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

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{\sqrt{f+g x}}{a+b \log \left (c (d+e x)^n\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0375366, 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 = {} \[ \int \frac{\sqrt{f+g x}}{a+b \log \left (c (d+e x)^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.856616, size = 0, normalized size = 0. \[ \int \frac{\sqrt{f+g x}}{a+b \log \left (c (d+e x)^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.69, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) }\sqrt{gx+f}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (g x + f\right )}^{\frac{3}{2}}}{3 \,{\left (b g \log \left ({\left (e x + d\right )}^{n}\right ) + b g \log \left (c\right ) + a g\right )}} + \int \frac{2 \,{\left (b e g n x + b e f n\right )} \sqrt{g x + f}}{3 \,{\left (b^{2} d g \log \left (c\right )^{2} + 2 \, a b d g \log \left (c\right ) + a^{2} d g +{\left (b^{2} e g x + b^{2} d g\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} +{\left (b^{2} e g \log \left (c\right )^{2} + 2 \, a b e g \log \left (c\right ) + a^{2} e g\right )} x + 2 \,{\left (b^{2} d g \log \left (c\right ) + a b d g +{\left (b^{2} e g \log \left (c\right ) + a b e g\right )} x\right )} \log \left ({\left (e x + d\right )}^{n}\right )\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(g*x + f)^(3/2)/(b*g*log((e*x + d)^n) + b*g*log(c) + a*g) + integrate(2/3*(b*e*g*n*x + b*e*f*n)*sqrt(g*x +
f)/(b^2*d*g*log(c)^2 + 2*a*b*d*g*log(c) + a^2*d*g + (b^2*e*g*x + b^2*d*g)*log((e*x + d)^n)^2 + (b^2*e*g*log(c
)^2 + 2*a*b*e*g*log(c) + a^2*e*g)*x + 2*(b^2*d*g*log(c) + a*b*d*g + (b^2*e*g*log(c) + a*b*e*g)*x)*log((e*x + d
)^n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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