3.241 \(\int \frac{(e x)^q}{(a+b \log (c (d x^m)^n))^2} \, dx\)

Optimal. Leaf size=127 \[ \frac{(q+1) (e x)^{q+1} e^{-\frac{a (q+1)}{b m n}} \left (c \left (d x^m\right )^n\right )^{-\frac{q+1}{m n}} \text{Ei}\left (\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )}{b^2 e m^2 n^2}-\frac{(e x)^{q+1}}{b e m n \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )} \]

[Out]

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

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Rubi [A]  time = 0.24181, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2306, 2310, 2178, 2445} \[ \frac{(q+1) (e x)^{q+1} e^{-\frac{a (q+1)}{b m n}} \left (c \left (d x^m\right )^n\right )^{-\frac{q+1}{m n}} \text{Ei}\left (\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )}{b^2 e m^2 n^2}-\frac{(e x)^{q+1}}{b e m n \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^q/(a + b*Log[c*(d*x^m)^n])^2,x]

[Out]

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

Rule 2306

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

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

Mathematica [A]  time = 0.296765, size = 112, normalized size = 0.88 \[ \frac{(e x)^q \left ((q+1) x^{-q} \exp \left (-\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )-b m n \log (x)\right )}{b m n}\right ) \text{Ei}\left (\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )-\frac{b m n x}{a+b \log \left (c \left (d x^m\right )^n\right )}\right )}{b^2 m^2 n^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^q/(a + b*Log[c*(d*x^m)^n])^2,x]

[Out]

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

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{q}}{ \left ( a+b\ln \left ( c \left ( d{x}^{m} \right ) ^{n} \right ) \right ) ^{2}}}\, 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))^2,x)

[Out]

int((e*x)^q/(a+b*ln(c*(d*x^m)^n))^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{q}{\left (q + 1\right )} \int \frac{x^{q}}{b^{2} m n \log \left ({\left (x^{m}\right )}^{n}\right ) + a b m n +{\left (m n \log \left (c\right ) + m n \log \left (d^{n}\right )\right )} b^{2}}\,{d x} - \frac{e^{q} x x^{q}}{b^{2} m n \log \left ({\left (x^{m}\right )}^{n}\right ) + a b m n +{\left (m n \log \left (c\right ) + m n \log \left (d^{n}\right )\right )} b^{2}} \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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.920079, size = 497, normalized size = 3.91 \begin{align*} -\frac{b m n x e^{\left (q \log \left (e\right ) + q \log \left (x\right )\right )} -{\left (a q +{\left (b q + b\right )} \log \left (c\right ) +{\left (b n q + b n\right )} \log \left (d\right ) +{\left (b m n q + b m n\right )} \log \left (x\right ) + a\right )}{\rm Ei}\left (\frac{a q +{\left (b q + b\right )} \log \left (c\right ) +{\left (b n q + b n\right )} \log \left (d\right ) +{\left (b m n q + b m n\right )} \log \left (x\right ) + a}{b m n}\right ) e^{\left (\frac{b m n q \log \left (e\right ) - a q -{\left (b q + b\right )} \log \left (c\right ) -{\left (b n q + b n\right )} \log \left (d\right ) - a}{b m n}\right )}}{b^{3} m^{3} n^{3} \log \left (x\right ) + b^{3} m^{2} n^{3} \log \left (d\right ) + b^{3} m^{2} n^{2} \log \left (c\right ) + a b^{2} m^{2} n^{2}} \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))^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{q}}{\left (a + b \log{\left (c \left (d x^{m}\right )^{n} \right )}\right )^{2}}\, dx \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))**2,x)

[Out]

Integral((e*x)**q/(a + b*log(c*(d*x**m)**n))**2, x)

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Giac [B]  time = 1.68669, size = 2079, normalized size = 16.37 \begin{align*} \text{result too large to display} \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))^2,x, algorithm="giac")

[Out]

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