3.160 \(\int \frac{1}{\sqrt{f+g x} \sqrt{a+b \log (c (d+e x)^n)}} \, dx\)

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{1}{\sqrt{f+g x} \sqrt{a+b \log \left (c (d+e x)^n\right )}},x\right ) \]

[Out]

Unintegrable[1/(Sqrt[f + g*x]*Sqrt[a + b*Log[c*(d + e*x)^n]]), x]

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Rubi [A]  time = 0.0556426, 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 = {} \[ \int \frac{1}{\sqrt{f+g x} \sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(Sqrt[f + g*x]*Sqrt[a + b*Log[c*(d + e*x)^n]]),x]

[Out]

Defer[Int][1/(Sqrt[f + g*x]*Sqrt[a + b*Log[c*(d + e*x)^n]]), x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{f+g x} \sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx &=\int \frac{1}{\sqrt{f+g x} \sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx\\ \end{align*}

Mathematica [A]  time = 2.80685, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{f+g x} \sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(Sqrt[f + g*x]*Sqrt[a + b*Log[c*(d + e*x)^n]]),x]

[Out]

Integrate[1/(Sqrt[f + g*x]*Sqrt[a + b*Log[c*(d + e*x)^n]]), x]

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Maple [A]  time = 0.841, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt{gx+f}}}{\frac{1}{\sqrt{a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{g x + f} \sqrt{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)^(1/2)/(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(g*x + f)*sqrt(b*log((e*x + d)^n*c) + a)), 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(1/(g*x+f)^(1/2)/(a+b*log(c*(e*x+d)^n))^(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{1}{\sqrt{a + b \log{\left (c \left (d + e x\right )^{n} \right )}} \sqrt{f + g x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)**(1/2)/(a+b*ln(c*(e*x+d)**n))**(1/2),x)

[Out]

Integral(1/(sqrt(a + b*log(c*(d + e*x)**n))*sqrt(f + g*x)), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{g x + f} \sqrt{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)^(1/2)/(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(g*x + f)*sqrt(b*log((e*x + d)^n*c) + a)), x)