3.61 \(\int \frac{\log (e (f (a+b x)^p (c+d x)^q)^r)}{x \log ^2(i (j (h x)^t)^u)} \, dx\)

Optimal. Leaf size=41 \[ \text{CannotIntegrate}\left (\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x \log ^2\left (i \left (j (h x)^t\right )^u\right )},x\right ) \]

[Out]

CannotIntegrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(x*Log[i*(j*(h*x)^t)^u]^2), x]

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Rubi [A]  time = 0.42531, 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 = {} \[ \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x \log ^2\left (i \left (j (h x)^t\right )^u\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x \log ^2\left (61 \left (j (h x)^t\right )^u\right )} \, dx &=\int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x \log ^2\left (61 \left (j (h x)^t\right )^u\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 2.46198, size = 0, normalized size = 0. \[ \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x \log ^2\left (i \left (j (h x)^t\right )^u\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.78, 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 ) }{x \left ( \ln \left ( i \left ( j \left ( hx \right ) ^{t} \right ) ^{u} \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)/x/ln(i*(j*(h*x)^t)^u)^2,x)

[Out]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x/ln(i*(j*(h*x)^t)^u)^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 )}{t u \log \left ({\left (h^{t}\right )}^{u}\right ) + t u \log \left (i\right ) + t u \log \left (j^{u}\right ) + t u \log \left ({\left (x^{t}\right )}^{u}\right )} + \int \frac{b c p r + a d q r +{\left (p r + q r\right )} b d x}{{\left (t u \log \left ({\left (h^{t}\right )}^{u}\right ) + t u \log \left (i\right ) + t u \log \left (j^{u}\right )\right )} b d x^{2} +{\left (t u \log \left ({\left (h^{t}\right )}^{u}\right ) + t u \log \left (i\right ) + t u \log \left (j^{u}\right )\right )} a c +{\left ({\left (t u \log \left ({\left (h^{t}\right )}^{u}\right ) + t u \log \left (i\right ) + t u \log \left (j^{u}\right )\right )} b c +{\left (t u \log \left ({\left (h^{t}\right )}^{u}\right ) + t u \log \left (i\right ) + t u \log \left (j^{u}\right )\right )} a d\right )} x +{\left (b d t u x^{2} + a c t u +{\left (b c t u + a d t u\right )} x\right )} \log \left ({\left (x^{t}\right )}^{u}\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)/x/log(i*(j*(h*x)^t)^u)^2,x, algorithm="maxima")

[Out]

-(log(((b*x + a)^p)^r) + log(((d*x + c)^q)^r) + log(e) + log(f^r))/(t*u*log((h^t)^u) + t*u*log(i) + t*u*log(j^
u) + t*u*log((x^t)^u)) + integrate((b*c*p*r + a*d*q*r + (p*r + q*r)*b*d*x)/((t*u*log((h^t)^u) + t*u*log(i) + t
*u*log(j^u))*b*d*x^2 + (t*u*log((h^t)^u) + t*u*log(i) + t*u*log(j^u))*a*c + ((t*u*log((h^t)^u) + t*u*log(i) +
t*u*log(j^u))*b*c + (t*u*log((h^t)^u) + t*u*log(i) + t*u*log(j^u))*a*d)*x + (b*d*t*u*x^2 + a*c*t*u + (b*c*t*u
+ a*d*t*u)*x)*log((x^t)^u)), 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 )}{x \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )^{2}}, 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)/x/log(i*(j*(h*x)^t)^u)^2,x, algorithm="fricas")

[Out]

integral(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/(x*log(((h*x)^t*j)^u*i)^2), 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)/x/ln(i*(j*(h*x)**t)**u)**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 )}{x \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\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)/x/log(i*(j*(h*x)^t)^u)^2,x, algorithm="giac")

[Out]

integrate(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/(x*log(((h*x)^t*j)^u*i)^2), x)