3.1100 \(\int e^{2 \tanh ^{-1}(a x)} x (c-a^2 c x^2)^{5/2} \, dx\)

Optimal. Leaf size=137 \[ \frac{c^2 x \sqrt{c-a^2 c x^2}}{8 a}+\frac{c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a^2}+\frac{c x \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac{(35 a x+27) \left (c-a^2 c x^2\right )^{5/2}}{105 a^2} \]

[Out]

(c^2*x*Sqrt[c - a^2*c*x^2])/(8*a) + (c*x*(c - a^2*c*x^2)^(3/2))/(12*a) - (x^2*(c - a^2*c*x^2)^(5/2))/7 - ((27
+ 35*a*x)*(c - a^2*c*x^2)^(5/2))/(105*a^2) + (c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a^2)

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Rubi [A]  time = 0.21575, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {6151, 1809, 780, 195, 217, 203} \[ \frac{c^2 x \sqrt{c-a^2 c x^2}}{8 a}+\frac{c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a^2}+\frac{c x \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac{(35 a x+27) \left (c-a^2 c x^2\right )^{5/2}}{105 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*x*Sqrt[c - a^2*c*x^2])/(8*a) + (c*x*(c - a^2*c*x^2)^(3/2))/(12*a) - (x^2*(c - a^2*c*x^2)^(5/2))/7 - ((27
+ 35*a*x)*(c - a^2*c*x^2)^(5/2))/(105*a^2) + (c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a^2)

Rule 6151

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

Rule 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[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 \tanh ^{-1}(a x)} x \left (c-a^2 c x^2\right )^{5/2} \, dx &=c \int x (1+a x)^2 \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\frac{1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac{\int x \left (-9 a^2 c-14 a^3 c x\right ) \left (c-a^2 c x^2\right )^{3/2} \, dx}{7 a^2}\\ &=-\frac{1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac{(27+35 a x) \left (c-a^2 c x^2\right )^{5/2}}{105 a^2}+\frac{c \int \left (c-a^2 c x^2\right )^{3/2} \, dx}{3 a}\\ &=\frac{c x \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac{(27+35 a x) \left (c-a^2 c x^2\right )^{5/2}}{105 a^2}+\frac{c^2 \int \sqrt{c-a^2 c x^2} \, dx}{4 a}\\ &=\frac{c^2 x \sqrt{c-a^2 c x^2}}{8 a}+\frac{c x \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac{(27+35 a x) \left (c-a^2 c x^2\right )^{5/2}}{105 a^2}+\frac{c^3 \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx}{8 a}\\ &=\frac{c^2 x \sqrt{c-a^2 c x^2}}{8 a}+\frac{c x \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac{(27+35 a x) \left (c-a^2 c x^2\right )^{5/2}}{105 a^2}+\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )}{8 a}\\ &=\frac{c^2 x \sqrt{c-a^2 c x^2}}{8 a}+\frac{c x \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac{(27+35 a x) \left (c-a^2 c x^2\right )^{5/2}}{105 a^2}+\frac{c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a^2}\\ \end{align*}

Mathematica [A]  time = 0.150981, size = 115, normalized size = 0.84 \[ -\frac{c^2 \left (\left (120 a^6 x^6+280 a^5 x^5-24 a^4 x^4-490 a^3 x^3-312 a^2 x^2+105 a x+216\right ) \sqrt{c-a^2 c x^2}+105 \sqrt{c} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )\right )}{840 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^2*(Sqrt[c - a^2*c*x^2]*(216 + 105*a*x - 312*a^2*x^2 - 490*a^3*x^3 - 24*a^4*x^4 + 280*a^5*x^5 + 120*a^6*x^6
) + 105*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))]))/(840*a^2)

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Maple [B]  time = 0.039, size = 284, normalized size = 2.1 \begin{align*}{\frac{1}{7\,{a}^{2}c} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{x}{3\,a} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{5\,cx}{12\,a} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,x{c}^{2}}{8\,a}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{5\,{c}^{3}}{8\,a}\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}^{2}} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{cx}{2\,a} \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\,a}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}+{\frac{3\,{c}^{3}}{4\,a}\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)^2/(-a^2*x^2+1)*x*(-a^2*c*x^2+c)^(5/2),x)

[Out]

1/7*(-a^2*c*x^2+c)^(7/2)/a^2/c-1/3*x/a*(-a^2*c*x^2+c)^(5/2)-5/12*c*x*(-a^2*c*x^2+c)^(3/2)/a-5/8*c^2*x*(-a^2*c*
x^2+c)^(1/2)/a-5/8/a*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^2*(-c*a^2*(x-1/a)^2-
2*a*c*(x-1/a))^(5/2)+1/2/a*c*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(3/2)*x+3/4/a*c^2*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a
))^(1/2)*x+3/4/a*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))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.85389, size = 609, normalized size = 4.45 \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 (120 \, a^{6} c^{2} x^{6} + 280 \, a^{5} c^{2} x^{5} - 24 \, a^{4} c^{2} x^{4} - 490 \, a^{3} c^{2} x^{3} - 312 \, a^{2} c^{2} x^{2} + 105 \, a c^{2} x + 216 \, c^{2}\right )} \sqrt{-a^{2} c x^{2} + c}}{1680 \, a^{2}}, -\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 (120 \, a^{6} c^{2} x^{6} + 280 \, a^{5} c^{2} x^{5} - 24 \, a^{4} c^{2} x^{4} - 490 \, a^{3} c^{2} x^{3} - 312 \, a^{2} c^{2} x^{2} + 105 \, a c^{2} x + 216 \, c^{2}\right )} \sqrt{-a^{2} c x^{2} + c}}{840 \, a^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/1680*(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*(120*a^6*c^2*x^6 + 28
0*a^5*c^2*x^5 - 24*a^4*c^2*x^4 - 490*a^3*c^2*x^3 - 312*a^2*c^2*x^2 + 105*a*c^2*x + 216*c^2)*sqrt(-a^2*c*x^2 +
c))/a^2, -1/840*(105*c^(5/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + (120*a^6*c^2*x^6 + 280
*a^5*c^2*x^5 - 24*a^4*c^2*x^4 - 490*a^3*c^2*x^3 - 312*a^2*c^2*x^2 + 105*a*c^2*x + 216*c^2)*sqrt(-a^2*c*x^2 + c
))/a^2]

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Sympy [A]  time = 18.4704, size = 586, normalized size = 4.28 \begin{align*} - a^{4} c^{2} \left (\begin{cases} \frac{x^{6} \sqrt{- a^{2} c x^{2} + c}}{7} - \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{35 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} c x^{2} + c}}{105 a^{4}} - \frac{8 \sqrt{- a^{2} c x^{2} + c}}{105 a^{6}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{6}}{6} & \text{otherwise} \end{cases}\right ) - 2 a^{3} 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 c^{2} \left (\begin{cases} \frac{i a^{2} \sqrt{c} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \sqrt{c} x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left (a x \right )}}{8 a^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{a^{2} \sqrt{c} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \sqrt{c} x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left (a x \right )}}{8 a^{3}} & \text{otherwise} \end{cases}\right ) + 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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**4*c**2*Piecewise((x**6*sqrt(-a**2*c*x**2 + c)/7 - x**4*sqrt(-a**2*c*x**2 + c)/(35*a**2) - 4*x**2*sqrt(-a**
2*c*x**2 + c)/(105*a**4) - 8*sqrt(-a**2*c*x**2 + c)/(105*a**6), Ne(a, 0)), (sqrt(c)*x**6/6, True)) - 2*a**3*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*c**2*Piecewise((I*a**2*sqrt(c)*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*sqrt(c)*x**3/(8
*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(8*a**3), Abs(a**2*x**
2) > 1), (-a**2*sqrt(c)*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*sqrt(c)*x**3/(8*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(8
*a**2*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(8*a**3), True)) + 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))

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Giac [A]  time = 1.18258, size = 177, normalized size = 1.29 \begin{align*} \frac{1}{840} \, \sqrt{-a^{2} c x^{2} + c}{\left ({\left (2 \,{\left (156 \, c^{2} +{\left (245 \, a c^{2} + 4 \,{\left (3 \, a^{2} c^{2} - 5 \,{\left (3 \, a^{4} c^{2} x + 7 \, a^{3} c^{2}\right )} x\right )} x\right )} x\right )} x - \frac{105 \, c^{2}}{a}\right )} x - \frac{216 \, c^{2}}{a^{2}}\right )} - \frac{c^{3} \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{8 \, a \sqrt{-c}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*sqrt(-a^2*c*x^2 + c)*((2*(156*c^2 + (245*a*c^2 + 4*(3*a^2*c^2 - 5*(3*a^4*c^2*x + 7*a^3*c^2)*x)*x)*x)*x -
105*c^2/a)*x - 216*c^2/a^2) - 1/8*c^3*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/(a*sqrt(-c)*abs(a))