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

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Rubi [A]  time = 0.06, antiderivative size = 79, normalized size of antiderivative = 1.00, 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 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

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 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]

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*}

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Mathematica [A]  time = 0.17, 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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fricas [A]  time = 0.45, size = 90, normalized size = 1.14 \[ \frac {b p x e^{\left (m \log \relax (e) + m \log \relax (x)\right )} \log \left (n \log \relax (x) + \log \relax (d)\right ) - b p {\rm Ei}\left (\frac {{\left (m + 1\right )} n \log \relax (x) + {\left (m + 1\right )} \log \relax (d)}{n}\right ) e^{\left (\frac {m n \log \relax (e) - {\left (m + 1\right )} \log \relax (d)}{n}\right )} + {\left (b x \log \relax (c) + a x\right )} e^{\left (m \log \relax (e) + m \log \relax (x)\right )}}{m + 1} \]

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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giac [A]  time = 0.21, size = 111, normalized size = 1.41 \[ \frac {b p x x^{m} e^{m} \log \left (n \log \relax (x) + \log \relax (d)\right )}{m + 1} - \frac {b n p {\rm Ei}\left (m \log \relax (x) + \frac {m \log \relax (d)}{n} + \frac {\log \relax (d)}{n} + \log \relax (x)\right ) e^{m}}{d^{\frac {m}{n}} d^{\left (\frac {1}{n}\right )} m n + d^{\frac {m}{n}} d^{\left (\frac {1}{n}\right )} n} + \frac {b x x^{m} e^{m} \log \relax (c)}{m + 1} + \frac {a x x^{m} e^{m}}{m + 1} \]

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)

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maple [F]  time = 2.59, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \ln \left (d \,x^{n}\right )^{p}\right )+a \right ) \left (e x \right )^{m}\, dx \]

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]  time = 0.00, size = 0, normalized size = 0.00 \[ -{\left (e^{m} n p \int \frac {x^{m}}{{\left (m + 1\right )} \log \relax (d) + {\left (m + 1\right )} \log \left (x^{n}\right )}\,{d x} - \frac {e^{m} x x^{m} \log \relax (c) + e^{m} x x^{m} \log \left ({\left (\log \relax (d) + \log \left (x^{n}\right )\right )}^{p}\right )}{m + 1}\right )} b + \frac {\left (e x\right )^{m + 1} a}{e {\left (m + 1\right )}} \]

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]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (e\,x\right )}^m\,\left (a+b\,\ln \left (c\,{\ln \left (d\,x^n\right )}^p\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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