3.2.53 \(\int \frac {\sqrt {f+g x}}{a+b \log (c (d+e x)^n)} \, dx\) [153]

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {\sqrt {f+g x}}{a+b \log \left (c (d+e x)^n\right )},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, 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 {\sqrt {f+g x}}{a+b \log \left (c (d+e x)^n\right )} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 0.59, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {f+g x}}{a+b \log \left (c (d+e x)^n\right )} \, dx \end {gather*}

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 = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {g x +f}}{a +b \ln \left (c \left (e x +d \right )^{n}\right )}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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

[Out]

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

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

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]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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