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

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n}{g+h x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0622584, 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 = {} \[ \int \frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n}{g+h x} \, dx \]

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*}

Mathematica [A]  time = 0.350357, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n}{g+h x} \, dx \]

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 = 0.678, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{n}}{hx+g}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{n}}{h x + g}, x\right ) \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{n}}{h x + g}\,{d x} \end{align*}

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)