3.1.21 \(\int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx\) [21]

3.1.21.1 Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx =\text {Too large to display} \]

output

4*(-b)^(3/2)*d*(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))/f^2+2*(-b 
)^(3/2)*d^2*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))^2/f^3+4*b^(3/2)*d*(d 
*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))/f^2+2*b^(3/2)*d^2*arctanh((b* 
tanh(f*x+e))^(1/2)/b^(1/2))^2/f^3-4*b^(3/2)*d^2*arctanh((b*tanh(f*x+e))^(1 
/2)/b^(1/2))*ln(2*b^(1/2)/(b^(1/2)-(b*tanh(f*x+e))^(1/2)))/f^3+4*b^(3/2)*d 
^2*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))*ln(2*b^(1/2)/(b^(1/2)+(b*tanh(f* 
x+e))^(1/2)))/f^3-2*b^(3/2)*d^2*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))*ln( 
2*b^(1/2)*((-b)^(1/2)-(b*tanh(f*x+e))^(1/2))/((-b)^(1/2)-b^(1/2))/(b^(1/2) 
+(b*tanh(f*x+e))^(1/2)))/f^3-2*b^(3/2)*d^2*arctanh((b*tanh(f*x+e))^(1/2)/b 
^(1/2))*ln(2*b^(1/2)*((-b)^(1/2)+(b*tanh(f*x+e))^(1/2))/((-b)^(1/2)+b^(1/2 
))/(b^(1/2)+(b*tanh(f*x+e))^(1/2)))/f^3-4*(-b)^(3/2)*d^2*arctanh((b*tanh(f 
*x+e))^(1/2)/(-b)^(1/2))*ln(2/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/f^3+2* 
(-b)^(3/2)*d^2*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))*ln(2*(b^(1/2)-(b* 
tanh(f*x+e))^(1/2))/((-b)^(1/2)+b^(1/2))/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1/ 
2)))/f^3+2*(-b)^(3/2)*d^2*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))*ln(-2* 
(b^(1/2)+(b*tanh(f*x+e))^(1/2))/((-b)^(1/2)-b^(1/2))/(1-(b*tanh(f*x+e))^(1 
/2)/(-b)^(1/2)))/f^3+4*(-b)^(3/2)*d^2*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^( 
1/2))*ln(2/(1+(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/f^3-2*b^(3/2)*d^2*polylog 
(2,1-2*b^(1/2)/(b^(1/2)-(b*tanh(f*x+e))^(1/2)))/f^3-2*b^(3/2)*d^2*polylog( 
2,1-2*b^(1/2)/(b^(1/2)+(b*tanh(f*x+e))^(1/2)))/f^3+b^(3/2)*d^2*polylog(...
 

3.1.21.2 Mathematica [N/A]

Not integrable

Time = 29.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx=\int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx \]

input

Integrate[(c + d*x)^2*(b*Tanh[e + f*x])^(3/2),x]
 

output

Integrate[(c + d*x)^2*(b*Tanh[e + f*x])^(3/2), x]
 

3.1.21.3 Rubi [N/A]

Not integrable

Time = 2.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {3042, 4203, 3042, 4219, 4223, 4853, 7267, 27, 7276, 2009}

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.

\(\displaystyle \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d x)^2 (-i b \tan (i e+i f x))^{3/2}dx\)

\(\Big \downarrow \) 4203

\(\displaystyle b^2 \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}}dx+\frac {4 b d \int (c+d x) \sqrt {b \tanh (e+f x)}dx}{f}-\frac {2 b (c+d x)^2 \sqrt {b \tanh (e+f x)}}{f}\)

\(\Big \downarrow \) 3042

\(\displaystyle b^2 \int \frac {(c+d x)^2}{\sqrt {-i b \tan (i e+i f x)}}dx+\frac {4 b d \int (c+d x) \sqrt {-i b \tan (i e+i f x)}dx}{f}-\frac {2 b (c+d x)^2 \sqrt {b \tanh (e+f x)}}{f}\)

\(\Big \downarrow \) 4219

\(\displaystyle \frac {4 b d \left (\frac {\sqrt {-b} d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )dx}{f}-\frac {\sqrt {b} d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )dx}{f}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}\right )}{f}+b^2 \int \frac {(c+d x)^2}{\sqrt {-i b \tan (i e+i f x)}}dx-\frac {2 b (c+d x)^2 \sqrt {b \tanh (e+f x)}}{f}\)

\(\Big \downarrow \) 4223

\(\displaystyle \frac {4 b d \left (\frac {\sqrt {-b} d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )dx}{f}-\frac {\sqrt {b} d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )dx}{f}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}\right )}{f}+b^2 \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}}dx-\frac {2 b (c+d x)^2 \sqrt {b \tanh (e+f x)}}{f}\)

\(\Big \downarrow \) 4853

\(\displaystyle \frac {4 b d \left (\frac {\sqrt {-b} d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{1-\tanh ^2(e+f x)}d\tanh (e+f x)}{f^2}-\frac {\sqrt {b} d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{1-\tanh ^2(e+f x)}d\tanh (e+f x)}{f^2}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}\right )}{f}+b^2 \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}}dx-\frac {2 b (c+d x)^2 \sqrt {b \tanh (e+f x)}}{f}\)

\(\Big \downarrow \) 7267

\(\displaystyle \frac {4 b d \left (\frac {2 \sqrt {-b} d \int \frac {b^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{b f^2}-\frac {2 d \int \frac {b^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{\sqrt {b} f^2}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}\right )}{f}+b^2 \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}}dx-\frac {2 b (c+d x)^2 \sqrt {b \tanh (e+f x)}}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 b d \left (\frac {2 \sqrt {-b} b d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{f^2}-\frac {2 b^{3/2} d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{f^2}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}\right )}{f}+b^2 \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}}dx-\frac {2 b (c+d x)^2 \sqrt {b \tanh (e+f x)}}{f}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {4 b d \left (-\frac {2 b^{3/2} d \int \left (\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (\tanh (e+f x) b+b)}-\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (b \tanh (e+f x)-b)}\right )d\sqrt {b \tanh (e+f x)}}{f^2}+\frac {2 \sqrt {-b} b d \int \left (\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (\tanh (e+f x) b+b)}-\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (b \tanh (e+f x)-b)}\right )d\sqrt {b \tanh (e+f x)}}{f^2}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}\right )}{f}+b^2 \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}}dx-\frac {2 b (c+d x)^2 \sqrt {b \tanh (e+f x)}}{f}\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}}dx b^2+\frac {4 d \left (-\frac {2 d \left (-\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{4 b}+\frac {\log \left (\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{2 b}-\frac {\log \left (\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{2 b}+\frac {\log \left (\frac {2 \sqrt {b} \left (\sqrt {-b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{4 b}+\frac {\log \left (\frac {2 \sqrt {b} \left (\sqrt {-b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{4 b}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b} \left (\sqrt {-b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right )}{8 b}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b} \left (\sqrt {-b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right )}{8 b}\right ) b^{3/2}}{f^2}+\frac {2 \sqrt {-b} d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{4 b}-\frac {\log \left (\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{2 b}+\frac {\log \left (\frac {2}{\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{2 b}-\frac {\log \left (-\frac {2 \left (\sqrt {b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1\right )}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{4 b}-\frac {\log \left (\frac {2 \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1\right )}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{4 b}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{4 b}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt {b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1\right )}+1\right )}{8 b}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1\right )}\right )}{8 b}\right ) b}{f^2}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b}}{f}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}\right ) b}{f}-\frac {2 (c+d x)^2 \sqrt {b \tanh (e+f x)} b}{f}\)

input

Int[(c + d*x)^2*(b*Tanh[e + f*x])^(3/2),x]
 

output

$Aborted
 

3.1.21.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4203

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb 
ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si 
mp[b*d*(m/(f*(n - 1)))   Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] 
, x] - Simp[b^2   Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free 
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
 

rule 4219

Int[((c_.) + (d_.)*(x_))*Sqrt[(a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Sym 
bol] :> Simp[(-I)*Rt[a - I*b, 2]*((c + d*x)/f)*ArcTanh[Sqrt[a + b*Tan[e + f 
*x]]/Rt[a - I*b, 2]], x] + (Simp[I*Rt[a + I*b, 2]*((c + d*x)/f)*ArcTanh[Sqr 
t[a + b*Tan[e + f*x]]/Rt[a + I*b, 2]], x] + Simp[I*d*(Rt[a - I*b, 2]/f)   I 
nt[ArcTanh[Sqrt[a + b*Tan[e + f*x]]/Rt[a - I*b, 2]], x], x] - Simp[I*d*(Rt[ 
a + I*b, 2]/f)   Int[ArcTanh[Sqrt[a + b*Tan[e + f*x]]/Rt[a + I*b, 2]], x], 
x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0]
 

rule 4223

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_. 
), x_Symbol] :> Unintegrable[(c + d*x)^m*(a + b*Tan[e + f*x])^n, x] /; Free 
Q[{a, b, c, d, e, f, m, n}, x]
 

rule 4853

Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFa 
ctors[Tan[v], x]}, d/Coefficient[v, x, 1]   Subst[Int[SubstFor[1/(1 + d^2*x 
^2), Tan[v]/d, u, x], x], x, Tan[v]/d]], x] /;  !FalseQ[v] && FunctionOfQ[N 
onfreeFactors[Tan[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], x 
]]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

3.1.21.4 Maple [N/A] (verified)

Not integrable

Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

input

int((d*x+c)^2*(b*tanh(f*x+e))^(3/2),x)
 

output

int((d*x+c)^2*(b*tanh(f*x+e))^(3/2),x)
 

3.1.21.5 Fricas [F(-2)]

Exception generated. \[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx=\text {Exception raised: TypeError} \]

input

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

output

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

3.1.21.6 Sympy [N/A]

Not integrable

Time = 3.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx=\int \left (b \tanh {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (c + d x\right )^{2}\, dx \]

input

integrate((d*x+c)**2*(b*tanh(f*x+e))**(3/2),x)
 

output

Integral((b*tanh(e + f*x))**(3/2)*(c + d*x)**2, x)
 

3.1.21.7 Maxima [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx=\int { {\left (d x + c\right )}^{2} \left (b \tanh \left (f x + e\right )\right )^{\frac {3}{2}} \,d x } \]

input

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

output

integrate((d*x + c)^2*(b*tanh(f*x + e))^(3/2), x)
 

3.1.21.8 Giac [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx=\int { {\left (d x + c\right )}^{2} \left (b \tanh \left (f x + e\right )\right )^{\frac {3}{2}} \,d x } \]

input

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

output

integrate((d*x + c)^2*(b*tanh(f*x + e))^(3/2), x)
 

3.1.21.9 Mupad [N/A]

Not integrable

Time = 2.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx=\int {\left (b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,x\right )}^2 \,d x \]

input

int((b*tanh(e + f*x))^(3/2)*(c + d*x)^2,x)
 

output

int((b*tanh(e + f*x))^(3/2)*(c + d*x)^2, x)