3.1099 \(\int e^{2 \tanh ^{-1}(a x)} x^2 (c-a^2 c x^2)^{5/2} \, dx\)

Optimal. Leaf size=162 \[ \frac{11 c^2 x \sqrt{c-a^2 c x^2}}{128 a^2}+\frac{11 c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{128 a^3}-\frac{1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}-\frac{2 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a}+\frac{11 c x \left (c-a^2 c x^2\right )^{3/2}}{192 a^2}-\frac{(385 a x+192) \left (c-a^2 c x^2\right )^{5/2}}{1680 a^3} \]

[Out]

(11*c^2*x*Sqrt[c - a^2*c*x^2])/(128*a^2) + (11*c*x*(c - a^2*c*x^2)^(3/2))/(192*a^2) - (2*x^2*(c - a^2*c*x^2)^(
5/2))/(7*a) - (x^3*(c - a^2*c*x^2)^(5/2))/8 - ((192 + 385*a*x)*(c - a^2*c*x^2)^(5/2))/(1680*a^3) + (11*c^(5/2)
*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(128*a^3)

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Rubi [A]  time = 0.325294, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6151, 1809, 833, 780, 195, 217, 203} \[ \frac{11 c^2 x \sqrt{c-a^2 c x^2}}{128 a^2}+\frac{11 c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{128 a^3}-\frac{1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}-\frac{2 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a}+\frac{11 c x \left (c-a^2 c x^2\right )^{3/2}}{192 a^2}-\frac{(385 a x+192) \left (c-a^2 c x^2\right )^{5/2}}{1680 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(11*c^2*x*Sqrt[c - a^2*c*x^2])/(128*a^2) + (11*c*x*(c - a^2*c*x^2)^(3/2))/(192*a^2) - (2*x^2*(c - a^2*c*x^2)^(
5/2))/(7*a) - (x^3*(c - a^2*c*x^2)^(5/2))/8 - ((192 + 385*a*x)*(c - a^2*c*x^2)^(5/2))/(1680*a^3) + (11*c^(5/2)
*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(128*a^3)

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 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

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^2 \left (c-a^2 c x^2\right )^{5/2} \, dx &=c \int x^2 (1+a x)^2 \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\frac{1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}-\frac{\int x^2 \left (-11 a^2 c-16 a^3 c x\right ) \left (c-a^2 c x^2\right )^{3/2} \, dx}{8 a^2}\\ &=-\frac{2 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a}-\frac{1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}+\frac{\int x \left (32 a^3 c^2+77 a^4 c^2 x\right ) \left (c-a^2 c x^2\right )^{3/2} \, dx}{56 a^4 c}\\ &=-\frac{2 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a}-\frac{1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}-\frac{(192+385 a x) \left (c-a^2 c x^2\right )^{5/2}}{1680 a^3}+\frac{(11 c) \int \left (c-a^2 c x^2\right )^{3/2} \, dx}{48 a^2}\\ &=\frac{11 c x \left (c-a^2 c x^2\right )^{3/2}}{192 a^2}-\frac{2 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a}-\frac{1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}-\frac{(192+385 a x) \left (c-a^2 c x^2\right )^{5/2}}{1680 a^3}+\frac{\left (11 c^2\right ) \int \sqrt{c-a^2 c x^2} \, dx}{64 a^2}\\ &=\frac{11 c^2 x \sqrt{c-a^2 c x^2}}{128 a^2}+\frac{11 c x \left (c-a^2 c x^2\right )^{3/2}}{192 a^2}-\frac{2 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a}-\frac{1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}-\frac{(192+385 a x) \left (c-a^2 c x^2\right )^{5/2}}{1680 a^3}+\frac{\left (11 c^3\right ) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx}{128 a^2}\\ &=\frac{11 c^2 x \sqrt{c-a^2 c x^2}}{128 a^2}+\frac{11 c x \left (c-a^2 c x^2\right )^{3/2}}{192 a^2}-\frac{2 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a}-\frac{1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}-\frac{(192+385 a x) \left (c-a^2 c x^2\right )^{5/2}}{1680 a^3}+\frac{\left (11 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 )}{128 a^2}\\ &=\frac{11 c^2 x \sqrt{c-a^2 c x^2}}{128 a^2}+\frac{11 c x \left (c-a^2 c x^2\right )^{3/2}}{192 a^2}-\frac{2 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a}-\frac{1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}-\frac{(192+385 a x) \left (c-a^2 c x^2\right )^{5/2}}{1680 a^3}+\frac{11 c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{128 a^3}\\ \end{align*}

Mathematica [A]  time = 0.173982, size = 123, normalized size = 0.76 \[ -\frac{c^2 \left (\left (1680 a^7 x^7+3840 a^6 x^6-280 a^5 x^5-6144 a^4 x^4-3710 a^3 x^3+768 a^2 x^2+1155 a x+1536\right ) \sqrt{c-a^2 c x^2}+1155 \sqrt{c} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )\right )}{13440 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^2*(Sqrt[c - a^2*c*x^2]*(1536 + 1155*a*x + 768*a^2*x^2 - 3710*a^3*x^3 - 6144*a^4*x^4 - 280*a^5*x^5 + 3840*a
^6*x^6 + 1680*a^7*x^7) + 1155*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))]))/(13440*a^3)

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Maple [B]  time = 0.04, size = 306, normalized size = 1.9 \begin{align*}{\frac{x}{8\,{a}^{2}c} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{17\,x}{48\,{a}^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{85\,cx}{192\,{a}^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{85\,x{c}^{2}}{128\,{a}^{2}}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{85\,{c}^{3}}{128\,{a}^{2}}\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}{7\,{a}^{3}c} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{2}{5\,{a}^{3}} \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}^{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\,{a}^{2}}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}+{\frac{3\,{c}^{3}}{4\,{a}^{2}}\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^2*(-a^2*c*x^2+c)^(5/2),x)

[Out]

1/8*x*(-a^2*c*x^2+c)^(7/2)/a^2/c-17/48/a^2*x*(-a^2*c*x^2+c)^(5/2)-85/192*c*x*(-a^2*c*x^2+c)^(3/2)/a^2-85/128*c
^2*x*(-a^2*c*x^2+c)^(1/2)/a^2-85/128/a^2*c^3/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))+2/7/a^
3*(-a^2*c*x^2+c)^(7/2)/c-2/5/a^3*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(5/2)+1/2/a^2*c*(-c*a^2*(x-1/a)^2-2*a*c*(x-1
/a))^(3/2)*x+3/4/a^2*c^2*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(1/2)*x+3/4/a^2*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^2*(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.8533, size = 683, normalized size = 4.22 \begin{align*} \left [\frac{1155 \, \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 (1680 \, a^{7} c^{2} x^{7} + 3840 \, a^{6} c^{2} x^{6} - 280 \, a^{5} c^{2} x^{5} - 6144 \, a^{4} c^{2} x^{4} - 3710 \, a^{3} c^{2} x^{3} + 768 \, a^{2} c^{2} x^{2} + 1155 \, a c^{2} x + 1536 \, c^{2}\right )} \sqrt{-a^{2} c x^{2} + c}}{26880 \, a^{3}}, -\frac{1155 \, 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 (1680 \, a^{7} c^{2} x^{7} + 3840 \, a^{6} c^{2} x^{6} - 280 \, a^{5} c^{2} x^{5} - 6144 \, a^{4} c^{2} x^{4} - 3710 \, a^{3} c^{2} x^{3} + 768 \, a^{2} c^{2} x^{2} + 1155 \, a c^{2} x + 1536 \, c^{2}\right )} \sqrt{-a^{2} c x^{2} + c}}{13440 \, a^{3}}\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^2*(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

[1/26880*(1155*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*(1680*a^7*c^2*x^7 +
3840*a^6*c^2*x^6 - 280*a^5*c^2*x^5 - 6144*a^4*c^2*x^4 - 3710*a^3*c^2*x^3 + 768*a^2*c^2*x^2 + 1155*a*c^2*x + 1
536*c^2)*sqrt(-a^2*c*x^2 + c))/a^3, -1/13440*(1155*c^(5/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2
- c)) + (1680*a^7*c^2*x^7 + 3840*a^6*c^2*x^6 - 280*a^5*c^2*x^5 - 6144*a^4*c^2*x^4 - 3710*a^3*c^2*x^3 + 768*a^2
*c^2*x^2 + 1155*a*c^2*x + 1536*c^2)*sqrt(-a^2*c*x^2 + c))/a^3]

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Sympy [C]  time = 20.3507, size = 687, normalized size = 4.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**4*c**2*Piecewise((I*a**2*sqrt(c)*x**9/(8*sqrt(a**2*x**2 - 1)) - 7*I*sqrt(c)*x**7/(48*sqrt(a**2*x**2 - 1))
- I*sqrt(c)*x**5/(192*a**2*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**3/(384*a**4*sqrt(a**2*x**2 - 1)) + 5*I*sqrt(c
)*x/(128*a**6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*acosh(a*x)/(128*a**7), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x*
*9/(8*sqrt(-a**2*x**2 + 1)) + 7*sqrt(c)*x**7/(48*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**5/(192*a**2*sqrt(-a**2*x**
2 + 1)) + 5*sqrt(c)*x**3/(384*a**4*sqrt(-a**2*x**2 + 1)) - 5*sqrt(c)*x/(128*a**6*sqrt(-a**2*x**2 + 1)) + 5*sqr
t(c)*asin(a*x)/(128*a**7), True)) - 2*a**3*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*c**2*Piecewise((x**4*sqrt(-a**2*c*x**2 + c)/5 - x**2*sqrt(-a**2*c*x**2 + c)/(1
5*a**2) - 2*sqrt(-a**2*c*x**2 + c)/(15*a**4), Ne(a, 0)), (sqrt(c)*x**4/4, True)) + 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))

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Giac [A]  time = 1.20321, size = 193, normalized size = 1.19 \begin{align*} \frac{1}{13440} \, \sqrt{-a^{2} c x^{2} + c}{\left ({\left (2 \,{\left ({\left (1855 \, c^{2} + 4 \,{\left (768 \, a c^{2} + 5 \,{\left (7 \, a^{2} c^{2} - 6 \,{\left (7 \, a^{4} c^{2} x + 16 \, a^{3} c^{2}\right )} x\right )} x\right )} x\right )} x - \frac{384 \, c^{2}}{a}\right )} x - \frac{1155 \, c^{2}}{a^{2}}\right )} x - \frac{1536 \, c^{2}}{a^{3}}\right )} - \frac{11 \, c^{3} \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{128 \, a^{2} \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^2*(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

1/13440*sqrt(-a^2*c*x^2 + c)*((2*((1855*c^2 + 4*(768*a*c^2 + 5*(7*a^2*c^2 - 6*(7*a^4*c^2*x + 16*a^3*c^2)*x)*x)
*x)*x - 384*c^2/a)*x - 1155*c^2/a^2)*x - 1536*c^2/a^3) - 11/128*c^3*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2
+ c)))/(a^2*sqrt(-c)*abs(a))