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=51 \[ \text {Int}\left (\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (t \log \left (i (g+h x)^n\right )+s\right )^2},x\right ) \]

[Out]

Unintegrable(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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Rubi [A]  time = 0.05, 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 = {} \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (i (g+h x)^n\right )\right )^2} \, dx \]

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*}

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Mathematica [A]  time = 2.87, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (i (g+h x)^n\right )\right )^2} \, dx \]

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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fricas [A]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h k s^{2} x + g k s^{2} + {\left (h k t^{2} x + g k t^{2}\right )} \log \left ({\left (h x + g\right )}^{n} i\right )^{2} + 2 \, {\left (h k s t x + g k s t\right )} \log \left ({\left (h x + g\right )}^{n} i\right )}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{{\left (h k x + g k\right )} {\left (t \log \left ({\left (h x + g\right )}^{n} i\right ) + s\right )}^{2}}\,{d x} \]

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)

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maple [A]  time = 0.53, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{\left (h k x +g k \right ) \left (t \ln \left (i \left (h x +g \right )^{n}\right )+s \right )^{2}}\, dx \]

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)/(t*ln(i*(h*x+g)^n)+s)^2,x)

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {r \log \relax (f) + \log \left ({\left ({\left (b x + a\right )}^{p}\right )}^{r}\right ) + \log \left ({\left ({\left (d x + c\right )}^{q}\right )}^{r}\right ) + \log \relax (e)}{h k n t^{2} \log \left ({\left (h x + g\right )}^{n}\right ) + {\left (k n t^{2} \log \relax (i) + k n s t\right )} h} + \int \frac {b c p r + a d q r + {\left (p r + q r\right )} b d x}{{\left (k n t^{2} \log \relax (i) + k n s t\right )} b d h x^{2} + {\left (k n t^{2} \log \relax (i) + k n s t\right )} a c h + {\left ({\left (k n t^{2} \log \relax (i) + k n s t\right )} b c h + {\left (k n t^{2} \log \relax (i) + k n s t\right )} a d h\right )} x + {\left (b d h k n t^{2} x^{2} + a c h k n t^{2} + {\left (b c h k n t^{2} + a d h k n t^{2}\right )} x\right )} \log \left ({\left (h x + g\right )}^{n}\right )}\,{d x} \]

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]

-(r*log(f) + log(((b*x + a)^p)^r) + log(((d*x + c)^q)^r) + log(e))/(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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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}{\left (g\,k+h\,k\,x\right )\,{\left (s+t\,\ln \left (i\,{\left (g+h\,x\right )}^n\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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