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

Optimal. Leaf size=108 \[ -\frac {\sin ^{-1}(a x)}{a^6 c^3}+\frac {x^4 (a x+1)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {15 a x+8}{15 a^6 c^3 \sqrt {1-a^2 x^2}}-\frac {x^2 (5 a x+4)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}} \]

[Out]

1/5*x^4*(a*x+1)/a^2/c^3/(-a^2*x^2+1)^(5/2)-1/15*x^2*(5*a*x+4)/a^4/c^3/(-a^2*x^2+1)^(3/2)-arcsin(a*x)/a^6/c^3+1
/15*(15*a*x+8)/a^6/c^3/(-a^2*x^2+1)^(1/2)

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Rubi [A]  time = 0.13, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6148, 819, 778, 216} \[ \frac {x^4 (a x+1)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^2 (5 a x+4)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {15 a x+8}{15 a^6 c^3 \sqrt {1-a^2 x^2}}-\frac {\sin ^{-1}(a x)}{a^6 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*(1 + a*x))/(5*a^2*c^3*(1 - a^2*x^2)^(5/2)) - (x^2*(4 + 5*a*x))/(15*a^4*c^3*(1 - a^2*x^2)^(3/2)) + (8 + 15
*a*x)/(15*a^6*c^3*Sqrt[1 - a^2*x^2]) - ArcSin[a*x]/(a^6*c^3)

Rule 216

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

Rule 778

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*(e*f + d*g) -
(c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 6148

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

Rubi steps

\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^5}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {x^5 (1+a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac {x^4 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {\int \frac {x^3 (4+5 a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 a^2 c^3}\\ &=\frac {x^4 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^2 (4+5 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {x (8+15 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 a^4 c^3}\\ &=\frac {x^4 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^2 (4+5 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8+15 a x}{15 a^6 c^3 \sqrt {1-a^2 x^2}}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^5 c^3}\\ &=\frac {x^4 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^2 (4+5 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8+15 a x}{15 a^6 c^3 \sqrt {1-a^2 x^2}}-\frac {\sin ^{-1}(a x)}{a^6 c^3}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 100, normalized size = 0.93 \[ \frac {23 a^4 x^4-8 a^3 x^3-27 a^2 x^2-15 (a x-1)^2 (a x+1) \sqrt {1-a^2 x^2} \sin ^{-1}(a x)+7 a x+8}{15 a^6 c^3 (a x-1)^2 (a x+1) \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8 + 7*a*x - 27*a^2*x^2 - 8*a^3*x^3 + 23*a^4*x^4 - 15*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(1
5*a^6*c^3*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2])

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fricas [B]  time = 0.72, size = 206, normalized size = 1.91 \[ \frac {8 \, a^{5} x^{5} - 8 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 8 \, a x + 30 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (23 \, a^{4} x^{4} - 8 \, a^{3} x^{3} - 27 \, a^{2} x^{2} + 7 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} - 8}{15 \, {\left (a^{11} c^{3} x^{5} - a^{10} c^{3} x^{4} - 2 \, a^{9} c^{3} x^{3} + 2 \, a^{8} c^{3} x^{2} + a^{7} c^{3} x - a^{6} c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

1/15*(8*a^5*x^5 - 8*a^4*x^4 - 16*a^3*x^3 + 16*a^2*x^2 + 8*a*x + 30*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2
+ a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (23*a^4*x^4 - 8*a^3*x^3 - 27*a^2*x^2 + 7*a*x + 8)*sqrt(-a^
2*x^2 + 1) - 8)/(a^11*c^3*x^5 - a^10*c^3*x^4 - 2*a^9*c^3*x^3 + 2*a^8*c^3*x^2 + a^7*c^3*x - a^6*c^3)

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

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maple [B]  time = 0.05, size = 243, normalized size = 2.25 \[ -\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{3} a^{5} \sqrt {a^{2}}}-\frac {3 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{10 c^{3} a^{8} \left (x -\frac {1}{a}\right )^{2}}-\frac {91 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{80 c^{3} a^{7} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{20 c^{3} a^{9} \left (x -\frac {1}{a}\right )^{3}}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{24 c^{3} a^{8} \left (x +\frac {1}{a}\right )^{2}}-\frac {19 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{48 c^{3} a^{7} \left (x +\frac {1}{a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/c^3/a^5/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-3/10/c^3/a^8/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-
1/a))^(1/2)-91/80/c^3/a^7/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-1/20/c^3/a^9/(x-1/a)^3*(-a^2*(x-1/a)^2-2*
a*(x-1/a))^(1/2)+1/24/c^3/a^8/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)-19/48/c^3/a^7/(x+1/a)*(-a^2*(x+1/a)
^2+2*a*(x+1/a))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -a \int \frac {x^{6}}{{\left (a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}\,{d x} + \frac {10 \, a^{2} x^{2} + 15 \, {\left (a^{2} x^{2} - 1\right )}^{2} - 7}{15 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{6} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

-a*integrate(x^6/((a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)*sqrt(a*x + 1)*sqrt(-a*x + 1)), x) + 1/15
*(10*a^2*x^2 + 15*(a^2*x^2 - 1)^2 - 7)/((-a^2*x^2 + 1)^(5/2)*a^6*c^3)

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mupad [B]  time = 0.93, size = 312, normalized size = 2.89 \[ \frac {\sqrt {1-a^2\,x^2}}{24\,\left (a^8\,c^3\,x^2+2\,a^7\,c^3\,x+a^6\,c^3\right )}-\frac {3\,\sqrt {1-a^2\,x^2}}{10\,\left (a^8\,c^3\,x^2-2\,a^7\,c^3\,x+a^6\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (a^4\,c^3\,\sqrt {-a^2}+3\,a^6\,c^3\,x^2\,\sqrt {-a^2}-a^7\,c^3\,x^3\,\sqrt {-a^2}-3\,a^5\,c^3\,x\,\sqrt {-a^2}\right )}+\frac {19\,\sqrt {1-a^2\,x^2}}{48\,\left (a^4\,c^3\,\sqrt {-a^2}+a^5\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {91\,\sqrt {1-a^2\,x^2}}{80\,\left (a^4\,c^3\,\sqrt {-a^2}-a^5\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^5\,c^3\,\sqrt {-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1 - a^2*x^2)^(1/2)/(24*(a^6*c^3 + 2*a^7*c^3*x + a^8*c^3*x^2)) - (3*(1 - a^2*x^2)^(1/2))/(10*(a^6*c^3 - 2*a^7*
c^3*x + a^8*c^3*x^2)) - (1 - a^2*x^2)^(1/2)/(20*(-a^2)^(1/2)*(a^4*c^3*(-a^2)^(1/2) + 3*a^6*c^3*x^2*(-a^2)^(1/2
) - a^7*c^3*x^3*(-a^2)^(1/2) - 3*a^5*c^3*x*(-a^2)^(1/2))) + (19*(1 - a^2*x^2)^(1/2))/(48*(a^4*c^3*(-a^2)^(1/2)
+ a^5*c^3*x*(-a^2)^(1/2))*(-a^2)^(1/2)) - (91*(1 - a^2*x^2)^(1/2))/(80*(a^4*c^3*(-a^2)^(1/2) - a^5*c^3*x*(-a^
2)^(1/2))*(-a^2)^(1/2)) - asinh(x*(-a^2)^(1/2))/(a^5*c^3*(-a^2)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{5}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{6}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(x**5/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x*
*2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**6/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**
2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**3

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