Integrand size = 23, antiderivative size = 231 \[ \int e^{3 \text {arctanh}(a x)} x^3 \sqrt {c-a c x} \, dx=-\frac {4 \sqrt {1+a x} \sqrt {c-a c x}}{a^4 \sqrt {1-a x}}-\frac {2 (1+a x)^{3/2} \sqrt {c-a c x}}{3 a^4 \sqrt {1-a x}}-\frac {2 (1+a x)^{5/2} \sqrt {c-a c x}}{5 a^4 \sqrt {1-a x}}+\frac {2 (1+a x)^{7/2} \sqrt {c-a c x}}{7 a^4 \sqrt {1-a x}}-\frac {2 (1+a x)^{9/2} \sqrt {c-a c x}}{9 a^4 \sqrt {1-a x}}+\frac {4 \sqrt {2} \sqrt {c-a c x} \text {arctanh}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )}{a^4 \sqrt {1-a x}} \] Output:
-4*(a*x+1)^(1/2)*(-a*c*x+c)^(1/2)/a^4/(-a*x+1)^(1/2)-2/3*(a*x+1)^(3/2)*(-a *c*x+c)^(1/2)/a^4/(-a*x+1)^(1/2)-2/5*(a*x+1)^(5/2)*(-a*c*x+c)^(1/2)/a^4/(- a*x+1)^(1/2)+2/7*(a*x+1)^(7/2)*(-a*c*x+c)^(1/2)/a^4/(-a*x+1)^(1/2)-2/9*(a* x+1)^(9/2)*(-a*c*x+c)^(1/2)/a^4/(-a*x+1)^(1/2)+4*2^(1/2)*(-a*c*x+c)^(1/2)* arctanh(1/2*(a*x+1)^(1/2)*2^(1/2))/a^4/(-a*x+1)^(1/2)
Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.40 \[ \int e^{3 \text {arctanh}(a x)} x^3 \sqrt {c-a c x} \, dx=-\frac {2 \sqrt {c-a c x} \left (\sqrt {1+a x} \left (788+236 a x+138 a^2 x^2+95 a^3 x^3+35 a^4 x^4\right )-630 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )\right )}{315 a^4 \sqrt {1-a x}} \] Input:
Integrate[E^(3*ArcTanh[a*x])*x^3*Sqrt[c - a*c*x],x]
Output:
(-2*Sqrt[c - a*c*x]*(Sqrt[1 + a*x]*(788 + 236*a*x + 138*a^2*x^2 + 95*a^3*x ^3 + 35*a^4*x^4) - 630*Sqrt[2]*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]]))/(315*a^4*S qrt[1 - a*x])
Time = 0.38 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.55, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6680, 37, 99, 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.
\(\displaystyle \int x^3 e^{3 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle \int \frac {x^3 (a x+1)^{3/2} \sqrt {c-a c x}}{(1-a x)^{3/2}}dx\) |
\(\Big \downarrow \) 37 |
\(\displaystyle \frac {\sqrt {c-a c x} \int \frac {x^3 (a x+1)^{3/2}}{1-a x}dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\sqrt {c-a c x} \int \left (-\frac {(a x+1)^{7/2}}{a^3}+\frac {(a x+1)^{5/2}}{a^3}+\frac {(a x+1)^{3/2}}{a^3 (1-a x)}-\frac {(a x+1)^{3/2}}{a^3}\right )dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )}{a^4}-\frac {2 (a x+1)^{9/2}}{9 a^4}+\frac {2 (a x+1)^{7/2}}{7 a^4}-\frac {2 (a x+1)^{5/2}}{5 a^4}-\frac {2 (a x+1)^{3/2}}{3 a^4}-\frac {4 \sqrt {a x+1}}{a^4}\right ) \sqrt {c-a c x}}{\sqrt {1-a x}}\) |
Input:
Int[E^(3*ArcTanh[a*x])*x^3*Sqrt[c - a*c*x],x]
Output:
(Sqrt[c - a*c*x]*((-4*Sqrt[1 + a*x])/a^4 - (2*(1 + a*x)^(3/2))/(3*a^4) - ( 2*(1 + a*x)^(5/2))/(5*a^4) + (2*(1 + a*x)^(7/2))/(7*a^4) - (2*(1 + a*x)^(9 /2))/(9*a^4) + (4*Sqrt[2]*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]])/a^4))/Sqrt[1 - a *x]
Int[(u_.)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> S imp[(a + b*x)^m/(c + d*x)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] && !SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
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])
Time = 0.22 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.63
method | result | size |
default | \(-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (-35 a^{4} x^{4} \sqrt {c \left (a x +1\right )}-95 a^{3} x^{3} \sqrt {c \left (a x +1\right )}-138 a^{2} x^{2} \sqrt {c \left (a x +1\right )}+630 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-236 a x \sqrt {c \left (a x +1\right )}-788 \sqrt {c \left (a x +1\right )}\right )}{315 \left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, a^{4}}\) | \(146\) |
risch | \(\frac {2 \left (35 a^{4} x^{4}+95 a^{3} x^{3}+138 a^{2} x^{2}+236 a x +788\right ) \left (a x +1\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{315 a^{4} \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}-\frac {4 \sqrt {2}\, \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right )}{a^{4} \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(178\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(-a*c*x+c)^(1/2),x,method=_RETURNVERB OSE)
Output:
-2/315*(-a^2*x^2+1)^(1/2)*(-c*(a*x-1))^(1/2)*(-35*a^4*x^4*(c*(a*x+1))^(1/2 )-95*a^3*x^3*(c*(a*x+1))^(1/2)-138*a^2*x^2*(c*(a*x+1))^(1/2)+630*c^(1/2)*2 ^(1/2)*arctanh(1/2*(c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))-236*a*x*(c*(a*x+1))^ (1/2)-788*(c*(a*x+1))^(1/2))/(a*x-1)/(c*(a*x+1))^(1/2)/a^4
Time = 0.08 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.17 \[ \int e^{3 \text {arctanh}(a x)} x^3 \sqrt {c-a c x} \, dx=\left [\frac {2 \, {\left (315 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + {\left (35 \, a^{4} x^{4} + 95 \, a^{3} x^{3} + 138 \, a^{2} x^{2} + 236 \, a x + 788\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}\right )}}{315 \, {\left (a^{5} x - a^{4}\right )}}, \frac {2 \, {\left (630 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{2 \, {\left (a c x - c\right )}}\right ) + {\left (35 \, a^{4} x^{4} + 95 \, a^{3} x^{3} + 138 \, a^{2} x^{2} + 236 \, a x + 788\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}\right )}}{315 \, {\left (a^{5} x - a^{4}\right )}}\right ] \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(-a*c*x+c)^(1/2),x, algorithm=" fricas")
Output:
[2/315*(315*sqrt(2)*(a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2 )*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1) ) + (35*a^4*x^4 + 95*a^3*x^3 + 138*a^2*x^2 + 236*a*x + 788)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^5*x - a^4), 2/315*(630*sqrt(2)*(a*x - 1)*sqrt(-c )*arctan(1/2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a*c*x - c)) + (35*a^4*x^4 + 95*a^3*x^3 + 138*a^2*x^2 + 236*a*x + 788)*sqrt(-a^2*x ^2 + 1)*sqrt(-a*c*x + c))/(a^5*x - a^4)]
\[ \int e^{3 \text {arctanh}(a x)} x^3 \sqrt {c-a c x} \, dx=\int \frac {x^{3} \sqrt {- c \left (a x - 1\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**3*(-a*c*x+c)**(1/2),x)
Output:
Integral(x**3*sqrt(-c*(a*x - 1))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2 ), x)
\[ \int e^{3 \text {arctanh}(a x)} x^3 \sqrt {c-a c x} \, dx=\int { \frac {\sqrt {-a c x + c} {\left (a x + 1\right )}^{3} x^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(-a*c*x+c)^(1/2),x, algorithm=" maxima")
Output:
integrate(sqrt(-a*c*x + c)*(a*x + 1)^3*x^3/(-a^2*x^2 + 1)^(3/2), x)
Exception generated. \[ \int e^{3 \text {arctanh}(a x)} x^3 \sqrt {c-a c x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(-a*c*x+c)^(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
Timed out. \[ \int e^{3 \text {arctanh}(a x)} x^3 \sqrt {c-a c x} \, dx=\int \frac {x^3\,\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int((x^3*(c - a*c*x)^(1/2)*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2),x)
Output:
int((x^3*(c - a*c*x)^(1/2)*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2), x)
Time = 0.16 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.40 \[ \int e^{3 \text {arctanh}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c}\, \left (-35 \sqrt {a x +1}\, a^{4} x^{4}-95 \sqrt {a x +1}\, a^{3} x^{3}-138 \sqrt {a x +1}\, a^{2} x^{2}-236 \sqrt {a x +1}\, a x -788 \sqrt {a x +1}-630 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right )+1292 \sqrt {2}\right )}{315 a^{4}} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(-a*c*x+c)^(1/2),x)
Output:
(2*sqrt(c)*( - 35*sqrt(a*x + 1)*a**4*x**4 - 95*sqrt(a*x + 1)*a**3*x**3 - 1 38*sqrt(a*x + 1)*a**2*x**2 - 236*sqrt(a*x + 1)*a*x - 788*sqrt(a*x + 1) - 6 30*sqrt(2)*log(tan(asin(sqrt( - a*x + 1)/sqrt(2))/2)) + 1292*sqrt(2)))/(31 5*a**4)