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

Optimal. Leaf size=60 \[ \frac{x \sqrt{a+b \log \left (c x^n\right )}}{d (d+e x)}-\frac{b n \text{Unintegrable}\left (\frac{1}{(d+e x) \sqrt{a+b \log \left (c x^n\right )}},x\right )}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.101237, 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 = {} \[ \int \frac{\sqrt{a+b \log \left (c x^n\right )}}{(d+e x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 6.42241, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \log \left (c x^n\right )}}{(d+e x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.5, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( ex+d \right ) ^{2}}\sqrt{a+b\ln \left ( c{x}^{n} \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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