3.1.70 \(\int \frac {\log (h (f+g x)^m)}{(a+b x) (c+d x) \log (e (\frac {a+b x}{c+d x})^n)} \, dx\) [70]

Optimal. Leaf size=108 \[ \frac {b \text {Int}\left (\frac {\log \left (h (f+g x)^m\right )}{(a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )},x\right )}{b c-a d}-\frac {d \text {Int}\left (\frac {\log \left (h (f+g x)^m\right )}{(c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )},x\right )}{b c-a d} \]

[Out]

b*Unintegrable(ln(h*(g*x+f)^m)/(b*x+a)/ln(e*((b*x+a)/(d*x+c))^n),x)/(-a*d+b*c)-d*Unintegrable(ln(h*(g*x+f)^m)/
(d*x+c)/ln(e*((b*x+a)/(d*x+c))^n),x)/(-a*d+b*c)

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Rubi [A]
time = 0.38, 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 (h (f+g x)^m\right )}{(a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(b*Defer[Int][Log[h*(f + g*x)^m]/((a + b*x)*Log[e*((a + b*x)/(c + d*x))^n]), x])/(b*c - a*d) - (d*Defer[Int][L
og[h*(f + g*x)^m]/((c + d*x)*Log[e*((a + b*x)/(c + d*x))^n]), x])/(b*c - a*d)

Rubi steps

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

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Mathematica [A]
time = 1.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (h \left (g x +f \right )^{m}\right )}{\left (b x +a \right ) \left (d x +c \right ) \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(h*(g*x+f)^m)/(b*x+a)/(d*x+c)/ln(e*((b*x+a)/(d*x+c))^n),x)

[Out]

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

[Out]

integrate(log((g*x + f)^m*h)/((b*x + a)*(d*x + c)*log(((b*x + a)/(d*x + c))^n*e)), 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(h*(g*x+f)^m)/(b*x+a)/(d*x+c)/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="fricas")

[Out]

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

[Out]

integrate(log((g*x + f)^m*h)/((b*x + a)*(d*x + c)*log(((b*x + a)/(d*x + c))^n*e)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(h*(f + g*x)^m)/(log(e*((a + b*x)/(c + d*x))^n)*(a + b*x)*(c + d*x)),x)

[Out]

int(log(h*(f + g*x)^m)/(log(e*((a + b*x)/(c + d*x))^n)*(a + b*x)*(c + d*x)), x)

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