3.1.61 \(\int \frac {\log (e (f (a+b x)^p (c+d x)^q)^r)}{x \log ^2(i (j (h x)^t)^u)} \, dx\) [61]

Optimal. Leaf size=42 \[ \text {Int}\left (\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x \log ^2\left (i \left (j (h x)^t\right )^u\right )},x\right ) \]

[Out]

CannotIntegrate(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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Rubi [A]
time = 0.30, 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 {\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 \end {gather*}

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*}

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Mathematica [A]
time = 1.51, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

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 = 1.06, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{x \ln \left (i \left (j \left (h x \right )^{t}\right )^{u}\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/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 = 0.00, 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(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]

-2*(r*log(f) + log(((b*x + a)^p)^r) + log(((d*x + c)^q)^r) + 1)/(2*t^2*u^2*log(h) + 2*t*u^2*log(j) + I*pi*t*u
+ 2*t*u*log((x^t)^u)) + integrate((b*c*p*r + a*d*q*r + (p*r + q*r)*b*d*x)/(1/2*I*pi*a*c*t*u + (t^2*u^2*log(h)
+ t*u^2*log(j))*a*c - 1/2*(-I*pi*b*d*t*u - 2*(t^2*u^2*log(h) + t*u^2*log(j))*b*d)*x^2 + 1/2*(2*(t^2*u^2*log(h)
+ t*u^2*log(j))*b*c + 2*(t^2*u^2*log(h) + t*u^2*log(j))*a*d + I*pi*(b*c*t*u + a*d*t*u))*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 = 0.00, 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(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]

-(2*p*r*log(b*x + a) + 2*q*r*log(d*x + c) - (2*t^2*u^2*log(h*x) + 2*t*u^2*log(j) + I*pi*t*u)*integral(((b*d*p
+ b*d*q)*r*x + (b*c*p + a*d*q)*r)/(1/2*I*pi*(b*d*t*u*x^2 + a*c*t*u + (b*c + a*d)*t*u*x) + (b*d*t^2*u^2*x^2 + a
*c*t^2*u^2 + (b*c + a*d)*t^2*u^2*x)*log(h*x) + (b*d*t*u^2*x^2 + a*c*t*u^2 + (b*c + a*d)*t*u^2*x)*log(j)), x) +
2*r*log(f) + 2)/(2*t^2*u^2*log(h*x) + 2*t*u^2*log(j) + I*pi*t*u)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 6439 deep

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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(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(I*((h*x)^t*j)^u)^2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}{x\,{\ln \left (i\,{\left (j\,{\left (h\,x\right )}^t\right )}^u\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

int(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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