Integrand size = 20, antiderivative size = 254 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=-\frac {32 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {2944 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^3}-\frac {256 (c-a c x)^{5/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {128 (c-a c x)^{5/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}} \]
-32/35*(a-1/x)^3*(-a*c*x+c)^(5/2)/a^4/(1-1/a/x)^(5/2)/(1+1/a/x)^(1/2)-2944 /35*(-a*c*x+c)^(5/2)/a^4/(1-1/a/x)^(5/2)/x^3/(1+1/a/x)^(1/2)-256/7*(-a*c*x +c)^(5/2)/a^3/(1-1/a/x)^(5/2)/x^2/(1+1/a/x)^(1/2)+128/35*(-a*c*x+c)^(5/2)/ a^2/(1-1/a/x)^(5/2)/x/(1+1/a/x)^(1/2)+2/7*(a-1/x)^4*x*(-a*c*x+c)^(5/2)/a^4 /(1-1/a/x)^(5/2)/(1+1/a/x)^(1/2)
Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.27 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 c^2 \sqrt {c-a c x} \left (-1451-708 a x+142 a^2 x^2-36 a^3 x^3+5 a^4 x^4\right )}{35 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
(2*c^2*Sqrt[c - a*c*x]*(-1451 - 708*a*x + 142*a^2*x^2 - 36*a^3*x^3 + 5*a^4 *x^4))/(35*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)
Time = 0.31 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.75, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6727, 27, 105, 105, 100, 27, 87, 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.
\(\displaystyle \int (c-a c x)^{5/2} e^{-3 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6727 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2} \int \frac {\left (a-\frac {1}{x}\right )^4}{a^4 \left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}}{\left (1-\frac {1}{a x}\right )^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2} \int \frac {\left (a-\frac {1}{x}\right )^4}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}}{a^4 \left (1-\frac {1}{a x}\right )^{5/2}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2} \left (-\frac {16}{7} \int \frac {\left (a-\frac {1}{x}\right )^3}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^4 \left (1-\frac {1}{a x}\right )^{5/2}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2} \left (-\frac {16}{7} \left (-\frac {12}{5} \int \frac {\left (a-\frac {1}{x}\right )^2}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^4 \left (1-\frac {1}{a x}\right )^{5/2}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2} \left (-\frac {16}{7} \left (-\frac {12}{5} \left (\frac {2}{3} \int -\frac {10 a-\frac {3}{x}}{2 \left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}-\frac {2 a^2}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^4 \left (1-\frac {1}{a x}\right )^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2} \left (-\frac {16}{7} \left (-\frac {12}{5} \left (-\frac {1}{3} \int \frac {10 a-\frac {3}{x}}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}-\frac {2 a^2}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^4 \left (1-\frac {1}{a x}\right )^{5/2}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2} \left (-\frac {16}{7} \left (-\frac {12}{5} \left (\frac {1}{3} \left (23 \int \frac {1}{\left (1+\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{x}}}d\frac {1}{x}+\frac {20 a}{\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^4 \left (1-\frac {1}{a x}\right )^{5/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{5/2} \left (-\frac {16}{7} \left (-\frac {12}{5} \left (\frac {1}{3} \left (\frac {20 a}{\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}+\frac {46 \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right ) (c-a c x)^{5/2}}{a^4 \left (1-\frac {1}{a x}\right )^{5/2}}\) |
-((((-16*((-12*(((20*a)/(Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)]) + (46*Sqrt[x^(-1) ])/Sqrt[1 + 1/(a*x)])/3 - (2*a^2)/(3*Sqrt[1 + 1/(a*x)]*(x^(-1))^(3/2))))/5 - (2*(a - x^(-1))^3)/(5*Sqrt[1 + 1/(a*x)]*(x^(-1))^(5/2))))/7 - (2*(a - x ^(-1))^4)/(7*Sqrt[1 + 1/(a*x)]*(x^(-1))^(7/2)))*(x^(-1))^(5/2)*(c - a*c*x) ^(5/2))/(a^4*(1 - 1/(a*x))^(5/2)))
3.3.72.3.1 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_))*((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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_), x_Symbol] :> Si mp[(-(1/x)^p)*((c + d*x)^p/(1 + c/(d*x))^p) Subst[Int[((1 + c*(x/d))^p*(( 1 + x/a)^(n/2)/x^(p + 2)))/(1 - x/a)^(n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !IntegerQ[p]
Time = 0.47 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.28
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \left (5 a^{4} x^{4}-36 a^{3} x^{3}+142 a^{2} x^{2}-708 a x -1451\right ) \left (-a c x +c \right )^{\frac {5}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{35 a \left (a x -1\right )^{4}}\) | \(72\) |
default | \(\frac {2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c^{2} \left (5 a^{4} x^{4}-36 a^{3} x^{3}+142 a^{2} x^{2}-708 a x -1451\right )}{35 \left (a x -1\right )^{2} a}\) | \(76\) |
risch | \(-\frac {2 \left (5 a^{3} x^{3}-41 a^{2} x^{2}+183 a x -891\right ) \left (a x +1\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}}{35 a \sqrt {-c \left (a x -1\right )}}+\frac {32 c^{3} \sqrt {\frac {a x -1}{a x +1}}}{a \sqrt {-c \left (a x -1\right )}}\) | \(95\) |
2/35*(a*x+1)*(5*a^4*x^4-36*a^3*x^3+142*a^2*x^2-708*a*x-1451)*(-a*c*x+c)^(5 /2)*((a*x-1)/(a*x+1))^(3/2)/a/(a*x-1)^4
Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.33 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 \, {\left (5 \, a^{4} c^{2} x^{4} - 36 \, a^{3} c^{2} x^{3} + 142 \, a^{2} c^{2} x^{2} - 708 \, a c^{2} x - 1451 \, c^{2}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{2} x - a\right )}} \]
2/35*(5*a^4*c^2*x^4 - 36*a^3*c^2*x^3 + 142*a^2*c^2*x^2 - 708*a*c^2*x - 145 1*c^2)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^2*x - a)
Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.47 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 \, {\left (5 \, a^{5} \sqrt {-c} c^{2} x^{5} - 31 \, a^{4} \sqrt {-c} c^{2} x^{4} + 106 \, a^{3} \sqrt {-c} c^{2} x^{3} - 566 \, a^{2} \sqrt {-c} c^{2} x^{2} - 2159 \, a \sqrt {-c} c^{2} x - 1451 \, \sqrt {-c} c^{2}\right )} {\left (a x - 1\right )}^{2}}{35 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]
2/35*(5*a^5*sqrt(-c)*c^2*x^5 - 31*a^4*sqrt(-c)*c^2*x^4 + 106*a^3*sqrt(-c)* c^2*x^3 - 566*a^2*sqrt(-c)*c^2*x^2 - 2159*a*sqrt(-c)*c^2*x - 1451*sqrt(-c) *c^2)*(a*x - 1)^2/((a^3*x^2 - 2*a^2*x + a)*(a*x + 1)^(3/2))
Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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 = 4.46 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.37 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2\,c^2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (5\,a^3\,x^3-31\,a^2\,x^2+111\,a\,x-597\right )}{35\,a}-\frac {4096\,c^2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{35\,a\,\left (a\,x-1\right )} \]