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

Optimal. Leaf size=82 \[ \frac{2 (f+g x)^{3/2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{3 g}-\frac{b e n \text{Unintegrable}\left (\frac{(f+g x)^{3/2}}{(d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}},x\right )}{3 g} \]

[Out]

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

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Rubi [A]  time = 0.264688, 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 = {} \[ \int \sqrt{f+g x} \sqrt{a+b \log \left (c (d+e x)^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.36404, size = 0, normalized size = 0. \[ \int \sqrt{f+g x} \sqrt{a+b \log \left (c (d+e x)^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.881, size = 0, normalized size = 0. \begin{align*} \int \sqrt{gx+f}\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((g*x+f)^(1/2)*(a+b*ln(c*(e*x+d)^n))^(1/2),x)

[Out]

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

[Out]

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

[Out]

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