\(\int \frac {\sqrt {b \tanh (e+f x)}}{c+d x} \, dx\) [26]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\sqrt {b \tanh (e+f x)}}{c+d x} \, dx=\text {Int}\left (\frac {\sqrt {b \tanh (e+f x)}}{c+d x},x\right ) \]

[Out]

Unintegrable((b*tanh(f*x+e))^(1/2)/(d*x+c),x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {\sqrt {b \tanh (e+f x)}}{c+d x} \, dx=\int \frac {\sqrt {b \tanh (e+f x)}}{c+d x} \, dx \]

[In]

Int[Sqrt[b*Tanh[e + f*x]]/(c + d*x),x]

[Out]

Defer[Int][Sqrt[b*Tanh[e + f*x]]/(c + d*x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {b \tanh (e+f x)}}{c+d x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {b \tanh (e+f x)}}{c+d x} \, dx=\int \frac {\sqrt {b \tanh (e+f x)}}{c+d x} \, dx \]

[In]

Integrate[Sqrt[b*Tanh[e + f*x]]/(c + d*x),x]

[Out]

Integrate[Sqrt[b*Tanh[e + f*x]]/(c + d*x), x]

Maple [N/A] (verified)

Not integrable

Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

\[\int \frac {\sqrt {b \tanh \left (f x +e \right )}}{d x +c}d x\]

[In]

int((b*tanh(f*x+e))^(1/2)/(d*x+c),x)

[Out]

int((b*tanh(f*x+e))^(1/2)/(d*x+c),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {b \tanh (e+f x)}}{c+d x} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((b*tanh(f*x+e))^(1/2)/(d*x+c),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.71 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {b \tanh (e+f x)}}{c+d x} \, dx=\int \frac {\sqrt {b \tanh {\left (e + f x \right )}}}{c + d x}\, dx \]

[In]

integrate((b*tanh(f*x+e))**(1/2)/(d*x+c),x)

[Out]

Integral(sqrt(b*tanh(e + f*x))/(c + d*x), x)

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {b \tanh (e+f x)}}{c+d x} \, dx=\int { \frac {\sqrt {b \tanh \left (f x + e\right )}}{d x + c} \,d x } \]

[In]

integrate((b*tanh(f*x+e))^(1/2)/(d*x+c),x, algorithm="maxima")

[Out]

integrate(sqrt(b*tanh(f*x + e))/(d*x + c), x)

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {b \tanh (e+f x)}}{c+d x} \, dx=\int { \frac {\sqrt {b \tanh \left (f x + e\right )}}{d x + c} \,d x } \]

[In]

integrate((b*tanh(f*x+e))^(1/2)/(d*x+c),x, algorithm="giac")

[Out]

integrate(sqrt(b*tanh(f*x + e))/(d*x + c), x)

Mupad [N/A]

Not integrable

Time = 1.71 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {b \tanh (e+f x)}}{c+d x} \, dx=\int \frac {\sqrt {b\,\mathrm {tanh}\left (e+f\,x\right )}}{c+d\,x} \,d x \]

[In]

int((b*tanh(e + f*x))^(1/2)/(c + d*x),x)

[Out]

int((b*tanh(e + f*x))^(1/2)/(c + d*x), x)