3.11.41 \(\int e^{2 \tanh ^{-1}(a x)} x^2 (c-a^2 c x^2)^3 \, dx\) [1041]

Optimal. Leaf size=84 \[ \frac {4 c^3 (1+a x)^5}{5 a^3}-\frac {2 c^3 (1+a x)^6}{a^3}+\frac {13 c^3 (1+a x)^7}{7 a^3}-\frac {3 c^3 (1+a x)^8}{4 a^3}+\frac {c^3 (1+a x)^9}{9 a^3} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6285, 90} \begin {gather*} \frac {c^3 (a x+1)^9}{9 a^3}-\frac {3 c^3 (a x+1)^8}{4 a^3}+\frac {13 c^3 (a x+1)^7}{7 a^3}-\frac {2 c^3 (a x+1)^6}{a^3}+\frac {4 c^3 (a x+1)^5}{5 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])*x^2*(c - a^2*c*x^2)^3,x]

[Out]

(4*c^3*(1 + a*x)^5)/(5*a^3) - (2*c^3*(1 + a*x)^6)/a^3 + (13*c^3*(1 + a*x)^7)/(7*a^3) - (3*c^3*(1 + a*x)^8)/(4*
a^3) + (c^3*(1 + a*x)^9)/(9*a^3)

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(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]))

Rule 6285

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[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])

Rubi steps

\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^2 \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int x^2 (1-a x)^2 (1+a x)^4 \, dx\\ &=c^3 \int \left (\frac {4 (1+a x)^4}{a^2}-\frac {12 (1+a x)^5}{a^2}+\frac {13 (1+a x)^6}{a^2}-\frac {6 (1+a x)^7}{a^2}+\frac {(1+a x)^8}{a^2}\right ) \, 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}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 70, normalized size = 0.83 \begin {gather*} 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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a*x])*x^2*(c - a^2*c*x^2)^3,x]

[Out]

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)

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Maple [A]
time = 0.08, size = 57, normalized size = 0.68

method	result	size
gosper	\(\frac {c^{3} x^{3} \left (140 x^{6} a^{6}+315 x^{5} a^{5}-180 a^{4} x^{4}-840 a^{3} x^{3}-252 a^{2} x^{2}+630 a x +420\right )}{1260}\)	\(55\)
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	\(\frac {1}{3} c^{3} x^{3}+\frac {1}{2} a \,c^{3} x^{4}-\frac {1}{5} a^{2} c^{3} x^{5}-\frac {2}{3} a^{3} c^{3} x^{6}-\frac {1}{7} a^{4} c^{3} x^{7}+\frac {1}{4} a^{5} c^{3} x^{8}+\frac {1}{9} a^{6} c^{3} x^{9}\)	\(74\)
risch	\(\frac {1}{3} c^{3} x^{3}+\frac {1}{2} a \,c^{3} x^{4}-\frac {1}{5} a^{2} c^{3} x^{5}-\frac {2}{3} a^{3} c^{3} x^{6}-\frac {1}{7} a^{4} c^{3} x^{7}+\frac {1}{4} a^{5} c^{3} x^{8}+\frac {1}{9} a^{6} c^{3} x^{9}\)	\(74\)
meijerg	\(\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {11}{2}} \left (385 x^{8} a^{8}+495 x^{6} a^{6}+693 a^{4} x^{4}+1155 a^{2} x^{2}+3465\right )}{3465 a^{10}}+\frac {2 \left (-a^{2}\right )^{\frac {11}{2}} \arctanh \left (a x \right )}{a^{11}}\right )}{2 a^{2} \sqrt {-a^{2}}}+\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {9}{2}} \left (45 x^{6} a^{6}+63 a^{4} x^{4}+105 a^{2} x^{2}+315\right )}{315 a^{8}}+\frac {2 \left (-a^{2}\right )^{\frac {9}{2}} \arctanh \left (a x \right )}{a^{9}}\right )}{a^{2} \sqrt {-a^{2}}}-\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {5}{2}} \left (5 a^{2} x^{2}+15\right )}{15 a^{4}}+\frac {2 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{a^{2} \sqrt {-a^{2}}}+\frac {c^{3} \left (\frac {x^{2} a^{2} \left (15 x^{6} a^{6}+20 a^{4} x^{4}+30 a^{2} x^{2}+60\right )}{60}+\ln \left (-a^{2} x^{2}+1\right )\right )}{a^{3}}+\frac {3 c^{3} \left (-\frac {x^{2} a^{2} \left (4 a^{4} x^{4}+6 a^{2} x^{2}+12\right )}{12}-\ln \left (-a^{2} x^{2}+1\right )\right )}{a^{3}}+\frac {3 c^{3} \left (\frac {x^{2} a^{2} \left (3 a^{2} x^{2}+6\right )}{6}+\ln \left (-a^{2} x^{2}+1\right )\right )}{a^{3}}+\frac {c^{3} \left (-a^{2} x^{2}-\ln \left (-a^{2} x^{2}+1\right )\right )}{a^{3}}-\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{2 a^{2} \sqrt {-a^{2}}}\)	\(419\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

c^3*(1/9*a^6*x^9+1/4*a^5*x^8-1/7*a^4*x^7-2/3*a^3*x^6-1/5*a^2*x^5+1/2*a*x^4+1/3*x^3)

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Maxima [A]
time = 0.25, size = 73, normalized size = 0.87 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.33, size = 73, normalized size = 0.87 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.03, size = 78, normalized size = 0.93 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*x**2*(-a**2*c*x**2+c)**3,x)

[Out]

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

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Giac [A]
time = 0.42, size = 73, normalized size = 0.87 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.03, size = 73, normalized size = 0.87 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2*(c - a^2*c*x^2)^3*(a*x + 1)^2)/(a^2*x^2 - 1),x)

[Out]

(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

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