3.6.17 \(\int \frac {(a+b \log (c (d (e+f x)^p)^q))^n}{g+h x} \, dx\) [517]

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n}{g+h x},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.04, 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 {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n}{g+h x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n}{g+h x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{n}}{h x +g}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

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

[Out]

integral((b*log(((f*x + e)^p*d)^q*c) + a)^n/(h*x + g), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))**n/(h*x+g),x)

[Out]

Integral((a + b*log(c*(d*(e + f*x)**p)**q))**n/(g + h*x), x)

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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((a+b*log(c*(d*(f*x+e)^p)^q))^n/(h*x+g),x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^n}{g+h\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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