3.11.40 \(\int e^{2 \tanh ^{-1}(a x)} x^3 (c-a^2 c x^2)^3 \, dx\) [1040]

Optimal. Leaf size=87 \[ \frac {c^3 x^4}{4}+\frac {2}{5} a c^3 x^5-\frac {1}{6} a^2 c^3 x^6-\frac {4}{7} a^3 c^3 x^7-\frac {1}{8} a^4 c^3 x^8+\frac {2}{9} a^5 c^3 x^9+\frac {1}{10} a^6 c^3 x^{10} \]

[Out]

1/4*c^3*x^4+2/5*a*c^3*x^5-1/6*a^2*c^3*x^6-4/7*a^3*c^3*x^7-1/8*a^4*c^3*x^8+2/9*a^5*c^3*x^9+1/10*a^6*c^3*x^10

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Rubi [A]
time = 0.07, antiderivative size = 87, 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 {1}{10} a^6 c^3 x^{10}+\frac {2}{9} a^5 c^3 x^9-\frac {1}{8} a^4 c^3 x^8-\frac {4}{7} a^3 c^3 x^7-\frac {1}{6} a^2 c^3 x^6+\frac {2}{5} a c^3 x^5+\frac {c^3 x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*x^4)/4 + (2*a*c^3*x^5)/5 - (a^2*c^3*x^6)/6 - (4*a^3*c^3*x^7)/7 - (a^4*c^3*x^8)/8 + (2*a^5*c^3*x^9)/9 + (a
^6*c^3*x^10)/10

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^3 \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int x^3 (1-a x)^2 (1+a x)^4 \, dx\\ &=c^3 \int \left (x^3+2 a x^4-a^2 x^5-4 a^3 x^6-a^4 x^7+2 a^5 x^8+a^6 x^9\right ) \, dx\\ &=\frac {c^3 x^4}{4}+\frac {2}{5} a c^3 x^5-\frac {1}{6} a^2 c^3 x^6-\frac {4}{7} a^3 c^3 x^7-\frac {1}{8} a^4 c^3 x^8+\frac {2}{9} a^5 c^3 x^9+\frac {1}{10} a^6 c^3 x^{10}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 70, normalized size = 0.80 \begin {gather*} c^3 \left (\frac {x^4}{4}+\frac {2 a x^5}{5}-\frac {a^2 x^6}{6}-\frac {4 a^3 x^7}{7}-\frac {a^4 x^8}{8}+\frac {2 a^5 x^9}{9}+\frac {a^6 x^{10}}{10}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

c^3*(x^4/4 + (2*a*x^5)/5 - (a^2*x^6)/6 - (4*a^3*x^7)/7 - (a^4*x^8)/8 + (2*a^5*x^9)/9 + (a^6*x^10)/10)

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Maple [A]
time = 0.07, size = 57, normalized size = 0.66

method	result	size
gosper	\(\frac {c^{3} x^{4} \left (252 x^{6} a^{6}+560 x^{5} a^{5}-315 a^{4} x^{4}-1440 a^{3} x^{3}-420 a^{2} x^{2}+1008 a x +630\right )}{2520}\)	\(55\)
default	\(c^{3} \left (\frac {1}{10} a^{6} x^{10}+\frac {2}{9} a^{5} x^{9}-\frac {1}{8} a^{4} x^{8}-\frac {4}{7} a^{3} x^{7}-\frac {1}{6} a^{2} x^{6}+\frac {2}{5} a \,x^{5}+\frac {1}{4} x^{4}\right )\)	\(57\)
norman	\(\frac {1}{4} c^{3} x^{4}+\frac {2}{5} a \,c^{3} x^{5}-\frac {1}{6} a^{2} c^{3} x^{6}-\frac {4}{7} a^{3} c^{3} x^{7}-\frac {1}{8} a^{4} c^{3} x^{8}+\frac {2}{9} a^{5} c^{3} x^{9}+\frac {1}{10} a^{6} c^{3} x^{10}\)	\(74\)
risch	\(\frac {1}{4} c^{3} x^{4}+\frac {2}{5} a \,c^{3} x^{5}-\frac {1}{6} a^{2} c^{3} x^{6}-\frac {4}{7} a^{3} c^{3} x^{7}-\frac {1}{8} a^{4} c^{3} x^{8}+\frac {2}{9} a^{5} c^{3} x^{9}+\frac {1}{10} a^{6} c^{3} x^{10}\)	\(74\)
meijerg	\(-\frac {c^{3} \left (-\frac {x^{2} a^{2} \left (12 x^{8} a^{8}+15 x^{6} a^{6}+20 a^{4} x^{4}+30 a^{2} x^{2}+60\right )}{60}-\ln \left (-a^{2} x^{2}+1\right )\right )}{2 a^{4}}-\frac {c^{3} \left (\frac {x^{2} a^{2} \left (15 x^{6} a^{6}+20 a^{4} x^{4}+30 a^{2} x^{2}+60\right )}{60}+\ln \left (-a^{2} x^{2}+1\right )\right )}{a^{4}}+\frac {c^{3} \left (\frac {x^{2} a^{2} \left (3 a^{2} x^{2}+6\right )}{6}+\ln \left (-a^{2} x^{2}+1\right )\right )}{a^{4}}+\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {11}{2}} \left (385 x^{8} a^{8}+495 x^{6} a^{6}+693 a^{4} x^{4}+1155 a^{2} x^{2}+3465\right )}{3465 a^{10}}+\frac {2 \left (-a^{2}\right )^{\frac {11}{2}} \arctanh \left (a x \right )}{a^{11}}\right )}{a^{3} \sqrt {-a^{2}}}+\frac {3 c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {9}{2}} \left (45 x^{6} a^{6}+63 a^{4} x^{4}+105 a^{2} x^{2}+315\right )}{315 a^{8}}+\frac {2 \left (-a^{2}\right )^{\frac {9}{2}} \arctanh \left (a x \right )}{a^{9}}\right )}{a^{3} \sqrt {-a^{2}}}+\frac {3 c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {7}{2}} \left (21 a^{4} x^{4}+35 a^{2} x^{2}+105\right )}{105 a^{6}}+\frac {2 \left (-a^{2}\right )^{\frac {7}{2}} \arctanh \left (a x \right )}{a^{7}}\right )}{a^{3} \sqrt {-a^{2}}}+\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {5}{2}} \left (5 a^{2} x^{2}+15\right )}{15 a^{4}}+\frac {2 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{a^{3} \sqrt {-a^{2}}}+\frac {c^{3} \left (-a^{2} x^{2}-\ln \left (-a^{2} x^{2}+1\right )\right )}{2 a^{4}}\)	\(453\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*(1/10*a^6*x^10+2/9*a^5*x^9-1/8*a^4*x^8-4/7*a^3*x^7-1/6*a^2*x^6+2/5*a*x^5+1/4*x^4)

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Maxima [A]
time = 0.26, size = 73, normalized size = 0.84 \begin {gather*} \frac {1}{10} \, a^{6} c^{3} x^{10} + \frac {2}{9} \, a^{5} c^{3} x^{9} - \frac {1}{8} \, a^{4} c^{3} x^{8} - \frac {4}{7} \, a^{3} c^{3} x^{7} - \frac {1}{6} \, a^{2} c^{3} x^{6} + \frac {2}{5} \, a c^{3} x^{5} + \frac {1}{4} \, c^{3} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*a^6*c^3*x^10 + 2/9*a^5*c^3*x^9 - 1/8*a^4*c^3*x^8 - 4/7*a^3*c^3*x^7 - 1/6*a^2*c^3*x^6 + 2/5*a*c^3*x^5 + 1/
4*c^3*x^4

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Fricas [A]
time = 0.33, size = 73, normalized size = 0.84 \begin {gather*} \frac {1}{10} \, a^{6} c^{3} x^{10} + \frac {2}{9} \, a^{5} c^{3} x^{9} - \frac {1}{8} \, a^{4} c^{3} x^{8} - \frac {4}{7} \, a^{3} c^{3} x^{7} - \frac {1}{6} \, a^{2} c^{3} x^{6} + \frac {2}{5} \, a c^{3} x^{5} + \frac {1}{4} \, c^{3} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*a^6*c^3*x^10 + 2/9*a^5*c^3*x^9 - 1/8*a^4*c^3*x^8 - 4/7*a^3*c^3*x^7 - 1/6*a^2*c^3*x^6 + 2/5*a*c^3*x^5 + 1/
4*c^3*x^4

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Sympy [A]
time = 0.04, size = 82, normalized size = 0.94 \begin {gather*} \frac {a^{6} c^{3} x^{10}}{10} + \frac {2 a^{5} c^{3} x^{9}}{9} - \frac {a^{4} c^{3} x^{8}}{8} - \frac {4 a^{3} c^{3} x^{7}}{7} - \frac {a^{2} c^{3} x^{6}}{6} + \frac {2 a c^{3} x^{5}}{5} + \frac {c^{3} x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**6*c**3*x**10/10 + 2*a**5*c**3*x**9/9 - a**4*c**3*x**8/8 - 4*a**3*c**3*x**7/7 - a**2*c**3*x**6/6 + 2*a*c**3*
x**5/5 + c**3*x**4/4

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Giac [A]
time = 0.41, size = 73, normalized size = 0.84 \begin {gather*} \frac {1}{10} \, a^{6} c^{3} x^{10} + \frac {2}{9} \, a^{5} c^{3} x^{9} - \frac {1}{8} \, a^{4} c^{3} x^{8} - \frac {4}{7} \, a^{3} c^{3} x^{7} - \frac {1}{6} \, a^{2} c^{3} x^{6} + \frac {2}{5} \, a c^{3} x^{5} + \frac {1}{4} \, c^{3} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*a^6*c^3*x^10 + 2/9*a^5*c^3*x^9 - 1/8*a^4*c^3*x^8 - 4/7*a^3*c^3*x^7 - 1/6*a^2*c^3*x^6 + 2/5*a*c^3*x^5 + 1/
4*c^3*x^4

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Mupad [B]
time = 0.03, size = 73, normalized size = 0.84 \begin {gather*} \frac {a^6\,c^3\,x^{10}}{10}+\frac {2\,a^5\,c^3\,x^9}{9}-\frac {a^4\,c^3\,x^8}{8}-\frac {4\,a^3\,c^3\,x^7}{7}-\frac {a^2\,c^3\,x^6}{6}+\frac {2\,a\,c^3\,x^5}{5}+\frac {c^3\,x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^3*x^4)/4 + (2*a*c^3*x^5)/5 - (a^2*c^3*x^6)/6 - (4*a^3*c^3*x^7)/7 - (a^4*c^3*x^8)/8 + (2*a^5*c^3*x^9)/9 + (a
^6*c^3*x^10)/10

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