3.242 \(\int (e x)^q (a+b \log (c (d x^m)^n))^p \, dx\)

Optimal. Leaf size=134 \[ \frac{(e x)^{q+1} e^{-\frac{a (q+1)}{b m n}} \left (c \left (d x^m\right )^n\right )^{-\frac{q+1}{m n}} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \left (-\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )}{e (q+1)} \]

[Out]

((e*x)^(1 + q)*Gamma[1 + p, -(((1 + q)*(a + b*Log[c*(d*x^m)^n]))/(b*m*n))]*(a + b*Log[c*(d*x^m)^n])^p)/(e*E^((
a*(1 + q))/(b*m*n))*(1 + q)*(c*(d*x^m)^n)^((1 + q)/(m*n))*(-(((1 + q)*(a + b*Log[c*(d*x^m)^n]))/(b*m*n)))^p)

________________________________________________________________________________________

Rubi [A]  time = 0.190116, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2310, 2181, 2445} \[ \frac{(e x)^{q+1} e^{-\frac{a (q+1)}{b m n}} \left (c \left (d x^m\right )^n\right )^{-\frac{q+1}{m n}} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \left (-\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )}{e (q+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((e*x)^(1 + q)*Gamma[1 + p, -(((1 + q)*(a + b*Log[c*(d*x^m)^n]))/(b*m*n))]*(a + b*Log[c*(d*x^m)^n])^p)/(e*E^((
a*(1 + q))/(b*m*n))*(1 + q)*(c*(d*x^m)^n)^((1 + q)/(m*n))*(-(((1 + q)*(a + b*Log[c*(d*x^m)^n]))/(b*m*n)))^p)

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 2445

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 )^p \, dx &=\operatorname{Subst}\left (\int (e x)^q \left (a+b \log \left (c d^n x^{m n}\right )\right )^p \, dx,c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=\operatorname{Subst}\left (\frac{\left ((e x)^{1+q} \left (c d^n x^{m n}\right )^{-\frac{1+q}{m n}}\right ) \operatorname{Subst}\left (\int e^{\frac{(1+q) x}{m n}} (a+b x)^p \, dx,x,\log \left (c d^n x^{m n}\right )\right )}{e m n},c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=\frac{e^{-\frac{a (1+q)}{b m n}} (e x)^{1+q} \left (c \left (d x^m\right )^n\right )^{-\frac{1+q}{m n}} \Gamma \left (1+p,-\frac{(1+q) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right ) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \left (-\frac{(1+q) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )^{-p}}{e (1+q)}\\ \end{align*}

Mathematica [A]  time = 0.193908, size = 133, normalized size = 0.99 \[ \frac{x^{-q} (e x)^q \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \exp \left (-\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )-b m n \log (x)\right )}{b m n}\right ) \left (-\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )}{q+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((e*x)^q*Gamma[1 + p, -(((1 + q)*(a + b*Log[c*(d*x^m)^n]))/(b*m*n))]*(a + b*Log[c*(d*x^m)^n])^p)/(E^(((1 + q)*
(a - b*m*n*Log[x] + b*Log[c*(d*x^m)^n]))/(b*m*n))*(1 + q)*x^q*(-(((1 + q)*(a + b*Log[c*(d*x^m)^n]))/(b*m*n)))^
p)

________________________________________________________________________________________

Maple [F]  time = 0.213, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{q} \left ( a+b\ln \left ( c \left ( d{x}^{m} \right ) ^{n} \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (e x\right )^{q}{\left (b \log \left (\left (d x^{m}\right )^{n} c\right ) + a\right )}^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{q}{\left (b \log \left (\left (d x^{m}\right )^{n} c\right ) + a\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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