### 3.625 $$\int e^{2 \coth ^{-1}(a x)} (c-a^2 c x^2)^{5/2} \, dx$$

Optimal. Leaf size=130 $-\frac{7}{16} c^2 x \sqrt{c-a^2 c x^2}-\frac{7 c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{16 a}-\frac{7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac{(a x+1) \left (c-a^2 c x^2\right )^{5/2}}{6 a}+\frac{7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}$

[Out]

(-7*c^2*x*Sqrt[c - a^2*c*x^2])/16 - (7*c*x*(c - a^2*c*x^2)^(3/2))/24 + (7*(c - a^2*c*x^2)^(5/2))/(30*a) + ((1
+ a*x)*(c - a^2*c*x^2)^(5/2))/(6*a) - (7*c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(16*a)

________________________________________________________________________________________

Rubi [A]  time = 0.137716, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.292, Rules used = {6167, 6141, 671, 641, 195, 217, 203} $-\frac{7}{16} c^2 x \sqrt{c-a^2 c x^2}-\frac{7 c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{16 a}-\frac{7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac{(a x+1) \left (c-a^2 c x^2\right )^{5/2}}{6 a}+\frac{7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-7*c^2*x*Sqrt[c - a^2*c*x^2])/16 - (7*c*x*(c - a^2*c*x^2)^(3/2))/24 + (7*(c - a^2*c*x^2)^(5/2))/(30*a) + ((1
+ a*x)*(c - a^2*c*x^2)^(5/2))/(6*a) - (7*c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(16*a)

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6141

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

Rule 671

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*(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, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

Rule 641

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=-\left (c \int (1+a x)^2 \left (c-a^2 c x^2\right )^{3/2} \, dx\right )\\ &=\frac{(1+a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}-\frac{1}{6} (7 c) \int (1+a x) \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=\frac{7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}-\frac{1}{6} (7 c) \int \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\frac{7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac{7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}-\frac{1}{8} \left (7 c^2\right ) \int \sqrt{c-a^2 c x^2} \, dx\\ &=-\frac{7}{16} c^2 x \sqrt{c-a^2 c x^2}-\frac{7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac{7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}-\frac{1}{16} \left (7 c^3\right ) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{7}{16} c^2 x \sqrt{c-a^2 c x^2}-\frac{7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac{7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}-\frac{1}{16} \left (7 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )\\ &=-\frac{7}{16} c^2 x \sqrt{c-a^2 c x^2}-\frac{7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac{7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}-\frac{7 c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{16 a}\\ \end{align*}

Mathematica [A]  time = 0.116158, size = 135, normalized size = 1.04 $\frac{c^2 \sqrt{c-a^2 c x^2} \left (\sqrt{a x+1} \left (-40 a^6 x^6-56 a^5 x^5+106 a^4 x^4+182 a^3 x^3-57 a^2 x^2-231 a x+96\right )+210 \sqrt{1-a x} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{240 a \sqrt{1-a x} \sqrt{1-a^2 x^2}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(96 - 231*a*x - 57*a^2*x^2 + 182*a^3*x^3 + 106*a^4*x^4 - 56*a^5*x^5 -
40*a^6*x^6) + 210*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(240*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])

________________________________________________________________________________________

Maple [B]  time = 0.049, size = 242, normalized size = 1.9 \begin{align*}{\frac{x}{6} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,cx}{24} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{5\,x{c}^{2}}{16}\sqrt{-{a}^{2}c{x}^{2}+c}}+{\frac{5\,{c}^{3}}{16}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}+{\frac{2}{5\,a} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{cx}{2} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{3\,x{c}^{2}}{4}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}-{\frac{3\,{c}^{3}}{4}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x*(-a^2*c*x^2+c)^(5/2)+5/24*c*x*(-a^2*c*x^2+c)^(3/2)+5/16*c^2*x*(-a^2*c*x^2+c)^(1/2)+5/16*c^3/(a^2*c)^(1/2
)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))+2/5/a*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(5/2)-1/2*c*(-c*a^2*(x-1
/a)^2-2*a*c*(x-1/a))^(3/2)*x-3/4*c^2*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(1/2)*x-3/4*c^3/(a^2*c)^(1/2)*arctan((a^
2*c)^(1/2)*x/(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(1/2))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.71476, size = 544, normalized size = 4.18 \begin{align*} \left [\frac{105 \, \sqrt{-c} c^{2} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) + 2 \,{\left (40 \, a^{5} c^{2} x^{5} + 96 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} - 192 \, a^{2} c^{2} x^{2} - 135 \, a c^{2} x + 96 \, c^{2}\right )} \sqrt{-a^{2} c x^{2} + c}}{480 \, a}, \frac{105 \, c^{\frac{5}{2}} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) +{\left (40 \, a^{5} c^{2} x^{5} + 96 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} - 192 \, a^{2} c^{2} x^{2} - 135 \, a c^{2} x + 96 \, c^{2}\right )} \sqrt{-a^{2} c x^{2} + c}}{240 \, a}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/480*(105*sqrt(-c)*c^2*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(40*a^5*c^2*x^5 + 96*a
^4*c^2*x^4 - 10*a^3*c^2*x^3 - 192*a^2*c^2*x^2 - 135*a*c^2*x + 96*c^2)*sqrt(-a^2*c*x^2 + c))/a, 1/240*(105*c^(5
/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + (40*a^5*c^2*x^5 + 96*a^4*c^2*x^4 - 10*a^3*c^2*x
^3 - 192*a^2*c^2*x^2 - 135*a*c^2*x + 96*c^2)*sqrt(-a^2*c*x^2 + c))/a]

________________________________________________________________________________________

Sympy [C]  time = 17.1193, size = 478, normalized size = 3.68 \begin{align*} a^{4} c^{2} \left (\begin{cases} \frac{i a^{2} \sqrt{c} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left (a x \right )}}{16 a^{5}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{a^{2} \sqrt{c} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left (a x \right )}}{16 a^{5}} & \text{otherwise} \end{cases}\right ) + 2 a^{3} c^{2} \left (\begin{cases} \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{5} - \frac{x^{2} \sqrt{- a^{2} c x^{2} + c}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} c x^{2} + c}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right ) - 2 a c^{2} \left (\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left (- a^{2} c x^{2} + c\right )^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right ) - c^{2} \left (\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left (a x \right )}}{2 a} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left (a x \right )}}{2 a} & \text{otherwise} \end{cases}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*c**2*Piecewise((I*a**2*sqrt(c)*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**5/(24*sqrt(a**2*x**2 - 1)) -
I*sqrt(c)*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*
x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*x**5/(24*sqrt(-a**
2*x**2 + 1)) + sqrt(c)*x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(16*a**4*sqrt(-a**2*x**2 + 1)) + sqrt(c
)*asin(a*x)/(16*a**5), True)) + 2*a**3*c**2*Piecewise((x**4*sqrt(-a**2*c*x**2 + c)/5 - x**2*sqrt(-a**2*c*x**2
+ c)/(15*a**2) - 2*sqrt(-a**2*c*x**2 + c)/(15*a**4), Ne(a, 0)), (sqrt(c)*x**4/4, True)) - 2*a*c**2*Piecewise((
0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2, 0)), (-(-a**2*c*x**2 + c)**(3/2)/(3*a**2*c), True)) - c**2*Piecewise((
I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(2*a)
, Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(-a**2*x**2 + 1)/2 + sqrt(c)*asin(a*x)/(2*a), True))

________________________________________________________________________________________

Giac [A]  time = 1.18673, size = 157, normalized size = 1.21 \begin{align*} \frac{7 \, c^{3} \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{16 \, \sqrt{-c}{\left | a \right |}} - \frac{1}{240} \, \sqrt{-a^{2} c x^{2} + c}{\left ({\left (135 \, c^{2} + 2 \,{\left (96 \, a c^{2} +{\left (5 \, a^{2} c^{2} - 4 \,{\left (5 \, a^{4} c^{2} x + 12 \, a^{3} c^{2}\right )} x\right )} x\right )} x\right )} x - \frac{96 \, c^{2}}{a}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/16*c^3*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/(sqrt(-c)*abs(a)) - 1/240*sqrt(-a^2*c*x^2 + c)*((135
*c^2 + 2*(96*a*c^2 + (5*a^2*c^2 - 4*(5*a^4*c^2*x + 12*a^3*c^2)*x)*x)*x)*x - 96*c^2/a)