3.227 \(\int e^{\tanh ^{-1}(a x)} (c-a c x)^{7/2} \, dx\)

Optimal. Leaf size=141 \[ \frac{256 c^5 \left (1-a^2 x^2\right )^{3/2}}{315 a (c-a c x)^{3/2}}+\frac{64 c^4 \left (1-a^2 x^2\right )^{3/2}}{105 a \sqrt{c-a c x}}+\frac{8 c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a c x}}{21 a}+\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{3/2}}{9 a} \]

[Out]

(256*c^5*(1 - a^2*x^2)^(3/2))/(315*a*(c - a*c*x)^(3/2)) + (64*c^4*(1 - a^2*x^2)^(3/2))/(105*a*Sqrt[c - a*c*x])
+ (8*c^3*Sqrt[c - a*c*x]*(1 - a^2*x^2)^(3/2))/(21*a) + (2*c^2*(c - a*c*x)^(3/2)*(1 - a^2*x^2)^(3/2))/(9*a)

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Rubi [A]  time = 0.104202, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6127, 657, 649} \[ \frac{256 c^5 \left (1-a^2 x^2\right )^{3/2}}{315 a (c-a c x)^{3/2}}+\frac{64 c^4 \left (1-a^2 x^2\right )^{3/2}}{105 a \sqrt{c-a c x}}+\frac{8 c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a c x}}{21 a}+\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{3/2}}{9 a} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*(c - a*c*x)^(7/2),x]

[Out]

(256*c^5*(1 - a^2*x^2)^(3/2))/(315*a*(c - a*c*x)^(3/2)) + (64*c^4*(1 - a^2*x^2)^(3/2))/(105*a*Sqrt[c - a*c*x])
+ (8*c^3*Sqrt[c - a*c*x]*(1 - a^2*x^2)^(3/2))/(21*a) + (2*c^2*(c - a*c*x)^(3/2)*(1 - a^2*x^2)^(3/2))/(9*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)^{7/2} \, dx &=c \int (c-a c x)^{5/2} \sqrt{1-a^2 x^2} \, dx\\ &=\frac{2 c^2 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{9 a}+\frac{1}{3} \left (4 c^2\right ) \int (c-a c x)^{3/2} \sqrt{1-a^2 x^2} \, dx\\ &=\frac{8 c^3 \sqrt{c-a c x} \left (1-a^2 x^2\right )^{3/2}}{21 a}+\frac{2 c^2 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{9 a}+\frac{1}{21} \left (32 c^3\right ) \int \sqrt{c-a c x} \sqrt{1-a^2 x^2} \, dx\\ &=\frac{64 c^4 \left (1-a^2 x^2\right )^{3/2}}{105 a \sqrt{c-a c x}}+\frac{8 c^3 \sqrt{c-a c x} \left (1-a^2 x^2\right )^{3/2}}{21 a}+\frac{2 c^2 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{9 a}+\frac{1}{105} \left (128 c^4\right ) \int \frac{\sqrt{1-a^2 x^2}}{\sqrt{c-a c x}} \, dx\\ &=\frac{256 c^5 \left (1-a^2 x^2\right )^{3/2}}{315 a (c-a c x)^{3/2}}+\frac{64 c^4 \left (1-a^2 x^2\right )^{3/2}}{105 a \sqrt{c-a c x}}+\frac{8 c^3 \sqrt{c-a c x} \left (1-a^2 x^2\right )^{3/2}}{21 a}+\frac{2 c^2 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{9 a}\\ \end{align*}

Mathematica [A]  time = 0.0413035, size = 62, normalized size = 0.44 \[ -\frac{2 c^3 (a x+1)^{3/2} \left (35 a^3 x^3-165 a^2 x^2+321 a x-319\right ) \sqrt{c-a c x}}{315 a \sqrt{1-a x}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcTanh[a*x]*(c - a*c*x)^(7/2),x]

[Out]

(-2*c^3*(1 + a*x)^(3/2)*Sqrt[c - a*c*x]*(-319 + 321*a*x - 165*a^2*x^2 + 35*a^3*x^3))/(315*a*Sqrt[1 - a*x])

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Maple [A]  time = 0.033, size = 63, normalized size = 0.5 \begin{align*}{\frac{2\, \left ( 35\,{x}^{3}{a}^{3}-165\,{a}^{2}{x}^{2}+321\,ax-319 \right ) \left ( ax+1 \right ) ^{2}}{315\, \left ( ax-1 \right ) ^{3}a} \left ( -acx+c \right ) ^{{\frac{7}{2}}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(7/2),x)

[Out]

2/315*(a*x+1)^2*(35*a^3*x^3-165*a^2*x^2+321*a*x-319)*(-a*c*x+c)^(7/2)/a/(a*x-1)^3/(-a^2*x^2+1)^(1/2)

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Maxima [A]  time = 1.02646, size = 173, normalized size = 1.23 \begin{align*} -\frac{2 \,{\left (5 \, a^{5} c^{\frac{7}{2}} x^{5} - 20 \, a^{4} c^{\frac{7}{2}} x^{4} + 32 \, a^{3} c^{\frac{7}{2}} x^{3} - 34 \, a^{2} c^{\frac{7}{2}} x^{2} + 91 \, a c^{\frac{7}{2}} x + 182 \, c^{\frac{7}{2}}\right )}}{45 \, \sqrt{a x + 1} a} - \frac{2 \,{\left (5 \, a^{4} c^{\frac{7}{2}} x^{4} - 22 \, a^{3} c^{\frac{7}{2}} x^{3} + 44 \, a^{2} c^{\frac{7}{2}} x^{2} - 106 \, a c^{\frac{7}{2}} x - 177 \, c^{\frac{7}{2}}\right )}}{35 \, \sqrt{a x + 1} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(7/2),x, algorithm="maxima")

[Out]

-2/45*(5*a^5*c^(7/2)*x^5 - 20*a^4*c^(7/2)*x^4 + 32*a^3*c^(7/2)*x^3 - 34*a^2*c^(7/2)*x^2 + 91*a*c^(7/2)*x + 182
*c^(7/2))/(sqrt(a*x + 1)*a) - 2/35*(5*a^4*c^(7/2)*x^4 - 22*a^3*c^(7/2)*x^3 + 44*a^2*c^(7/2)*x^2 - 106*a*c^(7/2
)*x - 177*c^(7/2))/(sqrt(a*x + 1)*a)

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Fricas [A]  time = 1.58907, size = 176, normalized size = 1.25 \begin{align*} \frac{2 \,{\left (35 \, a^{4} c^{3} x^{4} - 130 \, a^{3} c^{3} x^{3} + 156 \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x - 319 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{315 \,{\left (a^{2} x - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(7/2),x, algorithm="fricas")

[Out]

2/315*(35*a^4*c^3*x^4 - 130*a^3*c^3*x^3 + 156*a^2*c^3*x^2 + 2*a*c^3*x - 319*c^3)*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*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**(7/2),x)

[Out]

Timed out

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Giac [A]  time = 1.3117, size = 99, normalized size = 0.7 \begin{align*} -\frac{2 \,{\left (256 \, \sqrt{2} c^{\frac{5}{2}} + \frac{35 \,{\left (a c x + c\right )}^{\frac{9}{2}} - 270 \,{\left (a c x + c\right )}^{\frac{7}{2}} c + 756 \,{\left (a c x + c\right )}^{\frac{5}{2}} c^{2} - 840 \,{\left (a c x + c\right )}^{\frac{3}{2}} c^{3}}{c^{2}}\right )} c^{2}}{315 \, a{\left | c \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(7/2),x, algorithm="giac")

[Out]

-2/315*(256*sqrt(2)*c^(5/2) + (35*(a*c*x + c)^(9/2) - 270*(a*c*x + c)^(7/2)*c + 756*(a*c*x + c)^(5/2)*c^2 - 84
0*(a*c*x + c)^(3/2)*c^3)/c^2)*c^2/(a*abs(c))