3.3.42 \(\int (e x)^q (a+b \log (c (d x^m)^n))^p \, dx\) [242]

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

[Out]

(e*x)^(1+q)*GAMMA(1+p,-(1+q)*(a+b*ln(c*(d*x^m)^n))/b/m/n)*(a+b*ln(c*(d*x^m)^n))^p/e/exp(a*(1+q)/b/m/n)/(1+q)/(
(c*(d*x^m)^n)^((1+q)/m/n))/((-(1+q)*(a+b*ln(c*(d*x^m)^n))/b/m/n)^p)

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Rubi [A]
time = 0.13, antiderivative size = 134, normalized size of antiderivative = 1.00, 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 = {2347, 2212, 2495} \begin {gather*} \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)} \end {gather*}

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 2212

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

Rule 2347

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)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

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 )^p \, dx &=\text {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 )\\ &=\text {Subst}\left (\frac {\left ((e x)^{1+q} \left (c d^n x^{m n}\right )^{-\frac {1+q}{m n}}\right ) \text {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*}

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Mathematica [A]
time = 0.17, size = 133, normalized size = 0.99 \begin {gather*} \frac {e^{-\frac {(1+q) \left (a-b m n \log (x)+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}} x^{-q} (e x)^q \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}}{1+q} \end {gather*}

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)

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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{q} \left (a +b \ln \left (c \left (d \,x^{m}\right )^{n}\right )\right )^{p}\, dx\]

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)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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))^p,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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))^p,x, algorithm="fricas")

[Out]

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

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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 )^{p}\, 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))**p,x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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))^p,x, algorithm="giac")

[Out]

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

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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 )}^p \,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))^p,x)

[Out]

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

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