left (4\,a-b\,n\right )}{e}\right )}{4\,e}-\frac {3\,b\,f\,g\,\left (2\,a\,d\,g+2\,a\,e\,f-b\,e\,f\,n\right )}{e}\right )}{2}+\frac {b\,g^3\,x^4\,\left (4\,a-b\,n\right )}{8}\right )+\frac {\ln \left (d+e\,x\right )\,\left (25\,b^2\,d^4\,g^3\,n^2-88\,b^2\,d^3\,e\,f\,g^2\,n^2+108\,b^2\,d^2\,e^2\,f^2\,g\,n^2-48\,b^2\,d\,e^3\,f^3\,n^2-12\,a\,b\,d^4\,g^3\,n+48\,a\,b\,d^3\,e\,f\,g^2\,n-72\,a\,b\,d^2\,e^2\,f^2\,g\,n+48\,a\,b\,d\,e^3\,f^3\,n\right )}{24\,e^4}+\frac {g^3\,x^4\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((24*a^2*e^3*f^3 - 12*b^2*d^3*g^3*n^2 + 48*b^2*e^3*f^3*n^2 - 48*a*b*e^3*f^3*n + 72*a^2*d*e^2*f^2*g - 72*b^2*
d*e^2*f^2*g*n^2 + 48*b^2*d^2*e*f*g^2*n^2)/(24*e^3) + (d*((d*((g^2*(6*a^2*d*g + 18*a^2*e*f - b^2*d*g*n^2 + 4*b^
2*e*f*n^2 - 12*a*b*e*f*n))/(6*e) - (d*g^3*(8*a^2 + b^2*n^2 - 4*a*b*n))/(8*e)))/e - (g*(12*a^2*e^2*f^2 + b^2*d^
2*g^2*n^2 + 6*b^2*e^2*f^2*n^2 - 12*a*b*e^2*f^2*n + 12*a^2*d*e*f*g - 4*b^2*d*e*f*g*n^2))/(4*e^2)))/e) - x^2*((d
*((g^2*(6*a^2*d*g + 18*a^2*e*f - b^2*d*g*n^2 + 4*b^2*e*f*n^2 - 12*a*b*e*f*n))/(6*e) - (d*g^3*(8*a^2 + b^2*n^2
- 4*a*b*n))/(8*e)))/(2*e) - (g*(12*a^2*e^2*f^2 + b^2*d^2*g^2*n^2 + 6*b^2*e^2*f^2*n^2 - 12*a*b*e^2*f^2*n + 12*a
^2*d*e*f*g - 4*b^2*d*e*f*g*n^2))/(8*e^2)) + log(c*(d + e*x)^n)^2*(b^2*f^3*x - (d*(b^2*d^3*g^3 - 4*b^2*e^3*f^3
+ 6*b^2*d*e^2*f^2*g - 4*b^2*d^2*e*f*g^2))/(4*e^4) + (b^2*g^3*x^4)/4 + (3*b^2*f^2*g*x^2)/2 + b^2*f*g^2*x^3) + x
^3*((g^2*(6*a^2*d*g + 18*a^2*e*f - b^2*d*g*n^2 + 4*b^2*e*f*n^2 - 12*a*b*e*f*n))/(18*e) - (d*g^3*(8*a^2 + b^2*n
^2 - 4*a*b*n))/(24*e)) + log(c*(d + e*x)^n)*((x*((d*((d*((8*b*g^2*(a*d*g + 3*a*e*f - b*e*f*n))/e - (2*b*d*g^3*
(4*a - b*n))/e))/e - (12*b*f*g*(2*a*d*g + 2*a*e*f - b*e*f*n))/e))/(2*e) + (4*b*f^2*(3*a*d*g + a*e*f - b*e*f*n)
)/e))/2 + (x^3*((4*b*g^2*(a*d*g + 3*a*e*f - b*e*f*n))/(3*e) - (b*d*g^3*(4*a - b*n))/(3*e)))/2 - (x^2*((d*((8*b
*g^2*(a*d*g + 3*a*e*f - b*e*f*n))/e - (2*b*d*g^3*(4*a - b*n))/e))/(4*e) - (3*b*f*g*(2*a*d*g + 2*a*e*f - b*e*f*
n))/e))/2 + (b*g^3*x^4*(4*a - b*n))/8) + (log(d + e*x)*(25*b^2*d^4*g^3*n^2 - 12*a*b*d^4*g^3*n - 48*b^2*d*e^3*f
^3*n^2 - 88*b^2*d^3*e*f*g^2*n^2 + 48*a*b*d*e^3*f^3*n + 108*b^2*d^2*e^2*f^2*g*n^2 - 72*a*b*d^2*e^2*f^2*g*n + 48
*a*b*d^3*e*f*g^2*n))/(24*e^4) + (g^3*x^4*(8*a^2 + b^2*n^2 - 4*a*b*n))/32

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