3.236 \(\int \frac{1}{(f+g x) (h+i x) (a+b \log (c (d+e x)^n))} \, dx\)

Optimal. Leaf size=78 \[ \frac{g \text{Unintegrable}\left (\frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right )}{g h-f i}-\frac{i \text{Unintegrable}\left (\frac{1}{(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right )}{g h-f i} \]

[Out]

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

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Rubi [A]  time = 0.189458, 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 = {} \[ \int \frac{1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{1}{(h+236 x) (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx &=\int \left (\frac{236}{(236 f-g h) (h+236 x) \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac{g}{(236 f-g h) (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx\\ &=\frac{236 \int \frac{1}{(h+236 x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx}{236 f-g h}-\frac{g \int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx}{236 f-g h}\\ \end{align*}

Mathematica [A]  time = 0.843848, size = 0, normalized size = 0. \[ \int \frac{1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.563, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( ix+h \right ) \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(1/(g*x+f)/(i*x+h)/(a+b*ln(c*(e*x+d)^n)),x)

[Out]

int(1/(g*x+f)/(i*x+h)/(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{1}{{\left (g x + f\right )}{\left (i x + h\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(1/(g*x+f)/(i*x+h)/(a+b*log(c*(e*x+d)^n)),x, algorithm="maxima")

[Out]

integrate(1/((g*x + f)*(i*x + h)*(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{1}{a g i x^{2} + a f h +{\left (a g h + a f i\right )} x +{\left (b g i x^{2} + b f h +{\left (b g h + b f i\right )} x\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(1/(g*x+f)/(i*x+h)/(a+b*log(c*(e*x+d)^n)),x, algorithm="fricas")

[Out]

integral(1/(a*g*i*x^2 + a*f*h + (a*g*h + a*f*i)*x + (b*g*i*x^2 + b*f*h + (b*g*h + b*f*i)*x)*log((e*x + d)^n*c)
), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (g x + f\right )}{\left (i x + h\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(1/(g*x+f)/(i*x+h)/(a+b*log(c*(e*x+d)^n)),x, algorithm="giac")

[Out]

integrate(1/((g*x + f)*(i*x + h)*(b*log((e*x + d)^n*c) + a)), x)