Integrand size = 24, antiderivative size = 173 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=-\frac {a^2 \left (c-\frac {c}{a x}\right )^{5/2} x^3 \sqrt {1+a x}}{(1-a x)^{5/2}}+\frac {2 a \left (c-\frac {c}{a x}\right )^{5/2} x^2 (1+a x)^{3/2}}{3 (1-a x)^{5/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{5/2} x (1+a x)^{5/2}}{3 (1-a x)^{5/2}}-\frac {a^{3/2} \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{5/2}} \] Output:
-a^2*(c-c/a/x)^(5/2)*x^3*(a*x+1)^(1/2)/(-a*x+1)^(5/2)+2/3*a*(c-c/a/x)^(5/2 )*x^2*(a*x+1)^(3/2)/(-a*x+1)^(5/2)-2/3*(c-c/a/x)^(5/2)*x*(a*x+1)^(5/2)/(-a *x+1)^(5/2)-a^(3/2)*(c-c/a/x)^(5/2)*x^(5/2)*arcsinh(a^(1/2)*x^(1/2))/(-a*x +1)^(5/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.38 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=-\frac {2 c^2 \sqrt {c-\frac {c}{a x}} \left ((1+a x)^{5/2}-a x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{2},\frac {1}{2},-a x\right )\right )}{3 a^2 x \sqrt {1-a x}} \] Input:
Integrate[E^(3*ArcTanh[a*x])*(c - c/(a*x))^(5/2),x]
Output:
(-2*c^2*Sqrt[c - c/(a*x)]*((1 + a*x)^(5/2) - a*x*Hypergeometric2F1[-3/2, - 1/2, 1/2, -(a*x)]))/(3*a^2*x*Sqrt[1 - a*x])
Time = 0.45 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.63, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6684, 6678, 516, 87, 57, 60, 63, 222}
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.
| \(\displaystyle \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 6684 |
| \(\displaystyle \frac {x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2} \int \frac {e^{3 \text {arctanh}(a x)} (1-a x)^{5/2}}{x^{5/2}}dx}{(1-a x)^{5/2}}\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle \frac {x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2} \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^{5/2} \sqrt {1-a x}}dx}{(1-a x)^{5/2}}\) |
| \(\Big \downarrow \) 516 |
| \(\displaystyle \frac {x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2} \int \frac {(1-a x) (a x+1)^{3/2}}{x^{5/2}}dx}{(1-a x)^{5/2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2} \left (-\frac {1}{3} a \int \frac {(a x+1)^{3/2}}{x^{3/2}}dx-\frac {2 (a x+1)^{5/2}}{3 x^{3/2}}\right )}{(1-a x)^{5/2}}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2} \left (-\frac {1}{3} a \left (3 a \int \frac {\sqrt {a x+1}}{\sqrt {x}}dx-\frac {2 (a x+1)^{3/2}}{\sqrt {x}}\right )-\frac {2 (a x+1)^{5/2}}{3 x^{3/2}}\right )}{(1-a x)^{5/2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2} \left (-\frac {1}{3} a \left (3 a \left (\frac {1}{2} \int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx+\sqrt {x} \sqrt {a x+1}\right )-\frac {2 (a x+1)^{3/2}}{\sqrt {x}}\right )-\frac {2 (a x+1)^{5/2}}{3 x^{3/2}}\right )}{(1-a x)^{5/2}}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle \frac {x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2} \left (-\frac {1}{3} a \left (3 a \left (\int \frac {1}{\sqrt {a x+1}}d\sqrt {x}+\sqrt {x} \sqrt {a x+1}\right )-\frac {2 (a x+1)^{3/2}}{\sqrt {x}}\right )-\frac {2 (a x+1)^{5/2}}{3 x^{3/2}}\right )}{(1-a x)^{5/2}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {x^{5/2} \left (-\frac {1}{3} a \left (3 a \left (\frac {\text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}+\sqrt {x} \sqrt {a x+1}\right )-\frac {2 (a x+1)^{3/2}}{\sqrt {x}}\right )-\frac {2 (a x+1)^{5/2}}{3 x^{3/2}}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}{(1-a x)^{5/2}}\) |
Input:
Int[E^(3*ArcTanh[a*x])*(c - c/(a*x))^(5/2),x]
Output:
((c - c/(a*x))^(5/2)*x^(5/2)*((-2*(1 + a*x)^(5/2))/(3*x^(3/2)) - (a*((-2*( 1 + a*x)^(3/2))/Sqrt[x] + 3*a*(Sqrt[x]*Sqrt[1 + a*x] + ArcSinh[Sqrt[a]*Sqr t[x]]/Sqrt[a])))/3))/(1 - a*x)^(5/2)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b S ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x ] && GtQ[c, 0]
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; Free Q[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[n]))
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Simp[x^p*((c + d/x)^p/(1 + c*(x/d))^p) Int[u*(1 + c*(x/d))^p*(E^(n*Ar cTanh[a*x])/x^p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p]
Time = 0.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.79
| method | result | size |
| default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{2} \sqrt {-a^{2} x^{2}+1}\, \left (6 a^{\frac {5}{2}} x^{2} \sqrt {-x \left (a x +1\right )}-3 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a^{2} x^{2}+4 a^{\frac {3}{2}} x \sqrt {-x \left (a x +1\right )}+4 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}\right )}{6 x \,a^{\frac {5}{2}} \left (a x -1\right ) \sqrt {-x \left (a x +1\right )}}\) | \(136\) |
| risch | \(-\frac {\left (3 a^{3} x^{3}+5 a^{2} x^{2}+4 a x +2\right ) c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{3 x \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}\, a^{2}}-\frac {\arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right ) c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{2 \sqrt {a^{2} c}\, \sqrt {-a^{2} x^{2}+1}}\) | \(192\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/6*(c*(a*x-1)/a/x)^(1/2)*c^2*(-a^2*x^2+1)^(1/2)*(6*a^(5/2)*x^2*(-x*(a*x+1 ))^(1/2)-3*arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1/2))*a^2*x^2+4*a^(3 /2)*x*(-x*(a*x+1))^(1/2)+4*a^(1/2)*(-x*(a*x+1))^(1/2))/x/a^(5/2)/(a*x-1)/( -x*(a*x+1))^(1/2)
Time = 0.11 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.91 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\left [\frac {3 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (3 \, a^{2} c^{2} x^{2} + 2 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{12 \, {\left (a^{3} x^{2} - a^{2} x\right )}}, \frac {3 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (3 \, a^{2} c^{2} x^{2} + 2 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{6 \, {\left (a^{3} x^{2} - a^{2} x\right )}}\right ] \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(5/2),x, algorithm="frica s")
Output:
[1/12*(3*(a^2*c^2*x^2 - a*c^2*x)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4* (2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c) /(a*x - 1)) + 4*(3*a^2*c^2*x^2 + 2*a*c^2*x + 2*c^2)*sqrt(-a^2*x^2 + 1)*sqr t((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x), 1/6*(3*(a^2*c^2*x^2 - a*c^2*x)*sq rt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a ^2*c*x^2 - a*c*x - c)) + 2*(3*a^2*c^2*x^2 + 2*a*c^2*x + 2*c^2)*sqrt(-a^2*x ^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x)]
\[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**(5/2),x)
Output:
Integral((-c*(-1 + 1/(a*x)))**(5/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**( 3/2), x)
\[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int { \frac {{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(5/2),x, algorithm="maxim a")
Output:
integrate((a*x + 1)^3*(c - c/(a*x))^(5/2)/(-a^2*x^2 + 1)^(3/2), x)
\[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int { \frac {{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(5/2),x, algorithm="giac" )
Output:
integrate((a*x + 1)^3*(c - c/(a*x))^(5/2)/(-a^2*x^2 + 1)^(3/2), x)
Timed out. \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int(((c - c/(a*x))^(5/2)*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2),x)
Output:
int(((c - c/(a*x))^(5/2)*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2), x)
Time = 0.18 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.51 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\frac {\sqrt {c}\, c^{2} \left (3 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right ) a^{2} x^{2}+3 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} i \,x^{2}+2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a i x +2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i \right )}{3 a^{3} x^{2}} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(5/2),x)
Output:
(sqrt(c)*c**2*(3*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1))*a**2*x* *2 + 3*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**2*i*x**2 + 2*sqrt(x)*sqrt(a)*sqrt( a*x + 1)*a*i*x + 2*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i))/(3*a**3*x**2)