\(\int \frac {1}{(f+g x) (a+b \log (c (d+e x)^n))^{5/2}} \, dx\) [137]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\text {Int}\left (\frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}},x\right ) \]

[Out]

Unintegrable(1/(g*x+f)/(a+b*ln(c*(e*x+d)^n))^(5/2),x)

Rubi [N/A]

Not integrable

Time = 0.04 (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 = {} \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx \]

[In]

Int[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^(5/2)),x]

[Out]

Defer[Int][1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^(5/2)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx \]

[In]

Integrate[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^(5/2)),x]

[Out]

Integrate[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^(5/2)), x]

Maple [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

\[\int \frac {1}{\left (g x +f \right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{\frac {5}{2}}}d x\]

[In]

int(1/(g*x+f)/(a+b*ln(c*(e*x+d)^n))^(5/2),x)

[Out]

int(1/(g*x+f)/(a+b*ln(c*(e*x+d)^n))^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(g*x+f)/(a+b*log(c*(e*x+d)^n))^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(g*x+f)/(a+b*ln(c*(e*x+d)**n))**(5/2),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.64 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(g*x+f)/(a+b*log(c*(e*x+d)^n))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((g*x + f)*(b*log((e*x + d)^n*c) + a)^(5/2)), x)

Giac [N/A]

Not integrable

Time = 0.58 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(g*x+f)/(a+b*log(c*(e*x+d)^n))^(5/2),x, algorithm="giac")

[Out]

integrate(1/((g*x + f)*(b*log((e*x + d)^n*c) + a)^(5/2)), x)

Mupad [N/A]

Not integrable

Time = 1.69 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int \frac {1}{\left (f+g\,x\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{5/2}} \,d x \]

[In]

int(1/((f + g*x)*(a + b*log(c*(d + e*x)^n))^(5/2)),x)

[Out]

int(1/((f + g*x)*(a + b*log(c*(d + e*x)^n))^(5/2)), x)