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

Optimal. Leaf size=197 \[ -\frac{568 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{315 a^3 x^2 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{2 x \left (a-\frac{1}{x}\right )^3 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{9 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{48 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{35 a^2 x \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{8 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{21 a \left (1-\frac{1}{a x}\right )^{7/2}} \]

[Out]

(-8*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(7/2))/(21*a*(1 - 1/(a*x))^(7/2)) - (568*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(
7/2))/(315*a^3*(1 - 1/(a*x))^(7/2)*x^2) + (48*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(7/2))/(35*a^2*(1 - 1/(a*x))^(7/
2)*x) + (2*(a - x^(-1))^3*(1 + 1/(a*x))^(3/2)*x*(c - a*c*x)^(7/2))/(9*a^3*(1 - 1/(a*x))^(7/2))

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Rubi [A]  time = 0.189735, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6176, 6181, 94, 89, 78, 37} \[ -\frac{568 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{315 a^3 x^2 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{2 x \left (a-\frac{1}{x}\right )^3 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{9 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{48 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{35 a^2 x \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{8 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{21 a \left (1-\frac{1}{a x}\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(7/2))/(21*a*(1 - 1/(a*x))^(7/2)) - (568*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(
7/2))/(315*a^3*(1 - 1/(a*x))^(7/2)*x^2) + (48*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(7/2))/(35*a^2*(1 - 1/(a*x))^(7/
2)*x) + (2*(a - x^(-1))^3*(1 + 1/(a*x))^(3/2)*x*(c - a*c*x)^(7/2))/(9*a^3*(1 - 1/(a*x))^(7/2))

Rule 6176

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

Rule 6181

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

Rule 94

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

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx &=\frac{(c-a c x)^{7/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{7/2} x^{7/2} \, dx}{\left (1-\frac{1}{a x}\right )^{7/2} x^{7/2}}\\ &=-\frac{\left (\left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^3 \sqrt{1+\frac{x}{a}}}{x^{11/2}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{7/2}}\\ &=\frac{2 \left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} x (c-a c x)^{7/2}}{9 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (4 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2 \sqrt{1+\frac{x}{a}}}{x^{9/2}} \, dx,x,\frac{1}{x}\right )}{3 a \left (1-\frac{1}{a x}\right )^{7/2}}\\ &=-\frac{8 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{21 a \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{2 \left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} x (c-a c x)^{7/2}}{9 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (8 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{9}{a}+\frac{7 x}{2 a^2}\right ) \sqrt{1+\frac{x}{a}}}{x^{7/2}} \, dx,x,\frac{1}{x}\right )}{21 a \left (1-\frac{1}{a x}\right )^{7/2}}\\ &=-\frac{8 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{21 a \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{48 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{7/2} x}+\frac{2 \left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} x (c-a c x)^{7/2}}{9 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (284 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^{5/2}} \, dx,x,\frac{1}{x}\right )}{105 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}\\ &=-\frac{8 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{21 a \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{568 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{315 a^3 \left (1-\frac{1}{a x}\right )^{7/2} x^2}+\frac{48 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{7/2} x}+\frac{2 \left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} x (c-a c x)^{7/2}}{9 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0420794, size = 75, normalized size = 0.38 \[ -\frac{2 c^3 \sqrt{\frac{1}{a x}+1} (a x+1) \left (35 a^3 x^3-165 a^2 x^2+321 a x-319\right ) \sqrt{c-a c x}}{315 a \sqrt{1-\frac{1}{a x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.048, size = 64, normalized size = 0.3 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 35\,{x}^{3}{a}^{3}-165\,{a}^{2}{x}^{2}+321\,ax-319 \right ) }{315\, \left ( ax-1 \right ) ^{3}a} \left ( -acx+c \right ) ^{{\frac{7}{2}}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(a*x+1)*(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*x-1)/(a*x+1))^(1/2)

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Maxima [A]  time = 1.10513, size = 112, normalized size = 0.57 \begin{align*} -\frac{2 \,{\left (35 \, a^{4} \sqrt{-c} c^{3} x^{4} - 130 \, a^{3} \sqrt{-c} c^{3} x^{3} + 156 \, a^{2} \sqrt{-c} c^{3} x^{2} + 2 \, a \sqrt{-c} c^{3} x - 319 \, \sqrt{-c} c^{3}\right )} \sqrt{a x + 1}}{315 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(35*a^4*sqrt(-c)*c^3*x^4 - 130*a^3*sqrt(-c)*c^3*x^3 + 156*a^2*sqrt(-c)*c^3*x^2 + 2*a*sqrt(-c)*c^3*x - 3
19*sqrt(-c)*c^3)*sqrt(a*x + 1)/a

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Fricas [A]  time = 1.56228, size = 211, normalized size = 1.07 \begin{align*} -\frac{2 \,{\left (35 \, a^{5} c^{3} x^{5} - 95 \, a^{4} c^{3} x^{4} + 26 \, a^{3} c^{3} x^{3} + 158 \, a^{2} c^{3} x^{2} - 317 \, a c^{3} x - 319 \, c^{3}\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{315 \,{\left (a^{2} x - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(35*a^5*c^3*x^5 - 95*a^4*c^3*x^4 + 26*a^3*c^3*x^3 + 158*a^2*c^3*x^2 - 317*a*c^3*x - 319*c^3)*sqrt(-a*c*
x + c)*sqrt((a*x - 1)/(a*x + 1))/(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(1/((a*x-1)/(a*x+1))**(1/2)*(-a*c*x+c)**(7/2),x)

[Out]

Timed out

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Giac [A]  time = 1.25713, size = 167, normalized size = 0.85 \begin{align*} \frac{2 \,{\left (\frac{256 \, \sqrt{2} \sqrt{-c} c^{3}}{\mathrm{sgn}\left (c\right )} - \frac{35 \,{\left (a c x + c\right )}^{4} \sqrt{-a c x - c} - 270 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} c + 756 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c^{2} + 840 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{3}}{c \mathrm{sgn}\left (-a c x - c\right )}\right )}}{315 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(256*sqrt(2)*sqrt(-c)*c^3/sgn(c) - (35*(a*c*x + c)^4*sqrt(-a*c*x - c) - 270*(a*c*x + c)^3*sqrt(-a*c*x -
c)*c + 756*(a*c*x + c)^2*sqrt(-a*c*x - c)*c^2 + 840*(-a*c*x - c)^(3/2)*c^3)/(c*sgn(-a*c*x - c)))/a