3.3.37 \(\int (e x)^q (a+b \log (c (d x^m)^n))^3 \, dx\) [237]

Optimal. Leaf size=135 \[ -\frac {6 b^3 m^3 n^3 (e x)^{1+q}}{e (1+q)^4}+\frac {6 b^2 m^2 n^2 (e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{e (1+q)^3}-\frac {3 b m n (e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2}{e (1+q)^2}+\frac {(e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3}{e (1+q)} \]

[Out]

-6*b^3*m^3*n^3*(e*x)^(1+q)/e/(1+q)^4+6*b^2*m^2*n^2*(e*x)^(1+q)*(a+b*ln(c*(d*x^m)^n))/e/(1+q)^3-3*b*m*n*(e*x)^(
1+q)*(a+b*ln(c*(d*x^m)^n))^2/e/(1+q)^2+(e*x)^(1+q)*(a+b*ln(c*(d*x^m)^n))^3/e/(1+q)

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Rubi [A]
time = 0.15, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2342, 2341, 2495} \begin {gather*} \frac {6 b^2 m^2 n^2 (e x)^{q+1} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{e (q+1)^3}+\frac {(e x)^{q+1} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3}{e (q+1)}-\frac {3 b m n (e x)^{q+1} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2}{e (q+1)^2}-\frac {6 b^3 m^3 n^3 (e x)^{q+1}}{e (q+1)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*x)^q*(a + b*Log[c*(d*x^m)^n])^3,x]

[Out]

(-6*b^3*m^3*n^3*(e*x)^(1 + q))/(e*(1 + q)^4) + (6*b^2*m^2*n^2*(e*x)^(1 + q)*(a + b*Log[c*(d*x^m)^n]))/(e*(1 +
q)^3) - (3*b*m*n*(e*x)^(1 + q)*(a + b*Log[c*(d*x^m)^n])^2)/(e*(1 + q)^2) + ((e*x)^(1 + q)*(a + b*Log[c*(d*x^m)
^n])^3)/(e*(1 + q))

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 2495

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin {align*} \int (e x)^q \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3 \, dx &=\text {Subst}\left (\int (e x)^q \left (a+b \log \left (c d^n x^{m n}\right )\right )^3 \, dx,c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=\frac {(e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3}{e (1+q)}-\text {Subst}\left (\frac {(3 b m n) \int (e x)^q \left (a+b \log \left (c d^n x^{m n}\right )\right )^2 \, dx}{1+q},c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=-\frac {3 b m n (e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2}{e (1+q)^2}+\frac {(e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3}{e (1+q)}+\text {Subst}\left (\frac {\left (6 b^2 m^2 n^2\right ) \int (e x)^q \left (a+b \log \left (c d^n x^{m n}\right )\right ) \, dx}{(1+q)^2},c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=-\frac {6 b^3 m^3 n^3 (e x)^{1+q}}{e (1+q)^4}+\frac {6 b^2 m^2 n^2 (e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{e (1+q)^3}-\frac {3 b m n (e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2}{e (1+q)^2}+\frac {(e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3}{e (1+q)}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 91, normalized size = 0.67 \begin {gather*} \frac {x (e x)^q \left (\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3-\frac {3 b m n \left ((1+q)^2 \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2+2 b m n \left (b m n-(1+q) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )\right )\right )}{(1+q)^3}\right )}{1+q} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(e*x)^q*((a + b*Log[c*(d*x^m)^n])^3 - (3*b*m*n*((1 + q)^2*(a + b*Log[c*(d*x^m)^n])^2 + 2*b*m*n*(b*m*n - (1
+ q)*(a + b*Log[c*(d*x^m)^n]))))/(1 + q)^3))/(1 + q)

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{q} \left (a +b \ln \left (c \left (d \,x^{m}\right )^{n}\right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^q*(a+b*ln(c*(d*x^m)^n))^3,x)

[Out]

int((e*x)^q*(a+b*ln(c*(d*x^m)^n))^3,x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (135) = 270\).
time = 0.29, size = 285, normalized size = 2.11 \begin {gather*} \frac {\left (x e\right )^{q + 1} b^{3} e^{\left (-1\right )} \log \left (\left (d x^{m}\right )^{n} c\right )^{3}}{q + 1} + \frac {3 \, \left (x e\right )^{q + 1} a b^{2} e^{\left (-1\right )} \log \left (\left (d x^{m}\right )^{n} c\right )^{2}}{q + 1} - \frac {3 \, a^{2} b m n x e^{\left (q \log \left (x\right ) + q\right )}}{{\left (q + 1\right )}^{2}} + \frac {3 \, \left (x e\right )^{q + 1} a^{2} b e^{\left (-1\right )} \log \left (\left (d x^{m}\right )^{n} c\right )}{q + 1} + 6 \, {\left (\frac {m^{2} n^{2} x e^{\left (q \log \left (x\right ) + q\right )}}{{\left (q + 1\right )}^{3}} - \frac {m n x e^{\left (q \log \left (x\right ) + q\right )} \log \left (\left (d x^{m}\right )^{n} c\right )}{{\left (q + 1\right )}^{2}}\right )} a b^{2} - 3 \, {\left (\frac {m n x e^{\left (q \log \left (x\right ) + q\right )} \log \left (\left (d x^{m}\right )^{n} c\right )^{2}}{{\left (q + 1\right )}^{2}} + \frac {2 \, {\left (\frac {m^{2} n^{2} x e^{\left (q \log \left (x\right ) + q + 1\right )}}{{\left (q + 1\right )}^{3}} - \frac {m n x e^{\left (q \log \left (x\right ) + q + 1\right )} \log \left (\left (d x^{m}\right )^{n} c\right )}{{\left (q + 1\right )}^{2}}\right )} m n e^{\left (-1\right )}}{q + 1}\right )} b^{3} + \frac {\left (x e\right )^{q + 1} a^{3} e^{\left (-1\right )}}{q + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^q*(a+b*log(c*(d*x^m)^n))^3,x, algorithm="maxima")

[Out]

(x*e)^(q + 1)*b^3*e^(-1)*log((d*x^m)^n*c)^3/(q + 1) + 3*(x*e)^(q + 1)*a*b^2*e^(-1)*log((d*x^m)^n*c)^2/(q + 1)
- 3*a^2*b*m*n*x*e^(q*log(x) + q)/(q + 1)^2 + 3*(x*e)^(q + 1)*a^2*b*e^(-1)*log((d*x^m)^n*c)/(q + 1) + 6*(m^2*n^
2*x*e^(q*log(x) + q)/(q + 1)^3 - m*n*x*e^(q*log(x) + q)*log((d*x^m)^n*c)/(q + 1)^2)*a*b^2 - 3*(m*n*x*e^(q*log(
x) + q)*log((d*x^m)^n*c)^2/(q + 1)^2 + 2*(m^2*n^2*x*e^(q*log(x) + q + 1)/(q + 1)^3 - m*n*x*e^(q*log(x) + q + 1
)*log((d*x^m)^n*c)/(q + 1)^2)*m*n*e^(-1)/(q + 1))*b^3 + (x*e)^(q + 1)*a^3*e^(-1)/(q + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1355 vs. \(2 (135) = 270\).
time = 0.35, size = 1355, normalized size = 10.04 \begin {gather*} \frac {{\left ({\left (b^{3} q^{3} + 3 \, b^{3} q^{2} + 3 \, b^{3} q + b^{3}\right )} x \log \left (c\right )^{3} + {\left (b^{3} n^{3} q^{3} + 3 \, b^{3} n^{3} q^{2} + 3 \, b^{3} n^{3} q + b^{3} n^{3}\right )} x \log \left (d\right )^{3} + {\left (b^{3} m^{3} n^{3} q^{3} + 3 \, b^{3} m^{3} n^{3} q^{2} + 3 \, b^{3} m^{3} n^{3} q + b^{3} m^{3} n^{3}\right )} x \log \left (x\right )^{3} + 3 \, {\left (a b^{2} q^{3} - b^{3} m n + a b^{2} - {\left (b^{3} m n - 3 \, a b^{2}\right )} q^{2} - {\left (2 \, b^{3} m n - 3 \, a b^{2}\right )} q\right )} x \log \left (c\right )^{2} + 3 \, {\left (2 \, b^{3} m^{2} n^{2} + a^{2} b q^{3} - 2 \, a b^{2} m n + a^{2} b - {\left (2 \, a b^{2} m n - 3 \, a^{2} b\right )} q^{2} + {\left (2 \, b^{3} m^{2} n^{2} - 4 \, a b^{2} m n + 3 \, a^{2} b\right )} q\right )} x \log \left (c\right ) + 3 \, {\left ({\left (b^{3} n^{2} q^{3} + 3 \, b^{3} n^{2} q^{2} + 3 \, b^{3} n^{2} q + b^{3} n^{2}\right )} x \log \left (c\right ) + {\left (a b^{2} n^{2} q^{3} - b^{3} m n^{3} + a b^{2} n^{2} - {\left (b^{3} m n^{3} - 3 \, a b^{2} n^{2}\right )} q^{2} - {\left (2 \, b^{3} m n^{3} - 3 \, a b^{2} n^{2}\right )} q\right )} x\right )} \log \left (d\right )^{2} + 3 \, {\left ({\left (b^{3} m^{2} n^{2} q^{3} + 3 \, b^{3} m^{2} n^{2} q^{2} + 3 \, b^{3} m^{2} n^{2} q + b^{3} m^{2} n^{2}\right )} x \log \left (c\right ) + {\left (b^{3} m^{2} n^{3} q^{3} + 3 \, b^{3} m^{2} n^{3} q^{2} + 3 \, b^{3} m^{2} n^{3} q + b^{3} m^{2} n^{3}\right )} x \log \left (d\right ) + {\left (a b^{2} m^{2} n^{2} q^{3} - b^{3} m^{3} n^{3} + a b^{2} m^{2} n^{2} - {\left (b^{3} m^{3} n^{3} - 3 \, a b^{2} m^{2} n^{2}\right )} q^{2} - {\left (2 \, b^{3} m^{3} n^{3} - 3 \, a b^{2} m^{2} n^{2}\right )} q\right )} x\right )} \log \left (x\right )^{2} - {\left (6 \, b^{3} m^{3} n^{3} - 6 \, a b^{2} m^{2} n^{2} - a^{3} q^{3} + 3 \, a^{2} b m n - a^{3} + 3 \, {\left (a^{2} b m n - a^{3}\right )} q^{2} - 3 \, {\left (2 \, a b^{2} m^{2} n^{2} - 2 \, a^{2} b m n + a^{3}\right )} q\right )} x + 3 \, {\left ({\left (b^{3} n q^{3} + 3 \, b^{3} n q^{2} + 3 \, b^{3} n q + b^{3} n\right )} x \log \left (c\right )^{2} + 2 \, {\left (a b^{2} n q^{3} - b^{3} m n^{2} + a b^{2} n - {\left (b^{3} m n^{2} - 3 \, a b^{2} n\right )} q^{2} - {\left (2 \, b^{3} m n^{2} - 3 \, a b^{2} n\right )} q\right )} x \log \left (c\right ) + {\left (2 \, b^{3} m^{2} n^{3} + a^{2} b n q^{3} - 2 \, a b^{2} m n^{2} + a^{2} b n - {\left (2 \, a b^{2} m n^{2} - 3 \, a^{2} b n\right )} q^{2} + {\left (2 \, b^{3} m^{2} n^{3} - 4 \, a b^{2} m n^{2} + 3 \, a^{2} b n\right )} q\right )} x\right )} \log \left (d\right ) + 3 \, {\left ({\left (b^{3} m n q^{3} + 3 \, b^{3} m n q^{2} + 3 \, b^{3} m n q + b^{3} m n\right )} x \log \left (c\right )^{2} + {\left (b^{3} m n^{3} q^{3} + 3 \, b^{3} m n^{3} q^{2} + 3 \, b^{3} m n^{3} q + b^{3} m n^{3}\right )} x \log \left (d\right )^{2} + 2 \, {\left (a b^{2} m n q^{3} - b^{3} m^{2} n^{2} + a b^{2} m n - {\left (b^{3} m^{2} n^{2} - 3 \, a b^{2} m n\right )} q^{2} - {\left (2 \, b^{3} m^{2} n^{2} - 3 \, a b^{2} m n\right )} q\right )} x \log \left (c\right ) + {\left (2 \, b^{3} m^{3} n^{3} + a^{2} b m n q^{3} - 2 \, a b^{2} m^{2} n^{2} + a^{2} b m n - {\left (2 \, a b^{2} m^{2} n^{2} - 3 \, a^{2} b m n\right )} q^{2} + {\left (2 \, b^{3} m^{3} n^{3} - 4 \, a b^{2} m^{2} n^{2} + 3 \, a^{2} b m n\right )} q\right )} x + 2 \, {\left ({\left (b^{3} m n^{2} q^{3} + 3 \, b^{3} m n^{2} q^{2} + 3 \, b^{3} m n^{2} q + b^{3} m n^{2}\right )} x \log \left (c\right ) + {\left (a b^{2} m n^{2} q^{3} - b^{3} m^{2} n^{3} + a b^{2} m n^{2} - {\left (b^{3} m^{2} n^{3} - 3 \, a b^{2} m n^{2}\right )} q^{2} - {\left (2 \, b^{3} m^{2} n^{3} - 3 \, a b^{2} m n^{2}\right )} q\right )} x\right )} \log \left (d\right )\right )} \log \left (x\right )\right )} e^{\left (q \log \left (x\right ) + q\right )}}{q^{4} + 4 \, q^{3} + 6 \, q^{2} + 4 \, q + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^q*(a+b*log(c*(d*x^m)^n))^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e x\right )^{q} \left (a + b \log {\left (c \left (d x^{m}\right )^{n} \right )}\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**q*(a+b*ln(c*(d*x**m)**n))**3,x)

[Out]

Integral((e*x)**q*(a + b*log(c*(d*x**m)**n))**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1811 vs. \(2 (135) = 270\).
time = 4.39, size = 1811, normalized size = 13.41 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^q*(a+b*log(c*(d*x^m)^n))^3,x, algorithm="giac")

[Out]

b^3*m^3*n^3*q^3*x*x^q*e^q*log(x)^3/(q^4 + 4*q^3 + 6*q^2 + 4*q + 1) + 3*b^3*m^3*n^3*q^2*x*x^q*e^q*log(x)^3/(q^4
+ 4*q^3 + 6*q^2 + 4*q + 1) - 3*b^3*m^3*n^3*q^2*x*x^q*e^q*log(x)^2/(q^4 + 4*q^3 + 6*q^2 + 4*q + 1) + 3*b^3*m^2
*n^3*q^2*x*x^q*e^q*log(d)*log(x)^2/(q^3 + 3*q^2 + 3*q + 1) + 3*b^3*m^3*n^3*q*x*x^q*e^q*log(x)^3/(q^4 + 4*q^3 +
6*q^2 + 4*q + 1) - 6*b^3*m^3*n^3*q*x*x^q*e^q*log(x)^2/(q^4 + 4*q^3 + 6*q^2 + 4*q + 1) + 3*b^3*m^2*n^2*q^2*x*x
^q*e^q*log(c)*log(x)^2/(q^3 + 3*q^2 + 3*q + 1) + 6*b^3*m^2*n^3*q*x*x^q*e^q*log(d)*log(x)^2/(q^3 + 3*q^2 + 3*q
+ 1) + b^3*m^3*n^3*x*x^q*e^q*log(x)^3/(q^4 + 4*q^3 + 6*q^2 + 4*q + 1) + 6*b^3*m^3*n^3*q*x*x^q*e^q*log(x)/(q^4
+ 4*q^3 + 6*q^2 + 4*q + 1) - 6*b^3*m^2*n^3*q*x*x^q*e^q*log(d)*log(x)/(q^3 + 3*q^2 + 3*q + 1) + 3*b^3*m*n^3*q*x
*x^q*e^q*log(d)^2*log(x)/(q^2 + 2*q + 1) - 3*b^3*m^3*n^3*x*x^q*e^q*log(x)^2/(q^4 + 4*q^3 + 6*q^2 + 4*q + 1) +
3*a*b^2*m^2*n^2*q^2*x*x^q*e^q*log(x)^2/(q^3 + 3*q^2 + 3*q + 1) + 6*b^3*m^2*n^2*q*x*x^q*e^q*log(c)*log(x)^2/(q^
3 + 3*q^2 + 3*q + 1) + 3*b^3*m^2*n^3*x*x^q*e^q*log(d)*log(x)^2/(q^3 + 3*q^2 + 3*q + 1) + 6*b^3*m^3*n^3*x*x^q*e
^q*log(x)/(q^4 + 4*q^3 + 6*q^2 + 4*q + 1) - 6*b^3*m^2*n^2*q*x*x^q*e^q*log(c)*log(x)/(q^3 + 3*q^2 + 3*q + 1) -
6*b^3*m^2*n^3*x*x^q*e^q*log(d)*log(x)/(q^3 + 3*q^2 + 3*q + 1) + 6*b^3*m*n^2*q*x*x^q*e^q*log(c)*log(d)*log(x)/(
q^2 + 2*q + 1) + 3*b^3*m*n^3*x*x^q*e^q*log(d)^2*log(x)/(q^2 + 2*q + 1) + 6*a*b^2*m^2*n^2*q*x*x^q*e^q*log(x)^2/
(q^3 + 3*q^2 + 3*q + 1) + 3*b^3*m^2*n^2*x*x^q*e^q*log(c)*log(x)^2/(q^3 + 3*q^2 + 3*q + 1) - 6*b^3*m^3*n^3*x*x^
q*e^q/(q^4 + 4*q^3 + 6*q^2 + 4*q + 1) + 6*b^3*m^2*n^3*x*x^q*e^q*log(d)/(q^3 + 3*q^2 + 3*q + 1) - 3*b^3*m*n^3*x
*x^q*e^q*log(d)^2/(q^2 + 2*q + 1) + b^3*n^3*x*x^q*e^q*log(d)^3/(q + 1) - 6*a*b^2*m^2*n^2*q*x*x^q*e^q*log(x)/(q
^3 + 3*q^2 + 3*q + 1) - 6*b^3*m^2*n^2*x*x^q*e^q*log(c)*log(x)/(q^3 + 3*q^2 + 3*q + 1) + 3*b^3*m*n*q*x*x^q*e^q*
log(c)^2*log(x)/(q^2 + 2*q + 1) + 6*a*b^2*m*n^2*q*x*x^q*e^q*log(d)*log(x)/(q^2 + 2*q + 1) + 6*b^3*m*n^2*x*x^q*
e^q*log(c)*log(d)*log(x)/(q^2 + 2*q + 1) + 3*a*b^2*m^2*n^2*x*x^q*e^q*log(x)^2/(q^3 + 3*q^2 + 3*q + 1) + 6*b^3*
m^2*n^2*x*x^q*e^q*log(c)/(q^3 + 3*q^2 + 3*q + 1) - 6*b^3*m*n^2*x*x^q*e^q*log(c)*log(d)/(q^2 + 2*q + 1) + 3*b^3
*n^2*x*x^q*e^q*log(c)*log(d)^2/(q + 1) - 6*a*b^2*m^2*n^2*x*x^q*e^q*log(x)/(q^3 + 3*q^2 + 3*q + 1) + 6*a*b^2*m*
n*q*x*x^q*e^q*log(c)*log(x)/(q^2 + 2*q + 1) + 3*b^3*m*n*x*x^q*e^q*log(c)^2*log(x)/(q^2 + 2*q + 1) + 6*a*b^2*m*
n^2*x*x^q*e^q*log(d)*log(x)/(q^2 + 2*q + 1) + 6*a*b^2*m^2*n^2*x*x^q*e^q/(q^3 + 3*q^2 + 3*q + 1) - 3*b^3*m*n*x*
x^q*e^q*log(c)^2/(q^2 + 2*q + 1) - 6*a*b^2*m*n^2*x*x^q*e^q*log(d)/(q^2 + 2*q + 1) + 3*b^3*n*x*x^q*e^q*log(c)^2
*log(d)/(q + 1) + 3*a*b^2*n^2*x*x^q*e^q*log(d)^2/(q + 1) + 3*a^2*b*m*n*q*x*x^q*e^q*log(x)/(q^2 + 2*q + 1) + 6*
a*b^2*m*n*x*x^q*e^q*log(c)*log(x)/(q^2 + 2*q + 1) - 6*a*b^2*m*n*x*x^q*e^q*log(c)/(q^2 + 2*q + 1) + b^3*x*x^q*e
^q*log(c)^3/(q + 1) + 6*a*b^2*n*x*x^q*e^q*log(c)*log(d)/(q + 1) + 3*a^2*b*m*n*x*x^q*e^q*log(x)/(q^2 + 2*q + 1)
- 3*a^2*b*m*n*x*x^q*e^q/(q^2 + 2*q + 1) + 3*a*b^2*x*x^q*e^q*log(c)^2/(q + 1) + 3*a^2*b*n*x*x^q*e^q*log(d)/(q
+ 1) + 3*a^2*b*x*x^q*e^q*log(c)/(q + 1) + a^3*x*x^q*e^q/(q + 1)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (e\,x\right )}^q\,{\left (a+b\,\ln \left (c\,{\left (d\,x^m\right )}^n\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^q*(a + b*log(c*(d*x^m)^n))^3,x)

[Out]

int((e*x)^q*(a + b*log(c*(d*x^m)^n))^3, x)

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