3.11.39 \(\int e^{2 \tanh ^{-1}(a x)} x^4 (c-a^2 c x^2)^3 \, dx\) [1039]

Optimal. Leaf size=87 \[ \frac {c^3 x^5}{5}+\frac {1}{3} a c^3 x^6-\frac {1}{7} a^2 c^3 x^7-\frac {1}{2} a^3 c^3 x^8-\frac {1}{9} a^4 c^3 x^9+\frac {1}{5} a^5 c^3 x^{10}+\frac {1}{11} a^6 c^3 x^{11} \]

[Out]

1/5*c^3*x^5+1/3*a*c^3*x^6-1/7*a^2*c^3*x^7-1/2*a^3*c^3*x^8-1/9*a^4*c^3*x^9+1/5*a^5*c^3*x^10+1/11*a^6*c^3*x^11

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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}{11} a^6 c^3 x^{11}+\frac {1}{5} a^5 c^3 x^{10}-\frac {1}{9} a^4 c^3 x^9-\frac {1}{2} a^3 c^3 x^8-\frac {1}{7} a^2 c^3 x^7+\frac {1}{3} a c^3 x^6+\frac {c^3 x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])*x^4*(c - a^2*c*x^2)^3,x]

[Out]

(c^3*x^5)/5 + (a*c^3*x^6)/3 - (a^2*c^3*x^7)/7 - (a^3*c^3*x^8)/2 - (a^4*c^3*x^9)/9 + (a^5*c^3*x^10)/5 + (a^6*c^
3*x^11)/11

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^4 \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int x^4 (1-a x)^2 (1+a x)^4 \, dx\\ &=c^3 \int \left (x^4+2 a x^5-a^2 x^6-4 a^3 x^7-a^4 x^8+2 a^5 x^9+a^6 x^{10}\right ) \, dx\\ &=\frac {c^3 x^5}{5}+\frac {1}{3} a c^3 x^6-\frac {1}{7} a^2 c^3 x^7-\frac {1}{2} a^3 c^3 x^8-\frac {1}{9} a^4 c^3 x^9+\frac {1}{5} a^5 c^3 x^{10}+\frac {1}{11} a^6 c^3 x^{11}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 70, normalized size = 0.80 \begin {gather*} c^3 \left (\frac {x^5}{5}+\frac {a x^6}{3}-\frac {a^2 x^7}{7}-\frac {a^3 x^8}{2}-\frac {a^4 x^9}{9}+\frac {a^5 x^{10}}{5}+\frac {a^6 x^{11}}{11}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a*x])*x^4*(c - a^2*c*x^2)^3,x]

[Out]

c^3*(x^5/5 + (a*x^6)/3 - (a^2*x^7)/7 - (a^3*x^8)/2 - (a^4*x^9)/9 + (a^5*x^10)/5 + (a^6*x^11)/11)

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Maple [A]
time = 0.10, size = 57, normalized size = 0.66

method	result	size
gosper	\(\frac {c^{3} x^{5} \left (630 x^{6} a^{6}+1386 x^{5} a^{5}-770 a^{4} x^{4}-3465 a^{3} x^{3}-990 a^{2} x^{2}+2310 a x +1386\right )}{6930}\)	\(55\)
default	\(c^{3} \left (\frac {1}{11} a^{6} x^{11}+\frac {1}{5} a^{5} x^{10}-\frac {1}{9} a^{4} x^{9}-\frac {1}{2} a^{3} x^{8}-\frac {1}{7} a^{2} x^{7}+\frac {1}{3} a \,x^{6}+\frac {1}{5} x^{5}\right )\)	\(57\)
norman	\(\frac {1}{5} c^{3} x^{5}+\frac {1}{3} a \,c^{3} x^{6}-\frac {1}{7} a^{2} c^{3} x^{7}-\frac {1}{2} a^{3} c^{3} x^{8}-\frac {1}{9} a^{4} c^{3} x^{9}+\frac {1}{5} a^{5} c^{3} x^{10}+\frac {1}{11} a^{6} c^{3} x^{11}\)	\(74\)
risch	\(\frac {1}{5} c^{3} x^{5}+\frac {1}{3} a \,c^{3} x^{6}-\frac {1}{7} a^{2} c^{3} x^{7}-\frac {1}{2} a^{3} c^{3} x^{8}-\frac {1}{9} a^{4} c^{3} x^{9}+\frac {1}{5} a^{5} c^{3} x^{10}+\frac {1}{11} a^{6} c^{3} x^{11}\)	\(74\)
meijerg	\(-\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {13}{2}} \left (4095 x^{10} a^{10}+5005 x^{8} a^{8}+6435 x^{6} a^{6}+9009 a^{4} x^{4}+15015 a^{2} x^{2}+45045\right )}{45045 a^{12}}+\frac {2 \left (-a^{2}\right )^{\frac {13}{2}} \arctanh \left (a x \right )}{a^{13}}\right )}{2 a^{4} \sqrt {-a^{2}}}-\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {11}{2}} \left (385 x^{8} a^{8}+495 x^{6} a^{6}+693 a^{4} x^{4}+1155 a^{2} x^{2}+3465\right )}{3465 a^{10}}+\frac {2 \left (-a^{2}\right )^{\frac {11}{2}} \arctanh \left (a x \right )}{a^{11}}\right )}{a^{4} \sqrt {-a^{2}}}+\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {7}{2}} \left (21 a^{4} x^{4}+35 a^{2} x^{2}+105\right )}{105 a^{6}}+\frac {2 \left (-a^{2}\right )^{\frac {7}{2}} \arctanh \left (a x \right )}{a^{7}}\right )}{a^{4} \sqrt {-a^{2}}}-\frac {c^{3} \left (-\frac {x^{2} a^{2} \left (12 x^{8} a^{8}+15 x^{6} a^{6}+20 a^{4} x^{4}+30 a^{2} x^{2}+60\right )}{60}-\ln \left (-a^{2} x^{2}+1\right )\right )}{a^{5}}-\frac {3 c^{3} \left (\frac {x^{2} a^{2} \left (15 x^{6} a^{6}+20 a^{4} x^{4}+30 a^{2} x^{2}+60\right )}{60}+\ln \left (-a^{2} x^{2}+1\right )\right )}{a^{5}}-\frac {3 c^{3} \left (-\frac {x^{2} a^{2} \left (4 a^{4} x^{4}+6 a^{2} x^{2}+12\right )}{12}-\ln \left (-a^{2} x^{2}+1\right )\right )}{a^{5}}-\frac {c^{3} \left (\frac {x^{2} a^{2} \left (3 a^{2} x^{2}+6\right )}{6}+\ln \left (-a^{2} x^{2}+1\right )\right )}{a^{5}}+\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {5}{2}} \left (5 a^{2} x^{2}+15\right )}{15 a^{4}}+\frac {2 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{2 a^{4} \sqrt {-a^{2}}}\)	\(489\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*x^4*(-a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

c^3*(1/11*a^6*x^11+1/5*a^5*x^10-1/9*a^4*x^9-1/2*a^3*x^8-1/7*a^2*x^7+1/3*a*x^6+1/5*x^5)

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Maxima [A]
time = 0.26, size = 73, normalized size = 0.84 \begin {gather*} \frac {1}{11} \, a^{6} c^{3} x^{11} + \frac {1}{5} \, a^{5} c^{3} x^{10} - \frac {1}{9} \, a^{4} c^{3} x^{9} - \frac {1}{2} \, a^{3} c^{3} x^{8} - \frac {1}{7} \, a^{2} c^{3} x^{7} + \frac {1}{3} \, a c^{3} x^{6} + \frac {1}{5} \, c^{3} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^4*(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

1/11*a^6*c^3*x^11 + 1/5*a^5*c^3*x^10 - 1/9*a^4*c^3*x^9 - 1/2*a^3*c^3*x^8 - 1/7*a^2*c^3*x^7 + 1/3*a*c^3*x^6 + 1
/5*c^3*x^5

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Fricas [A]
time = 0.31, size = 73, normalized size = 0.84 \begin {gather*} \frac {1}{11} \, a^{6} c^{3} x^{11} + \frac {1}{5} \, a^{5} c^{3} x^{10} - \frac {1}{9} \, a^{4} c^{3} x^{9} - \frac {1}{2} \, a^{3} c^{3} x^{8} - \frac {1}{7} \, a^{2} c^{3} x^{7} + \frac {1}{3} \, a c^{3} x^{6} + \frac {1}{5} \, c^{3} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^4*(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

1/11*a^6*c^3*x^11 + 1/5*a^5*c^3*x^10 - 1/9*a^4*c^3*x^9 - 1/2*a^3*c^3*x^8 - 1/7*a^2*c^3*x^7 + 1/3*a*c^3*x^6 + 1
/5*c^3*x^5

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Sympy [A]
time = 0.03, size = 76, normalized size = 0.87 \begin {gather*} \frac {a^{6} c^{3} x^{11}}{11} + \frac {a^{5} c^{3} x^{10}}{5} - \frac {a^{4} c^{3} x^{9}}{9} - \frac {a^{3} c^{3} x^{8}}{2} - \frac {a^{2} c^{3} x^{7}}{7} + \frac {a c^{3} x^{6}}{3} + \frac {c^{3} x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*x**4*(-a**2*c*x**2+c)**3,x)

[Out]

a**6*c**3*x**11/11 + a**5*c**3*x**10/5 - a**4*c**3*x**9/9 - a**3*c**3*x**8/2 - a**2*c**3*x**7/7 + a*c**3*x**6/
3 + c**3*x**5/5

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Giac [A]
time = 0.43, size = 73, normalized size = 0.84 \begin {gather*} \frac {1}{11} \, a^{6} c^{3} x^{11} + \frac {1}{5} \, a^{5} c^{3} x^{10} - \frac {1}{9} \, a^{4} c^{3} x^{9} - \frac {1}{2} \, a^{3} c^{3} x^{8} - \frac {1}{7} \, a^{2} c^{3} x^{7} + \frac {1}{3} \, a c^{3} x^{6} + \frac {1}{5} \, c^{3} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^4*(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

1/11*a^6*c^3*x^11 + 1/5*a^5*c^3*x^10 - 1/9*a^4*c^3*x^9 - 1/2*a^3*c^3*x^8 - 1/7*a^2*c^3*x^7 + 1/3*a*c^3*x^6 + 1
/5*c^3*x^5

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Mupad [B]
time = 0.89, size = 73, normalized size = 0.84 \begin {gather*} \frac {a^6\,c^3\,x^{11}}{11}+\frac {a^5\,c^3\,x^{10}}{5}-\frac {a^4\,c^3\,x^9}{9}-\frac {a^3\,c^3\,x^8}{2}-\frac {a^2\,c^3\,x^7}{7}+\frac {a\,c^3\,x^6}{3}+\frac {c^3\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4*(c - a^2*c*x^2)^3*(a*x + 1)^2)/(a^2*x^2 - 1),x)

[Out]

(c^3*x^5)/5 + (a*c^3*x^6)/3 - (a^2*c^3*x^7)/7 - (a^3*c^3*x^8)/2 - (a^4*c^3*x^9)/9 + (a^5*c^3*x^10)/5 + (a^6*c^
3*x^11)/11

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