3.1348 \(\int \frac{e^{n \tanh ^{-1}(a x)} x^3}{(c-a^2 c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=407 \[ -\frac{3 (2-n) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{a^4 c^2 \left (9-n^2\right ) \sqrt{c-a^2 c x^2}}+\frac{3 \left (-n^2+2 n+1\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{a^4 c^2 (3-n) (n+1) (n+3) \sqrt{c-a^2 c x^2}}-\frac{3 \left (-n^2+2 n+1\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1-n}{2}}}{a^4 c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}}+\frac{x^3 \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-3)}}{a c^2 (n+3) \sqrt{c-a^2 c x^2}}-\frac{3 x \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{a^3 c^2 (n+3) \sqrt{c-a^2 c x^2}} \]

[Out]

(x^3*(1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(a*c^2*(3 + n)*Sqrt[c - a^2*c*x^2]) - (3
*(2 - n)*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(a^4*c^2*(9 - n^2)*Sqrt[c - a^2*c*x^
2]) - (3*x*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(a^3*c^2*(3 + n)*Sqrt[c - a^2*c*x^
2]) + (3*(1 + 2*n - n^2)*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[1 - a^2*x^2])/(a^4*c^2*(3 - n)*(1
+ n)*(3 + n)*Sqrt[c - a^2*c*x^2]) - (3*(1 + 2*n - n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[1 - a
^2*x^2])/(a^4*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])

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Rubi [A]  time = 0.486462, antiderivative size = 407, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6153, 6150, 94, 90, 79, 45, 37} \[ -\frac{3 (2-n) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{a^4 c^2 \left (9-n^2\right ) \sqrt{c-a^2 c x^2}}+\frac{3 \left (-n^2+2 n+1\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{a^4 c^2 (3-n) (n+1) (n+3) \sqrt{c-a^2 c x^2}}-\frac{3 \left (-n^2+2 n+1\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1-n}{2}}}{a^4 c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}}+\frac{x^3 \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-3)}}{a c^2 (n+3) \sqrt{c-a^2 c x^2}}-\frac{3 x \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{a^3 c^2 (n+3) \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(E^(n*ArcTanh[a*x])*x^3)/(c - a^2*c*x^2)^(5/2),x]

[Out]

(x^3*(1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(a*c^2*(3 + n)*Sqrt[c - a^2*c*x^2]) - (3
*(2 - n)*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(a^4*c^2*(9 - n^2)*Sqrt[c - a^2*c*x^
2]) - (3*x*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(a^3*c^2*(3 + n)*Sqrt[c - a^2*c*x^
2]) + (3*(1 + 2*n - n^2)*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[1 - a^2*x^2])/(a^4*c^2*(3 - n)*(1
+ n)*(3 + n)*Sqrt[c - a^2*c*x^2]) - (3*(1 + 2*n - n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[1 - a
^2*x^2])/(a^4*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])

Rule 6153

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

Rule 6150

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

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 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 79

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)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c,
d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimplerQ[p, 1]

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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 \frac{e^{n \tanh ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{n \tanh ^{-1}(a x)} x^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int x^3 (1-a x)^{-\frac{5}{2}-\frac{n}{2}} (1+a x)^{-\frac{5}{2}+\frac{n}{2}} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{x^3 (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{a c^2 (3+n) \sqrt{c-a^2 c x^2}}-\frac{\left (3 \sqrt{1-a^2 x^2}\right ) \int x^2 (1-a x)^{-\frac{3}{2}-\frac{n}{2}} (1+a x)^{-\frac{5}{2}+\frac{n}{2}} \, dx}{a c^2 (3+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{x^3 (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{a c^2 (3+n) \sqrt{c-a^2 c x^2}}-\frac{3 x (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{a^3 c^2 (3+n) \sqrt{c-a^2 c x^2}}-\frac{\left (3 \sqrt{1-a^2 x^2}\right ) \int (1-a x)^{-\frac{3}{2}-\frac{n}{2}} (1+a x)^{-\frac{5}{2}+\frac{n}{2}} (-1+a (1-n) x) \, dx}{a^3 c^2 (3+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{x^3 (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{a c^2 (3+n) \sqrt{c-a^2 c x^2}}-\frac{3 (2-n) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{a^4 c^2 \left (9-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{3 x (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{a^3 c^2 (3+n) \sqrt{c-a^2 c x^2}}+\frac{\left (3 \left (1+2 n-n^2\right ) \sqrt{1-a^2 x^2}\right ) \int (1-a x)^{-\frac{3}{2}-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \, dx}{a^3 c^2 (3-n) (3+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{x^3 (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{a c^2 (3+n) \sqrt{c-a^2 c x^2}}-\frac{3 (2-n) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{a^4 c^2 \left (9-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{3 x (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{a^3 c^2 (3+n) \sqrt{c-a^2 c x^2}}+\frac{3 \left (1+2 n-n^2\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{a^4 c^2 (3-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}+\frac{\left (3 \left (1+2 n-n^2\right ) \sqrt{1-a^2 x^2}\right ) \int (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \, dx}{a^3 c^2 (3-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{x^3 (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{a c^2 (3+n) \sqrt{c-a^2 c x^2}}-\frac{3 (2-n) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{a^4 c^2 \left (9-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{3 x (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{a^3 c^2 (3+n) \sqrt{c-a^2 c x^2}}+\frac{3 \left (1+2 n-n^2\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{a^4 c^2 (3-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}-\frac{3 \left (1+2 n-n^2\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{a^4 c^2 \left (9-10 n^2+n^4\right ) \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.197095, size = 112, normalized size = 0.28 \[ -\frac{\sqrt{1-a^2 x^2} (1-a x)^{\frac{1}{2} (-n-3)} (a x+1)^{\frac{n-3}{2}} \left (-a^3 n \left (n^2-7\right ) x^3+3 a^2 \left (n^2-3\right ) x^2-6 a n x+6\right )}{a^4 c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(n*ArcTanh[a*x])*x^3)/(c - a^2*c*x^2)^(5/2),x]

[Out]

-(((1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2]*(6 - 6*a*n*x + 3*a^2*(-3 + n^2)*x^2 - a^3*n
*(-7 + n^2)*x^3))/(a^4*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2]))

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Maple [A]  time = 0.028, size = 93, normalized size = 0.2 \begin{align*} -{\frac{ \left ({a}^{3}{n}^{3}{x}^{3}-7\,n{x}^{3}{a}^{3}-3\,{a}^{2}{n}^{2}{x}^{2}+9\,{a}^{2}{x}^{2}+6\,nax-6 \right ) \left ( ax-1 \right ) \left ( ax+1 \right ){{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{{a}^{4} \left ({n}^{4}-10\,{n}^{2}+9 \right ) } \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))*x^3/(-a^2*c*x^2+c)^(5/2),x)

[Out]

-(a*x-1)*(a*x+1)*(a^3*n^3*x^3-7*a^3*n*x^3-3*a^2*n^2*x^2+9*a^2*x^2+6*a*n*x-6)*exp(n*arctanh(a*x))/a^4/(n^4-10*n
^2+9)/(-a^2*c*x^2+c)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*x^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(x^3*((a*x + 1)/(a*x - 1))^(1/2*n)/(-a^2*c*x^2 + c)^(5/2), x)

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Fricas [A]  time = 2.41749, size = 352, normalized size = 0.86 \begin{align*} \frac{\sqrt{-a^{2} c x^{2} + c}{\left ({\left (a^{3} n^{3} - 7 \, a^{3} n\right )} x^{3} + 6 \, a n x - 3 \,{\left (a^{2} n^{2} - 3 \, a^{2}\right )} x^{2} - 6\right )} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{4} c^{3} n^{4} - 10 \, a^{4} c^{3} n^{2} + 9 \, a^{4} c^{3} +{\left (a^{8} c^{3} n^{4} - 10 \, a^{8} c^{3} n^{2} + 9 \, a^{8} c^{3}\right )} x^{4} - 2 \,{\left (a^{6} c^{3} n^{4} - 10 \, a^{6} c^{3} n^{2} + 9 \, a^{6} c^{3}\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*x^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

sqrt(-a^2*c*x^2 + c)*((a^3*n^3 - 7*a^3*n)*x^3 + 6*a*n*x - 3*(a^2*n^2 - 3*a^2)*x^2 - 6)*((a*x + 1)/(a*x - 1))^(
1/2*n)/(a^4*c^3*n^4 - 10*a^4*c^3*n^2 + 9*a^4*c^3 + (a^8*c^3*n^4 - 10*a^8*c^3*n^2 + 9*a^8*c^3)*x^4 - 2*(a^6*c^3
*n^4 - 10*a^6*c^3*n^2 + 9*a^6*c^3)*x^2)

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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(exp(n*atanh(a*x))*x**3/(-a**2*c*x**2+c)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*x^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

integrate(x^3*((a*x + 1)/(a*x - 1))^(1/2*n)/(-a^2*c*x^2 + c)^(5/2), x)