\(\int e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx\) [642]

Optimal result

Integrand size = 27, antiderivative size = 179 \[ \int e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=-\frac {11 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{8 a^2 \sqrt {1-a x}}+\frac {11 \sqrt {c-\frac {c}{a x}} x^2 \sqrt {1+a x}}{12 a \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} x^3 \sqrt {1+a x}}{3 \sqrt {1-a x}}+\frac {11 \sqrt {c-\frac {c}{a x}} \sqrt {x} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{8 a^{5/2} \sqrt {1-a x}} \] Output:

-11/8*(c-c/a/x)^(1/2)*x*(a*x+1)^(1/2)/a^2/(-a*x+1)^(1/2)+11/12*(c-c/a/x)^( 
1/2)*x^2*(a*x+1)^(1/2)/a/(-a*x+1)^(1/2)-1/3*(c-c/a/x)^(1/2)*x^3*(a*x+1)^(1 
/2)/(-a*x+1)^(1/2)+11/8*(c-c/a/x)^(1/2)*x^(1/2)*arcsinh(a^(1/2)*x^(1/2))/a 
^(5/2)/(-a*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.49 \[ \int e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {\sqrt {c-\frac {c}{a x}} \sqrt {x} \left (\sqrt {a} \sqrt {x} \sqrt {1+a x} \left (-33+22 a x-8 a^2 x^2\right )+33 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )\right )}{24 a^{5/2} \sqrt {1-a x}} \] Input:

Integrate[(Sqrt[c - c/(a*x)]*x^2)/E^ArcTanh[a*x],x]
 

Output:

(Sqrt[c - c/(a*x)]*Sqrt[x]*(Sqrt[a]*Sqrt[x]*Sqrt[1 + a*x]*(-33 + 22*a*x - 
8*a^2*x^2) + 33*ArcSinh[Sqrt[a]*Sqrt[x]]))/(24*a^(5/2)*Sqrt[1 - a*x])
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.68, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6684, 6678, 516, 90, 60, 60, 63, 222}

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.

\(\displaystyle \int x^2 e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx\)

\(\Big \downarrow \) 6684

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int e^{-\text {arctanh}(a x)} x^{3/2} \sqrt {1-a x}dx}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 6678

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

\(\Big \downarrow \) 516

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {11}{6} \int \frac {x^{3/2}}{\sqrt {a x+1}}dx-\frac {1}{3} x^{5/2} \sqrt {a x+1}\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {11}{6} \left (\frac {x^{3/2} \sqrt {a x+1}}{2 a}-\frac {3 \int \frac {\sqrt {x}}{\sqrt {a x+1}}dx}{4 a}\right )-\frac {1}{3} x^{5/2} \sqrt {a x+1}\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {11}{6} \left (\frac {x^{3/2} \sqrt {a x+1}}{2 a}-\frac {3 \left (\frac {\sqrt {x} \sqrt {a x+1}}{a}-\frac {\int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx}{2 a}\right )}{4 a}\right )-\frac {1}{3} x^{5/2} \sqrt {a x+1}\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 63

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {11}{6} \left (\frac {x^{3/2} \sqrt {a x+1}}{2 a}-\frac {3 \left (\frac {\sqrt {x} \sqrt {a x+1}}{a}-\frac {\int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{a}\right )}{4 a}\right )-\frac {1}{3} x^{5/2} \sqrt {a x+1}\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {\sqrt {x} \left (\frac {11}{6} \left (\frac {x^{3/2} \sqrt {a x+1}}{2 a}-\frac {3 \left (\frac {\sqrt {x} \sqrt {a x+1}}{a}-\frac {\text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2}}\right )}{4 a}\right )-\frac {1}{3} x^{5/2} \sqrt {a x+1}\right ) \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x}}\)

Input:

Int[(Sqrt[c - c/(a*x)]*x^2)/E^ArcTanh[a*x],x]
 

Output:

(Sqrt[c - c/(a*x)]*Sqrt[x]*(-1/3*(x^(5/2)*Sqrt[1 + a*x]) + (11*((x^(3/2)*S 
qrt[1 + a*x])/(2*a) - (3*((Sqrt[x]*Sqrt[1 + a*x])/a - ArcSinh[Sqrt[a]*Sqrt 
[x]]/a^(3/2)))/(4*a)))/6))/Sqrt[1 - a*x]
 

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 63

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b   S 
ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x 
] && GtQ[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 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 516

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; Free 
Q[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || 
(GtQ[a, 0] && GtQ[c, 0] &&  !IntegerQ[n]))
 

rule 6678

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* 
(x_))^(m_.), x_Symbol] :> Simp[c^n   Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - 
 a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 
0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 
, 0]) && IntegerQ[2*p]
 

rule 6684

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

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (16 a^{\frac {5}{2}} x^{2} \sqrt {-x \left (a x +1\right )}-44 a^{\frac {3}{2}} x \sqrt {-x \left (a x +1\right )}+66 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}+33 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right )\right )}{48 a^{\frac {5}{2}} \left (a x -1\right ) \sqrt {-x \left (a x +1\right )}}\)	\(125\)
risch	\(-\frac {\left (8 a^{2} x^{2}-22 a x +33\right ) \left (a x +1\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{24 a^{2} \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}}+\frac {11 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{16 a^{2} \sqrt {a^{2} c}\, \sqrt {-a^{2} x^{2}+1}}\)	\(184\)

Input:

int((c-c/a/x)^(1/2)*x^2/(a*x+1)*(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE 
)
 

Output:

1/48*(c*(a*x-1)/a/x)^(1/2)*x*(-a^2*x^2+1)^(1/2)/a^(5/2)*(16*a^(5/2)*x^2*(- 
x*(a*x+1))^(1/2)-44*a^(3/2)*x*(-x*(a*x+1))^(1/2)+66*a^(1/2)*(-x*(a*x+1))^( 
1/2)+33*arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1/2)))/(a*x-1)/(-x*(a*x 
+1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.63 \[ \int e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\left [\frac {33 \, {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (8 \, a^{3} x^{3} - 22 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{96 \, {\left (a^{4} x - a^{3}\right )}}, -\frac {33 \, {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (8 \, a^{3} x^{3} - 22 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{48 \, {\left (a^{4} x - a^{3}\right )}}\right ] \] Input:

integrate((c-c/a/x)^(1/2)*x^2/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="fri 
cas")
 

Output:

[1/96*(33*(a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + 
a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 
 4*(8*a^3*x^3 - 22*a^2*x^2 + 33*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/( 
a*x)))/(a^4*x - a^3), -1/48*(33*(a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 
 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(8* 
a^3*x^3 - 22*a^2*x^2 + 33*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x))) 
/(a^4*x - a^3)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((c-c/a/x)^(1/2)*x^2/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="max 
ima")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c-c/a/x)^(1/2)*x^2/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="gia 
c")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.40 \[ \int e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {\sqrt {c}\, i \left (8 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}-22 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x +33 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}-33 \,\mathrm {log}\left (\sqrt {a x +1}\, i +\sqrt {x}\, \sqrt {a}\, i \right )\right )}{24 a^{3}} \] Input:

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

Output:

(sqrt(c)*i*(8*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**2*x**2 - 22*sqrt(x)*sqrt(a) 
*sqrt(a*x + 1)*a*x + 33*sqrt(x)*sqrt(a)*sqrt(a*x + 1) - 33*log(sqrt(a*x + 
1)*i + sqrt(x)*sqrt(a)*i)))/(24*a**3)