\(\int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}} \, dx\) [23]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}} \, dx=\text {Int}\left (\frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}},x\right ) \] Output:

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

Mathematica [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}} \, dx=\text {\$Aborted} \] Input:

Integrate[(c + d*x)^2/Sqrt[b*Tanh[e + f*x]],x]
 

Output:

$Aborted
 

Rubi [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(c+d x)^2}{\sqrt {-i b \tan (i e+i f x)}}dx\)

\(\Big \downarrow \) 4223

\(\displaystyle \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}}dx\)

Input:

Int[(c + d*x)^2/Sqrt[b*Tanh[e + f*x]],x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

\[\int \frac {\left (d x +c \right )^{2}}{\sqrt {b \tanh \left (f x +e \right )}}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [N/A]

Not integrable

Time = 1.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}} \, dx=\int \frac {\left (c + d x\right )^{2}}{\sqrt {b \tanh {\left (e + f x \right )}}}\, dx \] Input:

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

Output:

Integral((c + d*x)**2/sqrt(b*tanh(e + f*x)), x)
 

Maxima [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{\sqrt {b \tanh \left (f x + e\right )}} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:Degree mismatch inside factorisation over extensionDegr 
ee mismatch inside factorisation over extensionindex.cc index_m i_lex_is_g 
reater Error
 

Mupad [N/A]

Not integrable

Time = 2.49 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{\sqrt {b\,\mathrm {tanh}\left (e+f\,x\right )}} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.85 \[ \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}} \, dx=\frac {\sqrt {b}\, \left (\left (\int \frac {\sqrt {\tanh \left (f x +e \right )}}{\tanh \left (f x +e \right )}d x \right ) c^{2}+\left (\int \frac {\sqrt {\tanh \left (f x +e \right )}\, x^{2}}{\tanh \left (f x +e \right )}d x \right ) d^{2}+2 \left (\int \frac {\sqrt {\tanh \left (f x +e \right )}\, x}{\tanh \left (f x +e \right )}d x \right ) c d \right )}{b} \] Input:

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

Output:

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