3.283 \(\int \frac{e^{\cosh ^{-1}(a+b x)}}{x^5} \, dx\)

Optimal. Leaf size=238 \[ \frac{\left (2 a^2+3\right ) b^2 \sqrt{a+b x-1} \sqrt{a+b x+1}}{24 \left (1-a^2\right )^2 x^2}+\frac{a \left (2 a^2+13\right ) b^3 \sqrt{a+b x-1} \sqrt{a+b x+1}}{24 \left (1-a^2\right )^3 x}-\frac{\left (4 a^2+1\right ) b^4 \tan ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{a+b x-1}}\right )}{4 \left (1-a^2\right )^{7/2}}+\frac{a b \sqrt{a+b x-1} \sqrt{a+b x+1}}{12 \left (1-a^2\right ) x^3}-\frac{\sqrt{a+b x-1} \sqrt{a+b x+1}}{4 x^4}-\frac{a}{4 x^4}-\frac{b}{3 x^3} \]

[Out]

-a/(4*x^4) - b/(3*x^3) - (Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(4*x^4) + (a*b*Sqrt[-1 + a + b*x]*Sqrt[1 + a +
b*x])/(12*(1 - a^2)*x^3) + ((3 + 2*a^2)*b^2*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(24*(1 - a^2)^2*x^2) + (a*(
13 + 2*a^2)*b^3*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(24*(1 - a^2)^3*x) - ((1 + 4*a^2)*b^4*ArcTan[(Sqrt[1 - a
]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[-1 + a + b*x])])/(4*(1 - a^2)^(7/2))

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Rubi [A]  time = 0.201372, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5909, 14, 97, 151, 12, 93, 205} \[ \frac{\left (2 a^2+3\right ) b^2 \sqrt{a+b x-1} \sqrt{a+b x+1}}{24 \left (1-a^2\right )^2 x^2}+\frac{a \left (2 a^2+13\right ) b^3 \sqrt{a+b x-1} \sqrt{a+b x+1}}{24 \left (1-a^2\right )^3 x}-\frac{\left (4 a^2+1\right ) b^4 \tan ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{a+b x-1}}\right )}{4 \left (1-a^2\right )^{7/2}}+\frac{a b \sqrt{a+b x-1} \sqrt{a+b x+1}}{12 \left (1-a^2\right ) x^3}-\frac{\sqrt{a+b x-1} \sqrt{a+b x+1}}{4 x^4}-\frac{a}{4 x^4}-\frac{b}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCosh[a + b*x]/x^5,x]

[Out]

-a/(4*x^4) - b/(3*x^3) - (Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(4*x^4) + (a*b*Sqrt[-1 + a + b*x]*Sqrt[1 + a +
b*x])/(12*(1 - a^2)*x^3) + ((3 + 2*a^2)*b^2*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(24*(1 - a^2)^2*x^2) + (a*(
13 + 2*a^2)*b^3*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(24*(1 - a^2)^3*x) - ((1 + 4*a^2)*b^4*ArcTan[(Sqrt[1 - a
]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[-1 + a + b*x])])/(4*(1 - a^2)^(7/2))

Rule 5909

Int[E^(ArcCosh[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(u + Sqrt[-1 + u]*Sqrt[1 + u])^n, x] /; RationalQ[m
] && IntegerQ[n] && PolynomialQ[u, x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 97

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)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (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 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 205

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

Rubi steps

\begin{align*} \int \frac{e^{\cosh ^{-1}(a+b x)}}{x^5} \, dx &=\int \frac{a+b x+\sqrt{-1+a+b x} \sqrt{1+a+b x}}{x^5} \, dx\\ &=\int \left (\frac{a}{x^5}+\frac{b}{x^4}+\frac{\sqrt{-1+a+b x} \sqrt{1+a+b x}}{x^5}\right ) \, dx\\ &=-\frac{a}{4 x^4}-\frac{b}{3 x^3}+\int \frac{\sqrt{-1+a+b x} \sqrt{1+a+b x}}{x^5} \, dx\\ &=-\frac{a}{4 x^4}-\frac{b}{3 x^3}-\frac{\sqrt{-1+a+b x} \sqrt{1+a+b x}}{4 x^4}+\frac{1}{4} \int \frac{a b+b^2 x}{x^4 \sqrt{-1+a+b x} \sqrt{1+a+b x}} \, dx\\ &=-\frac{a}{4 x^4}-\frac{b}{3 x^3}-\frac{\sqrt{-1+a+b x} \sqrt{1+a+b x}}{4 x^4}+\frac{a b \sqrt{-1+a+b x} \sqrt{1+a+b x}}{12 \left (1-a^2\right ) x^3}+\frac{\int \frac{\left (3+2 a^2\right ) b^2+2 a b^3 x}{x^3 \sqrt{-1+a+b x} \sqrt{1+a+b x}} \, dx}{12 \left (1-a^2\right )}\\ &=-\frac{a}{4 x^4}-\frac{b}{3 x^3}-\frac{\sqrt{-1+a+b x} \sqrt{1+a+b x}}{4 x^4}+\frac{a b \sqrt{-1+a+b x} \sqrt{1+a+b x}}{12 \left (1-a^2\right ) x^3}+\frac{\left (3+2 a^2\right ) b^2 \sqrt{-1+a+b x} \sqrt{1+a+b x}}{24 \left (1-a^2\right )^2 x^2}+\frac{\int \frac{a \left (13+2 a^2\right ) b^3+\left (3+2 a^2\right ) b^4 x}{x^2 \sqrt{-1+a+b x} \sqrt{1+a+b x}} \, dx}{24 \left (1-a^2\right )^2}\\ &=-\frac{a}{4 x^4}-\frac{b}{3 x^3}-\frac{\sqrt{-1+a+b x} \sqrt{1+a+b x}}{4 x^4}+\frac{a b \sqrt{-1+a+b x} \sqrt{1+a+b x}}{12 \left (1-a^2\right ) x^3}+\frac{\left (3+2 a^2\right ) b^2 \sqrt{-1+a+b x} \sqrt{1+a+b x}}{24 \left (1-a^2\right )^2 x^2}+\frac{a \left (13+2 a^2\right ) b^3 \sqrt{-1+a+b x} \sqrt{1+a+b x}}{24 \left (1-a^2\right )^3 x}+\frac{\int \frac{3 \left (1+4 a^2\right ) b^4}{x \sqrt{-1+a+b x} \sqrt{1+a+b x}} \, dx}{24 \left (1-a^2\right )^3}\\ &=-\frac{a}{4 x^4}-\frac{b}{3 x^3}-\frac{\sqrt{-1+a+b x} \sqrt{1+a+b x}}{4 x^4}+\frac{a b \sqrt{-1+a+b x} \sqrt{1+a+b x}}{12 \left (1-a^2\right ) x^3}+\frac{\left (3+2 a^2\right ) b^2 \sqrt{-1+a+b x} \sqrt{1+a+b x}}{24 \left (1-a^2\right )^2 x^2}+\frac{a \left (13+2 a^2\right ) b^3 \sqrt{-1+a+b x} \sqrt{1+a+b x}}{24 \left (1-a^2\right )^3 x}+\frac{\left (\left (1+4 a^2\right ) b^4\right ) \int \frac{1}{x \sqrt{-1+a+b x} \sqrt{1+a+b x}} \, dx}{8 \left (1-a^2\right )^3}\\ &=-\frac{a}{4 x^4}-\frac{b}{3 x^3}-\frac{\sqrt{-1+a+b x} \sqrt{1+a+b x}}{4 x^4}+\frac{a b \sqrt{-1+a+b x} \sqrt{1+a+b x}}{12 \left (1-a^2\right ) x^3}+\frac{\left (3+2 a^2\right ) b^2 \sqrt{-1+a+b x} \sqrt{1+a+b x}}{24 \left (1-a^2\right )^2 x^2}+\frac{a \left (13+2 a^2\right ) b^3 \sqrt{-1+a+b x} \sqrt{1+a+b x}}{24 \left (1-a^2\right )^3 x}+\frac{\left (\left (1+4 a^2\right ) b^4\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 )}{4 \left (1-a^2\right )^3}\\ &=-\frac{a}{4 x^4}-\frac{b}{3 x^3}-\frac{\sqrt{-1+a+b x} \sqrt{1+a+b x}}{4 x^4}+\frac{a b \sqrt{-1+a+b x} \sqrt{1+a+b x}}{12 \left (1-a^2\right ) x^3}+\frac{\left (3+2 a^2\right ) b^2 \sqrt{-1+a+b x} \sqrt{1+a+b x}}{24 \left (1-a^2\right )^2 x^2}+\frac{a \left (13+2 a^2\right ) b^3 \sqrt{-1+a+b x} \sqrt{1+a+b x}}{24 \left (1-a^2\right )^3 x}-\frac{\left (1+4 a^2\right ) b^4 \tan ^{-1}\left (\frac{\sqrt{1-a} \sqrt{1+a+b x}}{\sqrt{1+a} \sqrt{-1+a+b x}}\right )}{4 \left (1-a^2\right )^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.254085, size = 198, normalized size = 0.83 \[ \frac{1}{24} \left (-\frac{\sqrt{a+b x-1} \sqrt{a+b x+1} \left (\frac{a \left (2 a^2+13\right ) b^3 x^3}{\left (a^2-1\right )^3}-\frac{\left (2 a^2+3\right ) b^2 x^2}{\left (a^2-1\right )^2}+\frac{2 a b x}{a^2-1}+6\right )}{x^4}-\frac{3 i \left (4 a^2+1\right ) b^4 \log \left (\frac{16 i \left (1-a^2\right )^{5/2} \left (-i \sqrt{1-a^2} \sqrt{a+b x-1} \sqrt{a+b x+1}+a^2+a b x-1\right )}{b^4 \left (4 a^2 x+x\right )}\right )}{\left (1-a^2\right )^{7/2}}-\frac{6 a}{x^4}-\frac{8 b}{x^3}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCosh[a + b*x]/x^5,x]

[Out]

((-6*a)/x^4 - (8*b)/x^3 - (Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(6 + (2*a*b*x)/(-1 + a^2) - ((3 + 2*a^2)*b^2*x
^2)/(-1 + a^2)^2 + (a*(13 + 2*a^2)*b^3*x^3)/(-1 + a^2)^3))/x^4 - ((3*I)*(1 + 4*a^2)*b^4*Log[((16*I)*(1 - a^2)^
(5/2)*(-1 + a^2 + a*b*x - I*Sqrt[1 - a^2]*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]))/(b^4*(x + 4*a^2*x))])/(1 - a^
2)^(7/2))/24

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Maple [B]  time = 0.017, size = 603, normalized size = 2.5 \begin{align*}{\frac{1}{24\, \left ({a}^{2}-1 \right ) ^{4}{x}^{4}}\sqrt{bx+a-1}\sqrt{bx+a+1} \left ( 12\,\sqrt{{a}^{2}-1}\ln \left ( 2\,{\frac{xab+\sqrt{{a}^{2}-1}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}+{a}^{2}-1}{x}} \right ){x}^{4}{a}^{2}{b}^{4}-2\,{x}^{3}{a}^{5}{b}^{3}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}+2\,{x}^{2}{a}^{6}{b}^{2}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}+3\,\sqrt{{a}^{2}-1}\ln \left ( 2\,{\frac{xab+\sqrt{{a}^{2}-1}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}+{a}^{2}-1}{x}} \right ){x}^{4}{b}^{4}-11\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}{x}^{3}{a}^{3}{b}^{3}-2\,x{a}^{7}b\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}-{x}^{2}{a}^{4}{b}^{2}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}-6\,{a}^{8}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}+13\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}{x}^{3}a{b}^{3}+6\,x{a}^{5}b\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}-4\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}{x}^{2}{a}^{2}{b}^{2}+24\,{a}^{6}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}-6\,x{a}^{3}b\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}+3\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}{x}^{2}{b}^{2}-36\,{a}^{4}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}+2\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}xab+24\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}{a}^{2}-6\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1} \right ){\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}}}}-{\frac{a}{4\,{x}^{4}}}-{\frac{b}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)*(12*(a^2-1)^(1/2)*ln(2*(x*a*b+(a^2-1)^(1/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)
+a^2-1)/x)*x^4*a^2*b^4-2*x^3*a^5*b^3*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+2*x^2*a^6*b^2*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)
+3*(a^2-1)^(1/2)*ln(2*(x*a*b+(a^2-1)^(1/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a^2-1)/x)*x^4*b^4-11*(b^2*x^2+2*a*b*x
+a^2-1)^(1/2)*x^3*a^3*b^3-2*x*a^7*b*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)-x^2*a^4*b^2*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)-6*
a^8*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+13*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*x^3*a*b^3+6*x*a^5*b*(b^2*x^2+2*a*b*x+a^2-1)
^(1/2)-4*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*x^2*a^2*b^2+24*a^6*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)-6*x*a^3*b*(b^2*x^2+2*a
*b*x+a^2-1)^(1/2)+3*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*x^2*b^2-36*a^4*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+2*(b^2*x^2+2*a*
b*x+a^2-1)^(1/2)*x*a*b+24*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*a^2-6*(b^2*x^2+2*a*b*x+a^2-1)^(1/2))/(b^2*x^2+2*a*b*x+
a^2-1)^(1/2)/(a^2-1)^4/x^4-1/4*a/x^4-1/3*b/x^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.07963, size = 1337, normalized size = 5.62 \begin{align*} \left [\frac{3 \,{\left (4 \, a^{2} + 1\right )} \sqrt{a^{2} - 1} b^{4} x^{4} \log \left (\frac{a^{2} b x + a^{3} +{\left (a^{2} + \sqrt{a^{2} - 1} a - 1\right )} \sqrt{b x + a + 1} \sqrt{b x + a - 1} +{\left (a b x + a^{2} - 1\right )} \sqrt{a^{2} - 1} - a}{x}\right ) - 6 \, a^{9} -{\left (2 \, a^{5} + 11 \, a^{3} - 13 \, a\right )} b^{4} x^{4} + 24 \, a^{7} - 36 \, a^{5} + 24 \, a^{3} - 8 \,{\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} b x -{\left (6 \, a^{8} +{\left (2 \, a^{5} + 11 \, a^{3} - 13 \, a\right )} b^{3} x^{3} - 24 \, a^{6} -{\left (2 \, a^{6} - a^{4} - 4 \, a^{2} + 3\right )} b^{2} x^{2} + 36 \, a^{4} + 2 \,{\left (a^{7} - 3 \, a^{5} + 3 \, a^{3} - a\right )} b x - 24 \, a^{2} + 6\right )} \sqrt{b x + a + 1} \sqrt{b x + a - 1} - 6 \, a}{24 \,{\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} x^{4}}, -\frac{6 \,{\left (4 \, a^{2} + 1\right )} \sqrt{-a^{2} + 1} b^{4} x^{4} \arctan \left (-\frac{\sqrt{-a^{2} + 1} b x - \sqrt{-a^{2} + 1} \sqrt{b x + a + 1} \sqrt{b x + a - 1}}{a^{2} - 1}\right ) + 6 \, a^{9} +{\left (2 \, a^{5} + 11 \, a^{3} - 13 \, a\right )} b^{4} x^{4} - 24 \, a^{7} + 36 \, a^{5} - 24 \, a^{3} + 8 \,{\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} b x +{\left (6 \, a^{8} +{\left (2 \, a^{5} + 11 \, a^{3} - 13 \, a\right )} b^{3} x^{3} - 24 \, a^{6} -{\left (2 \, a^{6} - a^{4} - 4 \, a^{2} + 3\right )} b^{2} x^{2} + 36 \, a^{4} + 2 \,{\left (a^{7} - 3 \, a^{5} + 3 \, a^{3} - a\right )} b x - 24 \, a^{2} + 6\right )} \sqrt{b x + a + 1} \sqrt{b x + a - 1} + 6 \, a}{24 \,{\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/24*(3*(4*a^2 + 1)*sqrt(a^2 - 1)*b^4*x^4*log((a^2*b*x + a^3 + (a^2 + sqrt(a^2 - 1)*a - 1)*sqrt(b*x + a + 1)*
sqrt(b*x + a - 1) + (a*b*x + a^2 - 1)*sqrt(a^2 - 1) - a)/x) - 6*a^9 - (2*a^5 + 11*a^3 - 13*a)*b^4*x^4 + 24*a^7
- 36*a^5 + 24*a^3 - 8*(a^8 - 4*a^6 + 6*a^4 - 4*a^2 + 1)*b*x - (6*a^8 + (2*a^5 + 11*a^3 - 13*a)*b^3*x^3 - 24*a
^6 - (2*a^6 - a^4 - 4*a^2 + 3)*b^2*x^2 + 36*a^4 + 2*(a^7 - 3*a^5 + 3*a^3 - a)*b*x - 24*a^2 + 6)*sqrt(b*x + a +
1)*sqrt(b*x + a - 1) - 6*a)/((a^8 - 4*a^6 + 6*a^4 - 4*a^2 + 1)*x^4), -1/24*(6*(4*a^2 + 1)*sqrt(-a^2 + 1)*b^4*
x^4*arctan(-(sqrt(-a^2 + 1)*b*x - sqrt(-a^2 + 1)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1))/(a^2 - 1)) + 6*a^9 + (2*
a^5 + 11*a^3 - 13*a)*b^4*x^4 - 24*a^7 + 36*a^5 - 24*a^3 + 8*(a^8 - 4*a^6 + 6*a^4 - 4*a^2 + 1)*b*x + (6*a^8 + (
2*a^5 + 11*a^3 - 13*a)*b^3*x^3 - 24*a^6 - (2*a^6 - a^4 - 4*a^2 + 3)*b^2*x^2 + 36*a^4 + 2*(a^7 - 3*a^5 + 3*a^3
- a)*b*x - 24*a^2 + 6)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) + 6*a)/((a^8 - 4*a^6 + 6*a^4 - 4*a^2 + 1)*x^4)]

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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((b*x+a+(b*x+a-1)**(1/2)*(b*x+a+1)**(1/2))/x**5,x)

[Out]

Timed out

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Giac [B]  time = 2.06225, size = 1148, normalized size = 4.82 \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+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x^5,x, algorithm="giac")

[Out]

1/12*(3*(4*a^2*b^5 + b^5)*arctan(1/2*((sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 - 2*a)/sqrt(-a^2 + 1))/((a^6 -
3*a^4 + 3*a^2 - 1)*sqrt(-a^2 + 1)) - (a*b^5 + 4*b^5)/(a^4 + 4*a^3 + 6*a^2 + 4*a + 1) + 2*(128*a^6*b^5*(sqrt(b
*x + a + 1) - sqrt(b*x + a - 1))^10 + 12*a^2*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^14 - 128*a^7*b^5*(sqr
t(b*x + a + 1) - sqrt(b*x + a - 1))^8 - 168*a^3*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^12 + 448*a^4*b^5*(
sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^10 + 3*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^14 - 1216*a^5*b^5*(s
qrt(b*x + a + 1) - sqrt(b*x + a - 1))^8 - 42*a*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^12 + 512*a^6*b^5*(s
qrt(b*x + a + 1) - sqrt(b*x + a - 1))^6 + 768*a^2*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^10 - 2544*a^3*b^
5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^8 + 5632*a^4*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^6 - 84*b^5*
(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^10 - 1536*a^5*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^4 - 312*a*b^
5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^8 + 1920*a^2*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^6 - 7552*a^
3*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^4 + 1024*a^4*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 + 336
*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^6 - 992*a*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^4 + 5888*a^
2*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 - 256*a^3*b^5 - 192*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1)
)^2 - 1664*a*b^5)/((a^6 - 3*a^4 + 3*a^2 - 1)*((sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^4 - 4*a*(sqrt(b*x + a +
1) - sqrt(b*x + a - 1))^2 + 4)^4) - (4*(b*x + a + 1)*b^5 - a*b^5 - 4*b^5)/(b^4*x^4))/b