Integrand size = 23, antiderivative size = 97 \[ \int e^{-2 \text {arctanh}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {4 \sqrt {c-a c x}}{a^3}+\frac {2 (c-a c x)^{3/2}}{3 a^3 c}+\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}-\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^3} \] Output:
4*(-a*c*x+c)^(1/2)/a^3+2/3*(-a*c*x+c)^(3/2)/a^3/c+2/7*(-a*c*x+c)^(7/2)/a^3 /c^3-4*2^(1/2)*c^(1/2)*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))/a^3
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.80 \[ \int e^{-2 \text {arctanh}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c-a c x} \left (52-16 a x+9 a^2 x^2-3 a^3 x^3\right )-84 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{21 a^3} \] Input:
Integrate[(x^2*Sqrt[c - a*c*x])/E^(2*ArcTanh[a*x]),x]
Output:
(2*Sqrt[c - a*c*x]*(52 - 16*a*x + 9*a^2*x^2 - 3*a^3*x^3) - 84*Sqrt[2]*Sqrt [c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/(21*a^3)
Time = 0.38 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6680, 35, 99, 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.
\(\displaystyle \int x^2 e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle \int \frac {x^2 (1-a x) \sqrt {c-a c x}}{a x+1}dx\) |
\(\Big \downarrow \) 35 |
\(\displaystyle \frac {\int \frac {x^2 (c-a c x)^{3/2}}{a x+1}dx}{c}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\int \left (\frac {(c-a c x)^{3/2}}{a^2 (a x+1)}-\frac {(c-a c x)^{5/2}}{a^2 c}\right )dx}{c}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {4 \sqrt {2} c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^3}+\frac {2 (c-a c x)^{7/2}}{7 a^3 c^2}+\frac {2 (c-a c x)^{3/2}}{3 a^3}+\frac {4 c \sqrt {c-a c x}}{a^3}}{c}\) |
Input:
Int[(x^2*Sqrt[c - a*c*x])/E^(2*ArcTanh[a*x]),x]
Output:
((4*c*Sqrt[c - a*c*x])/a^3 + (2*(c - a*c*x)^(3/2))/(3*a^3) + (2*(c - a*c*x )^(7/2))/(7*a^3*c^2) - (4*Sqrt[2]*c^(3/2)*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2] *Sqrt[c])])/a^3)/c
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])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
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])
Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(\frac {-84 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-2 \sqrt {-c \left (a x -1\right )}\, \left (3 a^{3} x^{3}-9 a^{2} x^{2}+16 a x -52\right )}{21 a^{3}}\) | \(68\) |
risch | \(\frac {2 \left (3 a^{3} x^{3}-9 a^{2} x^{2}+16 a x -52\right ) \left (a x -1\right ) c}{21 a^{3} \sqrt {-c \left (a x -1\right )}}-\frac {4 \sqrt {2}\, \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a^{3}}\) | \(74\) |
derivativedivides | \(\frac {\frac {2 \left (-a c x +c \right )^{\frac {7}{2}}}{7}+\frac {2 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{3}+4 \sqrt {-a c x +c}\, c^{3}-4 c^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a^{3} c^{3}}\) | \(75\) |
default | \(-\frac {2 \left (-\frac {\left (-a c x +c \right )^{\frac {7}{2}}}{7}-\frac {c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{3}-2 \sqrt {-a c x +c}\, c^{3}+2 c^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{c^{3} a^{3}}\) | \(75\) |
Input:
int(x^2*(-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x,method=_RETURNVERBOSE)
Output:
1/21*(-84*c^(1/2)*2^(1/2)*arctanh(1/2*(-c*(a*x-1))^(1/2)*2^(1/2)/c^(1/2))- 2*(-c*(a*x-1))^(1/2)*(3*a^3*x^3-9*a^2*x^2+16*a*x-52))/a^3
Time = 0.09 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.64 \[ \int e^{-2 \text {arctanh}(a x)} x^2 \sqrt {c-a c x} \, dx=\left [\frac {2 \, {\left (21 \, \sqrt {2} \sqrt {c} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) - {\left (3 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 16 \, a x - 52\right )} \sqrt {-a c x + c}\right )}}{21 \, a^{3}}, -\frac {2 \, {\left (42 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) + {\left (3 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 16 \, a x - 52\right )} \sqrt {-a c x + c}\right )}}{21 \, a^{3}}\right ] \] Input:
integrate(x^2*(-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="fricas ")
Output:
[2/21*(21*sqrt(2)*sqrt(c)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) - (3*a^3*x^3 - 9*a^2*x^2 + 16*a*x - 52)*sqrt(-a*c*x + c) )/a^3, -2/21*(42*sqrt(2)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c) /(a*c*x - c)) + (3*a^3*x^3 - 9*a^2*x^2 + 16*a*x - 52)*sqrt(-a*c*x + c))/a^ 3]
Time = 7.34 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.40 \[ \int e^{-2 \text {arctanh}(a x)} x^2 \sqrt {c-a c x} \, dx=\begin {cases} - \frac {2 \left (- \frac {2 \sqrt {2} c^{4} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} - 2 c^{3} \sqrt {- a c x + c} - \frac {c^{2} \left (- a c x + c\right )^{\frac {3}{2}}}{3} - \frac {\left (- a c x + c\right )^{\frac {7}{2}}}{7}\right )}{a^{3} c^{3}} & \text {for}\: a c \neq 0 \\\sqrt {c} \left (- \frac {x^{3}}{3} + \frac {x^{2}}{a} - \frac {2 x}{a^{2}} + \frac {2 \left (\begin {cases} x & \text {for}\: a = 0 \\\frac {\log {\left (a x + 1 \right )}}{a} & \text {otherwise} \end {cases}\right )}{a^{2}}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(-a*c*x+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)
Output:
Piecewise((-2*(-2*sqrt(2)*c**4*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c))) /sqrt(-c) - 2*c**3*sqrt(-a*c*x + c) - c**2*(-a*c*x + c)**(3/2)/3 - (-a*c*x + c)**(7/2)/7)/(a**3*c**3), Ne(a*c, 0)), (sqrt(c)*(-x**3/3 + x**2/a - 2*x /a**2 + 2*Piecewise((x, Eq(a, 0)), (log(a*x + 1)/a, True))/a**2), True))
Time = 0.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int e^{-2 \text {arctanh}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \, {\left (21 \, \sqrt {2} c^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 3 \, {\left (-a c x + c\right )}^{\frac {7}{2}} + 7 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} + 42 \, \sqrt {-a c x + c} c^{3}\right )}}{21 \, a^{3} c^{3}} \] Input:
integrate(x^2*(-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="maxima ")
Output:
2/21*(21*sqrt(2)*c^(7/2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt(2 )*sqrt(c) + sqrt(-a*c*x + c))) + 3*(-a*c*x + c)^(7/2) + 7*(-a*c*x + c)^(3/ 2)*c^2 + 42*sqrt(-a*c*x + c)*c^3)/(a^3*c^3)
Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.08 \[ \int e^{-2 \text {arctanh}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {4 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a^{3} \sqrt {-c}} - \frac {2 \, {\left (3 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{18} c^{18} - 7 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{18} c^{20} - 42 \, \sqrt {-a c x + c} a^{18} c^{21}\right )}}{21 \, a^{21} c^{21}} \] Input:
integrate(x^2*(-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="giac")
Output:
4*sqrt(2)*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/(a^3*sqrt(-c)) - 2/21*(3*(a*c*x - c)^3*sqrt(-a*c*x + c)*a^18*c^18 - 7*(-a*c*x + c)^(3/2)*a ^18*c^20 - 42*sqrt(-a*c*x + c)*a^18*c^21)/(a^21*c^21)
Time = 15.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int e^{-2 \text {arctanh}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {4\,\sqrt {c-a\,c\,x}}{a^3}+\frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a^3\,c}+\frac {2\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a^3\,c^3}+\frac {\sqrt {2}\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i}}{a^3} \] Input:
int(-(x^2*(a^2*x^2 - 1)*(c - a*c*x)^(1/2))/(a*x + 1)^2,x)
Output:
(4*(c - a*c*x)^(1/2))/a^3 + (2*(c - a*c*x)^(3/2))/(3*a^3*c) + (2*(c - a*c* x)^(7/2))/(7*a^3*c^3) + (2^(1/2)*c^(1/2)*atan((2^(1/2)*(c - a*c*x)^(1/2)*1 i)/(2*c^(1/2)))*4i)/a^3
Time = 0.14 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int e^{-2 \text {arctanh}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c}\, \left (-3 \sqrt {-a x +1}\, a^{3} x^{3}+9 \sqrt {-a x +1}\, a^{2} x^{2}-16 \sqrt {-a x +1}\, a x +52 \sqrt {-a x +1}+21 \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right )-21 \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right )\right )}{21 a^{3}} \] Input:
int(x^2*(-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x)
Output:
(2*sqrt(c)*( - 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) + 21*sqrt(2)*log(sqrt( - a*x + 1) - sqrt(2)) - 21*sqrt(2)*log(sqrt( - a*x + 1) + sqrt(2))))/(21*a** 3)