3.270 \(\int e^{-3 \tanh ^{-1}(a x)} (c-a c x)^{5/2} \, dx\)

Optimal. Leaf size=171 \[ \frac{2 (c-a c x)^{9/2}}{7 a c^2 \sqrt{1-a^2 x^2}}-\frac{4096 c^2 \sqrt{c-a c x}}{35 a \sqrt{1-a^2 x^2}}+\frac{32 (c-a c x)^{7/2}}{35 a c \sqrt{1-a^2 x^2}}+\frac{128 (c-a c x)^{5/2}}{35 a \sqrt{1-a^2 x^2}}+\frac{1024 c (c-a c x)^{3/2}}{35 a \sqrt{1-a^2 x^2}} \]

[Out]

(-4096*c^2*Sqrt[c - a*c*x])/(35*a*Sqrt[1 - a^2*x^2]) + (1024*c*(c - a*c*x)^(3/2))/(35*a*Sqrt[1 - a^2*x^2]) + (
128*(c - a*c*x)^(5/2))/(35*a*Sqrt[1 - a^2*x^2]) + (32*(c - a*c*x)^(7/2))/(35*a*c*Sqrt[1 - a^2*x^2]) + (2*(c -
a*c*x)^(9/2))/(7*a*c^2*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.136879, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6127, 657, 649} \[ \frac{2 (c-a c x)^{9/2}}{7 a c^2 \sqrt{1-a^2 x^2}}-\frac{4096 c^2 \sqrt{c-a c x}}{35 a \sqrt{1-a^2 x^2}}+\frac{32 (c-a c x)^{7/2}}{35 a c \sqrt{1-a^2 x^2}}+\frac{128 (c-a c x)^{5/2}}{35 a \sqrt{1-a^2 x^2}}+\frac{1024 c (c-a c x)^{3/2}}{35 a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(c - a*c*x)^(5/2)/E^(3*ArcTanh[a*x]),x]

[Out]

(-4096*c^2*Sqrt[c - a*c*x])/(35*a*Sqrt[1 - a^2*x^2]) + (1024*c*(c - a*c*x)^(3/2))/(35*a*Sqrt[1 - a^2*x^2]) + (
128*(c - a*c*x)^(5/2))/(35*a*Sqrt[1 - a^2*x^2]) + (32*(c - a*c*x)^(7/2))/(35*a*c*Sqrt[1 - a^2*x^2]) + (2*(c -
a*c*x)^(9/2))/(7*a*c^2*Sqrt[1 - a^2*x^2])

Rule 6127

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

Rule 657

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[(2*c*d*Simplify[m + p])/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^
2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p]
, 0]

Rule 649

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p,
0]

Rubi steps

\begin{align*} \int e^{-3 \tanh ^{-1}(a x)} (c-a c x)^{5/2} \, dx &=\frac{\int \frac{(c-a c x)^{11/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac{2 (c-a c x)^{9/2}}{7 a c^2 \sqrt{1-a^2 x^2}}+\frac{16 \int \frac{(c-a c x)^{9/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{7 c^2}\\ &=\frac{32 (c-a c x)^{7/2}}{35 a c \sqrt{1-a^2 x^2}}+\frac{2 (c-a c x)^{9/2}}{7 a c^2 \sqrt{1-a^2 x^2}}+\frac{192 \int \frac{(c-a c x)^{7/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{35 c}\\ &=\frac{128 (c-a c x)^{5/2}}{35 a \sqrt{1-a^2 x^2}}+\frac{32 (c-a c x)^{7/2}}{35 a c \sqrt{1-a^2 x^2}}+\frac{2 (c-a c x)^{9/2}}{7 a c^2 \sqrt{1-a^2 x^2}}+\frac{512}{35} \int \frac{(c-a c x)^{5/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac{1024 c (c-a c x)^{3/2}}{35 a \sqrt{1-a^2 x^2}}+\frac{128 (c-a c x)^{5/2}}{35 a \sqrt{1-a^2 x^2}}+\frac{32 (c-a c x)^{7/2}}{35 a c \sqrt{1-a^2 x^2}}+\frac{2 (c-a c x)^{9/2}}{7 a c^2 \sqrt{1-a^2 x^2}}+\frac{1}{35} (2048 c) \int \frac{(c-a c x)^{3/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{4096 c^2 \sqrt{c-a c x}}{35 a \sqrt{1-a^2 x^2}}+\frac{1024 c (c-a c x)^{3/2}}{35 a \sqrt{1-a^2 x^2}}+\frac{128 (c-a c x)^{5/2}}{35 a \sqrt{1-a^2 x^2}}+\frac{32 (c-a c x)^{7/2}}{35 a c \sqrt{1-a^2 x^2}}+\frac{2 (c-a c x)^{9/2}}{7 a c^2 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0389899, size = 70, normalized size = 0.41 \[ \frac{2 c^3 \sqrt{1-a x} \left (5 a^4 x^4-36 a^3 x^3+142 a^2 x^2-708 a x-1451\right )}{35 a \sqrt{a x+1} \sqrt{c-a c x}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - a*c*x)^(5/2)/E^(3*ArcTanh[a*x]),x]

[Out]

(2*c^3*Sqrt[1 - a*x]*(-1451 - 708*a*x + 142*a^2*x^2 - 36*a^3*x^3 + 5*a^4*x^4))/(35*a*Sqrt[1 + a*x]*Sqrt[c - a*
c*x])

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Maple [A]  time = 0.032, size = 71, normalized size = 0.4 \begin{align*}{\frac{10\,{x}^{4}{a}^{4}-72\,{x}^{3}{a}^{3}+284\,{a}^{2}{x}^{2}-1416\,ax-2902}{35\, \left ( ax+1 \right ) ^{2} \left ( ax-1 \right ) ^{4}a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}} \left ( -acx+c \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*c*x+c)^(5/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)

[Out]

2/35*(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(5/2)*(5*a^4*x^4-36*a^3*x^3+142*a^2*x^2-708*a*x-1451)/(a*x+1)^2/(a*x-1)^4/a

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Maxima [A]  time = 1.0306, size = 99, normalized size = 0.58 \begin{align*} \frac{2 \,{\left (5 \, a^{4} c^{\frac{5}{2}} x^{4} - 36 \, a^{3} c^{\frac{5}{2}} x^{3} + 142 \, a^{2} c^{\frac{5}{2}} x^{2} - 708 \, a c^{\frac{5}{2}} x - 1451 \, c^{\frac{5}{2}}\right )} \sqrt{a x + 1}{\left (a x - 1\right )}}{35 \,{\left (a^{3} x^{2} - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(5/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="maxima")

[Out]

2/35*(5*a^4*c^(5/2)*x^4 - 36*a^3*c^(5/2)*x^3 + 142*a^2*c^(5/2)*x^2 - 708*a*c^(5/2)*x - 1451*c^(5/2))*sqrt(a*x
+ 1)*(a*x - 1)/(a^3*x^2 - a)

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Fricas [A]  time = 1.98562, size = 180, normalized size = 1.05 \begin{align*} -\frac{2 \,{\left (5 \, a^{4} c^{2} x^{4} - 36 \, a^{3} c^{2} x^{3} + 142 \, a^{2} c^{2} x^{2} - 708 \, a c^{2} x - 1451 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{35 \,{\left (a^{3} x^{2} - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(5/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="fricas")

[Out]

-2/35*(5*a^4*c^2*x^4 - 36*a^3*c^2*x^3 + 142*a^2*c^2*x^2 - 708*a*c^2*x - 1451*c^2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c
*x + c)/(a^3*x^2 - a)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)**(5/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.32841, size = 115, normalized size = 0.67 \begin{align*} \frac{2048 \, \sqrt{2} c^{\frac{3}{2}}{\left | c \right |}}{35 \, a} + \frac{2 \,{\left (5 \,{\left (a c x + c\right )}^{\frac{7}{2}} - 56 \,{\left (a c x + c\right )}^{\frac{5}{2}} c + 280 \,{\left (a c x + c\right )}^{\frac{3}{2}} c^{2} - 1120 \, \sqrt{a c x + c} c^{3} - \frac{560 \, c^{4}}{\sqrt{a c x + c}}\right )}{\left | c \right |}}{35 \, a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(5/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac")

[Out]

2048/35*sqrt(2)*c^(3/2)*abs(c)/a + 2/35*(5*(a*c*x + c)^(7/2) - 56*(a*c*x + c)^(5/2)*c + 280*(a*c*x + c)^(3/2)*
c^2 - 1120*sqrt(a*c*x + c)*c^3 - 560*c^4/sqrt(a*c*x + c))*abs(c)/(a*c^2)