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

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

[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

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Rubi [A]
time = 0.20, antiderivative size = 340, normalized size of antiderivative = 1.00, 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 = {2448, 2436, 2333, 2332, 2437, 2342, 2341} \begin {gather*} -\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} \end {gather*}

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 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*} \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 \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*}

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Mathematica [A]
time = 0.15, size = 258, normalized size = 0.76 \begin {gather*} \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} \end {gather*}

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)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 3.08, size = 37938, normalized size = 111.58

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

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,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. 1217 vs. \(2 (345) = 690\).
time = 0.34, size = 1217, normalized size = 3.58 \begin {gather*} \frac {1}{2} \, b^{4} g x^{2} \log \left ({\left (x e + d\right )}^{n} c\right )^{4} + 2 \, a b^{3} g x^{2} \log \left ({\left (x e + d\right )}^{n} c\right )^{3} + b^{4} f x \log \left ({\left (x e + d\right )}^{n} c\right )^{4} + 3 \, a^{2} b^{2} g x^{2} \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + 4 \, a b^{3} f x \log \left ({\left (x e + d\right )}^{n} c\right )^{3} + 4 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} a^{3} b f n e - {\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^{3} b g n e + 2 \, a^{3} b g x^{2} \log \left ({\left (x e + d\right )}^{n} c\right ) + 6 \, a^{2} b^{2} f x \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + \frac {1}{2} \, a^{4} g x^{2} + 4 \, a^{3} b f x \log \left ({\left (x e + d\right )}^{n} c\right ) - 6 \, {\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 )} a^{2} b^{2} f + 4 \, {\left (3 \, {\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 )^{2} + {\left ({\left (d \log \left (x e + d\right )^{3} + 3 \, d \log \left (x e + d\right )^{2} - 6 \, x e + 6 \, d \log \left (x e + d\right )\right )} n^{2} e^{\left (-2\right )} - 3 \, {\left (d \log \left (x e + d\right )^{2} - 2 \, x e + 2 \, d \log \left (x e + d\right )\right )} n e^{\left (-2\right )} \log \left ({\left (x e + d\right )}^{n} c\right )\right )} n e\right )} a b^{3} f + {\left (4 \, {\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 )^{3} - {\left (6 \, {\left (d \log \left (x e + d\right )^{2} - 2 \, x e + 2 \, d \log \left (x e + d\right )\right )} n e^{\left (-2\right )} \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + {\left ({\left (d \log \left (x e + d\right )^{4} + 4 \, d \log \left (x e + d\right )^{3} + 12 \, d \log \left (x e + d\right )^{2} - 24 \, x e + 24 \, d \log \left (x e + d\right )\right )} n^{2} e^{\left (-3\right )} - 4 \, {\left (d \log \left (x e + d\right )^{3} + 3 \, d \log \left (x e + d\right )^{2} - 6 \, x e + 6 \, d \log \left (x e + d\right )\right )} n e^{\left (-3\right )} \log \left ({\left (x e + d\right )}^{n} c\right )\right )} n e\right )} n e\right )} b^{4} f + \frac {3}{2} \, {\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 )} a^{2} b^{2} g - \frac {1}{2} \, {\left (6 \, {\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 )^{2} + {\left ({\left (4 \, d^{2} \log \left (x e + d\right )^{3} + 18 \, d^{2} \log \left (x e + d\right )^{2} + 3 \, x^{2} e^{2} - 42 \, d x e + 42 \, d^{2} \log \left (x e + d\right )\right )} n^{2} e^{\left (-3\right )} - 6 \, {\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 e^{\left (-3\right )} \log \left ({\left (x e + d\right )}^{n} c\right )\right )} n e\right )} a b^{3} g - \frac {1}{4} \, {\left (4 \, {\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 )^{3} - {\left (6 \, {\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 e^{\left (-3\right )} \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + {\left ({\left (2 \, d^{2} \log \left (x e + d\right )^{4} + 12 \, d^{2} \log \left (x e + d\right )^{3} + 42 \, d^{2} \log \left (x e + d\right )^{2} + 3 \, x^{2} e^{2} - 90 \, d x e + 90 \, d^{2} \log \left (x e + d\right )\right )} n^{2} e^{\left (-4\right )} - 2 \, {\left (4 \, d^{2} \log \left (x e + d\right )^{3} + 18 \, d^{2} \log \left (x e + d\right )^{2} + 3 \, x^{2} e^{2} - 42 \, d x e + 42 \, d^{2} \log \left (x e + d\right )\right )} n e^{\left (-4\right )} \log \left ({\left (x e + d\right )}^{n} c\right )\right )} n e\right )} n e\right )} b^{4} g + a^{4} f x \end {gather*}

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((x*e + d)^n*c)^4 + 2*a*b^3*g*x^2*log((x*e + d)^n*c)^3 + b^4*f*x*log((x*e + d)^n*c)^4 + 3*a^2
*b^2*g*x^2*log((x*e + d)^n*c)^2 + 4*a*b^3*f*x*log((x*e + d)^n*c)^3 + 4*(d*e^(-2)*log(x*e + d) - x*e^(-1))*a^3*
b*f*n*e - (2*d^2*e^(-3)*log(x*e + d) + (x^2*e - 2*d*x)*e^(-2))*a^3*b*g*n*e + 2*a^3*b*g*x^2*log((x*e + d)^n*c)
+ 6*a^2*b^2*f*x*log((x*e + d)^n*c)^2 + 1/2*a^4*g*x^2 + 4*a^3*b*f*x*log((x*e + d)^n*c) - 6*((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))*a^2*b^2*f
+ 4*(3*(d*e^(-2)*log(x*e + d) - x*e^(-1))*n*e*log((x*e + d)^n*c)^2 + ((d*log(x*e + d)^3 + 3*d*log(x*e + d)^2
- 6*x*e + 6*d*log(x*e + d))*n^2*e^(-2) - 3*(d*log(x*e + d)^2 - 2*x*e + 2*d*log(x*e + d))*n*e^(-2)*log((x*e + d
)^n*c))*n*e)*a*b^3*f + (4*(d*e^(-2)*log(x*e + d) - x*e^(-1))*n*e*log((x*e + d)^n*c)^3 - (6*(d*log(x*e + d)^2 -
2*x*e + 2*d*log(x*e + d))*n*e^(-2)*log((x*e + d)^n*c)^2 + ((d*log(x*e + d)^4 + 4*d*log(x*e + d)^3 + 12*d*log(
x*e + d)^2 - 24*x*e + 24*d*log(x*e + d))*n^2*e^(-3) - 4*(d*log(x*e + d)^3 + 3*d*log(x*e + d)^2 - 6*x*e + 6*d*l
og(x*e + d))*n*e^(-3)*log((x*e + d)^n*c))*n*e)*n*e)*b^4*f + 3/2*((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))
*a^2*b^2*g - 1/2*(6*(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)^2 + ((4*d^2*lo
g(x*e + d)^3 + 18*d^2*log(x*e + d)^2 + 3*x^2*e^2 - 42*d*x*e + 42*d^2*log(x*e + d))*n^2*e^(-3) - 6*(2*d^2*log(x
*e + d)^2 + x^2*e^2 - 6*d*x*e + 6*d^2*log(x*e + d))*n*e^(-3)*log((x*e + d)^n*c))*n*e)*a*b^3*g - 1/4*(4*(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)^3 - (6*(2*d^2*log(x*e + d)^2 + x^2*e^2 -
6*d*x*e + 6*d^2*log(x*e + d))*n*e^(-3)*log((x*e + d)^n*c)^2 + ((2*d^2*log(x*e + d)^4 + 12*d^2*log(x*e + d)^3 +
42*d^2*log(x*e + d)^2 + 3*x^2*e^2 - 90*d*x*e + 90*d^2*log(x*e + d))*n^2*e^(-4) - 2*(4*d^2*log(x*e + d)^3 + 18
*d^2*log(x*e + d)^2 + 3*x^2*e^2 - 42*d*x*e + 42*d^2*log(x*e + d))*n*e^(-4)*log((x*e + d)^n*c))*n*e)*n*e)*b^4*g
+ a^4*f*x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1635 vs. \(2 (345) = 690\).
time = 0.40, size = 1635, normalized size = 4.81 \begin {gather*} \frac {1}{4} \, {\left (2 \, {\left (b^{4} g x^{2} + 2 \, b^{4} f x\right )} e^{2} \log \left (c\right )^{4} - 2 \, {\left (b^{4} d^{2} g n^{4} - 2 \, b^{4} d f n^{4} e - {\left (b^{4} g n^{4} x^{2} + 2 \, b^{4} f n^{4} x\right )} e^{2}\right )} \log \left (x e + d\right )^{4} + 4 \, {\left (3 \, b^{4} d^{2} g n^{4} - 2 \, a b^{3} d^{2} g n^{3} - {\left ({\left (b^{4} g n^{4} - 2 \, a b^{3} g n^{3}\right )} x^{2} + 4 \, {\left (b^{4} f n^{4} - a b^{3} f n^{3}\right )} x\right )} e^{2} + 2 \, {\left (b^{4} d g n^{4} x - 2 \, b^{4} d f n^{4} + 2 \, a b^{3} d f n^{3}\right )} e - 2 \, {\left (b^{4} d^{2} g n^{3} - 2 \, b^{4} d f n^{3} e - {\left (b^{4} g n^{3} x^{2} + 2 \, b^{4} f n^{3} x\right )} e^{2}\right )} \log \left (c\right )\right )} \log \left (x e + d\right )^{3} + 4 \, {\left (2 \, b^{4} d g n x e - {\left ({\left (b^{4} g n - 2 \, a b^{3} g\right )} x^{2} + 4 \, {\left (b^{4} f n - a b^{3} f\right )} x\right )} e^{2}\right )} \log \left (c\right )^{3} - 2 \, {\left (45 \, b^{4} d g n^{4} - 42 \, a b^{3} d g n^{3} + 18 \, a^{2} b^{2} d g n^{2} - 4 \, a^{3} b d g n\right )} x e - 6 \, {\left (7 \, b^{4} d^{2} g n^{4} - 6 \, a b^{3} d^{2} g n^{3} + 2 \, a^{2} b^{2} d^{2} g n^{2} + 2 \, {\left (b^{4} d^{2} g n^{2} - 2 \, b^{4} d f n^{2} e - {\left (b^{4} g n^{2} x^{2} + 2 \, b^{4} f n^{2} x\right )} e^{2}\right )} \log \left (c\right )^{2} - {\left ({\left (b^{4} g n^{4} - 2 \, a b^{3} g n^{3} + 2 \, a^{2} b^{2} g n^{2}\right )} x^{2} + 4 \, {\left (2 \, b^{4} f n^{4} - 2 \, a b^{3} f n^{3} + a^{2} b^{2} f n^{2}\right )} x\right )} e^{2} - 2 \, {\left (4 \, b^{4} d f n^{4} - 4 \, a b^{3} d f n^{3} + 2 \, a^{2} b^{2} d f n^{2} - {\left (3 \, b^{4} d g n^{4} - 2 \, a b^{3} d g n^{3}\right )} x\right )} e - 2 \, {\left (3 \, b^{4} d^{2} g n^{3} - 2 \, a b^{3} d^{2} g n^{2} - {\left ({\left (b^{4} g n^{3} - 2 \, a b^{3} g n^{2}\right )} x^{2} + 4 \, {\left (b^{4} f n^{3} - a b^{3} f n^{2}\right )} x\right )} e^{2} + 2 \, {\left (b^{4} d g n^{3} x - 2 \, b^{4} d f n^{3} + 2 \, a b^{3} d f n^{2}\right )} e\right )} \log \left (c\right )\right )} \log \left (x e + d\right )^{2} - 6 \, {\left (2 \, {\left (3 \, b^{4} d g n^{2} - 2 \, a b^{3} d g n\right )} x e - {\left ({\left (b^{4} g n^{2} - 2 \, a b^{3} g n + 2 \, a^{2} b^{2} g\right )} x^{2} + 4 \, {\left (2 \, b^{4} f n^{2} - 2 \, a b^{3} f n + a^{2} b^{2} f\right )} x\right )} e^{2}\right )} \log \left (c\right )^{2} + {\left ({\left (3 \, b^{4} g n^{4} - 6 \, a b^{3} g n^{3} + 6 \, a^{2} b^{2} g n^{2} - 4 \, a^{3} b g n + 2 \, a^{4} g\right )} x^{2} + 4 \, {\left (24 \, b^{4} f n^{4} - 24 \, a b^{3} f n^{3} + 12 \, a^{2} b^{2} f n^{2} - 4 \, a^{3} b f n + a^{4} f\right )} x\right )} e^{2} + 2 \, {\left (45 \, b^{4} d^{2} g n^{4} - 42 \, a b^{3} d^{2} g n^{3} + 18 \, a^{2} b^{2} d^{2} g n^{2} - 4 \, a^{3} b d^{2} g n - 4 \, {\left (b^{4} d^{2} g n - 2 \, b^{4} d f n e - {\left (b^{4} g n x^{2} + 2 \, b^{4} f n x\right )} e^{2}\right )} \log \left (c\right )^{3} + 6 \, {\left (3 \, b^{4} d^{2} g n^{2} - 2 \, a b^{3} d^{2} g n - {\left ({\left (b^{4} g n^{2} - 2 \, a b^{3} g n\right )} x^{2} + 4 \, {\left (b^{4} f n^{2} - a b^{3} f n\right )} x\right )} e^{2} + 2 \, {\left (b^{4} d g n^{2} x - 2 \, b^{4} d f n^{2} + 2 \, a b^{3} d f n\right )} e\right )} \log \left (c\right )^{2} - {\left ({\left (3 \, b^{4} g n^{4} - 6 \, a b^{3} g n^{3} + 6 \, a^{2} b^{2} g n^{2} - 4 \, a^{3} b g n\right )} x^{2} + 8 \, {\left (6 \, b^{4} f n^{4} - 6 \, a b^{3} f n^{3} + 3 \, a^{2} b^{2} f n^{2} - a^{3} b f n\right )} x\right )} e^{2} - 2 \, {\left (24 \, b^{4} d f n^{4} - 24 \, a b^{3} d f n^{3} + 12 \, a^{2} b^{2} d f n^{2} - 4 \, a^{3} b d f n - 3 \, {\left (7 \, b^{4} d g n^{4} - 6 \, a b^{3} d g n^{3} + 2 \, a^{2} b^{2} d g n^{2}\right )} x\right )} e - 6 \, {\left (7 \, b^{4} d^{2} g n^{3} - 6 \, a b^{3} d^{2} g n^{2} + 2 \, a^{2} b^{2} d^{2} g n - {\left ({\left (b^{4} g n^{3} - 2 \, a b^{3} g n^{2} + 2 \, a^{2} b^{2} g n\right )} x^{2} + 4 \, {\left (2 \, b^{4} f n^{3} - 2 \, a b^{3} f n^{2} + a^{2} b^{2} f n\right )} x\right )} e^{2} - 2 \, {\left (4 \, b^{4} d f n^{3} - 4 \, a b^{3} d f n^{2} + 2 \, a^{2} b^{2} d f n - {\left (3 \, b^{4} d g n^{3} - 2 \, a b^{3} d g n^{2}\right )} x\right )} e\right )} \log \left (c\right )\right )} \log \left (x e + d\right ) + 2 \, {\left (6 \, {\left (7 \, b^{4} d g n^{3} - 6 \, a b^{3} d g n^{2} + 2 \, a^{2} b^{2} d g n\right )} x e - {\left ({\left (3 \, b^{4} g n^{3} - 6 \, a b^{3} g n^{2} + 6 \, a^{2} b^{2} g n - 4 \, a^{3} b g\right )} x^{2} + 8 \, {\left (6 \, b^{4} f n^{3} - 6 \, a b^{3} f n^{2} + 3 \, a^{2} b^{2} f n - a^{3} b f\right )} x\right )} e^{2}\right )} \log \left (c\right )\right )} e^{\left (-2\right )} \end {gather*}

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1372 vs. \(2 (332) = 664\).
time = 2.65, size = 1372, normalized size = 4.04 \begin {gather*} \begin {cases} a^{4} f x + \frac {a^{4} g x^{2}}{2} - \frac {2 a^{3} b d^{2} g \log {\left (c \left (d + e x\right )^{n} \right )}}{e^{2}} + \frac {4 a^{3} b d f \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {2 a^{3} b d g n x}{e} - 4 a^{3} b f n x + 4 a^{3} b f x \log {\left (c \left (d + e x\right )^{n} \right )} - a^{3} b g n x^{2} + 2 a^{3} b g x^{2} \log {\left (c \left (d + e x\right )^{n} \right )} + \frac {9 a^{2} b^{2} d^{2} g n \log {\left (c \left (d + e x\right )^{n} \right )}}{e^{2}} - \frac {3 a^{2} b^{2} d^{2} g \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e^{2}} - \frac {12 a^{2} b^{2} d f n \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {6 a^{2} b^{2} d f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} - \frac {9 a^{2} b^{2} d g n^{2} x}{e} + \frac {6 a^{2} b^{2} d g n x \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + 12 a^{2} b^{2} f n^{2} x - 12 a^{2} b^{2} f n x \log {\left (c \left (d + e x\right )^{n} \right )} + 6 a^{2} b^{2} f x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + \frac {3 a^{2} b^{2} g n^{2} x^{2}}{2} - 3 a^{2} b^{2} g n x^{2} \log {\left (c \left (d + e x\right )^{n} \right )} + 3 a^{2} b^{2} g x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2} - \frac {21 a b^{3} d^{2} g n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e^{2}} + \frac {9 a b^{3} d^{2} g n \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e^{2}} - \frac {2 a b^{3} d^{2} g \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e^{2}} + \frac {24 a b^{3} d f n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {12 a b^{3} d f n \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + \frac {4 a b^{3} d f \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e} + \frac {21 a b^{3} d g n^{3} x}{e} - \frac {18 a b^{3} d g n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {6 a b^{3} d g n x \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} - 24 a b^{3} f n^{3} x + 24 a b^{3} f n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )} - 12 a b^{3} f n x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + 4 a b^{3} f x \log {\left (c \left (d + e x\right )^{n} \right )}^{3} - \frac {3 a b^{3} g n^{3} x^{2}}{2} + 3 a b^{3} g n^{2} x^{2} \log {\left (c \left (d + e x\right )^{n} \right )} - 3 a b^{3} g n x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + 2 a b^{3} g x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{3} + \frac {45 b^{4} d^{2} g n^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{2 e^{2}} - \frac {21 b^{4} d^{2} g n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{2 e^{2}} + \frac {3 b^{4} d^{2} g n \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e^{2}} - \frac {b^{4} d^{2} g \log {\left (c \left (d + e x\right )^{n} \right )}^{4}}{2 e^{2}} - \frac {24 b^{4} d f n^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {12 b^{4} d f n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} - \frac {4 b^{4} d f n \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e} + \frac {b^{4} d f \log {\left (c \left (d + e x\right )^{n} \right )}^{4}}{e} - \frac {45 b^{4} d g n^{4} x}{2 e} + \frac {21 b^{4} d g n^{3} x \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {9 b^{4} d g n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + \frac {2 b^{4} d g n x \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e} + 24 b^{4} f n^{4} x - 24 b^{4} f n^{3} x \log {\left (c \left (d + e x\right )^{n} \right )} + 12 b^{4} f n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} - 4 b^{4} f n x \log {\left (c \left (d + e x\right )^{n} \right )}^{3} + b^{4} f x \log {\left (c \left (d + e x\right )^{n} \right )}^{4} + \frac {3 b^{4} g n^{4} x^{2}}{4} - \frac {3 b^{4} g n^{3} x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{2} + \frac {3 b^{4} g n^{2} x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{2} - b^{4} g n x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{3} + \frac {b^{4} g x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{4}}{2} & \text {for}\: e \neq 0 \\\left (a + b \log {\left (c d^{n} \right )}\right )^{4} \left (f x + \frac {g x^{2}}{2}\right ) & \text {otherwise} \end {cases} \end {gather*}

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*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))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2548 vs. \(2 (345) = 690\).
time = 6.45, size = 2548, normalized size = 7.49 \begin {gather*} \text {Too large to display} \end {gather*}

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)

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Mupad [B]
time = 0.80, size = 823, normalized size = 2.42 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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