Integrand size = 20, antiderivative size = 128 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {256 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x}{35 \sqrt {c-a c x}}+\frac {64}{35} c^2 \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a c x}+\frac {24}{35} c \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{3/2}+\frac {2}{7} \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{5/2} \] Output:
256/35*c^3*(1-1/a^2/x^2)^(1/2)*x/(-a*c*x+c)^(1/2)+64/35*c^2*(1-1/a^2/x^2)^ (1/2)*x*(-a*c*x+c)^(1/2)+24/35*c*(1-1/a^2/x^2)^(1/2)*x*(-a*c*x+c)^(3/2)+2/ 7*(1-1/a^2/x^2)^(1/2)*x*(-a*c*x+c)^(5/2)
Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.55 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 c^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x} \left (-177+71 a x-27 a^2 x^2+5 a^3 x^3\right )}{35 a \sqrt {1-\frac {1}{a x}}} \] Input:
Integrate[(c - a*c*x)^(5/2)/E^ArcCoth[a*x],x]
Output:
(2*c^2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(-177 + 71*a*x - 27*a^2*x^2 + 5*a ^3*x^3))/(35*a*Sqrt[1 - 1/(a*x)])
Time = 0.47 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.23, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6727, 27, 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.
\(\displaystyle \int (c-a c x)^{5/2} e^{-\coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6727 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2} \int \frac {\left (a-\frac {1}{x}\right )^3}{a^3 \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}}{\left (1-\frac {1}{a x}\right )^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2} \int \frac {\left (a-\frac {1}{x}\right )^3}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}}{a^3 \left (1-\frac {1}{a x}\right )^{5/2}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2} \left (-\frac {12}{7} \int \frac {\left (a-\frac {1}{x}\right )^2}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{7 \left (\frac {1}{x}\right )^{7/2}}\right )}{a^3 \left (1-\frac {1}{a x}\right )^{5/2}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2} \left (-\frac {12}{7} \left (\frac {2}{5} \int -\frac {14 a-\frac {5}{x}}{2 \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}-\frac {2 a^2 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{7 \left (\frac {1}{x}\right )^{7/2}}\right )}{a^3 \left (1-\frac {1}{a x}\right )^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2} \left (-\frac {12}{7} \left (-\frac {1}{5} \int \frac {14 a-\frac {5}{x}}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}-\frac {2 a^2 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{7 \left (\frac {1}{x}\right )^{7/2}}\right )}{a^3 \left (1-\frac {1}{a x}\right )^{5/2}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2} \left (-\frac {12}{7} \left (\frac {1}{5} \left (\frac {43}{3} \int \frac {1}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}+\frac {28 a \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )-\frac {2 a^2 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{7 \left (\frac {1}{x}\right )^{7/2}}\right )}{a^3 \left (1-\frac {1}{a x}\right )^{5/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{5/2} \left (-\frac {12}{7} \left (\frac {1}{5} \left (\frac {28 a \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}-\frac {86 \sqrt {\frac {1}{a x}+1}}{3 \sqrt {\frac {1}{x}}}\right )-\frac {2 a^2 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{7 \left (\frac {1}{x}\right )^{7/2}}\right ) (c-a c x)^{5/2}}{a^3 \left (1-\frac {1}{a x}\right )^{5/2}}\) |
Input:
Int[(c - a*c*x)^(5/2)/E^ArcCoth[a*x],x]
Output:
-((((-12*(((28*a*Sqrt[1 + 1/(a*x)])/(3*(x^(-1))^(3/2)) - (86*Sqrt[1 + 1/(a *x)])/(3*Sqrt[x^(-1)]))/5 - (2*a^2*Sqrt[1 + 1/(a*x)])/(5*(x^(-1))^(5/2)))) /7 - (2*(a - x^(-1))^3*Sqrt[1 + 1/(a*x)])/(7*(x^(-1))^(7/2)))*(x^(-1))^(5/ 2)*(c - a*c*x)^(5/2))/(a^3*(1 - 1/(a*x))^(5/2)))
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 = 0.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.48
method | result | size |
risch | \(-\frac {2 c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \left (5 a^{3} x^{3}-27 a^{2} x^{2}+71 a x -177\right ) \left (a x +1\right )}{35 \sqrt {-c \left (a x -1\right )}\, a}\) | \(61\) |
gosper | \(\frac {2 \left (a x +1\right ) \left (5 a^{3} x^{3}-27 a^{2} x^{2}+71 a x -177\right ) \left (-a c x +c \right )^{\frac {5}{2}} \sqrt {\frac {a x -1}{a x +1}}}{35 a \left (a x -1\right )^{3}}\) | \(64\) |
orering | \(\frac {2 \left (a x +1\right ) \left (5 a^{3} x^{3}-27 a^{2} x^{2}+71 a x -177\right ) \left (-a c x +c \right )^{\frac {5}{2}} \sqrt {\frac {a x -1}{a x +1}}}{35 a \left (a x -1\right )^{3}}\) | \(64\) |
default | \(\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c^{2} \left (5 a^{3} x^{3}-27 a^{2} x^{2}+71 a x -177\right )}{35 \left (a x -1\right ) a}\) | \(68\) |
Input:
int((-a*c*x+c)^(5/2)*((a*x-1)/(a*x+1))^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/35*c^3*((a*x-1)/(a*x+1))^(1/2)/(-c*(a*x-1))^(1/2)*(5*a^3*x^3-27*a^2*x^2 +71*a*x-177)/a*(a*x+1)
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.65 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 \, {\left (5 \, a^{4} c^{2} x^{4} - 22 \, a^{3} c^{2} x^{3} + 44 \, a^{2} c^{2} x^{2} - 106 \, a c^{2} x - 177 \, c^{2}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{2} x - a\right )}} \] Input:
integrate((-a*c*x+c)^(5/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fricas")
Output:
2/35*(5*a^4*c^2*x^4 - 22*a^3*c^2*x^3 + 44*a^2*c^2*x^2 - 106*a*c^2*x - 177* c^2)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^2*x - a)
Timed out. \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\text {Timed out} \] Input:
integrate((-a*c*x+c)**(5/2)*((a*x-1)/(a*x+1))**(1/2),x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.75 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 \, {\left (5 \, a^{4} \sqrt {-c} c^{2} x^{4} - 22 \, a^{3} \sqrt {-c} c^{2} x^{3} + 44 \, a^{2} \sqrt {-c} c^{2} x^{2} - 106 \, a \sqrt {-c} c^{2} x - 177 \, \sqrt {-c} c^{2}\right )} {\left (a x - 1\right )}}{35 \, {\left (a^{2} x - a\right )} \sqrt {a x + 1}} \] Input:
integrate((-a*c*x+c)^(5/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima")
Output:
2/35*(5*a^4*sqrt(-c)*c^2*x^4 - 22*a^3*sqrt(-c)*c^2*x^3 + 44*a^2*sqrt(-c)*c ^2*x^2 - 106*a*sqrt(-c)*c^2*x - 177*sqrt(-c)*c^2)*(a*x - 1)/((a^2*x - a)*s qrt(a*x + 1))
Exception generated. \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a*c*x+c)^(5/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="giac")
Output:
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 = 14.51 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.73 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2\,c^2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (5\,a^3\,x^3-17\,a^2\,x^2+27\,a\,x-79\right )}{35\,a}-\frac {512\,c^2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{35\,a\,\left (a\,x-1\right )} \] Input:
int((c - a*c*x)^(5/2)*((a*x - 1)/(a*x + 1))^(1/2),x)
Output:
(2*c^2*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(1/2)*(27*a*x - 17*a^2*x^2 + 5*a^3*x^3 - 79))/(35*a) - (512*c^2*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1 ))^(1/2))/(35*a*(a*x - 1))
Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.30 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 \sqrt {c}\, \sqrt {a x +1}\, c^{2} i \left (-5 a^{3} x^{3}+27 a^{2} x^{2}-71 a x +177\right )}{35 a} \] Input:
int((-a*c*x+c)^(5/2)*((a*x-1)/(a*x+1))^(1/2),x)
Output:
(2*sqrt(c)*sqrt(a*x + 1)*c**2*i*( - 5*a**3*x**3 + 27*a**2*x**2 - 71*a*x + 177))/(35*a)