Integrand size = 23, antiderivative size = 217 \[ \int e^{-3 \text {arctanh}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {8 \sqrt {c-a c x}}{a^4 \sqrt {1-a x} \sqrt {1+a x}}+\frac {32 \sqrt {1+a x} \sqrt {c-a c x}}{a^4 \sqrt {1-a x}}-\frac {50 (1+a x)^{3/2} \sqrt {c-a c x}}{3 a^4 \sqrt {1-a x}}+\frac {38 (1+a x)^{5/2} \sqrt {c-a c x}}{5 a^4 \sqrt {1-a x}}-\frac {2 (1+a x)^{7/2} \sqrt {c-a c x}}{a^4 \sqrt {1-a x}}+\frac {2 (1+a x)^{9/2} \sqrt {c-a c x}}{9 a^4 \sqrt {1-a x}} \] Output:
8*(-a*c*x+c)^(1/2)/a^4/(-a*x+1)^(1/2)/(a*x+1)^(1/2)+32*(a*x+1)^(1/2)*(-a*c *x+c)^(1/2)/a^4/(-a*x+1)^(1/2)-50/3*(a*x+1)^(3/2)*(-a*c*x+c)^(1/2)/a^4/(-a *x+1)^(1/2)+38/5*(a*x+1)^(5/2)*(-a*c*x+c)^(1/2)/a^4/(-a*x+1)^(1/2)-2*(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)
Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.35 \[ \int e^{-3 \text {arctanh}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {2 c \sqrt {1-a x} \left (656+328 a x-82 a^2 x^2+41 a^3 x^3-20 a^4 x^4+5 a^5 x^5\right )}{45 a^4 \sqrt {1+a x} \sqrt {c-a c x}} \] Input:
Integrate[(x^3*Sqrt[c - a*c*x])/E^(3*ArcTanh[a*x]),x]
Output:
(2*c*Sqrt[1 - a*x]*(656 + 328*a*x - 82*a^2*x^2 + 41*a^3*x^3 - 20*a^4*x^4 + 5*a^5*x^5))/(45*a^4*Sqrt[1 + a*x]*Sqrt[c - a*c*x])
Time = 0.38 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.52, 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 |
\(\displaystyle \int \frac {x^3 (1-a x)^{3/2} \sqrt {c-a c x}}{(a x+1)^{3/2}}dx\) |
\(\Big \downarrow \) 37 |
\(\displaystyle \frac {\sqrt {c-a c x} \int \frac {x^3 (1-a x)^2}{(a x+1)^{3/2}}dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\sqrt {c-a c x} \int \left (\frac {(a x+1)^{7/2}}{a^3}-\frac {7 (a x+1)^{5/2}}{a^3}+\frac {19 (a x+1)^{3/2}}{a^3}-\frac {25 \sqrt {a x+1}}{a^3}+\frac {16}{a^3 \sqrt {a x+1}}-\frac {4}{a^3 (a x+1)^{3/2}}\right )dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (\frac {2 (a x+1)^{9/2}}{9 a^4}-\frac {2 (a x+1)^{7/2}}{a^4}+\frac {38 (a x+1)^{5/2}}{5 a^4}-\frac {50 (a x+1)^{3/2}}{3 a^4}+\frac {32 \sqrt {a x+1}}{a^4}+\frac {8}{a^4 \sqrt {a x+1}}\right ) \sqrt {c-a c x}}{\sqrt {1-a x}}\) |
Input:
Int[(x^3*Sqrt[c - a*c*x])/E^(3*ArcTanh[a*x]),x]
Output:
(Sqrt[c - a*c*x]*(8/(a^4*Sqrt[1 + a*x]) + (32*Sqrt[1 + a*x])/a^4 - (50*(1 + a*x)^(3/2))/(3*a^4) + (38*(1 + a*x)^(5/2))/(5*a^4) - (2*(1 + a*x)^(7/2)) /a^4 + (2*(1 + a*x)^(9/2))/(9*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 = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.36
method | result | size |
gosper | \(\frac {2 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \sqrt {-a c x +c}\, \left (5 a^{5} x^{5}-20 a^{4} x^{4}+41 a^{3} x^{3}-82 a^{2} x^{2}+328 a x +656\right )}{45 \left (a x +1\right )^{2} \left (a x -1\right )^{2} a^{4}}\) | \(79\) |
orering | \(\frac {2 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \sqrt {-a c x +c}\, \left (5 a^{5} x^{5}-20 a^{4} x^{4}+41 a^{3} x^{3}-82 a^{2} x^{2}+328 a x +656\right )}{45 \left (a x +1\right )^{2} \left (a x -1\right )^{2} a^{4}}\) | \(79\) |
default | \(-\frac {2 \sqrt {-c \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (5 a^{5} x^{5}-20 a^{4} x^{4}+41 a^{3} x^{3}-82 a^{2} x^{2}+328 a x +656\right )}{45 \left (a x -1\right ) \left (a x +1\right ) a^{4}}\) | \(80\) |
risch | \(-\frac {2 \left (5 a^{4} x^{4}-25 a^{3} x^{3}+66 a^{2} x^{2}-148 a x +476\right ) \left (a x +1\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{45 a^{4} \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}-\frac {8 \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{a^{4} \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(165\) |
Input:
int(x^3*(-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x,method=_RETURNVERB OSE)
Output:
2/45*(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)*(5*a^5*x^5-20*a^4*x^4+41*a^3*x^3- 82*a^2*x^2+328*a*x+656)/(a*x+1)^2/(a*x-1)^2/a^4
Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.35 \[ \int e^{-3 \text {arctanh}(a x)} x^3 \sqrt {c-a c x} \, dx=-\frac {2 \, {\left (5 \, a^{5} x^{5} - 20 \, a^{4} x^{4} + 41 \, a^{3} x^{3} - 82 \, a^{2} x^{2} + 328 \, a x + 656\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{45 \, {\left (a^{6} x^{2} - a^{4}\right )}} \] Input:
integrate(x^3*(-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=" fricas")
Output:
-2/45*(5*a^5*x^5 - 20*a^4*x^4 + 41*a^3*x^3 - 82*a^2*x^2 + 328*a*x + 656)*s qrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^6*x^2 - a^4)
\[ \int e^{-3 \text {arctanh}(a x)} x^3 \sqrt {c-a c x} \, dx=\int \frac {x^{3} \sqrt {- c \left (a x - 1\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (a x + 1\right )^{3}}\, dx \] Input:
integrate(x**3*(-a*c*x+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)
Output:
Integral(x**3*sqrt(-c*(a*x - 1))*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)** 3, x)
Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.40 \[ \int e^{-3 \text {arctanh}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {2 \, {\left (5 \, a^{5} \sqrt {c} x^{5} - 20 \, a^{4} \sqrt {c} x^{4} + 41 \, a^{3} \sqrt {c} x^{3} - 82 \, a^{2} \sqrt {c} x^{2} + 328 \, a \sqrt {c} x + 656 \, \sqrt {c}\right )} \sqrt {a x + 1} {\left (a x - 1\right )}}{45 \, {\left (a^{6} x^{2} - a^{4}\right )}} \] Input:
integrate(x^3*(-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=" maxima")
Output:
2/45*(5*a^5*sqrt(c)*x^5 - 20*a^4*sqrt(c)*x^4 + 41*a^3*sqrt(c)*x^3 - 82*a^2 *sqrt(c)*x^2 + 328*a*sqrt(c)*x + 656*sqrt(c))*sqrt(a*x + 1)*(a*x - 1)/(a^6 *x^2 - a^4)
Exception generated. \[ \int e^{-3 \text {arctanh}(a x)} x^3 \sqrt {c-a c x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),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 = 15.79 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.53 \[ \int e^{-3 \text {arctanh}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {4\,\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}}{a^4\,\left (a\,x+1\right )}-\frac {928\,\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}}{45\,a^4\,\left (a\,x-1\right )}-\frac {2\,\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}\,\left (5\,a^3\,x^3-20\,a^2\,x^2+46\,a\,x-102\right )}{45\,a^4} \] Input:
int((x^3*(1 - a^2*x^2)^(3/2)*(c - a*c*x)^(1/2))/(a*x + 1)^3,x)
Output:
(4*(1 - a^2*x^2)^(1/2)*(c - a*c*x)^(1/2))/(a^4*(a*x + 1)) - (928*(1 - a^2* x^2)^(1/2)*(c - a*c*x)^(1/2))/(45*a^4*(a*x - 1)) - (2*(1 - a^2*x^2)^(1/2)* (c - a*c*x)^(1/2)*(46*a*x - 20*a^2*x^2 + 5*a^3*x^3 - 102))/(45*a^4)
Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.24 \[ \int e^{-3 \text {arctanh}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c}\, \left (5 a^{5} x^{5}-20 a^{4} x^{4}+41 a^{3} x^{3}-82 a^{2} x^{2}+328 a x +656\right )}{45 \sqrt {a x +1}\, a^{4}} \] Input:
int(x^3*(-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)
Output:
(2*sqrt(c)*(5*a**5*x**5 - 20*a**4*x**4 + 41*a**3*x**3 - 82*a**2*x**2 + 328 *a*x + 656))/(45*sqrt(a*x + 1)*a**4)