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

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

[Out]

-1/4*a/x^4-1/3*b/x^3-1/4*(4*a^2+1)*b^4*arctan((1-a)^(1/2)*(b*x+a+1)^(1/2)/(1+a)^(1/2)/(b*x+a-1)^(1/2))/(-a^2+1
)^(7/2)-1/4*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)/x^4+1/12*a*b*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)/(-a^2+1)/x^3+1/24*(2*
a^2+3)*b^2*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)/(-a^2+1)^2/x^2+1/24*a*(2*a^2+13)*b^3*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2
)/(-a^2+1)^3/x

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Rubi [A]  time = 0.20, antiderivative size = 238, normalized size of antiderivative = 1.00, 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 12

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

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]

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*}

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Mathematica [C]  time = 0.28, size = 198, normalized size = 0.83 \[ \frac {1}{24} \left (-\frac {3 i \left (4 a^2+1\right ) b^4 \log \left (\frac {16 i \left (1-a^2\right )^{5/2} \left (-i \sqrt {1-a^2} \sqrt {a+b x-1} \sqrt {a+b x+1}+a^2+a b x-1\right )}{b^4 \left (4 a^2 x+x\right )}\right )}{\left (1-a^2\right )^{7/2}}-\frac {\sqrt {a+b x-1} \sqrt {a+b x+1} \left (\frac {a \left (2 a^2+13\right ) b^3 x^3}{\left (a^2-1\right )^3}-\frac {\left (2 a^2+3\right ) b^2 x^2}{\left (a^2-1\right )^2}+\frac {2 a b x}{a^2-1}+6\right )}{x^4}-\frac {6 a}{x^4}-\frac {8 b}{x^3}\right ) \]

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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fricas [A]  time = 0.95, size = 569, normalized size = 2.39 \[ \left [\frac {3 \, {\left (4 \, a^{2} + 1\right )} \sqrt {a^{2} - 1} b^{4} x^{4} \log \left (\frac {a^{2} b x + a^{3} + {\left (a^{2} + \sqrt {a^{2} - 1} a - 1\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) - 6 \, a^{9} - {\left (2 \, a^{5} + 11 \, a^{3} - 13 \, a\right )} b^{4} x^{4} + 24 \, a^{7} - 36 \, a^{5} + 24 \, a^{3} - 8 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} b x - {\left (6 \, a^{8} + {\left (2 \, a^{5} + 11 \, a^{3} - 13 \, a\right )} b^{3} x^{3} - 24 \, a^{6} - {\left (2 \, a^{6} - a^{4} - 4 \, a^{2} + 3\right )} b^{2} x^{2} + 36 \, a^{4} + 2 \, {\left (a^{7} - 3 \, a^{5} + 3 \, a^{3} - a\right )} b x - 24 \, a^{2} + 6\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} - 6 \, a}{24 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} x^{4}}, -\frac {6 \, {\left (4 \, a^{2} + 1\right )} \sqrt {-a^{2} + 1} b^{4} x^{4} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {-a^{2} + 1} \sqrt {b x + a + 1} \sqrt {b x + a - 1}}{a^{2} - 1}\right ) + 6 \, a^{9} + {\left (2 \, a^{5} + 11 \, a^{3} - 13 \, a\right )} b^{4} x^{4} - 24 \, a^{7} + 36 \, a^{5} - 24 \, a^{3} + 8 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} b x + {\left (6 \, a^{8} + {\left (2 \, a^{5} + 11 \, a^{3} - 13 \, a\right )} b^{3} x^{3} - 24 \, a^{6} - {\left (2 \, a^{6} - a^{4} - 4 \, a^{2} + 3\right )} b^{2} x^{2} + 36 \, a^{4} + 2 \, {\left (a^{7} - 3 \, a^{5} + 3 \, a^{3} - a\right )} b x - 24 \, a^{2} + 6\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + 6 \, a}{24 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} x^{4}}\right ] \]

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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giac [B]  time = 1.00, size = 817, normalized size = 3.43 \[ \frac {\frac {3 \, {\left (4 \, a^{2} b^{5} + b^{5}\right )} \arctan \left (\frac {{\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} - 2 \, a}{2 \, \sqrt {-a^{2} + 1}}\right )}{{\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} \sqrt {-a^{2} + 1}} + \frac {2 \, {\left (128 \, a^{6} b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{10} + 12 \, a^{2} b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{14} - 128 \, a^{7} b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{8} - 168 \, a^{3} b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{12} + 448 \, a^{4} b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{10} + 3 \, b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{14} - 1216 \, a^{5} b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{8} - 42 \, a b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{12} + 512 \, a^{6} b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} + 768 \, a^{2} b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{10} - 2544 \, a^{3} b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{8} + 5632 \, a^{4} b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} - 84 \, b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{10} - 1536 \, a^{5} b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} - 312 \, a b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{8} + 1920 \, a^{2} b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} - 7552 \, a^{3} b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} + 1024 \, a^{4} b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} + 336 \, b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} - 992 \, a b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} + 5888 \, a^{2} b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} - 256 \, a^{3} b^{5} - 192 \, b^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} - 1664 \, a b^{5}\right )}}{{\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} {\left ({\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} - 4 \, a {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} + 4\right )}^{4}} - \frac {4 \, {\left (b x + a + 1\right )} b^{5} - a b^{5} - 4 \, b^{5}}{b^{4} x^{4}}}{12 \, b} \]

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)) + 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*(sqrt(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*(sqrt(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*(sqrt(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 - 9
92*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)*((sqr
t(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

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maple [B]  time = 0.02, size = 603, normalized size = 2.53 \[ \frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (12 \sqrt {a^{2}-1}\, \ln \left (\frac {2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+2 a^{2}-2}{x}\right ) x^{4} a^{2} b^{4}-2 x^{3} a^{5} b^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+2 x^{2} a^{6} b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+3 \sqrt {a^{2}-1}\, \ln \left (\frac {2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+2 a^{2}-2}{x}\right ) x^{4} b^{4}-11 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, x^{3} a^{3} b^{3}-2 x \,a^{7} b \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-x^{2} a^{4} b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-6 a^{8} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+13 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, x^{3} a \,b^{3}+6 x \,a^{5} b \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-4 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, x^{2} a^{2} b^{2}+24 a^{6} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-6 x \,a^{3} b \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+3 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, x^{2} b^{2}-36 a^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, x a b +24 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{2}-6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\right )}{24 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{4} x^{4}}-\frac {b}{3 x^{3}}-\frac {a}{4 x^{4}} \]

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*(a*b*x+(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*(a*b*x+(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^4-1/3*b/x^3-1/4*a/x^4

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

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a-1>0)', see `assume?` for mor
e details)Is a-1 zero or nonzero?

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mupad [B]  time = 28.03, size = 2347, normalized size = 9.86 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(((a - 1)^(1/2) - (a + b*x - 1)^(1/2))/((a + 1)^(1/2) - (a + b*x + 1)^(1/2)))*(b^4*(a - 1)^(1/2)*(a + 1)^(
1/2) + 4*a^2*b^4*(a - 1)^(1/2)*(a + 1)^(1/2)))/(48*a^4 - 32*a^2 - 32*a^6 + 8*a^8 + 8) - (a/4 + (b*x)/3)/x^4 -
(log(((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^2/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^2 - a^2 - (a^2*((a - 1)^(1/
2) - (a + b*x - 1)^(1/2))^2)/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^2 + (2*a*((a - 1)^(1/2) - (a + b*x - 1)^(1/
2))*(a - 1)^(1/2)*(a + 1)^(1/2))/((a + 1)^(1/2) - (a + b*x + 1)^(1/2)) + 1)*(b^4*(a - 1)^(1/2)*(a + 1)^(1/2) +
4*a^2*b^4*(a - 1)^(1/2)*(a + 1)^(1/2)))/(48*a^4 - 32*a^2 - 32*a^6 + 8*a^8 + 8) - ((((a - 1)^(1/2) - (a + b*x
- 1)^(1/2))^3*((17*a*b^4)/192 - (5*a^3*b^4)/192))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^3*(3*a^2 - 3*a^4 + a^
6 - 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^11*((7*a^3*b^4)/64 - (81*a*b^4)/128 + (3*a^5*b^4)/128))/(((a
+ 1)^(1/2) - (a + b*x + 1)^(1/2))^11*(6*a^4 - 4*a^2 - 4*a^6 + a^8 + 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2
))^5*((229*a^3*b^4)/64 - (119*a*b^4)/128 + (119*a^5*b^4)/384))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^5*(6*a^4
- 4*a^2 - 4*a^6 + a^8 + 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^9*((1025*a*b^4)/384 - (1745*a^3*b^4)/128
+ (385*a^5*b^4)/128 + (239*a^7*b^4)/384))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^9*(5*a^2 - 10*a^4 + 10*a^6 -
5*a^8 + a^10 - 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^7*((1103*a*b^4)/384 - (2199*a^3*b^4)/128 + (1039*
a^5*b^4)/128 + (521*a^7*b^4)/384))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^7*(5*a^2 - 10*a^4 + 10*a^6 - 5*a^8 +
a^10 - 1)) - (b^4*(a - 1)^(1/2)*(a + 1)^(1/2))/(1024*(a^4 - 2*a^2 + 1)) + (a*b^4*((a - 1)^(1/2) - (a + b*x -
1)^(1/2)))/(192*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))*(a^4 - 2*a^2 + 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/
2))^2*(b^4/128 - (a^2*b^4)/384)*(a - 1)^(1/2)*(a + 1)^(1/2))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^2*(3*a^2 -
3*a^4 + a^6 - 1)) - (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^4*(a - 1)^(1/2)*(a + 1)^(1/2)*((11*b^4)/512 + (249
*a^2*b^4)/256 + (a^4*b^4)/1536))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^4*(6*a^4 - 4*a^2 - 4*a^6 + a^8 + 1)) +
(((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^10*(a - 1)^(1/2)*(a + 1)^(1/2)*(b^4/256 + (1167*a^2*b^4)/256 - (225*a^
4*b^4)/256 - (47*a^6*b^4)/256))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^10*(5*a^2 - 10*a^4 + 10*a^6 - 5*a^8 + a
^10 - 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^6*(a - 1)^(1/2)*(a + 1)^(1/2)*((7*b^4)/256 + (1665*a^2*b^4)
/256 - (5365*a^4*b^4)/768 - (707*a^6*b^4)/768))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^6*(5*a^2 - 10*a^4 + 10*
a^6 - 5*a^8 + a^10 - 1)) - (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^8*(a - 1)^(1/2)*(a + 1)^(1/2)*((8765*a^2*b^4
)/768 - (239*b^4)/1024 - (11789*a^4*b^4)/512 + (1471*a^6*b^4)/256 + (3635*a^8*b^4)/3072))/(((a + 1)^(1/2) - (a
+ b*x + 1)^(1/2))^8*(15*a^4 - 6*a^2 - 20*a^6 + 15*a^8 - 6*a^10 + a^12 + 1)))/(((a - 1)^(1/2) - (a + b*x - 1)^
(1/2))^4/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^4 + ((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^12/((a + 1)^(1/2) - (
a + b*x + 1)^(1/2))^12 + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^6*(28*a^2 - 4))/((a^2 - 1)*((a + 1)^(1/2) - (a
+ b*x + 1)^(1/2))^6) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^10*(28*a^2 - 4))/((a^2 - 1)*((a + 1)^(1/2) - (a
+ b*x + 1)^(1/2))^10) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^8*(70*a^4 - 60*a^2 + 6))/(((a + 1)^(1/2) - (a
+ b*x + 1)^(1/2))^8*(a^4 - 2*a^2 + 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^7*(24*a - 56*a^3)*(a - 1)^(1/2
)*(a + 1)^(1/2))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^7*(a^4 - 2*a^2 + 1)) + (((a - 1)^(1/2) - (a + b*x - 1)
^(1/2))^9*(24*a - 56*a^3)*(a - 1)^(1/2)*(a + 1)^(1/2))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^9*(a^4 - 2*a^2 +
1)) - (8*a*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^5*(a - 1)^(1/2)*(a + 1)^(1/2))/((a^2 - 1)*((a + 1)^(1/2) - (
a + b*x + 1)^(1/2))^5) - (8*a*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^11*(a - 1)^(1/2)*(a + 1)^(1/2))/((a^2 - 1)
*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^11)) - (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^2*((5*a*(a^2 - 1)^2*((a*b
^4)/(32*(a - 1)^2*(a + 1)^2) - (5*a*b^4*(a^2 - 1)^3)/(128*(a - 1)^5*(a + 1)^5)))/((a - 1)^(5/2)*(a + 1)^(5/2))
- (b^4*(23*a^2 - 7))/(512*(a - 1)^(5/2)*(a + 1)^(5/2)) + (5*b^4*(a^2 - 1)*(9*a^2 - 1))/(512*(a - 1)^(7/2)*(a
+ 1)^(7/2))))/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^2 - (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^3*((a*b^4)/(96*
(a - 1)^2*(a + 1)^2) - (5*a*b^4*(a^2 - 1)^3)/(384*(a - 1)^5*(a + 1)^5)))/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))
^3 + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))*((5*((a*b^4)/(32*(a - 1)^2*(a + 1)^2) - (5*a*b^4*(a^2 - 1)^3)/(128
*(a - 1)^5*(a + 1)^5))*(9*a^2 - 1))/((a - 1)*(a + 1)) - (a*b^4)/(16*(a - 1)^2*(a + 1)^2) - (10*a*(a^2 - 1)^2*(
(10*a*(a^2 - 1)^2*((a*b^4)/(32*(a - 1)^2*(a + 1)^2) - (5*a*b^4*(a^2 - 1)^3)/(128*(a - 1)^5*(a + 1)^5)))/((a -
1)^(5/2)*(a + 1)^(5/2)) - (b^4*(23*a^2 - 7))/(256*(a - 1)^(5/2)*(a + 1)^(5/2)) + (5*b^4*(a^2 - 1)*(9*a^2 - 1))
/(256*(a - 1)^(7/2)*(a + 1)^(7/2))))/((a - 1)^(5/2)*(a + 1)^(5/2)) + (5*a*b^4*(a^2 - 1)*(3*a^4 - 4*a^2 + 1))/(
32*(a - 1)^5*(a + 1)^5)))/((a + 1)^(1/2) - (a + b*x + 1)^(1/2)) + (b^4*(a^2 - 1)*((a - 1)^(1/2) - (a + b*x - 1
)^(1/2))^4)/(1024*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^4*(a - 1)^(5/2)*(a + 1)^(5/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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