3.234 \(\int \frac{h+i x}{(f+g x) (a+b \log (c (d+e x)^n))} \, dx\)

Optimal. Leaf size=106 \[ \frac{(g h-f i) \text{Unintegrable}\left (\frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right )}{g}+\frac{i e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e g n} \]

[Out]

(i*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(b*e*E^(a/(b*n))*g*n*(c*(d + e*x)^n)^n^(-1)) + (
(g*h - f*i)*Unintegrable[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])), x])/g

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Rubi [A]  time = 0.176109, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(h + i*x)/((f + g*x)*(a + b*Log[c*(d + e*x)^n])),x]

[Out]

(i*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(b*e*E^(a/(b*n))*g*n*(c*(d + e*x)^n)^n^(-1)) + (
(g*h - f*i)*Defer[Int][1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])), x])/g

Rubi steps

\begin{align*} \int \frac{h+234 x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx &=\int \left (\frac{234}{g \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{-234 f+g h}{g (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx\\ &=\frac{234 \int \frac{1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{g}+\frac{(-234 f+g h) \int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx}{g}\\ &=\frac{234 \operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e g}+\frac{(-234 f+g h) \int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx}{g}\\ &=\frac{(-234 f+g h) \int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx}{g}+\frac{\left (234 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e g n}\\ &=\frac{234 e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e g n}+\frac{(-234 f+g h) \int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx}{g}\\ \end{align*}

Mathematica [A]  time = 0.229699, size = 0, normalized size = 0. \[ \int \frac{h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(h + i*x)/((f + g*x)*(a + b*Log[c*(d + e*x)^n])),x]

[Out]

Integrate[(h + i*x)/((f + g*x)*(a + b*Log[c*(d + e*x)^n])), x]

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Maple [A]  time = 0.872, size = 0, normalized size = 0. \begin{align*} \int{\frac{ix+h}{ \left ( gx+f \right ) \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((i*x+h)/(g*x+f)/(a+b*ln(c*(e*x+d)^n)),x)

[Out]

int((i*x+h)/(g*x+f)/(a+b*ln(c*(e*x+d)^n)),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{i x + h}{{\left (g x + f\right )}{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)/(g*x+f)/(a+b*log(c*(e*x+d)^n)),x, algorithm="maxima")

[Out]

integrate((i*x + h)/((g*x + f)*(b*log((e*x + d)^n*c) + a)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{i x + h}{a g x + a f +{\left (b g x + b f\right )} \log \left ({\left (e x + d\right )}^{n} c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)/(g*x+f)/(a+b*log(c*(e*x+d)^n)),x, algorithm="fricas")

[Out]

integral((i*x + h)/(a*g*x + a*f + (b*g*x + b*f)*log((e*x + d)^n*c)), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{h + i x}{\left (a + b \log{\left (c \left (d + e x\right )^{n} \right )}\right ) \left (f + g x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)/(g*x+f)/(a+b*ln(c*(e*x+d)**n)),x)

[Out]

Integral((h + i*x)/((a + b*log(c*(d + e*x)**n))*(f + g*x)), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{i x + h}{{\left (g x + f\right )}{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)/(g*x+f)/(a+b*log(c*(e*x+d)^n)),x, algorithm="giac")

[Out]

integrate((i*x + h)/((g*x + f)*(b*log((e*x + d)^n*c) + a)), x)