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

Optimal. Leaf size=34 \[ \text{Unintegrable}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{g+h x}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.309956, 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 = {} \[ \int \frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{g+h x}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.96321, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{g+h x}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.657, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) }{\frac{1}{\sqrt{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))^(1/2)/(h*x+g)^(1/2),x)

[Out]

int((a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2)/(h*x+g)^(1/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}}{\sqrt{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))^(1/2)/(h*x+g)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b \log{\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}}{\sqrt{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))**(1/2)/(h*x+g)**(1/2),x)

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}}{\sqrt{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))^(1/2)/(h*x+g)^(1/2),x, algorithm="giac")

[Out]

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