Integrand size = 23, antiderivative size = 231 \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=-\frac {2672 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{105 a^3 \sqrt {1-\frac {1}{a x}}}-\frac {334 x \sqrt {c-a c x}}{35 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {1336 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {44 x^2 \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}} \]
-334/35*x*(-a*c*x+c)^(1/2)/a^2/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)-44/35*x^2*( -a*c*x+c)^(1/2)/a/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)+2/7*x^3*(-a*c*x+c)^(1/2) /(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)-2672/105*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2) /a^3/(1-1/a/x)^(1/2)+1336/105*x*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/a^2/(1-1/ a/x)^(1/2)
Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.28 \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c-a c x} \left (-1336-668 a x+167 a^2 x^2-66 a^3 x^3+15 a^4 x^4\right )}{105 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
(2*Sqrt[c - a*c*x]*(-1336 - 668*a*x + 167*a^2*x^2 - 66*a^3*x^3 + 15*a^4*x^ 4))/(105*a^4*Sqrt[1 - 1/(a^2*x^2)]*x)
Time = 0.36 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.76, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6730, 27, 100, 27, 87, 55, 55, 48}
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 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {\left (a-\frac {1}{x}\right )^2}{a^2 \left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {\left (a-\frac {1}{x}\right )^2}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {2}{7} \int -\frac {22 a-\frac {7}{x}}{2 \left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}-\frac {2 a^2}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (-\frac {1}{7} \int \frac {22 a-\frac {7}{x}}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}-\frac {2 a^2}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{7} \left (\frac {167}{5} \int \frac {1}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}+\frac {44 a}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{7} \left (\frac {167}{5} \left (4 \int \frac {1}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}+\frac {2}{\left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {44 a}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{7} \left (\frac {167}{5} \left (4 \left (-\frac {2 \int \frac {1}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}}{3 a}-\frac {2 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )+\frac {2}{\left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {44 a}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{7} \left (\frac {44 a}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}+\frac {167}{5} \left (4 \left (\frac {4 \sqrt {\frac {1}{a x}+1}}{3 a \sqrt {\frac {1}{x}}}-\frac {2 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )+\frac {2}{\left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )\right )-\frac {2 a^2}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right ) \sqrt {c-a c x}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
-(((((167*(4*((-2*Sqrt[1 + 1/(a*x)])/(3*(x^(-1))^(3/2)) + (4*Sqrt[1 + 1/(a *x)])/(3*a*Sqrt[x^(-1)])) + 2/(Sqrt[1 + 1/(a*x)]*(x^(-1))^(3/2))))/5 + (44 *a)/(5*Sqrt[1 + 1/(a*x)]*(x^(-1))^(5/2)))/7 - (2*a^2)/(7*Sqrt[1 + 1/(a*x)] *(x^(-1))^(7/2)))*Sqrt[x^(-1)]*Sqrt[c - a*c*x])/(a^2*Sqrt[1 - 1/(a*x)]))
3.4.51.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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])))
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]
Time = 0.46 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.31
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \left (15 a^{4} x^{4}-66 a^{3} x^{3}+167 a^{2} x^{2}-668 a x -1336\right ) \sqrt {-a c x +c}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{105 a^{3} \left (a x -1\right )^{2}}\) | \(72\) |
default | \(\frac {2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (15 a^{4} x^{4}-66 a^{3} x^{3}+167 a^{2} x^{2}-668 a x -1336\right )}{105 \left (a x -1\right )^{2} a^{3}}\) | \(73\) |
risch | \(-\frac {2 \left (15 a^{3} x^{3}-81 a^{2} x^{2}+248 a x -916\right ) \left (a x +1\right ) c \sqrt {\frac {a x -1}{a x +1}}}{105 a^{3} \sqrt {-c \left (a x -1\right )}}+\frac {8 c \sqrt {\frac {a x -1}{a x +1}}}{a^{3} \sqrt {-c \left (a x -1\right )}}\) | \(91\) |
2/105*(a*x+1)*(15*a^4*x^4-66*a^3*x^3+167*a^2*x^2-668*a*x-1336)*(-a*c*x+c)^ (1/2)*((a*x-1)/(a*x+1))^(3/2)/a^3/(a*x-1)^2
Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.30 \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \, {\left (15 \, a^{4} x^{4} - 66 \, a^{3} x^{3} + 167 \, a^{2} x^{2} - 668 \, a x - 1336\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{105 \, {\left (a^{4} x - a^{3}\right )}} \]
2/105*(15*a^4*x^4 - 66*a^3*x^3 + 167*a^2*x^2 - 668*a*x - 1336)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^4*x - a^3)
Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\text {Timed out} \]
Time = 0.24 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.45 \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \, {\left (15 \, a^{5} \sqrt {-c} x^{5} - 51 \, a^{4} \sqrt {-c} x^{4} + 101 \, a^{3} \sqrt {-c} x^{3} - 501 \, a^{2} \sqrt {-c} x^{2} - 2004 \, a \sqrt {-c} x - 1336 \, \sqrt {-c}\right )} {\left (a x - 1\right )}^{2}}{105 \, {\left (a^{5} x^{2} - 2 \, a^{4} x + a^{3}\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]
2/105*(15*a^5*sqrt(-c)*x^5 - 51*a^4*sqrt(-c)*x^4 + 101*a^3*sqrt(-c)*x^3 - 501*a^2*sqrt(-c)*x^2 - 2004*a*sqrt(-c)*x - 1336*sqrt(-c))*(a*x - 1)^2/((a^ 5*x^2 - 2*a^4*x + a^3)*(a*x + 1)^(3/2))
Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\text {Exception raised: TypeError} \]
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
Time = 4.58 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.38 \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (15\,a^3\,x^3-51\,a^2\,x^2+116\,a\,x-552\right )}{105\,a^3}-\frac {3776\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{105\,a^3\,\left (a\,x-1\right )} \]