3.1.54 \(\int (f+g x) (a+b \log (c (d+e x)^n))^3 \, dx\) [54]

Optimal. Leaf size=265 \[ \frac {6 a b^2 (e f-d g) n^2 x}{e}-\frac {6 b^3 (e f-d g) n^3 x}{e}-\frac {3 b^3 g n^3 (d+e x)^2}{8 e^2}+\frac {6 b^3 (e f-d g) n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e^2}+\frac {3 b^2 g n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2}-\frac {3 b (e f-d g) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}-\frac {3 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}+\frac {g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 11, 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^2 g n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2}+\frac {6 a b^2 n^2 x (e f-d g)}{e}-\frac {3 b n (d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {(d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}-\frac {3 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2}+\frac {6 b^3 n^2 (d+e x) (e f-d g) \log \left (c (d+e x)^n\right )}{e^2}-\frac {3 b^3 g n^3 (d+e x)^2}{8 e^2}-\frac {6 b^3 n^3 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])^3,x]

[Out]

(6*a*b^2*(e*f - d*g)*n^2*x)/e - (6*b^3*(e*f - d*g)*n^3*x)/e - (3*b^3*g*n^3*(d + e*x)^2)/(8*e^2) + (6*b^3*(e*f
- d*g)*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e^2 + (3*b^2*g*n^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(4*e^2) -
(3*b*(e*f - d*g)*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e^2 - (3*b*g*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n
])^2)/(4*e^2) + ((e*f - d*g)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e^2 + (g*(d + e*x)^2*(a + b*Log[c*(d + e*
x)^n])^3)/(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 )^3 \, dx &=\int \left (\frac {(e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}\right ) \, dx\\ &=\frac {g \int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e}+\frac {(e f-d g) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e}\\ &=\frac {g \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 {(e f-d g) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, 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 )^3}{e^2}+\frac {g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2}-\frac {(3 b g n) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e^2}-\frac {(3 b (e f-d g) n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^2}\\ &=-\frac {3 b (e f-d g) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}-\frac {3 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {(e f-d g) (d+e x) \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 )^3}{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 ) \, dx,x,d+e x\right )}{2 e^2}+\frac {\left (6 b^2 (e f-d g) n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2}\\ &=\frac {6 a b^2 (e f-d g) n^2 x}{e}-\frac {3 b^3 g n^3 (d+e x)^2}{8 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 )}{4 e^2}-\frac {3 b (e f-d g) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}-\frac {3 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {(e f-d g) (d+e x) \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 )^3}{2 e^2}+\frac {\left (6 b^3 (e f-d g) n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2}\\ &=\frac {6 a b^2 (e f-d g) n^2 x}{e}-\frac {6 b^3 (e f-d g) n^3 x}{e}-\frac {3 b^3 g n^3 (d+e x)^2}{8 e^2}+\frac {6 b^3 (e f-d g) n^2 (d+e x) \log \left (c (d+e x)^n\right )}{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 )}{4 e^2}-\frac {3 b (e f-d g) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}-\frac {3 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {(e f-d g) (d+e x) \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 )^3}{2 e^2}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 201, normalized size = 0.76 \begin {gather*} \frac {8 (e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3+4 g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3-24 b (e f-d g) 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 )-3 b g 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 )}{8 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*(e*f - d*g)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3 + 4*g*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^3 - 24*b*(e
*f - d*g)*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]))
- 3*b*g*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]))))/(8*e^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.62, size = 11547, normalized size = 43.57

method	result	size
risch	\(\text {Expression too large to display}\)	\(11547\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)*(a+b*ln(c*(e*x+d)^n))^3,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. 687 vs. \(2 (267) = 534\).
time = 0.30, size = 687, normalized size = 2.59 \begin {gather*} \frac {1}{2} \, b^{3} g x^{2} \log \left ({\left (x e + d\right )}^{n} c\right )^{3} + \frac {3}{2} \, a b^{2} g x^{2} \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + b^{3} f x \log \left ({\left (x e + d\right )}^{n} c\right )^{3} + 3 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} a^{2} b f n e - \frac {3}{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 )} a^{2} b g n e + \frac {3}{2} \, a^{2} b g x^{2} \log \left ({\left (x e + d\right )}^{n} c\right ) + 3 \, a b^{2} f x \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + \frac {1}{2} \, a^{3} g x^{2} + 3 \, a^{2} b f x \log \left ({\left (x e + d\right )}^{n} c\right ) - 3 \, {\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 b^{2} f + {\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 )} b^{3} f + \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 )} a b^{2} g - \frac {1}{8} \, {\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 )} b^{3} g + a^{3} 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))^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 859 vs. \(2 (267) = 534\).
time = 0.38, size = 859, normalized size = 3.24 \begin {gather*} \frac {1}{8} \, {\left (4 \, {\left (b^{3} g x^{2} + 2 \, b^{3} f x\right )} e^{2} \log \left (c\right )^{3} - 4 \, {\left (b^{3} d^{2} g n^{3} - 2 \, b^{3} d f n^{3} e - {\left (b^{3} g n^{3} x^{2} + 2 \, b^{3} f n^{3} x\right )} e^{2}\right )} \log \left (x e + d\right )^{3} + 6 \, {\left (7 \, b^{3} d g n^{3} - 6 \, a b^{2} d g n^{2} + 2 \, a^{2} b d g n\right )} x e + 6 \, {\left (3 \, b^{3} d^{2} g n^{3} - 2 \, a b^{2} d^{2} g n^{2} - {\left ({\left (b^{3} g n^{3} - 2 \, a b^{2} g n^{2}\right )} x^{2} + 4 \, {\left (b^{3} f n^{3} - a b^{2} f n^{2}\right )} x\right )} e^{2} + 2 \, {\left (b^{3} d g n^{3} x - 2 \, b^{3} d f n^{3} + 2 \, a b^{2} d f n^{2}\right )} e - 2 \, {\left (b^{3} d^{2} g n^{2} - 2 \, b^{3} d f n^{2} e - {\left (b^{3} g n^{2} x^{2} + 2 \, b^{3} f n^{2} x\right )} e^{2}\right )} \log \left (c\right )\right )} \log \left (x e + d\right )^{2} + 6 \, {\left (2 \, b^{3} d g n x e - {\left ({\left (b^{3} g n - 2 \, a b^{2} g\right )} x^{2} + 4 \, {\left (b^{3} f n - a b^{2} f\right )} x\right )} e^{2}\right )} \log \left (c\right )^{2} - {\left ({\left (3 \, b^{3} g n^{3} - 6 \, a b^{2} g n^{2} + 6 \, a^{2} b g n - 4 \, a^{3} g\right )} x^{2} + 8 \, {\left (6 \, b^{3} f n^{3} - 6 \, a b^{2} f n^{2} + 3 \, a^{2} b f n - a^{3} f\right )} x\right )} e^{2} - 6 \, {\left (7 \, b^{3} d^{2} g n^{3} - 6 \, a b^{2} d^{2} g n^{2} + 2 \, a^{2} b d^{2} g n + 2 \, {\left (b^{3} d^{2} g n - 2 \, b^{3} d f n e - {\left (b^{3} g n x^{2} + 2 \, b^{3} f n x\right )} e^{2}\right )} \log \left (c\right )^{2} - {\left ({\left (b^{3} g n^{3} - 2 \, a b^{2} g n^{2} + 2 \, a^{2} b g n\right )} x^{2} + 4 \, {\left (2 \, b^{3} f n^{3} - 2 \, a b^{2} f n^{2} + a^{2} b f n\right )} x\right )} e^{2} - 2 \, {\left (4 \, b^{3} d f n^{3} - 4 \, a b^{2} d f n^{2} + 2 \, a^{2} b d f n - {\left (3 \, b^{3} d g n^{3} - 2 \, a b^{2} d g n^{2}\right )} x\right )} e - 2 \, {\left (3 \, b^{3} d^{2} g n^{2} - 2 \, a b^{2} d^{2} g n - {\left ({\left (b^{3} g n^{2} - 2 \, a b^{2} g n\right )} x^{2} + 4 \, {\left (b^{3} f n^{2} - a b^{2} f n\right )} x\right )} e^{2} + 2 \, {\left (b^{3} d g n^{2} x - 2 \, b^{3} d f n^{2} + 2 \, a b^{2} d f n\right )} e\right )} \log \left (c\right )\right )} \log \left (x e + d\right ) - 6 \, {\left (2 \, {\left (3 \, b^{3} d g n^{2} - 2 \, a b^{2} d g n\right )} x e - {\left ({\left (b^{3} g n^{2} - 2 \, a b^{2} g n + 2 \, a^{2} b g\right )} x^{2} + 4 \, {\left (2 \, b^{3} f n^{2} - 2 \, a b^{2} f n + a^{2} 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))^3,x, algorithm="fricas")

[Out]

1/8*(4*(b^3*g*x^2 + 2*b^3*f*x)*e^2*log(c)^3 - 4*(b^3*d^2*g*n^3 - 2*b^3*d*f*n^3*e - (b^3*g*n^3*x^2 + 2*b^3*f*n^
3*x)*e^2)*log(x*e + d)^3 + 6*(7*b^3*d*g*n^3 - 6*a*b^2*d*g*n^2 + 2*a^2*b*d*g*n)*x*e + 6*(3*b^3*d^2*g*n^3 - 2*a*
b^2*d^2*g*n^2 - ((b^3*g*n^3 - 2*a*b^2*g*n^2)*x^2 + 4*(b^3*f*n^3 - a*b^2*f*n^2)*x)*e^2 + 2*(b^3*d*g*n^3*x - 2*b
^3*d*f*n^3 + 2*a*b^2*d*f*n^2)*e - 2*(b^3*d^2*g*n^2 - 2*b^3*d*f*n^2*e - (b^3*g*n^2*x^2 + 2*b^3*f*n^2*x)*e^2)*lo
g(c))*log(x*e + d)^2 + 6*(2*b^3*d*g*n*x*e - ((b^3*g*n - 2*a*b^2*g)*x^2 + 4*(b^3*f*n - a*b^2*f)*x)*e^2)*log(c)^
2 - ((3*b^3*g*n^3 - 6*a*b^2*g*n^2 + 6*a^2*b*g*n - 4*a^3*g)*x^2 + 8*(6*b^3*f*n^3 - 6*a*b^2*f*n^2 + 3*a^2*b*f*n
- a^3*f)*x)*e^2 - 6*(7*b^3*d^2*g*n^3 - 6*a*b^2*d^2*g*n^2 + 2*a^2*b*d^2*g*n + 2*(b^3*d^2*g*n - 2*b^3*d*f*n*e -
(b^3*g*n*x^2 + 2*b^3*f*n*x)*e^2)*log(c)^2 - ((b^3*g*n^3 - 2*a*b^2*g*n^2 + 2*a^2*b*g*n)*x^2 + 4*(2*b^3*f*n^3 -
2*a*b^2*f*n^2 + a^2*b*f*n)*x)*e^2 - 2*(4*b^3*d*f*n^3 - 4*a*b^2*d*f*n^2 + 2*a^2*b*d*f*n - (3*b^3*d*g*n^3 - 2*a*
b^2*d*g*n^2)*x)*e - 2*(3*b^3*d^2*g*n^2 - 2*a*b^2*d^2*g*n - ((b^3*g*n^2 - 2*a*b^2*g*n)*x^2 + 4*(b^3*f*n^2 - a*b
^2*f*n)*x)*e^2 + 2*(b^3*d*g*n^2*x - 2*b^3*d*f*n^2 + 2*a*b^2*d*f*n)*e)*log(c))*log(x*e + d) - 6*(2*(3*b^3*d*g*n
^2 - 2*a*b^2*d*g*n)*x*e - ((b^3*g*n^2 - 2*a*b^2*g*n + 2*a^2*b*g)*x^2 + 4*(2*b^3*f*n^2 - 2*a*b^2*f*n + a^2*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. 836 vs. \(2 (258) = 516\).
time = 1.36, size = 836, normalized size = 3.15 \begin {gather*} \begin {cases} a^{3} f x + \frac {a^{3} g x^{2}}{2} - \frac {3 a^{2} b d^{2} g \log {\left (c \left (d + e x\right )^{n} \right )}}{2 e^{2}} + \frac {3 a^{2} b d f \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {3 a^{2} b d g n x}{2 e} - 3 a^{2} b f n x + 3 a^{2} b f x \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {3 a^{2} b g n x^{2}}{4} + \frac {3 a^{2} b g x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{2} + \frac {9 a b^{2} d^{2} g n \log {\left (c \left (d + e x\right )^{n} \right )}}{2 e^{2}} - \frac {3 a b^{2} d^{2} g \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{2 e^{2}} - \frac {6 a b^{2} d f n \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {3 a b^{2} d f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} - \frac {9 a b^{2} d g n^{2} x}{2 e} + \frac {3 a b^{2} d g n x \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + 6 a b^{2} f n^{2} x - 6 a b^{2} f n x \log {\left (c \left (d + e x\right )^{n} \right )} + 3 a b^{2} f x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + \frac {3 a b^{2} g n^{2} x^{2}}{4} - \frac {3 a b^{2} g n x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{2} + \frac {3 a b^{2} g x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{2} - \frac {21 b^{3} d^{2} g n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{4 e^{2}} + \frac {9 b^{3} d^{2} g n \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{4 e^{2}} - \frac {b^{3} d^{2} g \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{2 e^{2}} + \frac {6 b^{3} d f n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {3 b^{3} d f n \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + \frac {b^{3} d f \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e} + \frac {21 b^{3} d g n^{3} x}{4 e} - \frac {9 b^{3} d g n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}}{2 e} + \frac {3 b^{3} d g n x \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{2 e} - 6 b^{3} f n^{3} x + 6 b^{3} f n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )} - 3 b^{3} f n x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + b^{3} f x \log {\left (c \left (d + e x\right )^{n} \right )}^{3} - \frac {3 b^{3} g n^{3} x^{2}}{8} + \frac {3 b^{3} g n^{2} x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{4} - \frac {3 b^{3} g n x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{4} + \frac {b^{3} g x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{2} & \text {for}\: e \neq 0 \\\left (a + b \log {\left (c d^{n} \right )}\right )^{3} \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))**3,x)

[Out]

Piecewise((a**3*f*x + a**3*g*x**2/2 - 3*a**2*b*d**2*g*log(c*(d + e*x)**n)/(2*e**2) + 3*a**2*b*d*f*log(c*(d + e
*x)**n)/e + 3*a**2*b*d*g*n*x/(2*e) - 3*a**2*b*f*n*x + 3*a**2*b*f*x*log(c*(d + e*x)**n) - 3*a**2*b*g*n*x**2/4 +
3*a**2*b*g*x**2*log(c*(d + e*x)**n)/2 + 9*a*b**2*d**2*g*n*log(c*(d + e*x)**n)/(2*e**2) - 3*a*b**2*d**2*g*log(
c*(d + e*x)**n)**2/(2*e**2) - 6*a*b**2*d*f*n*log(c*(d + e*x)**n)/e + 3*a*b**2*d*f*log(c*(d + e*x)**n)**2/e - 9
*a*b**2*d*g*n**2*x/(2*e) + 3*a*b**2*d*g*n*x*log(c*(d + e*x)**n)/e + 6*a*b**2*f*n**2*x - 6*a*b**2*f*n*x*log(c*(
d + e*x)**n) + 3*a*b**2*f*x*log(c*(d + e*x)**n)**2 + 3*a*b**2*g*n**2*x**2/4 - 3*a*b**2*g*n*x**2*log(c*(d + e*x
)**n)/2 + 3*a*b**2*g*x**2*log(c*(d + e*x)**n)**2/2 - 21*b**3*d**2*g*n**2*log(c*(d + e*x)**n)/(4*e**2) + 9*b**3
*d**2*g*n*log(c*(d + e*x)**n)**2/(4*e**2) - b**3*d**2*g*log(c*(d + e*x)**n)**3/(2*e**2) + 6*b**3*d*f*n**2*log(
c*(d + e*x)**n)/e - 3*b**3*d*f*n*log(c*(d + e*x)**n)**2/e + b**3*d*f*log(c*(d + e*x)**n)**3/e + 21*b**3*d*g*n*
*3*x/(4*e) - 9*b**3*d*g*n**2*x*log(c*(d + e*x)**n)/(2*e) + 3*b**3*d*g*n*x*log(c*(d + e*x)**n)**2/(2*e) - 6*b**
3*f*n**3*x + 6*b**3*f*n**2*x*log(c*(d + e*x)**n) - 3*b**3*f*n*x*log(c*(d + e*x)**n)**2 + b**3*f*x*log(c*(d + e
*x)**n)**3 - 3*b**3*g*n**3*x**2/8 + 3*b**3*g*n**2*x**2*log(c*(d + e*x)**n)/4 - 3*b**3*g*n*x**2*log(c*(d + e*x)
**n)**2/4 + b**3*g*x**2*log(c*(d + e*x)**n)**3/2, Ne(e, 0)), ((a + b*log(c*d**n))**3*(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. 1351 vs. \(2 (267) = 534\).
time = 6.18, size = 1351, normalized size = 5.10 \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))^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.57, size = 511, normalized size = 1.93 \begin {gather*} {\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^3\,\left (\frac {b^3\,g\,x^2}{2}-\frac {d\,\left (b^3\,d\,g-2\,b^3\,e\,f\right )}{2\,e^2}+b^3\,f\,x\right )+\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (\frac {x\,\left (\frac {12\,a^2\,b\,d\,g+12\,a^2\,b\,e\,f-12\,b^3\,d\,g\,n^2+24\,b^3\,e\,f\,n^2-24\,a\,b^2\,e\,f\,n}{2\,e}-\frac {3\,b\,d\,g\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{e}\right )}{2}+\frac {3\,b\,g\,x^2\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{4}\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (\frac {x\,\left (\frac {6\,b^2\,\left (a\,d\,g+a\,e\,f-b\,e\,f\,n\right )}{e}-\frac {3\,b^2\,d\,g\,\left (2\,a-b\,n\right )}{e}\right )}{2}-\frac {3\,d\,\left (2\,a\,b^2\,d\,g-4\,a\,b^2\,e\,f-3\,b^3\,d\,g\,n+4\,b^3\,e\,f\,n\right )}{4\,e^2}+\frac {3\,b^2\,g\,x^2\,\left (2\,a-b\,n\right )}{4}\right )+x\,\left (\frac {4\,a^3\,d\,g+4\,a^3\,e\,f+18\,b^3\,d\,g\,n^3-24\,b^3\,e\,f\,n^3-12\,a\,b^2\,d\,g\,n^2+24\,a\,b^2\,e\,f\,n^2-12\,a^2\,b\,e\,f\,n}{4\,e}-\frac {d\,g\,\left (4\,a^3-6\,a^2\,b\,n+6\,a\,b^2\,n^2-3\,b^3\,n^3\right )}{4\,e}\right )+\frac {g\,x^2\,\left (4\,a^3-6\,a^2\,b\,n+6\,a\,b^2\,n^2-3\,b^3\,n^3\right )}{8}-\frac {\ln \left (d+e\,x\right )\,\left (6\,g\,a^2\,b\,d^2\,n-12\,e\,f\,a^2\,b\,d\,n-18\,g\,a\,b^2\,d^2\,n^2+24\,e\,f\,a\,b^2\,d\,n^2+21\,g\,b^3\,d^2\,n^3-24\,e\,f\,b^3\,d\,n^3\right )}{4\,e^2} \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))^3,x)

[Out]

log(c*(d + e*x)^n)^3*((b^3*g*x^2)/2 - (d*(b^3*d*g - 2*b^3*e*f))/(2*e^2) + b^3*f*x) + log(c*(d + e*x)^n)*((x*((
12*a^2*b*d*g + 12*a^2*b*e*f - 12*b^3*d*g*n^2 + 24*b^3*e*f*n^2 - 24*a*b^2*e*f*n)/(2*e) - (3*b*d*g*(2*a^2 + b^2*
n^2 - 2*a*b*n))/e))/2 + (3*b*g*x^2*(2*a^2 + b^2*n^2 - 2*a*b*n))/4) + log(c*(d + e*x)^n)^2*((x*((6*b^2*(a*d*g +
a*e*f - b*e*f*n))/e - (3*b^2*d*g*(2*a - b*n))/e))/2 - (3*d*(2*a*b^2*d*g - 4*a*b^2*e*f - 3*b^3*d*g*n + 4*b^3*e
*f*n))/(4*e^2) + (3*b^2*g*x^2*(2*a - b*n))/4) + x*((4*a^3*d*g + 4*a^3*e*f + 18*b^3*d*g*n^3 - 24*b^3*e*f*n^3 -
12*a*b^2*d*g*n^2 + 24*a*b^2*e*f*n^2 - 12*a^2*b*e*f*n)/(4*e) - (d*g*(4*a^3 - 3*b^3*n^3 + 6*a*b^2*n^2 - 6*a^2*b*
n))/(4*e)) + (g*x^2*(4*a^3 - 3*b^3*n^3 + 6*a*b^2*n^2 - 6*a^2*b*n))/8 - (log(d + e*x)*(21*b^3*d^2*g*n^3 + 6*a^2
*b*d^2*g*n - 24*b^3*d*e*f*n^3 - 18*a*b^2*d^2*g*n^2 - 12*a^2*b*d*e*f*n + 24*a*b^2*d*e*f*n^2))/(4*e^2)

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