Integrand size = 23, antiderivative size = 80 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {4 \sqrt {c-a c x}}{a^3}-\frac {10 (c-a c x)^{3/2}}{3 a^3 c}+\frac {8 (c-a c x)^{5/2}}{5 a^3 c^2}-\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3} \] Output:
4*(-a*c*x+c)^(1/2)/a^3-10/3*(-a*c*x+c)^(3/2)/a^3/c+8/5*(-a*c*x+c)^(5/2)/a^ 3/c^2-2/7*(-a*c*x+c)^(7/2)/a^3/c^3
Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.50 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c-a c x} \left (104+52 a x+39 a^2 x^2+15 a^3 x^3\right )}{105 a^3} \] Input:
Integrate[E^(2*ArcCoth[a*x])*x^2*Sqrt[c - a*c*x],x]
Output:
(2*Sqrt[c - a*c*x]*(104 + 52*a*x + 39*a^2*x^2 + 15*a^3*x^3))/(105*a^3)
Time = 0.77 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6717, 6680, 35, 86, 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.
\(\displaystyle \int x^2 \sqrt {c-a c x} e^{2 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int e^{2 \text {arctanh}(a x)} x^2 \sqrt {c-a c x}dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle -\int \frac {x^2 (a x+1) \sqrt {c-a c x}}{1-a x}dx\) |
\(\Big \downarrow \) 35 |
\(\displaystyle -c \int \frac {x^2 (a x+1)}{\sqrt {c-a c x}}dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle -c \int \left (-\frac {(c-a c x)^{5/2}}{a^2 c^3}+\frac {4 (c-a c x)^{3/2}}{a^2 c^2}-\frac {5 \sqrt {c-a c x}}{a^2 c}+\frac {2}{a^2 \sqrt {c-a c x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -c \left (\frac {2 (c-a c x)^{7/2}}{7 a^3 c^4}-\frac {8 (c-a c x)^{5/2}}{5 a^3 c^3}+\frac {10 (c-a c x)^{3/2}}{3 a^3 c^2}-\frac {4 \sqrt {c-a c x}}{a^3 c}\right )\) |
Input:
Int[E^(2*ArcCoth[a*x])*x^2*Sqrt[c - a*c*x],x]
Output:
-(c*((-4*Sqrt[c - a*c*x])/(a^3*c) + (10*(c - a*c*x)^(3/2))/(3*a^3*c^2) - ( 8*(c - a*c*x)^(5/2))/(5*a^3*c^3) + (2*(c - a*c*x)^(7/2))/(7*a^3*c^4)))
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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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])
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.46
method | result | size |
gosper | \(\frac {2 \sqrt {-a c x +c}\, \left (15 a^{3} x^{3}+39 a^{2} x^{2}+52 a x +104\right )}{105 a^{3}}\) | \(37\) |
trager | \(\frac {2 \sqrt {-a c x +c}\, \left (15 a^{3} x^{3}+39 a^{2} x^{2}+52 a x +104\right )}{105 a^{3}}\) | \(37\) |
orering | \(\frac {2 \sqrt {-a c x +c}\, \left (15 a^{3} x^{3}+39 a^{2} x^{2}+52 a x +104\right )}{105 a^{3}}\) | \(37\) |
pseudoelliptic | \(\frac {2 \sqrt {-c \left (a x -1\right )}\, \left (15 a^{3} x^{3}+39 a^{2} x^{2}+52 a x +104\right )}{105 a^{3}}\) | \(38\) |
risch | \(-\frac {2 c \left (15 a^{3} x^{3}+39 a^{2} x^{2}+52 a x +104\right ) \left (a x -1\right )}{105 a^{3} \sqrt {-c \left (a x -1\right )}}\) | \(44\) |
derivativedivides | \(-\frac {2 \left (\frac {\left (-a c x +c \right )^{\frac {7}{2}}}{7}-\frac {4 \left (-a c x +c \right )^{\frac {5}{2}} c}{5}+\frac {5 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{3}-2 \sqrt {-a c x +c}\, c^{3}\right )}{c^{3} a^{3}}\) | \(61\) |
default | \(\frac {-\frac {2 \left (-a c x +c \right )^{\frac {7}{2}}}{7}+\frac {8 \left (-a c x +c \right )^{\frac {5}{2}} c}{5}-\frac {10 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{3}+4 \sqrt {-a c x +c}\, c^{3}}{a^{3} c^{3}}\) | \(61\) |
Input:
int((-a*c*x+c)^(1/2)*(a*x+1)*x^2/(a*x-1),x,method=_RETURNVERBOSE)
Output:
2/105*(-a*c*x+c)^(1/2)*(15*a^3*x^3+39*a^2*x^2+52*a*x+104)/a^3
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.45 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \, {\left (15 \, a^{3} x^{3} + 39 \, a^{2} x^{2} + 52 \, a x + 104\right )} \sqrt {-a c x + c}}{105 \, a^{3}} \] Input:
integrate(1/(a*x-1)*(a*x+1)*x^2*(-a*c*x+c)^(1/2),x, algorithm="fricas")
Output:
2/105*(15*a^3*x^3 + 39*a^2*x^2 + 52*a*x + 104)*sqrt(-a*c*x + c)/a^3
Time = 2.42 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.36 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\begin {cases} - \frac {2 \left (- 2 c^{3} \sqrt {- a c x + c} + \frac {5 c^{2} \left (- a c x + c\right )^{\frac {3}{2}}}{3} - \frac {4 c \left (- a c x + c\right )^{\frac {5}{2}}}{5} + \frac {\left (- a c x + c\right )^{\frac {7}{2}}}{7}\right )}{a^{3} c^{3}} & \text {for}\: a c \neq 0 \\\sqrt {c} \left (\frac {x^{3}}{3} + \frac {x^{2}}{a} + \frac {2 x}{a^{2}} + \frac {2 \left (\begin {cases} - x & \text {for}\: a = 0 \\\frac {\log {\left (a x - 1 \right )}}{a} & \text {otherwise} \end {cases}\right )}{a^{2}}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(1/(a*x-1)*(a*x+1)*x**2*(-a*c*x+c)**(1/2),x)
Output:
Piecewise((-2*(-2*c**3*sqrt(-a*c*x + c) + 5*c**2*(-a*c*x + c)**(3/2)/3 - 4 *c*(-a*c*x + c)**(5/2)/5 + (-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 = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=-\frac {2 \, {\left (15 \, {\left (-a c x + c\right )}^{\frac {7}{2}} - 84 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c + 175 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} - 210 \, \sqrt {-a c x + c} c^{3}\right )}}{105 \, a^{3} c^{3}} \] Input:
integrate(1/(a*x-1)*(a*x+1)*x^2*(-a*c*x+c)^(1/2),x, algorithm="maxima")
Output:
-2/105*(15*(-a*c*x + c)^(7/2) - 84*(-a*c*x + c)^(5/2)*c + 175*(-a*c*x + c) ^(3/2)*c^2 - 210*sqrt(-a*c*x + c)*c^3)/(a^3*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (66) = 132\).
Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.78 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \, {\left (\frac {7 \, {\left (3 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} - 10 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {-a c x + c} c^{2}\right )}}{a^{2} c^{2}} + \frac {3 \, {\left (5 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} + 21 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} c - 35 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} + 35 \, \sqrt {-a c x + c} c^{3}\right )}}{a^{2} c^{3}}\right )}}{105 \, a} \] Input:
integrate(1/(a*x-1)*(a*x+1)*x^2*(-a*c*x+c)^(1/2),x, algorithm="giac")
Output:
2/105*(7*(3*(a*c*x - c)^2*sqrt(-a*c*x + c) - 10*(-a*c*x + c)^(3/2)*c + 15* sqrt(-a*c*x + c)*c^2)/(a^2*c^2) + 3*(5*(a*c*x - c)^3*sqrt(-a*c*x + c) + 21 *(a*c*x - c)^2*sqrt(-a*c*x + c)*c - 35*(-a*c*x + c)^(3/2)*c^2 + 35*sqrt(-a *c*x + c)*c^3)/(a^2*c^3))/a
Time = 13.63 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.82 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {4\,\sqrt {c-a\,c\,x}}{a^3}-\frac {10\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a^3\,c}+\frac {8\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a^3\,c^2}-\frac {2\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a^3\,c^3} \] Input:
int((x^2*(c - a*c*x)^(1/2)*(a*x + 1))/(a*x - 1),x)
Output:
(4*(c - a*c*x)^(1/2))/a^3 - (10*(c - a*c*x)^(3/2))/(3*a^3*c) + (8*(c - a*c *x)^(5/2))/(5*a^3*c^2) - (2*(c - a*c*x)^(7/2))/(7*a^3*c^3)
Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.45 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c}\, \sqrt {-a x +1}\, \left (15 a^{3} x^{3}+39 a^{2} x^{2}+52 a x +104\right )}{105 a^{3}} \] Input:
int(1/(a*x-1)*(a*x+1)*x^2*(-a*c*x+c)^(1/2),x)
Output:
(2*sqrt(c)*sqrt( - a*x + 1)*(15*a**3*x**3 + 39*a**2*x**2 + 52*a*x + 104))/ (105*a**3)