3.55 \(\int \frac{\log (e (f (a+b x)^p (c+d x)^q)^r)}{(g k+h k x) (s+t \log (i (g+h x)^n))^2} \, dx\)

Optimal. Leaf size=50 \[ \text{Unintegrable}\left (\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (t \log \left (i (g+h x)^n\right )+s\right )^2},x\right ) \]

[Out]

Unintegrable[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/((g*k + h*k*x)*(s + t*Log[i*(g + h*x)^n])^2), x]

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Rubi [A]  time = 0.0525536, 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 = {} \[ \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (i (g+h x)^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/((g*k + h*k*x)*(s + t*Log[i*(g + h*x)^n])^2),x]

[Out]

Defer[Int][Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/((g*k + h*k*x)*(s + t*Log[i*(g + h*x)^n])^2), x]

Rubi steps

\begin{align*} \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (55 (g+h x)^n\right )\right )^2} \, dx &=\int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (55 (g+h x)^n\right )\right )^2} \, dx\\ \end{align*}

Mathematica [A]  time = 2.74292, size = 0, normalized size = 0. \[ \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (i (g+h x)^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/((g*k + h*k*x)*(s + t*Log[i*(g + h*x)^n])^2),x]

[Out]

Integrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/((g*k + h*k*x)*(s + t*Log[i*(g + h*x)^n])^2), x]

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Maple [A]  time = 0.739, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) }{ \left ( hkx+gk \right ) \left ( s+t\ln \left ( i \left ( hx+g \right ) ^{n} \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*k*x+g*k)/(s+t*ln(i*(h*x+g)^n))^2,x)

[Out]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*k*x+g*k)/(s+t*ln(i*(h*x+g)^n))^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\log \left ({\left ({\left (b x + a\right )}^{p}\right )}^{r}\right ) + \log \left ({\left ({\left (d x + c\right )}^{q}\right )}^{r}\right ) + \log \left (e\right ) + \log \left (f^{r}\right )}{h k n t^{2} \log \left ({\left (h x + g\right )}^{n}\right ) +{\left (k n t^{2} \log \left (i\right ) + k n s t\right )} h} + \int \frac{b c p r + a d q r +{\left (p r + q r\right )} b d x}{{\left (k n t^{2} \log \left (i\right ) + k n s t\right )} b d h x^{2} +{\left (k n t^{2} \log \left (i\right ) + k n s t\right )} a c h +{\left ({\left (k n t^{2} \log \left (i\right ) + k n s t\right )} b c h +{\left (k n t^{2} \log \left (i\right ) + k n s t\right )} a d h\right )} x +{\left (b d h k n t^{2} x^{2} + a c h k n t^{2} +{\left (b c h k n t^{2} + a d h k n t^{2}\right )} x\right )} \log \left ({\left (h x + g\right )}^{n}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*k*x+g*k)/(s+t*log(i*(h*x+g)^n))^2,x, algorithm="maxima")

[Out]

-(log(((b*x + a)^p)^r) + log(((d*x + c)^q)^r) + log(e) + log(f^r))/(h*k*n*t^2*log((h*x + g)^n) + (k*n*t^2*log(
i) + k*n*s*t)*h) + integrate((b*c*p*r + a*d*q*r + (p*r + q*r)*b*d*x)/((k*n*t^2*log(i) + k*n*s*t)*b*d*h*x^2 + (
k*n*t^2*log(i) + k*n*s*t)*a*c*h + ((k*n*t^2*log(i) + k*n*s*t)*b*c*h + (k*n*t^2*log(i) + k*n*s*t)*a*d*h)*x + (b
*d*h*k*n*t^2*x^2 + a*c*h*k*n*t^2 + (b*c*h*k*n*t^2 + a*d*h*k*n*t^2)*x)*log((h*x + g)^n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*k*x+g*k)/(s+t*log(i*(h*x+g)^n))^2,x, algorithm="fricas")

[Out]

integral(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/(h*k*s^2*x + g*k*s^2 + (h*k*t^2*x + g*k*t^2)*log((h*x + g)^n*i)^
2 + 2*(h*k*s*t*x + g*k*s*t)*log((h*x + g)^n*i)), 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(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)/(h*k*x+g*k)/(s+t*ln(i*(h*x+g)**n))**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )}{{\left (h k x + g k\right )}{\left (t \log \left ({\left (h x + g\right )}^{n} i\right ) + s\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*k*x+g*k)/(s+t*log(i*(h*x+g)^n))^2,x, algorithm="giac")

[Out]

integrate(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/((h*k*x + g*k)*(t*log((h*x + g)^n*i) + s)^2), x)