3.161 \(\int \frac{1}{(f+g x)^{3/2} \sqrt{a+b \log (c (d+e x)^n)}} \, dx\)

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{1}{(f+g x)^{3/2} \sqrt{a+b \log \left (c (d+e x)^n\right )}},x\right ) \]

[Out]

Unintegrable[1/((f + g*x)^(3/2)*Sqrt[a + b*Log[c*(d + e*x)^n]]), x]

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Rubi [A]  time = 0.0578403, 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 = {} \[ \int \frac{1}{(f+g x)^{3/2} \sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((f + g*x)^(3/2)*Sqrt[a + b*Log[c*(d + e*x)^n]]),x]

[Out]

Defer[Int][1/((f + g*x)^(3/2)*Sqrt[a + b*Log[c*(d + e*x)^n]]), x]

Rubi steps

\begin{align*} \int \frac{1}{(f+g x)^{3/2} \sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx &=\int \frac{1}{(f+g x)^{3/2} \sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx\\ \end{align*}

Mathematica [A]  time = 0.74405, size = 0, normalized size = 0. \[ \int \frac{1}{(f+g x)^{3/2} \sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((f + g*x)^(3/2)*Sqrt[a + b*Log[c*(d + e*x)^n]]),x]

[Out]

Integrate[1/((f + g*x)^(3/2)*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{ \left ( gx+f \right ) ^{-{\frac{3}{2}}}{\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)^(3/2)/(a+b*ln(c*(e*x+d)^n))^(1/2),x)

[Out]

int(1/(g*x+f)^(3/2)/(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}{{\left (g x + f\right )}^{\frac{3}{2}} \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)^(3/2)/(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((g*x + f)^(3/2)*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)^(3/2)/(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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(1/(g*x+f)**(3/2)/(a+b*ln(c*(e*x+d)**n))**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (g x + f\right )}^{\frac{3}{2}} \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)^(3/2)/(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="giac")

[Out]

integrate(1/((g*x + f)^(3/2)*sqrt(b*log((e*x + d)^n*c) + a)), x)