3.2.56 \(\int \sqrt {f+g x} \sqrt {a+b \log (c (d+e x)^n)} \, dx\) [156]

Optimal. Leaf size=83 \[ \frac {2 (f+g x)^{3/2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{3 g}-\frac {b e n \text {Int}\left (\frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}},x\right )}{3 g} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.17, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx \end {gather*}

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 = 0.56, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx \end {gather*}

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]

________________________________________________________________________________________

Maple [A]
time = 0.13, size = 0, normalized size = 0.00 \[\int \sqrt {g x +f}\, \sqrt {a +b \ln \left (c \left (e x +d \right )^{n}\right )}\, dx\]

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)

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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((x*e + d)^n*c) + a), x)

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x*e + d)^n*c) + a), x)

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {f+g\,x}\,\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________