Integrand size = 18, antiderivative size = 141 \[ \int e^{\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\frac {256 c^5 \left (1-a^2 x^2\right )^{3/2}}{315 a (c-a c x)^{3/2}}+\frac {64 c^4 \left (1-a^2 x^2\right )^{3/2}}{105 a \sqrt {c-a c x}}+\frac {8 c^3 \sqrt {c-a c x} \left (1-a^2 x^2\right )^{3/2}}{21 a}+\frac {2 c^2 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{9 a} \] Output:
256/315*c^5*(-a^2*x^2+1)^(3/2)/a/(-a*c*x+c)^(3/2)+64/105*c^4*(-a^2*x^2+1)^ (3/2)/a/(-a*c*x+c)^(1/2)+8/21*c^3*(-a*c*x+c)^(1/2)*(-a^2*x^2+1)^(3/2)/a+2/ 9*c^2*(-a*c*x+c)^(3/2)*(-a^2*x^2+1)^(3/2)/a
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.44 \[ \int e^{\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 c^3 (1+a x)^{3/2} \sqrt {c-a c x} \left (-319+321 a x-165 a^2 x^2+35 a^3 x^3\right )}{315 a \sqrt {1-a x}} \] Input:
Integrate[E^ArcTanh[a*x]*(c - a*c*x)^(7/2),x]
Output:
(-2*c^3*(1 + a*x)^(3/2)*Sqrt[c - a*c*x]*(-319 + 321*a*x - 165*a^2*x^2 + 35 *a^3*x^3))/(315*a*Sqrt[1 - a*x])
Time = 0.32 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6677, 459, 459, 459, 458}
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.
\(\displaystyle \int e^{\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx\) |
\(\Big \downarrow \) 6677 |
\(\displaystyle c \int (c-a c x)^{5/2} \sqrt {1-a^2 x^2}dx\) |
\(\Big \downarrow \) 459 |
\(\displaystyle c \left (\frac {4}{3} c \int (c-a c x)^{3/2} \sqrt {1-a^2 x^2}dx+\frac {2 c \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{3/2}}{9 a}\right )\) |
\(\Big \downarrow \) 459 |
\(\displaystyle c \left (\frac {4}{3} c \left (\frac {8}{7} c \int \sqrt {c-a c x} \sqrt {1-a^2 x^2}dx+\frac {2 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a c x}}{7 a}\right )+\frac {2 c \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{3/2}}{9 a}\right )\) |
\(\Big \downarrow \) 459 |
\(\displaystyle c \left (\frac {4}{3} c \left (\frac {8}{7} c \left (\frac {4}{5} c \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}dx+\frac {2 c \left (1-a^2 x^2\right )^{3/2}}{5 a \sqrt {c-a c x}}\right )+\frac {2 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a c x}}{7 a}\right )+\frac {2 c \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{3/2}}{9 a}\right )\) |
\(\Big \downarrow \) 458 |
\(\displaystyle c \left (\frac {4}{3} c \left (\frac {8}{7} c \left (\frac {8 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 a (c-a c x)^{3/2}}+\frac {2 c \left (1-a^2 x^2\right )^{3/2}}{5 a \sqrt {c-a c x}}\right )+\frac {2 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a c x}}{7 a}\right )+\frac {2 c \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{3/2}}{9 a}\right )\) |
Input:
Int[E^ArcTanh[a*x]*(c - a*c*x)^(7/2),x]
Output:
c*((2*c*(c - a*c*x)^(3/2)*(1 - a^2*x^2)^(3/2))/(9*a) + (4*c*((2*c*Sqrt[c - a*c*x]*(1 - a^2*x^2)^(3/2))/(7*a) + (8*c*((8*c^2*(1 - a^2*x^2)^(3/2))/(15 *a*(c - a*c*x)^(3/2)) + (2*c*(1 - a^2*x^2)^(3/2))/(5*a*Sqrt[c - a*c*x])))/ 7))/3)
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c , d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* (Simplify[n + p]/(n + 2*p + 1)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[Simplif y[n + p], 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S imp[c^n Int[(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Time = 0.18 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.45
method | result | size |
gosper | \(\frac {2 \left (a x +1\right )^{2} \left (35 a^{3} x^{3}-165 a^{2} x^{2}+321 a x -319\right ) \left (-a c x +c \right )^{\frac {7}{2}}}{315 a \left (a x -1\right )^{3} \sqrt {-a^{2} x^{2}+1}}\) | \(63\) |
orering | \(\frac {2 \left (a x +1\right )^{2} \left (35 a^{3} x^{3}-165 a^{2} x^{2}+321 a x -319\right ) \left (-a c x +c \right )^{\frac {7}{2}}}{315 a \left (a x -1\right )^{3} \sqrt {-a^{2} x^{2}+1}}\) | \(63\) |
default | \(\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, c^{3} \left (a x +1\right ) \left (35 a^{3} x^{3}-165 a^{2} x^{2}+321 a x -319\right )}{315 \left (a x -1\right ) a}\) | \(65\) |
risch | \(\frac {2 \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c^{4} \left (35 a^{4} x^{4}-130 a^{3} x^{3}+156 a^{2} x^{2}+2 a x -319\right ) \left (a x +1\right )}{315 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, a \sqrt {c \left (a x +1\right )}}\) | \(102\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(7/2),x,method=_RETURNVERBOSE)
Output:
2/315*(a*x+1)^2*(35*a^3*x^3-165*a^2*x^2+321*a*x-319)*(-a*c*x+c)^(7/2)/a/(a *x-1)^3/(-a^2*x^2+1)^(1/2)
Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.57 \[ \int e^{\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\frac {2 \, {\left (35 \, a^{4} c^{3} x^{4} - 130 \, a^{3} c^{3} x^{3} + 156 \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x - 319 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{315 \, {\left (a^{2} x - a\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(7/2),x, algorithm="fricas ")
Output:
2/315*(35*a^4*c^3*x^4 - 130*a^3*c^3*x^3 + 156*a^2*c^3*x^2 + 2*a*c^3*x - 31 9*c^3)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)
\[ \int e^{\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\int \frac {\left (- c \left (a x - 1\right )\right )^{\frac {7}{2}} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**(7/2),x)
Output:
Integral((-c*(a*x - 1))**(7/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)
Time = 0.05 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91 \[ \int e^{\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 \, {\left (5 \, a^{5} c^{\frac {7}{2}} x^{5} - 20 \, a^{4} c^{\frac {7}{2}} x^{4} + 32 \, a^{3} c^{\frac {7}{2}} x^{3} - 34 \, a^{2} c^{\frac {7}{2}} x^{2} + 91 \, a c^{\frac {7}{2}} x + 182 \, c^{\frac {7}{2}}\right )}}{45 \, \sqrt {a x + 1} a} - \frac {2 \, {\left (5 \, a^{4} c^{\frac {7}{2}} x^{4} - 22 \, a^{3} c^{\frac {7}{2}} x^{3} + 44 \, a^{2} c^{\frac {7}{2}} x^{2} - 106 \, a c^{\frac {7}{2}} x - 177 \, c^{\frac {7}{2}}\right )}}{35 \, \sqrt {a x + 1} a} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(7/2),x, algorithm="maxima ")
Output:
-2/45*(5*a^5*c^(7/2)*x^5 - 20*a^4*c^(7/2)*x^4 + 32*a^3*c^(7/2)*x^3 - 34*a^ 2*c^(7/2)*x^2 + 91*a*c^(7/2)*x + 182*c^(7/2))/(sqrt(a*x + 1)*a) - 2/35*(5* a^4*c^(7/2)*x^4 - 22*a^3*c^(7/2)*x^3 + 44*a^2*c^(7/2)*x^2 - 106*a*c^(7/2)* x - 177*c^(7/2))/(sqrt(a*x + 1)*a)
Exception generated. \[ \int e^{\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(7/2),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 = 14.71 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.56 \[ \int e^{\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\frac {\sqrt {c-a\,c\,x}\,\left (\frac {634\,c^3\,x}{315}+\frac {638\,c^3}{315\,a}-\frac {316\,a\,c^3\,x^2}{315}-\frac {52\,a^2\,c^3\,x^3}{315}+\frac {38\,a^3\,c^3\,x^4}{63}-\frac {2\,a^4\,c^3\,x^5}{9}\right )}{\sqrt {1-a^2\,x^2}} \] Input:
int(((c - a*c*x)^(7/2)*(a*x + 1))/(1 - a^2*x^2)^(1/2),x)
Output:
((c - a*c*x)^(1/2)*((634*c^3*x)/315 + (638*c^3)/(315*a) - (316*a*c^3*x^2)/ 315 - (52*a^2*c^3*x^3)/315 + (38*a^3*c^3*x^4)/63 - (2*a^4*c^3*x^5)/9))/(1 - a^2*x^2)^(1/2)
Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.33 \[ \int e^{\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\frac {2 \sqrt {c}\, \sqrt {a x +1}\, c^{3} \left (-35 a^{4} x^{4}+130 a^{3} x^{3}-156 a^{2} x^{2}-2 a x +319\right )}{315 a} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(7/2),x)
Output:
(2*sqrt(c)*sqrt(a*x + 1)*c**3*( - 35*a**4*x**4 + 130*a**3*x**3 - 156*a**2* x**2 - 2*a*x + 319))/(315*a)