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

Optimal. Leaf size=260 \[ \frac{\left (2 a^2+51 a+52\right ) b^3 \sqrt{a+b x+1}}{6 (1-a)^4 (a+1) \sqrt{-a-b x+1}}-\frac{\left (6 a^2+18 a+11\right ) b^3 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{(1-a)^4 (a+1) \sqrt{1-a^2}}-\frac{(16 a+19) b^2 \sqrt{a+b x+1}}{6 (1-a)^3 (a+1) x \sqrt{-a-b x+1}}-\frac{7 b \sqrt{a+b x+1}}{6 (1-a)^2 x^2 \sqrt{-a-b x+1}}-\frac{(a+1) \sqrt{a+b x+1}}{3 (1-a) x^3 \sqrt{-a-b x+1}} \]

[Out]

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

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Rubi [A]  time = 0.209521, antiderivative size = 260, 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} \[ \frac{\left (2 a^2+51 a+52\right ) b^3 \sqrt{a+b x+1}}{6 (1-a)^4 (a+1) \sqrt{-a-b x+1}}-\frac{\left (6 a^2+18 a+11\right ) b^3 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{(1-a)^4 (a+1) \sqrt{1-a^2}}-\frac{(16 a+19) b^2 \sqrt{a+b x+1}}{6 (1-a)^3 (a+1) x \sqrt{-a-b x+1}}-\frac{7 b \sqrt{a+b x+1}}{6 (1-a)^2 x^2 \sqrt{-a-b x+1}}-\frac{(a+1) \sqrt{a+b x+1}}{3 (1-a) x^3 \sqrt{-a-b x+1}} \]

Antiderivative was successfully verified.

[In]

Int[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)^4*(1 + a)*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 + 1
6*a)*b^2*Sqrt[1 + a + b*x])/(6*(1 - a)^3*(1 + a)*x*Sqrt[1 - a - b*x]) - ((11 + 18*a + 6*a^2)*b^3*ArcTanh[(Sqrt
[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)^4*(1 + a)*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 (1-a-b x)^{3/2} \sqrt{1+a+b x}} \, 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{(1+a) (19+16 a) b^2+14 (1+a) b^3 x}{x^2 (1-a-b x)^{3/2} \sqrt{1+a+b x}} \, dx}{6 (1-a)^2 (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{(19+16 a) b^2 \sqrt{1+a+b x}}{6 (1-a)^3 (1+a) x \sqrt{1-a-b x}}-\frac{\int \frac{-3 (1+a) \left (11+18 a+6 a^2\right ) b^3-(1+a) (19+16 a) b^4 x}{x (1-a-b x)^{3/2} \sqrt{1+a+b x}} \, dx}{6 (1-a)^3 (1+a)^2}\\ &=\frac{\left (52+51 a+2 a^2\right ) b^3 \sqrt{1+a+b x}}{6 (1-a)^4 (1+a) \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)^3 (1+a) 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)^4 (1+a)^2 b}\\ &=\frac{\left (52+51 a+2 a^2\right ) b^3 \sqrt{1+a+b x}}{6 (1-a)^4 (1+a) \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)^3 (1+a) 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)^4 (1+a)}\\ &=\frac{\left (52+51 a+2 a^2\right ) b^3 \sqrt{1+a+b x}}{6 (1-a)^4 (1+a) \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)^3 (1+a) 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)^4 (1+a)}\\ &=\frac{\left (52+51 a+2 a^2\right ) b^3 \sqrt{1+a+b x}}{6 (1-a)^4 (1+a) \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)^3 (1+a) 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)^4 (1+a) \sqrt{1-a^2}}\\ \end{align*}

Mathematica [A]  time = 0.337401, size = 194, normalized size = 0.75 \[ -\frac{-\left (6 a^2+18 a+11\right ) b^2 x^2 \left (\sqrt{a-1} \sqrt{a+b x+1} \left (a^2+a b x+5 b x-1\right )+6 \sqrt{a+1} b x \sqrt{-a-b x+1} \tan ^{-1}\left (\frac{\sqrt{-a-b x+1}}{\sqrt{\frac{a-1}{a+1}} \sqrt{a+b x+1}}\right )\right )-2 (a+1) (a-1)^{7/2} (a+b x+1)^{5/2}+(4 a+3) (a-1)^{5/2} b x (a+b x+1)^{5/2}}{6 (a-1)^{5/2} \left (a^2-1\right )^2 x^3 \sqrt{-a-b x+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(-2*(-1 + a)^(7/2)*(1 + a)*(1 + a + b*x)^(5/2) + (-1 + a)^(5/2)*(3 + 4*a)*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])

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Maple [B]  time = 0.047, size = 2947, normalized size = 11.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)-35/2*a^6*b^3/(-a^
2+1)^(9/2)*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)-105/2*a^5*b^3/(-a^2+1)^(9/
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)-105/2*a^4*b^3/(-a^2+1)^(9/2)*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/(-a^2+1)/x^3/(-b^2*x^2-2*a*b*x-a^2+1)^
(1/2)*a^3-1/(-a^2+1)/x^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a^2-1/(-a^2+1)/x^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a+70
*a^6*b^3/(-a^2+1)^4/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+105/2*a^4*b^3/(-a^2+1)^4/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+35/
2*a^8*b^3/(-a^2+1)^4/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-4/3*b^2/(-a^2+1)^2/x/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+8/3*b^
4/(-a^2+1)^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x+35/2*a^3*b^3/(-a^2+1)^4/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+70*a^5*b^
3/(-a^2+1)^4/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+15/2*a*b^3/(-a^2+1)^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+45*a^2*b^3/(-
a^2+1)^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+110*a^4*b^3/(-a^2+1)^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-3/2*b/(-a^2+1)/x
^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+260/3*a^3*b^3/(-a^2+1)^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+125/3*a^6*b^3/(-a^2+
1)^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+205/2*a^5*b^3/(-a^2+1)^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-45*a^2*b^3/(-a^2+1
)^(7/2)*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*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)+9/2*b^3/(-a^2+1)^2/(-b^2*x^2-2*a*b*x-a^2+1)
^(1/2)+b^3/(-a^2+1)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-1/3/(-a^2+1)/x^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+41*b^3*a^2/
(-a^2+1)^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+187/6*b^3*a^4/(-a^2+1)^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+7*b^3/(-a^2+
1)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a^2-3*b^2/(-a^2+1)/x/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+62/3*b^3*a/(-a^2+1)^2/(-
b^2*x^2-2*a*b*x-a^2+1)^(1/2)+56*b^3*a^3/(-a^2+1)^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+6*b^4/(-a^2+1)/(-b^2*x^2-2*a
*b*x-a^2+1)^(1/2)*x+6*b^3/(-a^2+1)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a-35/2*a^3*b^3/(-a^2+1)^(9/2)*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)-15/2*a*b^3/(-a^2+1)^(7/2)*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)+105/2*a^7*b^3/(-a^2+1)^4/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-23/2*a
*b^2/(-a^2+1)^2/x/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-3/2*b/(-a^2+1)/x^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a^2-30*a^4*
b^3/(-a^2+1)^(7/2)*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)-27/2*a^2*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)-135/2*a^3*b^3/(-a^2+1)^(7/2
)*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)-18*a*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)-3*b/(-a^2+1)/x^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*
a-53/6*a^3*b^2/(-a^2+1)^2/x/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-19*a^2*b^2/(-a^2+1)^2/x/(-b^2*x^2-2*a*b*x-a^2+1)^(1
/2)+125/3*a^5*b^4/(-a^2+1)^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x+205/2*a^4*b^4/(-a^2+1)^3/(-b^2*x^2-2*a*b*x-a^2+1
)^(1/2)*x+55/2*a*b^4/(-a^2+1)^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x+80*a^3*b^4/(-a^2+1)^3/(-b^2*x^2-2*a*b*x-a^2+1
)^(1/2)*x+35/2*a^7*b^4/(-a^2+1)^4/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x+105/2*a^6*b^4/(-a^2+1)^4/(-b^2*x^2-2*a*b*x-
a^2+1)^(1/2)*x-7/2*a^3*b/(-a^2+1)^2/x^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-7/2*a^2*b/(-a^2+1)^2/x^2/(-b^2*x^2-2*a*
b*x-a^2+1)^(1/2)-35/6*a^5*b^2/(-a^2+1)^3/x/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-35/2*a^4*b^2/(-a^2+1)^3/x/(-b^2*x^2-
2*a*b*x-a^2+1)^(1/2)-35/2*a^3*b^2/(-a^2+1)^3/x/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+115/6*a^2*b^4/(-a^2+1)^3/(-b^2*x
^2-2*a*b*x-a^2+1)^(1/2)*x-7/6*a*b/(-a^2+1)^2/x^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-35/6*a^2*b^2/(-a^2+1)^3/x/(-b^
2*x^2-2*a*b*x-a^2+1)^(1/2)+35/2*a^4*b^4/(-a^2+1)^4/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x+7*b^4*a/(-a^2+1)/(-b^2*x^2
-2*a*b*x-a^2+1)^(1/2)*x+56*b^4*a^2/(-a^2+1)^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x-3*b^2*a/(-a^2+1)/x/(-b^2*x^2-2*
a*b*x-a^2+1)^(1/2)+187/6*b^4*a^3/(-a^2+1)^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x+105/2*a^5*b^4/(-a^2+1)^4/(-b^2*x^
2-2*a*b*x-a^2+1)^(1/2)*x-7/6*a^4*b/(-a^2+1)^2/x^2/(-b^2*x^2-2*a*b*x-a^2+1)^(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((b*x+a+1)^3/(1-(b*x+a)^2)^(3/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.76915, size = 1611, normalized size = 6.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.44147, size = 2053, normalized size = 7.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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*x + a)^2 + 1)*abs(b) + b)/(
b^2*x + a*b) - 1)) + (6*a^2*b^4 + 18*a*b^4 + 11*b^4)*arctan(((sqrt(-(b*x + a)^2 + 1)*abs(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*x + a)^2 + 1)*abs(b) + b)^2*a^7*b^4/(b^2*x + a*b)^2 + 6*(sqrt(-(b*x + a)^2 + 1)*a
bs(b) + b)^4*a^7*b^4/(b^2*x + a*b)^4 + 6*a^7*b^4 - 24*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)*a^6*b^4/(b^2*x + a*b
) + 72*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^2*a^6*b^4/(b^2*x + a*b)^2 - 36*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^
3*a^6*b^4/(b^2*x + a*b)^3 + 36*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^4*a^6*b^4/(b^2*x + a*b)^4 - 12*(sqrt(-(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*x + a)^2 + 1)*abs(b) + b)*a^5*b
^4/(b^2*x + a*b) + 84*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^2*a^5*b^4/(b^2*x + a*b)^2 - 216*(sqrt(-(b*x + a)^2 +
1)*abs(b) + b)^3*a^5*b^4/(b^2*x + a*b)^3 + 54*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^4*a^5*b^4/(b^2*x + a*b)^4 -
45*(sqrt(-(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*x + a)^2 + 1)*a
bs(b) + b)*a^4*b^4/(b^2*x + a*b) + 252*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^2*a^4*b^4/(b^2*x + a*b)^2 - 156*(sq
rt(-(b*x + a)^2 + 1)*abs(b) + b)^3*a^4*b^4/(b^2*x + a*b)^3 + 153*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^4*a^4*b^4
/(b^2*x + a*b)^4 - 12*(sqrt(-(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
*x + a)^2 + 1)*abs(b) + b)*a^3*b^4/(b^2*x + a*b) + 192*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^2*a^3*b^4/(b^2*x +
a*b)^2 - 90*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^3*a^3*b^4/(b^2*x + a*b)^3 + 78*(sqrt(-(b*x + a)^2 + 1)*abs(b)
+ b)^4*a^3*b^4/(b^2*x + a*b)^4 + 18*(sqrt(-(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*x + a)^2 + 1)*abs(b) + b)*a^2*b^4/(b^2*x + a*b) - 54*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^2*a^2*b
^4/(b^2*x + a*b)^2 - 100*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^3*a^2*b^4/(b^2*x + a*b)^3 - 54*(sqrt(-(b*x + a)^2
+ 1)*abs(b) + b)^4*a^2*b^4/(b^2*x + a*b)^4 - 6*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^5*a^2*b^4/(b^2*x + a*b)^5
+ 12*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^2*a*b^4/(b^2*x + a*b)^2 + 36*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^3*a*
b^4/(b^2*x + a*b)^3 + 12*(sqrt(-(b*x + a)^2 + 1)*abs(b) + b)^4*a*b^4/(b^2*x + a*b)^4 - 8*(sqrt(-(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*x + a)^2 + 1)*abs(b) + b)^2*a/(b^2*x + a*b)^2 + a - 2*(sqrt(-(b*x + a)^2 + 1)*abs(b)
+ b)/(b^2*x + a*b))^3)