3.2.96 \(\int \frac {1}{(d e+d f x) (h+i x) (a+b \log (c (e+f x)))} \, dx\) [196]

Optimal. Leaf size=72 \[ \frac {\log (a+b \log (c (e+f x)))}{b d (f h-e i)}-\frac {i \text {Int}\left (\frac {1}{(h+i x) (a+b \log (c (e+f x)))},x\right )}{d (f h-e i)} \]

[Out]

ln(a+b*ln(c*(f*x+e)))/b/d/(-e*i+f*h)-i*Unintegrable(1/(i*x+h)/(a+b*ln(c*(f*x+e))),x)/d/(-e*i+f*h)

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Rubi [A]
time = 0.17, 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 {1}{(d e+d f x) (h+i x) (a+b \log (c (e+f x)))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/((d*e + d*f*x)*(h + i*x)*(a + b*Log[c*(e + f*x)])),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {1}{(h+196 x) (d e+d f x) (a+b \log (c (e+f x)))} \, dx &=\int \left (\frac {196}{d (196 e-f h) (h+196 x) (a+b \log (c (e+f x)))}-\frac {f}{d (196 e-f h) (e+f x) (a+b \log (c (e+f x)))}\right ) \, dx\\ &=\frac {196 \int \frac {1}{(h+196 x) (a+b \log (c (e+f x)))} \, dx}{d (196 e-f h)}-\frac {f \int \frac {1}{(e+f x) (a+b \log (c (e+f x)))} \, dx}{d (196 e-f h)}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d (196 e-f h)}+\frac {196 \int \frac {1}{(h+196 x) (a+b \log (c (e+f x)))} \, dx}{d (196 e-f h)}\\ &=\frac {196 \int \frac {1}{(h+196 x) (a+b \log (c (e+f x)))} \, dx}{d (196 e-f h)}-\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,a+b \log (c (e+f x))\right )}{b d (196 e-f h)}\\ &=-\frac {\log (a+b \log (c (e+f x)))}{b d (196 e-f h)}+\frac {196 \int \frac {1}{(h+196 x) (a+b \log (c (e+f x)))} \, dx}{d (196 e-f h)}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(d e+d f x) (h+i x) (a+b \log (c (e+f x)))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((d*e + d*f*x)*(h + i*x)*(a + b*Log[c*(e + f*x)])),x]

[Out]

Integrate[1/((d*e + d*f*x)*(h + i*x)*(a + b*Log[c*(e + f*x)])), x]

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Maple [A]
time = 1.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d f x +e d \right ) \left (i x +h \right ) \left (a +b \ln \left (c \left (f x +e \right )\right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*f*x+d*e)/(i*x+h)/(a+b*ln(c*(f*x+e))),x)

[Out]

int(1/(d*f*x+d*e)/(i*x+h)/(a+b*ln(c*(f*x+e))),x)

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

[Out]

integrate(1/((d*f*x + d*e)*(b*log((f*x + e)*c) + a)*(h + I*x)), x)

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

[Out]

integral(-I/(-I*a*d*f*h*x + a*d*f*x^2 + (-I*a*d*h + a*d*x)*e + (-I*b*d*f*h*x + b*d*f*x^2 + (-I*b*d*h + b*d*x)*
e)*log(c*f*x + c*e)), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{a e h + a e i x + a f h x + a f i x^{2} + b e h \log {\left (c e + c f x \right )} + b e i x \log {\left (c e + c f x \right )} + b f h x \log {\left (c e + c f x \right )} + b f i x^{2} \log {\left (c e + c f x \right )}}\, dx}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*f*x+d*e)/(i*x+h)/(a+b*ln(c*(f*x+e))),x)

[Out]

Integral(1/(a*e*h + a*e*i*x + a*f*h*x + a*f*i*x**2 + b*e*h*log(c*e + c*f*x) + b*e*i*x*log(c*e + c*f*x) + b*f*h
*x*log(c*e + c*f*x) + b*f*i*x**2*log(c*e + c*f*x)), x)/d

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

[Out]

integrate(1/((d*f*x + d*e)*(b*log((f*x + e)*c) + a)*(h + I*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((h + i*x)*(d*e + d*f*x)*(a + b*log(c*(e + f*x)))),x)

[Out]

int(1/((h + i*x)*(d*e + d*f*x)*(a + b*log(c*(e + f*x)))), x)

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