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

Optimal. Leaf size=77 \[ \frac{b e n \text{Unintegrable}\left (\frac{1}{(d+e x) \sqrt{f+g x} \sqrt{a+b \log \left (c (d+e x)^n\right )}},x\right )}{g}-\frac{2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{g \sqrt{f+g x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.256699, 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 = {} \[ \int \frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.28669, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.777, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) } \left ( gx+f \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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