\(\int \frac {(f+g x)^{3/2}}{a+b \log (c (d+e x)^n)} \, dx\) [152]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {(f+g x)^{3/2}}{a+b \log \left (c (d+e x)^n\right )} \, dx=\text {Int}\left (\frac {(f+g x)^{3/2}}{a+b \log \left (c (d+e x)^n\right )},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(f+g x)^{3/2}}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int \frac {(f+g x)^{3/2}}{a+b \log \left (c (d+e x)^n\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {(f+g x)^{3/2}}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int \frac {(f+g x)^{3/2}}{a+b \log \left (c (d+e x)^n\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

\[\int \frac {\left (g x +f \right )^{\frac {3}{2}}}{a +b \ln \left (c \left (e x +d \right )^{n}\right )}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {(f+g x)^{3/2}}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {3}{2}}}{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 21.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {(f+g x)^{3/2}}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int \frac {\left (f + g x\right )^{\frac {3}{2}}}{a + b \log {\left (c \left (d + e x\right )^{n} \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 204, normalized size of antiderivative = 7.85 \[ \int \frac {(f+g x)^{3/2}}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {3}{2}}}{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]

[In]

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

[Out]

2/5*(g^2*x^2 + 2*f*g*x + f^2)*sqrt(g*x + f)/(b*g*log((e*x + d)^n) + b*g*log(c) + a*g) + integrate(2/5*(b*e*g^2
*n*x^2 + 2*b*e*f*g*n*x + b*e*f^2*n)*sqrt(g*x + f)/(b^2*d*g*log(c)^2 + 2*a*b*d*g*log(c) + a^2*d*g + (b^2*e*g*x
+ b^2*d*g)*log((e*x + d)^n)^2 + (b^2*e*g*log(c)^2 + 2*a*b*e*g*log(c) + a^2*e*g)*x + 2*(b^2*d*g*log(c) + a*b*d*
g + (b^2*e*g*log(c) + a*b*e*g)*x)*log((e*x + d)^n)), x)

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {(f+g x)^{3/2}}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {3}{2}}}{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {(f+g x)^{3/2}}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int \frac {{\left (f+g\,x\right )}^{3/2}}{a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )} \,d x \]

[In]

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

[Out]

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