3.252 \(\int e^{-\tanh ^{-1}(a x)} (c-a c x)^{9/2} \, dx\)

Optimal. Leaf size=206 \[ \frac{16384 c^5 \sqrt{1-a^2 x^2}}{693 a \sqrt{c-a c x}}+\frac{4096 c^4 \sqrt{1-a^2 x^2} \sqrt{c-a c x}}{693 a}+\frac{512 c^3 \sqrt{1-a^2 x^2} (c-a c x)^{3/2}}{231 a}+\frac{640 c^2 \sqrt{1-a^2 x^2} (c-a c x)^{5/2}}{693 a}+\frac{40 c \sqrt{1-a^2 x^2} (c-a c x)^{7/2}}{99 a}+\frac{2 \sqrt{1-a^2 x^2} (c-a c x)^{9/2}}{11 a} \]

[Out]

(16384*c^5*Sqrt[1 - a^2*x^2])/(693*a*Sqrt[c - a*c*x]) + (4096*c^4*Sqrt[c - a*c*x]*Sqrt[1 - a^2*x^2])/(693*a) +
(512*c^3*(c - a*c*x)^(3/2)*Sqrt[1 - a^2*x^2])/(231*a) + (640*c^2*(c - a*c*x)^(5/2)*Sqrt[1 - a^2*x^2])/(693*a)
+ (40*c*(c - a*c*x)^(7/2)*Sqrt[1 - a^2*x^2])/(99*a) + (2*(c - a*c*x)^(9/2)*Sqrt[1 - a^2*x^2])/(11*a)

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Rubi [A]  time = 0.161547, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6127, 657, 649} \[ \frac{16384 c^5 \sqrt{1-a^2 x^2}}{693 a \sqrt{c-a c x}}+\frac{4096 c^4 \sqrt{1-a^2 x^2} \sqrt{c-a c x}}{693 a}+\frac{512 c^3 \sqrt{1-a^2 x^2} (c-a c x)^{3/2}}{231 a}+\frac{640 c^2 \sqrt{1-a^2 x^2} (c-a c x)^{5/2}}{693 a}+\frac{40 c \sqrt{1-a^2 x^2} (c-a c x)^{7/2}}{99 a}+\frac{2 \sqrt{1-a^2 x^2} (c-a c x)^{9/2}}{11 a} \]

Antiderivative was successfully verified.

[In]

Int[(c - a*c*x)^(9/2)/E^ArcTanh[a*x],x]

[Out]

(16384*c^5*Sqrt[1 - a^2*x^2])/(693*a*Sqrt[c - a*c*x]) + (4096*c^4*Sqrt[c - a*c*x]*Sqrt[1 - a^2*x^2])/(693*a) +
(512*c^3*(c - a*c*x)^(3/2)*Sqrt[1 - a^2*x^2])/(231*a) + (640*c^2*(c - a*c*x)^(5/2)*Sqrt[1 - a^2*x^2])/(693*a)
+ (40*c*(c - a*c*x)^(7/2)*Sqrt[1 - a^2*x^2])/(99*a) + (2*(c - a*c*x)^(9/2)*Sqrt[1 - a^2*x^2])/(11*a)

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^{-\tanh ^{-1}(a x)} (c-a c x)^{9/2} \, dx &=\frac{\int \frac{(c-a c x)^{11/2}}{\sqrt{1-a^2 x^2}} \, dx}{c}\\ &=\frac{2 (c-a c x)^{9/2} \sqrt{1-a^2 x^2}}{11 a}+\frac{20}{11} \int \frac{(c-a c x)^{9/2}}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{40 c (c-a c x)^{7/2} \sqrt{1-a^2 x^2}}{99 a}+\frac{2 (c-a c x)^{9/2} \sqrt{1-a^2 x^2}}{11 a}+\frac{1}{99} (320 c) \int \frac{(c-a c x)^{7/2}}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{640 c^2 (c-a c x)^{5/2} \sqrt{1-a^2 x^2}}{693 a}+\frac{40 c (c-a c x)^{7/2} \sqrt{1-a^2 x^2}}{99 a}+\frac{2 (c-a c x)^{9/2} \sqrt{1-a^2 x^2}}{11 a}+\frac{1}{231} \left (1280 c^2\right ) \int \frac{(c-a c x)^{5/2}}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{512 c^3 (c-a c x)^{3/2} \sqrt{1-a^2 x^2}}{231 a}+\frac{640 c^2 (c-a c x)^{5/2} \sqrt{1-a^2 x^2}}{693 a}+\frac{40 c (c-a c x)^{7/2} \sqrt{1-a^2 x^2}}{99 a}+\frac{2 (c-a c x)^{9/2} \sqrt{1-a^2 x^2}}{11 a}+\frac{1}{231} \left (2048 c^3\right ) \int \frac{(c-a c x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{4096 c^4 \sqrt{c-a c x} \sqrt{1-a^2 x^2}}{693 a}+\frac{512 c^3 (c-a c x)^{3/2} \sqrt{1-a^2 x^2}}{231 a}+\frac{640 c^2 (c-a c x)^{5/2} \sqrt{1-a^2 x^2}}{693 a}+\frac{40 c (c-a c x)^{7/2} \sqrt{1-a^2 x^2}}{99 a}+\frac{2 (c-a c x)^{9/2} \sqrt{1-a^2 x^2}}{11 a}+\frac{1}{693} \left (8192 c^4\right ) \int \frac{\sqrt{c-a c x}}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{16384 c^5 \sqrt{1-a^2 x^2}}{693 a \sqrt{c-a c x}}+\frac{4096 c^4 \sqrt{c-a c x} \sqrt{1-a^2 x^2}}{693 a}+\frac{512 c^3 (c-a c x)^{3/2} \sqrt{1-a^2 x^2}}{231 a}+\frac{640 c^2 (c-a c x)^{5/2} \sqrt{1-a^2 x^2}}{693 a}+\frac{40 c (c-a c x)^{7/2} \sqrt{1-a^2 x^2}}{99 a}+\frac{2 (c-a c x)^{9/2} \sqrt{1-a^2 x^2}}{11 a}\\ \end{align*}

Mathematica [A]  time = 0.0553282, size = 73, normalized size = 0.35 \[ -\frac{2 c^5 \sqrt{1-a^2 x^2} \left (63 a^5 x^5-455 a^4 x^4+1510 a^3 x^3-3198 a^2 x^2+5419 a x-11531\right )}{693 a \sqrt{c-a c x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c - a*c*x)^(9/2)/E^ArcTanh[a*x],x]

[Out]

(-2*c^5*Sqrt[1 - a^2*x^2]*(-11531 + 5419*a*x - 3198*a^2*x^2 + 1510*a^3*x^3 - 455*a^4*x^4 + 63*a^5*x^5))/(693*a
*Sqrt[c - a*c*x])

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Maple [A]  time = 0.031, size = 72, normalized size = 0.4 \begin{align*}{\frac{126\,{x}^{5}{a}^{5}-910\,{x}^{4}{a}^{4}+3020\,{x}^{3}{a}^{3}-6396\,{a}^{2}{x}^{2}+10838\,ax-23062}{693\, \left ( ax-1 \right ) ^{5}a}\sqrt{-{a}^{2}{x}^{2}+1} \left ( -acx+c \right ) ^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*c*x+c)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x)

[Out]

2/693*(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(9/2)*(63*a^5*x^5-455*a^4*x^4+1510*a^3*x^3-3198*a^2*x^2+5419*a*x-11531)/(a
*x-1)^5/a

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Maxima [A]  time = 1.01987, size = 111, normalized size = 0.54 \begin{align*} -\frac{2 \,{\left (63 \, a^{5} c^{\frac{9}{2}} x^{5} - 455 \, a^{4} c^{\frac{9}{2}} x^{4} + 1510 \, a^{3} c^{\frac{9}{2}} x^{3} - 3198 \, a^{2} c^{\frac{9}{2}} x^{2} + 5419 \, a c^{\frac{9}{2}} x - 11531 \, c^{\frac{9}{2}}\right )} \sqrt{a x + 1}{\left (a x - 1\right )}}{693 \,{\left (a^{2} x - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

-2/693*(63*a^5*c^(9/2)*x^5 - 455*a^4*c^(9/2)*x^4 + 1510*a^3*c^(9/2)*x^3 - 3198*a^2*c^(9/2)*x^2 + 5419*a*c^(9/2
)*x - 11531*c^(9/2))*sqrt(a*x + 1)*(a*x - 1)/(a^2*x - a)

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Fricas [A]  time = 2.19806, size = 209, normalized size = 1.01 \begin{align*} \frac{2 \,{\left (63 \, a^{5} c^{4} x^{5} - 455 \, a^{4} c^{4} x^{4} + 1510 \, a^{3} c^{4} x^{3} - 3198 \, a^{2} c^{4} x^{2} + 5419 \, a c^{4} x - 11531 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{693 \,{\left (a^{2} x - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

2/693*(63*a^5*c^4*x^5 - 455*a^4*c^4*x^4 + 1510*a^3*c^4*x^3 - 3198*a^2*c^4*x^2 + 5419*a*c^4*x - 11531*c^4)*sqrt
(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - 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)**(9/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.32767, size = 132, normalized size = 0.64 \begin{align*} -\frac{16384 \, \sqrt{2} c^{\frac{7}{2}}{\left | c \right |}}{693 \, a} - \frac{2 \,{\left (63 \,{\left (a c x + c\right )}^{\frac{11}{2}} - 770 \,{\left (a c x + c\right )}^{\frac{9}{2}} c + 3960 \,{\left (a c x + c\right )}^{\frac{7}{2}} c^{2} - 11088 \,{\left (a c x + c\right )}^{\frac{5}{2}} c^{3} + 18480 \,{\left (a c x + c\right )}^{\frac{3}{2}} c^{4} - 22176 \, \sqrt{a c x + c} c^{5}\right )}{\left | c \right |}}{693 \, a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

-16384/693*sqrt(2)*c^(7/2)*abs(c)/a - 2/693*(63*(a*c*x + c)^(11/2) - 770*(a*c*x + c)^(9/2)*c + 3960*(a*c*x + c
)^(7/2)*c^2 - 11088*(a*c*x + c)^(5/2)*c^3 + 18480*(a*c*x + c)^(3/2)*c^4 - 22176*sqrt(a*c*x + c)*c^5)*abs(c)/(a
*c^2)