\(\int e^{-4 \text {arctanh}(a x)} (e x)^m \sqrt {c+a c x} \, dx\) [478]

Optimal result

Integrand size = 24, antiderivative size = 121 \[ \int e^{-4 \text {arctanh}(a x)} (e x)^m \sqrt {c+a c x} \, dx=\frac {8 c (e x)^{1+m}}{e \sqrt {c+a c x}}+\frac {2 (e x)^{1+m} \sqrt {c+a c x}}{e (3+2 m)}-\frac {c \left (23+40 m+16 m^2\right ) (e x)^{1+m} \sqrt {1+a x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+m,2+m,-a x\right )}{e (1+m) (3+2 m) \sqrt {c+a c x}} \] Output:

8*c*(e*x)^(1+m)/e/(a*c*x+c)^(1/2)+2*(e*x)^(1+m)*(a*c*x+c)^(1/2)/e/(3+2*m)- 
c*(16*m^2+40*m+23)*(e*x)^(1+m)*(a*x+1)^(1/2)*hypergeom([1/2, 1+m],[2+m],-a 
*x)/e/(1+m)/(3+2*m)/(a*c*x+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.02 \[ \int e^{-4 \text {arctanh}(a x)} (e x)^m \sqrt {c+a c x} \, dx=(e x)^m \sqrt {c+a c x} \left (4 (3+4 m) x (-a x)^{-1-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},1+a x\right )+x \left (\frac {8}{1+a x}-\frac {\operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+m,2+m,-a x\right )}{(1+m) \sqrt {1+a x}}+\frac {a x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},2+m,3+m,-a x\right )}{(2+m) \sqrt {1+a x}}\right )\right ) \] Input:

Integrate[((e*x)^m*Sqrt[c + a*c*x])/E^(4*ArcTanh[a*x]),x]
 

Output:

(e*x)^m*Sqrt[c + a*c*x]*(4*(3 + 4*m)*x*(-(a*x))^(-1 - m)*Hypergeometric2F1 
[1/2, -m, 3/2, 1 + a*x] + x*(8/(1 + a*x) - Hypergeometric2F1[1/2, 1 + m, 2 
 + m, -(a*x)]/((1 + m)*Sqrt[1 + a*x]) + (a*x*Hypergeometric2F1[1/2, 2 + m, 
 3 + m, -(a*x)])/((2 + m)*Sqrt[1 + a*x])))
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6680, 35, 100, 27, 90, 77, 75}

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.

\(\displaystyle \int e^{-4 \text {arctanh}(a x)} \sqrt {a c x+c} (e x)^m \, dx\)

\(\Big \downarrow \) 6680

\(\displaystyle \int \frac {(1-a x)^2 \sqrt {a c x+c} (e x)^m}{(a x+1)^2}dx\)

\(\Big \downarrow \) 35

\(\displaystyle c^2 \int \frac {(e x)^m (1-a x)^2}{(a x c+c)^{3/2}}dx\)

\(\Big \downarrow \) 100

\(\displaystyle c^2 \left (\frac {8 (e x)^{m+1}}{c e \sqrt {a c x+c}}-\frac {2 \int \frac {a^2 c^2 e (e x)^m (8 m-a x+7)}{2 \sqrt {a x c+c}}dx}{a^2 c^3 e}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c^2 \left (\frac {8 (e x)^{m+1}}{c e \sqrt {a c x+c}}-\frac {\int \frac {(e x)^m (8 m-a x+7)}{\sqrt {a x c+c}}dx}{c}\right )\)

\(\Big \downarrow \) 90

\(\displaystyle c^2 \left (\frac {8 (e x)^{m+1}}{c e \sqrt {a c x+c}}-\frac {\frac {\left (16 m^2+40 m+23\right ) \int \frac {(e x)^m}{\sqrt {a x c+c}}dx}{2 m+3}-\frac {2 \sqrt {a c x+c} (e x)^{m+1}}{c e (2 m+3)}}{c}\right )\)

\(\Big \downarrow \) 77

\(\displaystyle c^2 \left (\frac {8 (e x)^{m+1}}{c e \sqrt {a c x+c}}-\frac {\frac {\left (16 m^2+40 m+23\right ) (-a x)^{-m} (e x)^m \int \frac {(-a x)^m}{\sqrt {a x c+c}}dx}{2 m+3}-\frac {2 \sqrt {a c x+c} (e x)^{m+1}}{c e (2 m+3)}}{c}\right )\)

\(\Big \downarrow \) 75

\(\displaystyle c^2 \left (\frac {8 (e x)^{m+1}}{c e \sqrt {a c x+c}}-\frac {\frac {2 \left (16 m^2+40 m+23\right ) \sqrt {a c x+c} (-a x)^{-m} (e x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},a x+1\right )}{a c (2 m+3)}-\frac {2 \sqrt {a c x+c} (e x)^{m+1}}{c e (2 m+3)}}{c}\right )\)

Input:

Int[((e*x)^m*Sqrt[c + a*c*x])/E^(4*ArcTanh[a*x]),x]
 

Output:

c^2*((8*(e*x)^(1 + m))/(c*e*Sqrt[c + a*c*x]) - ((-2*(e*x)^(1 + m)*Sqrt[c + 
 a*c*x])/(c*e*(3 + 2*m)) + (2*(23 + 40*m + 16*m^2)*(e*x)^m*Sqrt[c + a*c*x] 
*Hypergeometric2F1[1/2, -m, 3/2, 1 + a*x])/(a*c*(3 + 2*m)*(-(a*x))^m))/c)
 

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 35

Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> 
 Simp[(b/d)^m   Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} 
, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] &&  !(IntegerQ[n] && SimplerQ[a + 
b*x, c + d*x])
 

rule 75

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x 
)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + 
 d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (IntegerQ[m] 
 || GtQ[-d/(b*c), 0])
 

rule 77

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((-b)*(c/ 
d))^IntPart[m]*((b*x)^FracPart[m]/((-d)*(x/c))^FracPart[m])   Int[((-d)*(x/ 
c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && 
 !IntegerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

rule 6680

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol 
] :> Int[u*(c + d*x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c 
, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] &&  !(IntegerQ[p] || GtQ[c, 0])
 

Maple [F]

\[\int \frac {\left (e x \right )^{m} \sqrt {a c x +c}\, \left (-a^{2} x^{2}+1\right )^{2}}{\left (a x +1\right )^{4}}d x\]

Input:

int((e*x)^m*(a*c*x+c)^(1/2)/(a*x+1)^4*(-a^2*x^2+1)^2,x)
 

Output:

int((e*x)^m*(a*c*x+c)^(1/2)/(a*x+1)^4*(-a^2*x^2+1)^2,x)
 

Fricas [F]

\[ \int e^{-4 \text {arctanh}(a x)} (e x)^m \sqrt {c+a c x} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {a c x + c} \left (e x\right )^{m}}{{\left (a x + 1\right )}^{4}} \,d x } \] Input:

integrate((e*x)^m*(a*c*x+c)^(1/2)/(a*x+1)^4*(-a^2*x^2+1)^2,x, algorithm="f 
ricas")
 

Output:

integral((a^2*x^2 - 2*a*x + 1)*sqrt(a*c*x + c)*(e*x)^m/(a^2*x^2 + 2*a*x + 
1), x)
 

Sympy [F]

\[ \int e^{-4 \text {arctanh}(a x)} (e x)^m \sqrt {c+a c x} \, dx=\int \frac {\sqrt {c \left (a x + 1\right )} \left (e x\right )^{m} \left (a x - 1\right )^{2}}{\left (a x + 1\right )^{2}}\, dx \] Input:

integrate((e*x)**m*(a*c*x+c)**(1/2)/(a*x+1)**4*(-a**2*x**2+1)**2,x)
 

Output:

Integral(sqrt(c*(a*x + 1))*(e*x)**m*(a*x - 1)**2/(a*x + 1)**2, x)
 

Maxima [F]

\[ \int e^{-4 \text {arctanh}(a x)} (e x)^m \sqrt {c+a c x} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {a c x + c} \left (e x\right )^{m}}{{\left (a x + 1\right )}^{4}} \,d x } \] Input:

integrate((e*x)^m*(a*c*x+c)^(1/2)/(a*x+1)^4*(-a^2*x^2+1)^2,x, algorithm="m 
axima")
 

Output:

integrate((a^2*x^2 - 1)^2*sqrt(a*c*x + c)*(e*x)^m/(a*x + 1)^4, x)
 

Giac [F]

\[ \int e^{-4 \text {arctanh}(a x)} (e x)^m \sqrt {c+a c x} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {a c x + c} \left (e x\right )^{m}}{{\left (a x + 1\right )}^{4}} \,d x } \] Input:

integrate((e*x)^m*(a*c*x+c)^(1/2)/(a*x+1)^4*(-a^2*x^2+1)^2,x, algorithm="g 
iac")
 

Output:

integrate((a^2*x^2 - 1)^2*sqrt(a*c*x + c)*(e*x)^m/(a*x + 1)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{-4 \text {arctanh}(a x)} (e x)^m \sqrt {c+a c x} \, dx=\int \frac {{\left (e\,x\right )}^m\,{\left (a^2\,x^2-1\right )}^2\,\sqrt {c+a\,c\,x}}{{\left (a\,x+1\right )}^4} \,d x \] Input:

int(((e*x)^m*(a^2*x^2 - 1)^2*(c + a*c*x)^(1/2))/(a*x + 1)^4,x)
 

Output:

int(((e*x)^m*(a^2*x^2 - 1)^2*(c + a*c*x)^(1/2))/(a*x + 1)^4, x)
 

Reduce [F]

\[ \int e^{-4 \text {arctanh}(a x)} (e x)^m \sqrt {c+a c x} \, dx =\text {Too large to display} \] Input:

int((e*x)^m*(a*c*x+c)^(1/2)/(a*x+1)^4*(-a^2*x^2+1)^2,x)
 

Output:

(2*e**m*sqrt(c)*(4*x**m*sqrt(a*x + 1)*a**2*m**2*x**2 - x**m*sqrt(a*x + 1)* 
a**2*x**2 - 12*x**m*sqrt(a*x + 1)*a*m**2*x - 14*x**m*sqrt(a*x + 1)*a*m*x + 
 10*x**m*sqrt(a*x + 1)*a*x + 16*x**m*sqrt(a*x + 1)*m**2 + 40*x**m*sqrt(a*x 
 + 1)*m + 23*x**m*sqrt(a*x + 1) - 128*int((x**m*sqrt(a*x + 1))/(8*a**2*m** 
3*x**3 + 12*a**2*m**2*x**3 - 2*a**2*m*x**3 - 3*a**2*x**3 + 16*a*m**3*x**2 
+ 24*a*m**2*x**2 - 4*a*m*x**2 - 6*a*x**2 + 8*m**3*x + 12*m**2*x - 2*m*x - 
3*x),x)*a*m**6*x - 512*int((x**m*sqrt(a*x + 1))/(8*a**2*m**3*x**3 + 12*a** 
2*m**2*x**3 - 2*a**2*m*x**3 - 3*a**2*x**3 + 16*a*m**3*x**2 + 24*a*m**2*x** 
2 - 4*a*m*x**2 - 6*a*x**2 + 8*m**3*x + 12*m**2*x - 2*m*x - 3*x),x)*a*m**5* 
x - 632*int((x**m*sqrt(a*x + 1))/(8*a**2*m**3*x**3 + 12*a**2*m**2*x**3 - 2 
*a**2*m*x**3 - 3*a**2*x**3 + 16*a*m**3*x**2 + 24*a*m**2*x**2 - 4*a*m*x**2 
- 6*a*x**2 + 8*m**3*x + 12*m**2*x - 2*m*x - 3*x),x)*a*m**4*x - 148*int((x* 
*m*sqrt(a*x + 1))/(8*a**2*m**3*x**3 + 12*a**2*m**2*x**3 - 2*a**2*m*x**3 - 
3*a**2*x**3 + 16*a*m**3*x**2 + 24*a*m**2*x**2 - 4*a*m*x**2 - 6*a*x**2 + 8* 
m**3*x + 12*m**2*x - 2*m*x - 3*x),x)*a*m**3*x + 166*int((x**m*sqrt(a*x + 1 
))/(8*a**2*m**3*x**3 + 12*a**2*m**2*x**3 - 2*a**2*m*x**3 - 3*a**2*x**3 + 1 
6*a*m**3*x**2 + 24*a*m**2*x**2 - 4*a*m*x**2 - 6*a*x**2 + 8*m**3*x + 12*m** 
2*x - 2*m*x - 3*x),x)*a*m**2*x + 69*int((x**m*sqrt(a*x + 1))/(8*a**2*m**3* 
x**3 + 12*a**2*m**2*x**3 - 2*a**2*m*x**3 - 3*a**2*x**3 + 16*a*m**3*x**2 + 
24*a*m**2*x**2 - 4*a*m*x**2 - 6*a*x**2 + 8*m**3*x + 12*m**2*x - 2*m*x -...