\(\int e^{3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx\) [334]

Optimal result

Integrand size = 23, antiderivative size = 262 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {104 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{21 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {32 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{21 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {6 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{7 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {\frac {1}{a x}} \sqrt {c-a c x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{a^3 \sqrt {1-\frac {1}{a x}}} \] Output:

104/21*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/a^3/(1-1/a/x)^(1/2)+32/21*(1+1/a/x 
)^(1/2)*x*(-a*c*x+c)^(1/2)/a^2/(1-1/a/x)^(1/2)+6/7*(1+1/a/x)^(1/2)*x^2*(-a 
*c*x+c)^(1/2)/a/(1-1/a/x)^(1/2)+2/7*(1+1/a/x)^(1/2)*x^3*(-a*c*x+c)^(1/2)/( 
1-1/a/x)^(1/2)-4*2^(1/2)*(1/a/x)^(1/2)*(-a*c*x+c)^(1/2)*arctanh(2^(1/2)*(1 
/a/x)^(1/2)/(1+1/a/x)^(1/2))/a^3/(1-1/a/x)^(1/2)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.47 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c-a c x} \left (\sqrt {a} \sqrt {1+\frac {1}{a x}} \left (52+16 a x+9 a^2 x^2+3 a^3 x^3\right )-42 \sqrt {2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{21 a^{7/2} \sqrt {1-\frac {1}{a x}}} \] Input:

Integrate[E^(3*ArcCoth[a*x])*x^2*Sqrt[c - a*c*x],x]
 

Output:

(2*Sqrt[c - a*c*x]*(Sqrt[a]*Sqrt[1 + 1/(a*x)]*(52 + 16*a*x + 9*a^2*x^2 + 3 
*a^3*x^3) - 42*Sqrt[2]*Sqrt[x^(-1)]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a 
]*Sqrt[1 + 1/(a*x)])]))/(21*a^(7/2)*Sqrt[1 - 1/(a*x)])
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.75, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6730, 27, 109, 27, 169, 27, 169, 27, 169, 27, 104, 219}

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.

\(\displaystyle \int x^2 \sqrt {c-a c x} e^{3 \coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6730

\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {a \left (1+\frac {1}{a x}\right )^{3/2}}{\left (a-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {\left (1+\frac {1}{a x}\right )^{3/2}}{\left (a-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 109

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (-\frac {2 \int -\frac {15 a+\frac {13}{x}}{2 a \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}}{7 a}-\frac {2 \sqrt {\frac {1}{a x}+1}}{7 a \left (\frac {1}{x}\right )^{7/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\int \frac {15 a+\frac {13}{x}}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}}{7 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{7 a \left (\frac {1}{x}\right )^{7/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {-\frac {2 \int -\frac {10 \left (4 a+\frac {3}{x}\right )}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}}{5 a}-\frac {6 \sqrt {\frac {1}{a x}+1}}{\left (\frac {1}{x}\right )^{5/2}}}{7 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{7 a \left (\frac {1}{x}\right )^{7/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\frac {4 \int \frac {4 a+\frac {3}{x}}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}}{a}-\frac {6 \sqrt {\frac {1}{a x}+1}}{\left (\frac {1}{x}\right )^{5/2}}}{7 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{7 a \left (\frac {1}{x}\right )^{7/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\frac {4 \left (-\frac {2 \int -\frac {13 a+\frac {8}{x}}{2 \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}}{3 a}-\frac {8 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )}{a}-\frac {6 \sqrt {\frac {1}{a x}+1}}{\left (\frac {1}{x}\right )^{5/2}}}{7 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{7 a \left (\frac {1}{x}\right )^{7/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\frac {4 \left (\frac {\int \frac {13 a+\frac {8}{x}}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}}{3 a}-\frac {8 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )}{a}-\frac {6 \sqrt {\frac {1}{a x}+1}}{\left (\frac {1}{x}\right )^{5/2}}}{7 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{7 a \left (\frac {1}{x}\right )^{7/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\frac {4 \left (\frac {-\frac {2 \int -\frac {21 a}{2 \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}}{a}-\frac {26 \sqrt {\frac {1}{a x}+1}}{\sqrt {\frac {1}{x}}}}{3 a}-\frac {8 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )}{a}-\frac {6 \sqrt {\frac {1}{a x}+1}}{\left (\frac {1}{x}\right )^{5/2}}}{7 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{7 a \left (\frac {1}{x}\right )^{7/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\frac {4 \left (\frac {21 \int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}-\frac {26 \sqrt {\frac {1}{a x}+1}}{\sqrt {\frac {1}{x}}}}{3 a}-\frac {8 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )}{a}-\frac {6 \sqrt {\frac {1}{a x}+1}}{\left (\frac {1}{x}\right )^{5/2}}}{7 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{7 a \left (\frac {1}{x}\right )^{7/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\frac {4 \left (\frac {42 \int \frac {1}{a-\frac {2}{x^2}}d\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {26 \sqrt {\frac {1}{a x}+1}}{\sqrt {\frac {1}{x}}}}{3 a}-\frac {8 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )}{a}-\frac {6 \sqrt {\frac {1}{a x}+1}}{\left (\frac {1}{x}\right )^{5/2}}}{7 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{7 a \left (\frac {1}{x}\right )^{7/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \left (\frac {\frac {4 \left (\frac {\frac {21 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\sqrt {a}}-\frac {26 \sqrt {\frac {1}{a x}+1}}{\sqrt {\frac {1}{x}}}}{3 a}-\frac {8 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )}{a}-\frac {6 \sqrt {\frac {1}{a x}+1}}{\left (\frac {1}{x}\right )^{5/2}}}{7 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{7 a \left (\frac {1}{x}\right )^{7/2}}\right ) \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}\)

Input:

Int[E^(3*ArcCoth[a*x])*x^2*Sqrt[c - a*c*x],x]
 

Output:

-((a*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*((-2*Sqrt[1 + 1/(a*x)])/(7*a*(x^(-1))^(7 
/2)) + ((-6*Sqrt[1 + 1/(a*x)])/(x^(-1))^(5/2) + (4*((-8*Sqrt[1 + 1/(a*x)]) 
/(3*(x^(-1))^(3/2)) + ((-26*Sqrt[1 + 1/(a*x)])/Sqrt[x^(-1)] + (21*Sqrt[2]* 
ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/Sqrt[a])/(3*a 
)))/a)/(7*a^2)))/Sqrt[1 - 1/(a*x)])
 

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 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 109

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

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 6730

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

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.53

method	result	size
risch	\(-\frac {2 \left (3 a^{3} x^{3}+9 a^{2} x^{2}+16 a x +52\right ) c \left (a x -1\right )}{21 a^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}-\frac {4 \sqrt {2}\, \sqrt {c}\, \arctan \left (\frac {\sqrt {-a c x -c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {-c \left (a x +1\right )}\, \left (a x -1\right )}{a^{3} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}}\)	\(138\)
default	\(\frac {2 \left (a x -1\right ) \sqrt {-c \left (a x -1\right )}\, \left (3 a^{3} x^{3} \sqrt {-c \left (a x +1\right )}+9 a^{2} x^{2} \sqrt {-c \left (a x +1\right )}+16 a x \sqrt {-c \left (a x +1\right )}-42 \sqrt {c}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )+52 \sqrt {-c \left (a x +1\right )}\right )}{21 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x +1\right )}\, a^{3}}\)	\(143\)

Input:

int(1/((a*x-1)/(a*x+1))^(3/2)*x^2*(-a*c*x+c)^(1/2),x,method=_RETURNVERBOSE 
)
 

Output:

-2/21*(3*a^3*x^3+9*a^2*x^2+16*a*x+52)/a^3*c/((a*x-1)/(a*x+1))^(1/2)/(-c*(a 
*x-1))^(1/2)*(a*x-1)-4/a^3*2^(1/2)*c^(1/2)*arctan(1/2*(-a*c*x-c)^(1/2)*2^( 
1/2)/c^(1/2))/((a*x-1)/(a*x+1))^(1/2)/(a*x+1)*(-c*(a*x+1))^(1/2)/(-c*(a*x- 
1))^(1/2)*(a*x-1)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.12 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\left [\frac {2 \, {\left (21 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + {\left (3 \, a^{4} x^{4} + 12 \, a^{3} x^{3} + 25 \, a^{2} x^{2} + 68 \, a x + 52\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{21 \, {\left (a^{4} x - a^{3}\right )}}, -\frac {2 \, {\left (42 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a c x - c\right )}}\right ) - {\left (3 \, a^{4} x^{4} + 12 \, a^{3} x^{3} + 25 \, a^{2} x^{2} + 68 \, a x + 52\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{21 \, {\left (a^{4} x - a^{3}\right )}}\right ] \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*x^2*(-a*c*x+c)^(1/2),x, algorithm="fri 
cas")
 

Output:

[2/21*(21*sqrt(2)*(a*x - 1)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x + 2*sqrt(2) 
*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 3*c)/(a^2 
*x^2 - 2*a*x + 1)) + (3*a^4*x^4 + 12*a^3*x^3 + 25*a^2*x^2 + 68*a*x + 52)*s 
qrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^4*x - a^3), -2/21*(42*sqrt(2 
)*(a*x - 1)*sqrt(c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(c)* 
sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - (3*a^4*x^4 + 12*a^3*x^3 + 25*a^2* 
x^2 + 68*a*x + 52)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^4*x - a^ 
3)]
 

Sympy [F]

\[ \int e^{3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\int \frac {x^{2} \sqrt {- c \left (a x - 1\right )}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/((a*x-1)/(a*x+1))**(3/2)*x**2*(-a*c*x+c)**(1/2),x)
 

Output:

Integral(x**2*sqrt(-c*(a*x - 1))/((a*x - 1)/(a*x + 1))**(3/2), x)
 

Maxima [F]

\[ \int e^{3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\int { \frac {\sqrt {-a c x + c} x^{2}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*x^2*(-a*c*x+c)^(1/2),x, algorithm="max 
ima")
 

Output:

integrate(sqrt(-a*c*x + c)*x^2/((a*x - 1)/(a*x + 1))^(3/2), x)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.55 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=-\frac {2 \, c^{2} {\left (\frac {2 \, \sqrt {2} {\left (21 \, \sqrt {c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {c}}\right ) - 40 \, \sqrt {-c}\right )}}{a^{2} c} - \frac {42 \, \sqrt {2} c^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right ) - 3 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} + 7 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c^{2} - 42 \, \sqrt {-a c x - c} c^{3}}{a^{2} c^{4}}\right )}}{21 \, a {\left | c \right |} \mathrm {sgn}\left (a x + 1\right )} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*x^2*(-a*c*x+c)^(1/2),x, algorithm="gia 
c")
 

Output:

-2/21*c^2*(2*sqrt(2)*(21*sqrt(c)*arctan(sqrt(-c)/sqrt(c)) - 40*sqrt(-c))/( 
a^2*c) - (42*sqrt(2)*c^(7/2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c)) 
- 3*(a*c*x + c)^3*sqrt(-a*c*x - c) + 7*(-a*c*x - c)^(3/2)*c^2 - 42*sqrt(-a 
*c*x - c)*c^3)/(a^2*c^4))/(a*abs(c)*sgn(a*x + 1))
 

Mupad [F(-1)]

Timed out. \[ \int e^{3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\int \frac {x^2\,\sqrt {c-a\,c\,x}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \] Input:

int((x^2*(c - a*c*x)^(1/2))/((a*x - 1)/(a*x + 1))^(3/2),x)
 

Output:

int((x^2*(c - a*c*x)^(1/2))/((a*x - 1)/(a*x + 1))^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.31 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c}\, i \left (-3 \sqrt {a x +1}\, a^{3} x^{3}-9 \sqrt {a x +1}\, a^{2} x^{2}-16 \sqrt {a x +1}\, a x -52 \sqrt {a x +1}-42 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right )+80 \sqrt {2}\right )}{21 a^{3}} \] Input:

int(1/((a*x-1)/(a*x+1))^(3/2)*x^2*(-a*c*x+c)^(1/2),x)
 

Output:

(2*sqrt(c)*i*( - 3*sqrt(a*x + 1)*a**3*x**3 - 9*sqrt(a*x + 1)*a**2*x**2 - 1 
6*sqrt(a*x + 1)*a*x - 52*sqrt(a*x + 1) - 42*sqrt(2)*log(tan(asin(sqrt( - a 
*x + 1)/sqrt(2))/2)) + 80*sqrt(2)))/(21*a**3)