Integrand size = 25, antiderivative size = 84 \[ \int e^{2 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^3 \, dx=\frac {4 c^3 (1+a x)^5}{5 a^3}-\frac {2 c^3 (1+a x)^6}{a^3}+\frac {13 c^3 (1+a x)^7}{7 a^3}-\frac {3 c^3 (1+a x)^8}{4 a^3}+\frac {c^3 (1+a x)^9}{9 a^3} \] Output:
4/5*c^3*(a*x+1)^5/a^3-2*c^3*(a*x+1)^6/a^3+13/7*c^3*(a*x+1)^7/a^3-3/4*c^3*( a*x+1)^8/a^3+1/9*c^3*(a*x+1)^9/a^3
Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83 \[ \int e^{2 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^3 \, dx=c^3 \left (\frac {x^3}{3}+\frac {a x^4}{2}-\frac {a^2 x^5}{5}-\frac {2 a^3 x^6}{3}-\frac {a^4 x^7}{7}+\frac {a^5 x^8}{4}+\frac {a^6 x^9}{9}\right ) \] Input:
Integrate[E^(2*ArcTanh[a*x])*x^2*(c - a^2*c*x^2)^3,x]
Output:
c^3*(x^3/3 + (a*x^4)/2 - (a^2*x^5)/5 - (2*a^3*x^6)/3 - (a^4*x^7)/7 + (a^5* x^8)/4 + (a^6*x^9)/9)
Time = 0.54 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.87, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6700, 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.
\(\displaystyle \int x^2 e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle c^3 \int x^2 (1-a x)^2 (a x+1)^4dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle c^3 \int \left (\frac {(a x+1)^8}{a^2}-\frac {6 (a x+1)^7}{a^2}+\frac {13 (a x+1)^6}{a^2}-\frac {12 (a x+1)^5}{a^2}+\frac {4 (a x+1)^4}{a^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle c^3 \left (\frac {(a x+1)^9}{9 a^3}-\frac {3 (a x+1)^8}{4 a^3}+\frac {13 (a x+1)^7}{7 a^3}-\frac {2 (a x+1)^6}{a^3}+\frac {4 (a x+1)^5}{5 a^3}\right )\) |
Input:
Int[E^(2*ArcTanh[a*x])*x^2*(c - a^2*c*x^2)^3,x]
Output:
c^3*((4*(1 + a*x)^5)/(5*a^3) - (2*(1 + a*x)^6)/a^3 + (13*(1 + a*x)^7)/(7*a ^3) - (3*(1 + a*x)^8)/(4*a^3) + (1 + a*x)^9/(9*a^3))
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_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Time = 0.14 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.65
method | result | size |
gosper | \(\frac {c^{3} x^{3} \left (140 x^{6} a^{6}+315 a^{5} x^{5}-180 a^{4} x^{4}-840 a^{3} x^{3}-252 a^{2} x^{2}+630 a x +420\right )}{1260}\) | \(55\) |
default | \(c^{3} \left (\frac {1}{9} a^{6} x^{9}+\frac {1}{4} a^{5} x^{8}-\frac {1}{7} a^{4} x^{7}-\frac {2}{3} a^{3} x^{6}-\frac {1}{5} a^{2} x^{5}+\frac {1}{2} a \,x^{4}+\frac {1}{3} x^{3}\right )\) | \(57\) |
norman | \(\frac {1}{3} c^{3} x^{3}+\frac {1}{2} a \,c^{3} x^{4}-\frac {1}{5} a^{2} c^{3} x^{5}-\frac {2}{3} a^{3} c^{3} x^{6}+\frac {1}{4} a^{5} c^{3} x^{8}+\frac {1}{9} a^{6} c^{3} x^{9}-\frac {1}{7} c^{3} a^{4} x^{7}\) | \(74\) |
risch | \(\frac {1}{3} c^{3} x^{3}+\frac {1}{2} a \,c^{3} x^{4}-\frac {1}{5} a^{2} c^{3} x^{5}-\frac {2}{3} a^{3} c^{3} x^{6}+\frac {1}{4} a^{5} c^{3} x^{8}+\frac {1}{9} a^{6} c^{3} x^{9}-\frac {1}{7} c^{3} a^{4} x^{7}\) | \(74\) |
parallelrisch | \(\frac {1}{3} c^{3} x^{3}+\frac {1}{2} a \,c^{3} x^{4}-\frac {1}{5} a^{2} c^{3} x^{5}-\frac {2}{3} a^{3} c^{3} x^{6}+\frac {1}{4} a^{5} c^{3} x^{8}+\frac {1}{9} a^{6} c^{3} x^{9}-\frac {1}{7} c^{3} a^{4} x^{7}\) | \(74\) |
orering | \(\frac {x^{3} \left (140 x^{6} a^{6}+315 a^{5} x^{5}-180 a^{4} x^{4}-840 a^{3} x^{3}-252 a^{2} x^{2}+630 a x +420\right ) \left (-a^{2} c \,x^{2}+c \right )^{3}}{1260 \left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (-a^{2} x^{2}+1\right )}\) | \(91\) |
meijerg | \(\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {11}{2}} \left (385 a^{8} x^{8}+495 x^{6} a^{6}+693 a^{4} x^{4}+1155 a^{2} x^{2}+3465\right )}{3465 a^{10}}+\frac {2 \left (-a^{2}\right )^{\frac {11}{2}} \operatorname {arctanh}\left (a x \right )}{a^{11}}\right )}{2 a^{2} \sqrt {-a^{2}}}+\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {9}{2}} \left (45 x^{6} a^{6}+63 a^{4} x^{4}+105 a^{2} x^{2}+315\right )}{315 a^{8}}+\frac {2 \left (-a^{2}\right )^{\frac {9}{2}} \operatorname {arctanh}\left (a x \right )}{a^{9}}\right )}{a^{2} \sqrt {-a^{2}}}-\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {5}{2}} \left (5 a^{2} x^{2}+15\right )}{15 a^{4}}+\frac {2 \left (-a^{2}\right )^{\frac {5}{2}} \operatorname {arctanh}\left (a x \right )}{a^{5}}\right )}{a^{2} \sqrt {-a^{2}}}+\frac {c^{3} \left (\frac {a^{2} x^{2} \left (15 x^{6} a^{6}+20 a^{4} x^{4}+30 a^{2} x^{2}+60\right )}{60}+\ln \left (-a^{2} x^{2}+1\right )\right )}{a^{3}}+\frac {3 c^{3} \left (-\frac {a^{2} x^{2} \left (4 a^{4} x^{4}+6 a^{2} x^{2}+12\right )}{12}-\ln \left (-a^{2} x^{2}+1\right )\right )}{a^{3}}+\frac {3 c^{3} \left (\frac {x^{2} a^{2} \left (3 a^{2} x^{2}+6\right )}{6}+\ln \left (-a^{2} x^{2}+1\right )\right )}{a^{3}}+\frac {c^{3} \left (-a^{2} x^{2}-\ln \left (-a^{2} x^{2}+1\right )\right )}{a^{3}}-\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \operatorname {arctanh}\left (a x \right )}{a^{3}}\right )}{2 a^{2} \sqrt {-a^{2}}}\) | \(419\) |
Input:
int((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)
Output:
1/1260*c^3*x^3*(140*a^6*x^6+315*a^5*x^5-180*a^4*x^4-840*a^3*x^3-252*a^2*x^ 2+630*a*x+420)
Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.87 \[ \int e^{2 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^3 \, dx=\frac {1}{9} \, a^{6} c^{3} x^{9} + \frac {1}{4} \, a^{5} c^{3} x^{8} - \frac {1}{7} \, a^{4} c^{3} x^{7} - \frac {2}{3} \, a^{3} c^{3} x^{6} - \frac {1}{5} \, a^{2} c^{3} x^{5} + \frac {1}{2} \, a c^{3} x^{4} + \frac {1}{3} \, c^{3} x^{3} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^3,x, algorithm="fricas ")
Output:
1/9*a^6*c^3*x^9 + 1/4*a^5*c^3*x^8 - 1/7*a^4*c^3*x^7 - 2/3*a^3*c^3*x^6 - 1/ 5*a^2*c^3*x^5 + 1/2*a*c^3*x^4 + 1/3*c^3*x^3
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.93 \[ \int e^{2 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^3 \, dx=\frac {a^{6} c^{3} x^{9}}{9} + \frac {a^{5} c^{3} x^{8}}{4} - \frac {a^{4} c^{3} x^{7}}{7} - \frac {2 a^{3} c^{3} x^{6}}{3} - \frac {a^{2} c^{3} x^{5}}{5} + \frac {a c^{3} x^{4}}{2} + \frac {c^{3} x^{3}}{3} \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)*x**2*(-a**2*c*x**2+c)**3,x)
Output:
a**6*c**3*x**9/9 + a**5*c**3*x**8/4 - a**4*c**3*x**7/7 - 2*a**3*c**3*x**6/ 3 - a**2*c**3*x**5/5 + a*c**3*x**4/2 + c**3*x**3/3
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.87 \[ \int e^{2 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^3 \, dx=\frac {1}{9} \, a^{6} c^{3} x^{9} + \frac {1}{4} \, a^{5} c^{3} x^{8} - \frac {1}{7} \, a^{4} c^{3} x^{7} - \frac {2}{3} \, a^{3} c^{3} x^{6} - \frac {1}{5} \, a^{2} c^{3} x^{5} + \frac {1}{2} \, a c^{3} x^{4} + \frac {1}{3} \, c^{3} x^{3} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^3,x, algorithm="maxima ")
Output:
1/9*a^6*c^3*x^9 + 1/4*a^5*c^3*x^8 - 1/7*a^4*c^3*x^7 - 2/3*a^3*c^3*x^6 - 1/ 5*a^2*c^3*x^5 + 1/2*a*c^3*x^4 + 1/3*c^3*x^3
Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.87 \[ \int e^{2 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^3 \, dx=\frac {1}{9} \, a^{6} c^{3} x^{9} + \frac {1}{4} \, a^{5} c^{3} x^{8} - \frac {1}{7} \, a^{4} c^{3} x^{7} - \frac {2}{3} \, a^{3} c^{3} x^{6} - \frac {1}{5} \, a^{2} c^{3} x^{5} + \frac {1}{2} \, a c^{3} x^{4} + \frac {1}{3} \, c^{3} x^{3} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^3,x, algorithm="giac")
Output:
1/9*a^6*c^3*x^9 + 1/4*a^5*c^3*x^8 - 1/7*a^4*c^3*x^7 - 2/3*a^3*c^3*x^6 - 1/ 5*a^2*c^3*x^5 + 1/2*a*c^3*x^4 + 1/3*c^3*x^3
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.87 \[ \int e^{2 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^3 \, dx=\frac {a^6\,c^3\,x^9}{9}+\frac {a^5\,c^3\,x^8}{4}-\frac {a^4\,c^3\,x^7}{7}-\frac {2\,a^3\,c^3\,x^6}{3}-\frac {a^2\,c^3\,x^5}{5}+\frac {a\,c^3\,x^4}{2}+\frac {c^3\,x^3}{3} \] Input:
int(-(x^2*(c - a^2*c*x^2)^3*(a*x + 1)^2)/(a^2*x^2 - 1),x)
Output:
(c^3*x^3)/3 + (a*c^3*x^4)/2 - (a^2*c^3*x^5)/5 - (2*a^3*c^3*x^6)/3 - (a^4*c ^3*x^7)/7 + (a^5*c^3*x^8)/4 + (a^6*c^3*x^9)/9
Time = 0.15 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.64 \[ \int e^{2 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^3 \, dx=\frac {c^{3} x^{3} \left (140 a^{6} x^{6}+315 a^{5} x^{5}-180 a^{4} x^{4}-840 a^{3} x^{3}-252 a^{2} x^{2}+630 a x +420\right )}{1260} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^3,x)
Output:
(c**3*x**3*(140*a**6*x**6 + 315*a**5*x**5 - 180*a**4*x**4 - 840*a**3*x**3 - 252*a**2*x**2 + 630*a*x + 420))/1260