3.3.34 \(\int \frac {h+i x}{(f+g x) (a+b \log (c (d+e x)^n))} \, dx\) [234]

Optimal. Leaf size=107 \[ \frac {e^{-\frac {a}{b n}} i (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e g n}+\frac {(g h-f i) \text {Int}\left (\frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right )}{g} \]

[Out]

i*(e*x+d)*Ei((a+b*ln(c*(e*x+d)^n))/b/n)/b/e/exp(a/b/n)/g/n/((c*(e*x+d)^n)^(1/n))+(-f*i+g*h)*Unintegrable(1/(g*
x+f)/(a+b*ln(c*(e*x+d)^n)),x)/g

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \end {gather*}

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 \text {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 ) \text {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 = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \end {gather*}

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]

________________________________________________________________________________________

Maple [A]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {i x +h}{\left (g x +f \right ) \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}\, dx\]

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)

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {h+i\,x}{\left (f+g\,x\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________