3.244 \(\int e^{3 \tanh ^{-1}(a x)} (c-a c x)^{7/2} \, dx\)

Optimal. Leaf size=106 \[ \frac{64 c^6 \left (1-a^2 x^2\right )^{5/2}}{315 a (c-a c x)^{5/2}}+\frac{16 c^5 \left (1-a^2 x^2\right )^{5/2}}{63 a (c-a c x)^{3/2}}+\frac{2 c^4 \left (1-a^2 x^2\right )^{5/2}}{9 a \sqrt{c-a c x}} \]

[Out]

(64*c^6*(1 - a^2*x^2)^(5/2))/(315*a*(c - a*c*x)^(5/2)) + (16*c^5*(1 - a^2*x^2)^(5/2))/(63*a*(c - a*c*x)^(3/2))
+ (2*c^4*(1 - a^2*x^2)^(5/2))/(9*a*Sqrt[c - a*c*x])

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Rubi [A]  time = 0.087507, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6127, 657, 649} \[ \frac{64 c^6 \left (1-a^2 x^2\right )^{5/2}}{315 a (c-a c x)^{5/2}}+\frac{16 c^5 \left (1-a^2 x^2\right )^{5/2}}{63 a (c-a c x)^{3/2}}+\frac{2 c^4 \left (1-a^2 x^2\right )^{5/2}}{9 a \sqrt{c-a c x}} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*(c - a*c*x)^(7/2),x]

[Out]

(64*c^6*(1 - a^2*x^2)^(5/2))/(315*a*(c - a*c*x)^(5/2)) + (16*c^5*(1 - a^2*x^2)^(5/2))/(63*a*(c - a*c*x)^(3/2))
+ (2*c^4*(1 - a^2*x^2)^(5/2))/(9*a*Sqrt[c - a*c*x])

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)^{7/2} \, dx &=c^3 \int \sqrt{c-a c x} \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{2 c^4 \left (1-a^2 x^2\right )^{5/2}}{9 a \sqrt{c-a c x}}+\frac{1}{9} \left (8 c^4\right ) \int \frac{\left (1-a^2 x^2\right )^{3/2}}{\sqrt{c-a c x}} \, dx\\ &=\frac{16 c^5 \left (1-a^2 x^2\right )^{5/2}}{63 a (c-a c x)^{3/2}}+\frac{2 c^4 \left (1-a^2 x^2\right )^{5/2}}{9 a \sqrt{c-a c x}}+\frac{1}{63} \left (32 c^5\right ) \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{3/2}} \, dx\\ &=\frac{64 c^6 \left (1-a^2 x^2\right )^{5/2}}{315 a (c-a c x)^{5/2}}+\frac{16 c^5 \left (1-a^2 x^2\right )^{5/2}}{63 a (c-a c x)^{3/2}}+\frac{2 c^4 \left (1-a^2 x^2\right )^{5/2}}{9 a \sqrt{c-a c x}}\\ \end{align*}

Mathematica [A]  time = 0.0358382, size = 54, normalized size = 0.51 \[ \frac{2 c^3 (a x+1)^{5/2} \left (35 a^2 x^2-110 a x+107\right ) \sqrt{c-a c x}}{315 a \sqrt{1-a x}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])*(c - a*c*x)^(7/2),x]

[Out]

(2*c^3*(1 + a*x)^(5/2)*Sqrt[c - a*c*x]*(107 - 110*a*x + 35*a^2*x^2))/(315*a*Sqrt[1 - a*x])

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Maple [A]  time = 0.032, size = 55, normalized size = 0.5 \begin{align*}{\frac{2\, \left ( 35\,{a}^{2}{x}^{2}-110\,ax+107 \right ) \left ( ax+1 \right ) ^{4}}{315\, \left ( ax-1 \right ) ^{2}a} \left ( -acx+c \right ) ^{{\frac{7}{2}}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(a*x+1)^4*(35*a^2*x^2-110*a*x+107)*(-a*c*x+c)^(7/2)/a/(a*x-1)^2/(-a^2*x^2+1)^(3/2)

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Maxima [B]  time = 1.13235, size = 284, normalized size = 2.68 \begin{align*} \frac{2 \,{\left (5 \, a^{5} c^{\frac{7}{2}} x^{5} - 20 \, a^{4} c^{\frac{7}{2}} x^{4} + 41 \, a^{3} c^{\frac{7}{2}} x^{3} - 82 \, a^{2} c^{\frac{7}{2}} x^{2} + 328 \, a c^{\frac{7}{2}} x + 656 \, c^{\frac{7}{2}}\right )}}{45 \, \sqrt{a x + 1} a} + \frac{2 \,{\left (15 \, a^{4} c^{\frac{7}{2}} x^{4} - 66 \, a^{3} c^{\frac{7}{2}} x^{3} + 167 \, a^{2} c^{\frac{7}{2}} x^{2} - 668 \, a c^{\frac{7}{2}} x - 1336 \, c^{\frac{7}{2}}\right )}}{35 \, \sqrt{a x + 1} a} + \frac{2 \,{\left (3 \, a^{3} c^{\frac{7}{2}} x^{3} - 16 \, a^{2} c^{\frac{7}{2}} x^{2} + 79 \, a c^{\frac{7}{2}} x + 158 \, c^{\frac{7}{2}}\right )}}{5 \, \sqrt{a x + 1} a} + \frac{2 \,{\left (a^{2} c^{\frac{7}{2}} x^{2} - 10 \, a c^{\frac{7}{2}} x - 23 \, c^{\frac{7}{2}}\right )}}{3 \, \sqrt{a x + 1} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-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 + 41*a^3*c^(7/2)*x^3 - 82*a^2*c^(7/2)*x^2 + 328*a*c^(7/2)*x + 656
*c^(7/2))/(sqrt(a*x + 1)*a) + 2/35*(15*a^4*c^(7/2)*x^4 - 66*a^3*c^(7/2)*x^3 + 167*a^2*c^(7/2)*x^2 - 668*a*c^(7
/2)*x - 1336*c^(7/2))/(sqrt(a*x + 1)*a) + 2/5*(3*a^3*c^(7/2)*x^3 - 16*a^2*c^(7/2)*x^2 + 79*a*c^(7/2)*x + 158*c
^(7/2))/(sqrt(a*x + 1)*a) + 2/3*(a^2*c^(7/2)*x^2 - 10*a*c^(7/2)*x - 23*c^(7/2))/(sqrt(a*x + 1)*a)

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Fricas [A]  time = 2.22248, size = 177, normalized size = 1.67 \begin{align*} -\frac{2 \,{\left (35 \, a^{4} c^{3} x^{4} - 40 \, a^{3} c^{3} x^{3} - 78 \, a^{2} c^{3} x^{2} + 104 \, a c^{3} x + 107 \, 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)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(7/2),x, algorithm="fricas")

[Out]

-2/315*(35*a^4*c^3*x^4 - 40*a^3*c^3*x^3 - 78*a^2*c^3*x^2 + 104*a*c^3*x + 107*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)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**(7/2),x)

[Out]

Timed out

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Giac [A]  time = 1.27557, size = 82, normalized size = 0.77 \begin{align*} -\frac{2 \,{\left (128 \, \sqrt{2} c^{\frac{5}{2}} - \frac{35 \,{\left (a c x + c\right )}^{\frac{9}{2}} - 180 \,{\left (a c x + c\right )}^{\frac{7}{2}} c + 252 \,{\left (a c x + c\right )}^{\frac{5}{2}} c^{2}}{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)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(7/2),x, algorithm="giac")

[Out]

-2/315*(128*sqrt(2)*c^(5/2) - (35*(a*c*x + c)^(9/2) - 180*(a*c*x + c)^(7/2)*c + 252*(a*c*x + c)^(5/2)*c^2)/c^2
)*c^2/(a*abs(c))