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

Optimal. Leaf size=79 \[ \frac{(e x)^{m+1} \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )}{e (m+1)}-\frac{b p (e x)^{m+1} \left (d x^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+1) \log \left (d x^n\right )}{n}\right )}{e (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0608507, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2522, 2310, 2178} \[ \frac{(e x)^{m+1} \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )}{e (m+1)}-\frac{b p (e x)^{m+1} \left (d x^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+1) \log \left (d x^n\right )}{n}\right )}{e (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2522

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

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 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

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

Mathematica [A]  time = 0.156762, size = 59, normalized size = 0.75 \[ \frac{x (e x)^m \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )-b p \left (d x^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+1) \log \left (d x^n\right )}{n}\right )\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.185, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( a+b\ln \left ( c \left ( \ln \left ( d{x}^{n} \right ) \right ) ^{p} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.97843, size = 261, normalized size = 3.3 \begin{align*} \frac{b p x e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) - b p{\rm Ei}\left (\frac{{\left (m + 1\right )} n \log \left (x\right ) +{\left (m + 1\right )} \log \left (d\right )}{n}\right ) e^{\left (\frac{m n \log \left (e\right ) -{\left (m + 1\right )} \log \left (d\right )}{n}\right )} +{\left (b x \log \left (c\right ) + a x\right )} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )}}{m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*p*x*e^(m*log(e) + m*log(x))*log(n*log(x) + log(d)) - b*p*Ei(((m + 1)*n*log(x) + (m + 1)*log(d))/n)*e^((m*n*
log(e) - (m + 1)*log(d))/n) + (b*x*log(c) + a*x)*e^(m*log(e) + m*log(x)))/(m + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.42738, size = 150, normalized size = 1.9 \begin{align*} \frac{b p x x^{m} e^{m} \log \left (n \log \left (x\right ) + \log \left (d\right )\right )}{m + 1} - \frac{b n p{\rm Ei}\left (m \log \left (x\right ) + \frac{m \log \left (d\right )}{n} + \frac{\log \left (d\right )}{n} + \log \left (x\right )\right ) e^{m}}{d^{\frac{m}{n}} d^{\left (\frac{1}{n}\right )} m n + d^{\frac{m}{n}} d^{\left (\frac{1}{n}\right )} n} + \frac{b x x^{m} e^{m} \log \left (c\right )}{m + 1} + \frac{a x x^{m} e^{m}}{m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*p*x*x^m*e^m*log(n*log(x) + log(d))/(m + 1) - b*n*p*Ei(m*log(x) + m*log(d)/n + log(d)/n + log(x))*e^m/(d^(m/n
)*d^(1/n)*m*n + d^(m/n)*d^(1/n)*n) + b*x*x^m*e^m*log(c)/(m + 1) + a*x*x^m*e^m/(m + 1)