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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.49, antiderivative size = 407, normalized size of antiderivative = 1.00, 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 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]

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 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 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 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 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 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]

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 = 0.22, 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]))

________________________________________________________________________________________

fricas [A]  time = 0.56, size = 175, normalized size = 0.43 \[ \frac {\sqrt {-a^{2} c x^{2} + c} {\left ({\left (a^{3} n^{3} - 7 \, a^{3} n\right )} x^{3} + 6 \, a n x - 3 \, {\left (a^{2} n^{2} - 3 \, a^{2}\right )} x^{2} - 6\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{4} c^{3} n^{4} - 10 \, a^{4} c^{3} n^{2} + 9 \, a^{4} c^{3} + {\left (a^{8} c^{3} n^{4} - 10 \, a^{8} c^{3} n^{2} + 9 \, a^{8} c^{3}\right )} x^{4} - 2 \, {\left (a^{6} c^{3} n^{4} - 10 \, a^{6} c^{3} n^{2} + 9 \, a^{6} c^{3}\right )} x^{2}} \]

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)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 0.03, size = 93, normalized size = 0.23 \[ -\frac {\left (a x -1\right ) \left (a x +1\right ) \left (a^{3} n^{3} x^{3}-7 x^{3} a^{3} n -3 a^{2} n^{2} x^{2}+9 a^{2} x^{2}+6 n a x -6\right ) {\mathrm e}^{n \arctanh \left (a x \right )}}{a^{4} \left (n^{4}-10 n^{2}+9\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}} \]

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)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]

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)

________________________________________________________________________________________

mupad [B]  time = 1.28, size = 162, normalized size = 0.40 \[ -\frac {{\left (a\,x+1\right )}^{n/2}\,\left (\frac {6}{a^6\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {6\,n\,x}{a^5\,c^2\,\left (n^4-10\,n^2+9\right )}+\frac {x^2\,\left (3\,n^2-9\right )}{a^4\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {n\,x^3\,\left (n^2-7\right )}{a^3\,c^2\,\left (n^4-10\,n^2+9\right )}\right )}{{\left (1-a\,x\right )}^{n/2}\,\left (\frac {\sqrt {c-a^2\,c\,x^2}}{a^2}-x^2\,\sqrt {c-a^2\,c\,x^2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} e^{n \operatorname {atanh}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]

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]

Integral(x**3*exp(n*atanh(a*x))/(-c*(a*x - 1)*(a*x + 1))**(5/2), x)

________________________________________________________________________________________