3.3.62 \(\int \frac {1}{(a f h+b g h x^2+h (b f x+a g x)) (A+B \log (e (a+b x)^n (c+d x)^{-n}))} \, dx\) [262]

Optimal. Leaf size=83 \[ \frac {\text {Subst}\left (\text {Int}\left (\frac {1}{(a+b x) (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )},x\right ),e \left (\frac {a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right )}{h} \]

[Out]

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

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Rubi [A]
time = 0.19, 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}{\left (a f h+b g h x^2+h (b f x+a g x)\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/((a*f*h + b*g*h*x^2 + h*(b*f*x + a*g*x))*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])),x]

[Out]

Defer[Subst][Defer[Int][1/((a + b*x)*(f + g*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]/h

Rubi steps

\begin {align*} \int \frac {1}{\left (a f h+b g h x^2+h (b f x+a g x)\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx &=\int \frac {1}{h (a+b x) (f+g x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx\\ &=\frac {\int \frac {1}{(a+b x) (f+g x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx}{h}\\ &=\frac {\int \left (\frac {b}{(b f-a g) (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) (f+g x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}\right ) \, dx}{h}\\ &=\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*}

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Mathematica [A]
time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a f h+b g h x^2+h (b f x+a g x)\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((a*f*h + b*g*h*x^2 + h*(b*f*x + a*g*x))*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])),x]

[Out]

Integrate[1/((a*f*h + b*g*h*x^2 + h*(b*f*x + a*g*x))*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])), x]

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Maple [A]
time = 0.47, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a f h +b g h \,x^{2}+h \left (a g x +b f x \right )\right ) \left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x))/(A+B*ln(e*(b*x+a)^n/((d*x+c)^n))),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/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x))/(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="maxima")

[Out]

integrate(1/((b*g*h*x^2 + a*f*h + (b*f*x + a*g*x)*h)*(B*log((b*x + a)^n*e/(d*x + c)^n) + A)), 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/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x))/(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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x))/(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )\,\left (h\,\left (a\,g\,x+b\,f\,x\right )+a\,f\,h+b\,g\,h\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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