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

Optimal. Leaf size=87 \[ \frac{(d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{(f+g x) (e f-d g)}-\frac{b e n \text{Unintegrable}\left (\frac{1}{(f+g x) \sqrt{a+b \log \left (c (d+e x)^n\right )}},x\right )}{2 (e f-d g)} \]

[Out]

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

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Rubi [A]  time = 0.102329, 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 = {} \[ \int \frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.318953, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.889, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) ^{2}}\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((a+b*ln(c*(e*x+d)^n))^(1/2)/(g*x+f)^2,x)

[Out]

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

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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 )}^{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)^2,x, algorithm="maxima")

[Out]

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

[Out]

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

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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 )}^{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)^2,x, algorithm="giac")

[Out]

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