3.420 \(\int e^{-2 \tanh ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx\)

Optimal. Leaf size=97 \[ \frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}+\frac {2 (c-a c x)^{3/2}}{3 a^3 c}+\frac {4 \sqrt {c-a c x}}{a^3}-\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^3} \]

[Out]

2/3*(-a*c*x+c)^(3/2)/a^3/c+2/7*(-a*c*x+c)^(7/2)/a^3/c^3-4*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))*2^(1/2
)*c^(1/2)/a^3+4*(-a*c*x+c)^(1/2)/a^3

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Rubi [A]  time = 0.16, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6130, 21, 88, 50, 63, 206} \[ \frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}+\frac {2 (c-a c x)^{3/2}}{3 a^3 c}+\frac {4 \sqrt {c-a c x}}{a^3}-\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[c - a*c*x])/a^3 + (2*(c - a*c*x)^(3/2))/(3*a^3*c) + (2*(c - a*c*x)^(7/2))/(7*a^3*c^3) - (4*Sqrt[2]*Sqr
t[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a^3

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 88

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6130

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Int[(u*(c + d*x)^p*(1 + a*x)^(
n/2))/(1 - a*x)^(n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] &&  !(IntegerQ[p] || GtQ[c, 0]
)

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 78, normalized size = 0.80 \[ \frac {2 \left (-3 a^3 x^3+9 a^2 x^2-16 a x+52\right ) \sqrt {c-a c x}-84 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{21 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c - a*c*x]*(52 - 16*a*x + 9*a^2*x^2 - 3*a^3*x^3) - 84*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]
*Sqrt[c])])/(21*a^3)

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fricas [A]  time = 0.55, size = 154, normalized size = 1.59 \[ \left [\frac {2 \, {\left (21 \, \sqrt {2} \sqrt {c} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) - {\left (3 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 16 \, a x - 52\right )} \sqrt {-a c x + c}\right )}}{21 \, a^{3}}, \frac {2 \, {\left (42 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - {\left (3 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 16 \, a x - 52\right )} \sqrt {-a c x + c}\right )}}{21 \, a^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2/21*(21*sqrt(2)*sqrt(c)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) - (3*a^3*x^3 - 9*a
^2*x^2 + 16*a*x - 52)*sqrt(-a*c*x + c))/a^3, 2/21*(42*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqr
t(-c)/c) - (3*a^3*x^3 - 9*a^2*x^2 + 16*a*x - 52)*sqrt(-a*c*x + c))/a^3]

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giac [A]  time = 0.17, size = 105, normalized size = 1.08 \[ \frac {4 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a^{3} \sqrt {-c}} - \frac {2 \, {\left (3 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{18} c^{18} - 7 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{18} c^{20} - 42 \, \sqrt {-a c x + c} a^{18} c^{21}\right )}}{21 \, a^{21} c^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*sqrt(2)*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/(a^3*sqrt(-c)) - 2/21*(3*(a*c*x - c)^3*sqrt(-a*c*x +
c)*a^18*c^18 - 7*(-a*c*x + c)^(3/2)*a^18*c^20 - 42*sqrt(-a*c*x + c)*a^18*c^21)/(a^21*c^21)

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maple [A]  time = 0.04, size = 75, normalized size = 0.77 \[ \frac {\frac {2 \left (-a c x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-a c x +c \right )^{\frac {3}{2}} c^{2}}{3}+4 \sqrt {-a c x +c}\, c^{3}-4 c^{\frac {7}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{3} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/c^3/a^3*(1/7*(-a*c*x+c)^(7/2)+1/3*(-a*c*x+c)^(3/2)*c^2+2*(-a*c*x+c)^(1/2)*c^3-2*c^(7/2)*2^(1/2)*arctanh(1/2*
(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2)))

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maxima [A]  time = 0.42, size = 97, normalized size = 1.00 \[ \frac {2 \, {\left (21 \, \sqrt {2} c^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 3 \, {\left (-a c x + c\right )}^{\frac {7}{2}} + 7 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} + 42 \, \sqrt {-a c x + c} c^{3}\right )}}{21 \, a^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(21*sqrt(2)*c^(7/2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt(2)*sqrt(c) + sqrt(-a*c*x + c))) + 3*(
-a*c*x + c)^(7/2) + 7*(-a*c*x + c)^(3/2)*c^2 + 42*sqrt(-a*c*x + c)*c^3)/(a^3*c^3)

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mupad [B]  time = 0.10, size = 80, normalized size = 0.82 \[ \frac {4\,\sqrt {c-a\,c\,x}}{a^3}+\frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a^3\,c}+\frac {2\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a^3\,c^3}+\frac {\sqrt {2}\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i}}{a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*(c - a*c*x)^(1/2))/a^3 + (2*(c - a*c*x)^(3/2))/(3*a^3*c) + (2*(c - a*c*x)^(7/2))/(7*a^3*c^3) + (2^(1/2)*c^(
1/2)*atan((2^(1/2)*(c - a*c*x)^(1/2)*1i)/(2*c^(1/2)))*4i)/a^3

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sympy [A]  time = 14.87, size = 97, normalized size = 1.00 \[ - \frac {2 \left (- \frac {2 \sqrt {2} c^{4} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} - 2 c^{3} \sqrt {- a c x + c} - \frac {c^{2} \left (- a c x + c\right )^{\frac {3}{2}}}{3} - \frac {\left (- a c x + c\right )^{\frac {7}{2}}}{7}\right )}{a^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(-2*sqrt(2)*c**4*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) - 2*c**3*sqrt(-a*c*x + c) - c**2*(-a*
c*x + c)**(3/2)/3 - (-a*c*x + c)**(7/2)/7)/(a**3*c**3)

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