3.867 \(\int \frac{e^{-3 \tanh ^{-1}(a+b x)}}{x^4} \, dx\)

Optimal. Leaf size=257 \[ -\frac{\left (2 a^2-51 a+52\right ) b^3 \sqrt{-a-b x+1}}{6 (1-a) (a+1)^4 \sqrt{a+b x+1}}+\frac{\left (6 a^2-18 a+11\right ) b^3 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{(1-a) (a+1)^4 \sqrt{1-a^2}}-\frac{(19-16 a) b^2 \sqrt{-a-b x+1}}{6 (1-a) (a+1)^3 x \sqrt{a+b x+1}}+\frac{7 b \sqrt{-a-b x+1}}{6 (a+1)^2 x^2 \sqrt{a+b x+1}}-\frac{(1-a) \sqrt{-a-b x+1}}{3 (a+1) x^3 \sqrt{a+b x+1}} \]

[Out]

-((52 - 51*a + 2*a^2)*b^3*Sqrt[1 - a - b*x])/(6*(1 - a)*(1 + a)^4*Sqrt[1 + a + b*x]) - ((1 - a)*Sqrt[1 - a - b
*x])/(3*(1 + a)*x^3*Sqrt[1 + a + b*x]) + (7*b*Sqrt[1 - a - b*x])/(6*(1 + a)^2*x^2*Sqrt[1 + a + b*x]) - ((19 -
16*a)*b^2*Sqrt[1 - a - b*x])/(6*(1 - a)*(1 + a)^3*x*Sqrt[1 + a + b*x]) + ((11 - 18*a + 6*a^2)*b^3*ArcTanh[(Sqr
t[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)*(1 + a)^4*Sqrt[1 - a^2])

________________________________________________________________________________________

Rubi [A]  time = 0.216336, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6163, 98, 151, 152, 12, 93, 208} \[ -\frac{\left (2 a^2-51 a+52\right ) b^3 \sqrt{-a-b x+1}}{6 (1-a) (a+1)^4 \sqrt{a+b x+1}}+\frac{\left (6 a^2-18 a+11\right ) b^3 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{(1-a) (a+1)^4 \sqrt{1-a^2}}-\frac{(19-16 a) b^2 \sqrt{-a-b x+1}}{6 (1-a) (a+1)^3 x \sqrt{a+b x+1}}+\frac{7 b \sqrt{-a-b x+1}}{6 (a+1)^2 x^2 \sqrt{a+b x+1}}-\frac{(1-a) \sqrt{-a-b x+1}}{3 (a+1) x^3 \sqrt{a+b x+1}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcTanh[a + b*x])*x^4),x]

[Out]

-((52 - 51*a + 2*a^2)*b^3*Sqrt[1 - a - b*x])/(6*(1 - a)*(1 + a)^4*Sqrt[1 + a + b*x]) - ((1 - a)*Sqrt[1 - a - b
*x])/(3*(1 + a)*x^3*Sqrt[1 + a + b*x]) + (7*b*Sqrt[1 - a - b*x])/(6*(1 + a)^2*x^2*Sqrt[1 + a + b*x]) - ((19 -
16*a)*b^2*Sqrt[1 - a - b*x])/(6*(1 - a)*(1 + a)^3*x*Sqrt[1 + a + b*x]) + ((11 - 18*a + 6*a^2)*b^3*ArcTanh[(Sqr
t[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)*(1 + a)^4*Sqrt[1 - a^2])

Rule 6163

Int[E^(ArcTanh[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[((d + e*x)^m*(1
+ a*c + b*c*x)^(n/2))/(1 - a*c - b*c*x)^(n/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]

Rule 98

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

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a+b x)}}{x^4} \, dx &=\int \frac{(1-a-b x)^{3/2}}{x^4 (1+a+b x)^{3/2}} \, dx\\ &=-\frac{(1-a) \sqrt{1-a-b x}}{3 (1+a) x^3 \sqrt{1+a+b x}}-\frac{\int \frac{7 (1-a) b-6 b^2 x}{x^3 \sqrt{1-a-b x} (1+a+b x)^{3/2}} \, dx}{3 (1+a)}\\ &=-\frac{(1-a) \sqrt{1-a-b x}}{3 (1+a) x^3 \sqrt{1+a+b x}}+\frac{7 b \sqrt{1-a-b x}}{6 (1+a)^2 x^2 \sqrt{1+a+b x}}+\frac{\int \frac{(19-16 a) (1-a) b^2-14 (1-a) b^3 x}{x^2 \sqrt{1-a-b x} (1+a+b x)^{3/2}} \, dx}{6 (1-a) (1+a)^2}\\ &=-\frac{(1-a) \sqrt{1-a-b x}}{3 (1+a) x^3 \sqrt{1+a+b x}}+\frac{7 b \sqrt{1-a-b x}}{6 (1+a)^2 x^2 \sqrt{1+a+b x}}-\frac{(19-16 a) b^2 \sqrt{1-a-b x}}{6 (1-a) (1+a)^3 x \sqrt{1+a+b x}}-\frac{\int \frac{3 (1-a) \left (11-18 a+6 a^2\right ) b^3-(19-16 a) (1-a) b^4 x}{x \sqrt{1-a-b x} (1+a+b x)^{3/2}} \, dx}{6 (1-a)^2 (1+a)^3}\\ &=-\frac{\left (52-51 a+2 a^2\right ) b^3 \sqrt{1-a-b x}}{6 (1-a) (1+a)^4 \sqrt{1+a+b x}}-\frac{(1-a) \sqrt{1-a-b x}}{3 (1+a) x^3 \sqrt{1+a+b x}}+\frac{7 b \sqrt{1-a-b x}}{6 (1+a)^2 x^2 \sqrt{1+a+b x}}-\frac{(19-16 a) b^2 \sqrt{1-a-b x}}{6 (1-a) (1+a)^3 x \sqrt{1+a+b x}}-\frac{\int \frac{3 (1-a) \left (11-18 a+6 a^2\right ) b^4}{x \sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx}{6 (1-a)^2 (1+a)^4 b}\\ &=-\frac{\left (52-51 a+2 a^2\right ) b^3 \sqrt{1-a-b x}}{6 (1-a) (1+a)^4 \sqrt{1+a+b x}}-\frac{(1-a) \sqrt{1-a-b x}}{3 (1+a) x^3 \sqrt{1+a+b x}}+\frac{7 b \sqrt{1-a-b x}}{6 (1+a)^2 x^2 \sqrt{1+a+b x}}-\frac{(19-16 a) b^2 \sqrt{1-a-b x}}{6 (1-a) (1+a)^3 x \sqrt{1+a+b x}}-\frac{\left (\left (11-18 a+6 a^2\right ) b^3\right ) \int \frac{1}{x \sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx}{2 (1-a) (1+a)^4}\\ &=-\frac{\left (52-51 a+2 a^2\right ) b^3 \sqrt{1-a-b x}}{6 (1-a) (1+a)^4 \sqrt{1+a+b x}}-\frac{(1-a) \sqrt{1-a-b x}}{3 (1+a) x^3 \sqrt{1+a+b x}}+\frac{7 b \sqrt{1-a-b x}}{6 (1+a)^2 x^2 \sqrt{1+a+b x}}-\frac{(19-16 a) b^2 \sqrt{1-a-b x}}{6 (1-a) (1+a)^3 x \sqrt{1+a+b x}}-\frac{\left (\left (11-18 a+6 a^2\right ) b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-1-a-(-1+a) x^2} \, dx,x,\frac{\sqrt{1+a+b x}}{\sqrt{1-a-b x}}\right )}{(1-a) (1+a)^4}\\ &=-\frac{\left (52-51 a+2 a^2\right ) b^3 \sqrt{1-a-b x}}{6 (1-a) (1+a)^4 \sqrt{1+a+b x}}-\frac{(1-a) \sqrt{1-a-b x}}{3 (1+a) x^3 \sqrt{1+a+b x}}+\frac{7 b \sqrt{1-a-b x}}{6 (1+a)^2 x^2 \sqrt{1+a+b x}}-\frac{(19-16 a) b^2 \sqrt{1-a-b x}}{6 (1-a) (1+a)^3 x \sqrt{1+a+b x}}+\frac{\left (11-18 a+6 a^2\right ) b^3 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{1+a+b x}}{\sqrt{1+a} \sqrt{1-a-b x}}\right )}{(1-a) (1+a)^4 \sqrt{1-a^2}}\\ \end{align*}

Mathematica [A]  time = 0.3427, size = 200, normalized size = 0.78 \[ \frac{-\left (6 a^2-18 a+11\right ) b^2 x^2 \left (-\sqrt{a+1} \sqrt{-a-b x+1} \left (a^2+a b x-5 b x-1\right )-6 \sqrt{a-1} b x \sqrt{a+b x+1} \tan ^{-1}\left (\frac{\sqrt{-a-b x+1}}{\sqrt{\frac{a-1}{a+1}} \sqrt{a+b x+1}}\right )\right )-2 (1-a) (a+1)^{7/2} (-a-b x+1)^{5/2}+(3-4 a) (a+1)^{5/2} b x (-a-b x+1)^{5/2}}{6 (a+1)^{5/2} \left (a^2-1\right )^2 x^3 \sqrt{a+b x+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcTanh[a + b*x])*x^4),x]

[Out]

(-2*(1 - a)*(1 + a)^(7/2)*(1 - a - b*x)^(5/2) + (3 - 4*a)*(1 + a)^(5/2)*b*x*(1 - a - b*x)^(5/2) - (11 - 18*a +
6*a^2)*b^2*x^2*(-(Sqrt[1 + a]*Sqrt[1 - a - b*x]*(-1 + a^2 - 5*b*x + a*b*x)) - 6*Sqrt[-1 + a]*b*x*Sqrt[1 + a +
b*x]*ArcTan[Sqrt[1 - a - b*x]/(Sqrt[(-1 + a)/(1 + a)]*Sqrt[1 + a + b*x])]))/(6*(1 + a)^(5/2)*(-1 + a^2)^2*x^3
*Sqrt[1 + a + b*x])

________________________________________________________________________________________

Maple [B]  time = 0.067, size = 4212, normalized size = 16.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2)/x^4,x)

[Out]

-10/3/(1+a)^6*b^3*(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)-10/(1+a)^6*b^3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+10/3/(1+a)^6*b^
3*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(3/2)-1/(1+a)^4/(x+1/b+a/b)^3*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(5/2
)-2/(1+a)^4*b^3*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(3/2)+4/(1+a)^5*b^3*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^
(3/2)+5/(1+a)^6*b^4*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2)*x+5/(1+a)^6*b^3*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a
)/b))^(1/2)*a+4/(1+a)^5*b/(x+1/b+a/b)^2*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(5/2)+6/(1+a)^5*b^4*(-(x+(1+a)/b)
^2*b^2+2*b*(x+(1+a)/b))^(1/2)*x+6/(1+a)^5*b^3*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2)*a+6/(1+a)^5*b^4/(b^2)
^(1/2)*arctan((b^2)^(1/2)*(x+(1+a)/b-1/b)/(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2))+15/(1+a)^6*b^3*a^2*(-b^2
*x^2-2*a*b*x-a^2+1)^(1/2)+10/(1+a)^6*b^3*(-a^2+1)^(1/2)*ln((-2*a^2+2-2*x*a*b+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*
x-a^2+1)^(1/2))/x)-1/3/(1+a)^3/(-a^2+1)/x^3*(-b^2*x^2-2*a*b*x-a^2+1)^(5/2)+5/(1+a)^6*b^4/(b^2)^(1/2)*arctan((b
^2)^(1/2)*(x+(1+a)/b-1/b)/(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2))-2/(1+a)^4*b/(x+1/b+a/b)^2*(-(x+(1+a)/b)^
2*b^2+2*b*(x+(1+a)/b))^(5/2)+3/2/(1+a)^4*b^3/(-a^2+1)*(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)+9/2/(1+a)^4*b^3/(-a^2+1)*
(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-9/2/(1+a)^4*b^3/(-a^2+1)^(1/2)*ln((-2*a^2+2-2*x*a*b+2*(-a^2+1)^(1/2)*(-b^2*x^2-
2*a*b*x-a^2+1)^(1/2))/x)+7/6/(1+a)^3*a*b^3/(-a^2+1)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)+1/2/(1+a)^3*a^5*b^3/(-a^2
+1)^(5/2)*ln((-2*a^2+2-2*x*a*b+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x)+2/3/(1+a)^3*b^2/(-a^2+1)^2/
x*(-b^2*x^2-2*a*b*x-a^2+1)^(5/2)+15/(1+a)^6*b^4*a/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2
+1)^(1/2))-10/(1+a)^6*b^4*a^3/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-10/(1+a)^
6*b^3*(-a^2+1)^(1/2)*ln((-2*a^2+2-2*x*a*b+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x)*a^2-3/(1+a)^4*b^
4/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+(1+a)/b-1/b)/(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2))-3/4/(1+a)^3*a^4*b
^4/(-a^2+1)^3/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))+1/2/(1+a)^3*a^6*b^4/(-a^2
+1)^3/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))+1/6/(1+a)^3*a^2*b^4/(-a^2+1)^3*(-
b^2*x^2-2*a*b*x-a^2+1)^(3/2)*x-3/2/(1+a)^4*b^2*a/(-a^2+1)^2/x*(-b^2*x^2-2*a*b*x-a^2+1)^(5/2)+9/4/(1+a)^4*b^4*a
^3/(-a^2+1)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x+27/4/(1+a)^4*b^4*a^3/(-a^2+1)^2/(b^2)^(1/2)*arctan((b^2)^(1/2)*
(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-9/2/(1+a)^4*b^4*a^5/(-a^2+1)^2/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a/b)/
(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-3/2/(1+a)^4*b^4*a/(-a^2+1)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*x-9/4/(1+a)^4*b^4*
a/(-a^2+1)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x-9/4/(1+a)^4*b^4*a/(-a^2+1)^2/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a
/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-9/4/(1+a)^4*b^4/(-a^2+1)*a*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x-27/4/(1+a)^4*b
^4/(-a^2+1)*a/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))+9/2/(1+a)^4*b^4/(-a^2+1)*
a^3/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))+27/(1+a)^5*b^4*a^2/(-a^2+1)/(b^2)^(
1/2)*arctan((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-18/(1+a)^5*b^4*a^4/(-a^2+1)/(b^2)^(1/2)*arctan
((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))+9/(1+a)^5*b^4*a^2/(-a^2+1)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)
*x+1/4/(1+a)^3*a^2*b^4/(-a^2+1)^3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x-1/6/(1+a)^3*a*b/(-a^2+1)^2/x^2*(-b^2*x^2-2*
a*b*x-a^2+1)^(5/2)+1/4/(1+a)^3*a^2*b^4/(-a^2+1)^3/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2
+1)^(1/2))-3/4/(1+a)^3*a^2*b^4/(-a^2+1)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x-9/4/(1+a)^3*a^2*b^4/(-a^2+1)^2/(b^2
)^(1/2)*arctan((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))+3/2/(1+a)^3*a^4*b^4/(-a^2+1)^2/(b^2)^(1/2)*
arctan((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-1/4/(1+a)^3*a^4*b^4/(-a^2+1)^3*(-b^2*x^2-2*a*b*x-a^
2+1)^(1/2)*x-3/(1+a)^4*b^4*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2)*x-3/(1+a)^4*b^3*(-(x+(1+a)/b)^2*b^2+2*b*
(x+(1+a)/b))^(1/2)*a+3/2/(1+a)^3*a^3*b^3/(-a^2+1)^(3/2)*ln((-2*a^2+2-2*x*a*b+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*
x-a^2+1)^(1/2))/x)-3/2/(1+a)^3*a*b^3/(-a^2+1)^(3/2)*ln((-2*a^2+2-2*x*a*b+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^
2+1)^(1/2))/x)-9/4/(1+a)^3*a^3*b^3/(-a^2+1)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+5/2/(1+a)^3*a*b^3/(-a^2+1)^2*(-b^
2*x^2-2*a*b*x-a^2+1)^(1/2)+1/3/(1+a)^3*a^3*b^3/(-a^2+1)^3*(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)-3/4/(1+a)^3*a^5*b^3/(
-a^2+1)^3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+3/4/(1+a)^3*a^3*b^3/(-a^2+1)^3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-18/(1+a
)^5*b^3*a^3/(-a^2+1)^(1/2)*ln((-2*a^2+2-2*x*a*b+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x)+3/2/(1+a)^
4*b/(-a^2+1)/x^2*(-b^2*x^2-2*a*b*x-a^2+1)^(5/2)-3/(1+a)^4*b^3*a^2/(-a^2+1)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)+27
/4/(1+a)^4*b^3*a^4/(-a^2+1)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-27/4/(1+a)^4*b^3*a^2/(-a^2+1)^2*(-b^2*x^2-2*a*b*x
-a^2+1)^(1/2)-9/2/(1+a)^4*b^3*a^4/(-a^2+1)^(3/2)*ln((-2*a^2+2-2*x*a*b+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1
)^(1/2))/x)+9/2/(1+a)^4*b^3*a^2/(-a^2+1)^(3/2)*ln((-2*a^2+2-2*x*a*b+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^
(1/2))/x)-27/4/(1+a)^4*b^3/(-a^2+1)*a^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+9/2/(1+a)^4*b^3/(-a^2+1)^(1/2)*ln((-2*a
^2+2-2*x*a*b+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x)*a^2-6/(1+a)^5*b^4/(-a^2+1)*(-b^2*x^2-2*a*b*x-
a^2+1)^(3/2)*x-9/(1+a)^5*b^4/(-a^2+1)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x-9/(1+a)^5*b^4/(-a^2+1)/(b^2)^(1/2)*arct
an((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))+2/3/(1+a)^3*b^4/(-a^2+1)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(3/
2)*x+1/(1+a)^3*b^4/(-a^2+1)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x+1/(1+a)^3*b^4/(-a^2+1)^2/(b^2)^(1/2)*arctan((b^
2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))+18/(1+a)^5*b^3*a/(-a^2+1)^(1/2)*ln((-2*a^2+2-2*x*a*b+2*(-a^2+
1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x)-6/(1+a)^5*b^2/(-a^2+1)/x*(-b^2*x^2-2*a*b*x-a^2+1)^(5/2)-12/(1+a)^5
*b^3*a/(-a^2+1)*(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)+27/(1+a)^5*b^3*a^3/(-a^2+1)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-27/(
1+a)^5*b^3*a/(-a^2+1)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-1/2/(1+a)^3*a^3*b^3/(-a^2+1)^(5/2)*ln((-2*a^2+2-2*x*a*b+2
*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x)+5/(1+a)^6*b^4*a*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x+1/6/(1+a)^
3*a^2*b^2/(-a^2+1)^3/x*(-b^2*x^2-2*a*b*x-a^2+1)^(5/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac{3}{2}}}{{\left (b x + a + 1\right )}^{3} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2)/x^4,x, algorithm="maxima")

[Out]

integrate((-(b*x + a)^2 + 1)^(3/2)/((b*x + a + 1)^3*x^4), x)

________________________________________________________________________________________

Fricas [A]  time = 2.38553, size = 1611, normalized size = 6.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2)/x^4,x, algorithm="fricas")

[Out]

[-1/12*(3*((6*a^2 - 18*a + 11)*b^4*x^4 + (6*a^3 - 12*a^2 - 7*a + 11)*b^3*x^3)*sqrt(-a^2 + 1)*log(((2*a^2 - 1)*
b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*
a^2 + 2)/x^2) - 2*(2*a^7 + (2*a^4 - 51*a^3 + 50*a^2 + 51*a - 52)*b^3*x^3 + 2*a^6 - 6*a^5 - (16*a^4 - 3*a^3 - 3
5*a^2 + 3*a + 19)*b^2*x^2 - 6*a^4 + 6*a^3 + 7*(a^5 + a^4 - 2*a^3 - 2*a^2 + a + 1)*b*x + 6*a^2 - 2*a - 2)*sqrt(
-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^7 + 3*a^6 + a^5 - 5*a^4 - 5*a^3 + a^2 + 3*a + 1)*b*x^4 + (a^8 + 4*a^7 + 4*a
^6 - 4*a^5 - 10*a^4 - 4*a^3 + 4*a^2 + 4*a + 1)*x^3), 1/6*(3*((6*a^2 - 18*a + 11)*b^4*x^4 + (6*a^3 - 12*a^2 - 7
*a + 11)*b^3*x^3)*sqrt(a^2 - 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^
2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) + (2*a^7 + (2*a^4 - 51*a^3 + 50*a^2 + 51*a - 52)*b^3*x^3
+ 2*a^6 - 6*a^5 - (16*a^4 - 3*a^3 - 35*a^2 + 3*a + 19)*b^2*x^2 - 6*a^4 + 6*a^3 + 7*(a^5 + a^4 - 2*a^3 - 2*a^2
+ a + 1)*b*x + 6*a^2 - 2*a - 2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^7 + 3*a^6 + a^5 - 5*a^4 - 5*a^3 + a^2
+ 3*a + 1)*b*x^4 + (a^8 + 4*a^7 + 4*a^6 - 4*a^5 - 10*a^4 - 4*a^3 + 4*a^2 + 4*a + 1)*x^3)]

________________________________________________________________________________________

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(1/(b*x+a+1)**3*(1-(b*x+a)**2)**(3/2)/x**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.32481, size = 2483, normalized size = 9.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2)/x^4,x, algorithm="giac")

[Out]

8*b^4/((a^4*abs(b) + 4*a^3*abs(b) + 6*a^2*abs(b) + 4*a*abs(b) + abs(b))*((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a
bs(b) + b)/(b^2*x + a*b) + 1)) + (6*a^2*b^4 - 18*a*b^4 + 11*b^4)*arctan(((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a
bs(b) + b)*a/(b^2*x + a*b) - 1)/sqrt(a^2 - 1))/((a^5*abs(b) + 3*a^4*abs(b) + 2*a^3*abs(b) - 2*a^2*abs(b) - 3*a
*abs(b) - abs(b))*sqrt(a^2 - 1)) + 1/3*(12*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^7*b^4/(b^2*x +
a*b)^2 + 6*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^4*a^7*b^4/(b^2*x + a*b)^4 + 6*a^7*b^4 - 24*(sqrt(-b
^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a^6*b^4/(b^2*x + a*b) - 72*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b)
+ b)^2*a^6*b^4/(b^2*x + a*b)^2 - 36*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a^6*b^4/(b^2*x + a*b)^3
- 36*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^4*a^6*b^4/(b^2*x + a*b)^4 - 12*(sqrt(-b^2*x^2 - 2*a*b*x -
a^2 + 1)*abs(b) + b)^5*a^6*b^4/(b^2*x + a*b)^5 - 36*a^6*b^4 + 171*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b)
+ b)*a^5*b^4/(b^2*x + a*b) + 84*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^5*b^4/(b^2*x + a*b)^2 + 21
6*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a^5*b^4/(b^2*x + a*b)^3 + 54*(sqrt(-b^2*x^2 - 2*a*b*x - a^
2 + 1)*abs(b) + b)^4*a^5*b^4/(b^2*x + a*b)^4 + 45*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^5*a^5*b^4/(b
^2*x + a*b)^5 + 22*a^5*b^4 - 120*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a^4*b^4/(b^2*x + a*b) - 252*(
sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^4*b^4/(b^2*x + a*b)^2 - 156*(sqrt(-b^2*x^2 - 2*a*b*x - a^2
+ 1)*abs(b) + b)^3*a^4*b^4/(b^2*x + a*b)^3 - 153*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^4*a^4*b^4/(b^
2*x + a*b)^4 - 12*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^5*a^4*b^4/(b^2*x + a*b)^5 + 9*a^4*b^4 - 36*(
sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a^3*b^4/(b^2*x + a*b) + 192*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)
*abs(b) + b)^2*a^3*b^4/(b^2*x + a*b)^2 + 90*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a^3*b^4/(b^2*x +
a*b)^3 + 78*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^4*a^3*b^4/(b^2*x + a*b)^4 - 18*(sqrt(-b^2*x^2 - 2
*a*b*x - a^2 + 1)*abs(b) + b)^5*a^3*b^4/(b^2*x + a*b)^5 + 2*a^3*b^4 - 6*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*ab
s(b) + b)*a^2*b^4/(b^2*x + a*b) + 54*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^2*b^4/(b^2*x + a*b)^2
- 100*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a^2*b^4/(b^2*x + a*b)^3 + 54*(sqrt(-b^2*x^2 - 2*a*b*x
- a^2 + 1)*abs(b) + b)^4*a^2*b^4/(b^2*x + a*b)^4 - 6*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^5*a^2*b^
4/(b^2*x + a*b)^5 + 12*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a*b^4/(b^2*x + a*b)^2 - 36*(sqrt(-b^2
*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a*b^4/(b^2*x + a*b)^3 + 12*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b)
+ b)^4*a*b^4/(b^2*x + a*b)^4 - 8*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*b^4/(b^2*x + a*b)^3)/((a^8*
abs(b) + 3*a^7*abs(b) + 2*a^6*abs(b) - 2*a^5*abs(b) - 3*a^4*abs(b) - a^3*abs(b))*((sqrt(-b^2*x^2 - 2*a*b*x - a
^2 + 1)*abs(b) + b)^2*a/(b^2*x + a*b)^2 + a - 2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)/(b^2*x + a*b))
^3)