3.253 \(\int \frac{1}{(f+g x) (a h+b h x) (A+B \log (e (a+b x)^n (c+d x)^{-n}))} \, dx\)

Optimal. Leaf size=81 \[ \operatorname{Subst}\left (\text{Unintegrable}\left (\frac{1}{(f+g x) (a h+b h x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )},x\right ),e \left (\frac{a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right ) \]

[Out]

Defer[Subst][Unintegrable[1/((f + g*x)*(a*h + b*h*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x], e*((a + b*x)
/(c + d*x))^n, (e*(a + b*x)^n)/(c + d*x)^n]

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Rubi [A]  time = 0.426943, 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 = {} \[ \int \frac{1}{(f+g x) (a h+b h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.16094, size = 0, normalized size = 0. \[ \int \frac{1}{(f+g x) (a h+b h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 7.096, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) \left ( bhx+ah \right ) } \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(g*x+f)/(b*h*x+a*h)/(A+B*ln(e*(b*x+a)^n/((d*x+c)^n))),x)

[Out]

int(1/(g*x+f)/(b*h*x+a*h)/(A+B*ln(e*(b*x+a)^n/((d*x+c)^n))),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)/(b*h*x+a*h)/(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{A b g h x^{2} + A a f h +{\left (A b f + A a g\right )} h x +{\left (B b g h x^{2} + B a f h +{\left (B b f + B a g\right )} h x\right )} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)/(b*h*x+a*h)/(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)/(b*h*x+a*h)/(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="giac")

[Out]

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