3.281 \(\int \frac {e^{\cosh ^{-1}(a+b x)}}{x^3} \, dx\)

Optimal. Leaf size=138 \[ -\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {a+b x-1}}\right )}{\left (1-a^2\right )^{3/2}}+\frac {b \sqrt {a+b x-1} \sqrt {a+b x+1}}{2 \left (1-a^2\right ) x}-\frac {\sqrt {a+b x-1} (a+b x+1)^{3/2}}{2 (a+1) x^2}-\frac {a}{2 x^2}-\frac {b}{x} \]

[Out]

-1/2*a/x^2-b/x-b^2*arctan((1-a)^(1/2)*(b*x+a+1)^(1/2)/(1+a)^(1/2)/(b*x+a-1)^(1/2))/(-a^2+1)^(3/2)-1/2*(b*x+a+1
)^(3/2)*(b*x+a-1)^(1/2)/(1+a)/x^2+1/2*b*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)/(-a^2+1)/x

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Rubi [A]  time = 0.11, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5909, 14, 94, 93, 205} \[ -\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {a+b x-1}}\right )}{\left (1-a^2\right )^{3/2}}+\frac {b \sqrt {a+b x-1} \sqrt {a+b x+1}}{2 \left (1-a^2\right ) x}-\frac {\sqrt {a+b x-1} (a+b x+1)^{3/2}}{2 (a+1) x^2}-\frac {a}{2 x^2}-\frac {b}{x} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCosh[a + b*x]/x^3,x]

[Out]

-a/(2*x^2) - b/x + (b*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(2*(1 - a^2)*x) - (Sqrt[-1 + a + b*x]*(1 + a + b*x
)^(3/2))/(2*(1 + a)*x^2) - (b^2*ArcTan[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[-1 + a + b*x])])/(1 -
a^2)^(3/2)

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 94

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 + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

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^3} \, dx &=\int \frac {a+b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}}{x^3} \, dx\\ &=\int \left (\frac {a}{x^3}+\frac {b}{x^2}+\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x}}{x^3}\right ) \, dx\\ &=-\frac {a}{2 x^2}-\frac {b}{x}+\int \frac {\sqrt {-1+a+b x} \sqrt {1+a+b x}}{x^3} \, dx\\ &=-\frac {a}{2 x^2}-\frac {b}{x}-\frac {\sqrt {-1+a+b x} (1+a+b x)^{3/2}}{2 (1+a) x^2}+\frac {b \int \frac {\sqrt {1+a+b x}}{x^2 \sqrt {-1+a+b x}} \, dx}{2 (1+a)}\\ &=-\frac {a}{2 x^2}-\frac {b}{x}+\frac {b \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x}-\frac {\sqrt {-1+a+b x} (1+a+b x)^{3/2}}{2 (1+a) x^2}+\frac {b^2 \int \frac {1}{x \sqrt {-1+a+b x} \sqrt {1+a+b x}} \, dx}{2 \left (1-a^2\right )}\\ &=-\frac {a}{2 x^2}-\frac {b}{x}+\frac {b \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x}-\frac {\sqrt {-1+a+b x} (1+a+b x)^{3/2}}{2 (1+a) x^2}+\frac {b^2 \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^2}\\ &=-\frac {a}{2 x^2}-\frac {b}{x}+\frac {b \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x}-\frac {\sqrt {-1+a+b x} (1+a+b x)^{3/2}}{2 (1+a) x^2}-\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {-1+a+b x}}\right )}{\left (1-a^2\right )^{3/2}}\\ \end {align*}

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Mathematica [C]  time = 0.27, size = 142, normalized size = 1.03 \[ \frac {1}{2} \left (-\frac {i b^2 \log \left (\frac {4 i \sqrt {1-a^2} \left (-i \sqrt {1-a^2} \sqrt {a+b x-1} \sqrt {a+b x+1}+a^2+a b x-1\right )}{b^2 x}\right )}{\left (1-a^2\right )^{3/2}}-\frac {\sqrt {a+b x-1} \sqrt {a+b x+1} \left (a^2+a b x-1\right )}{\left (a^2-1\right ) x^2}-\frac {a}{x^2}-\frac {2 b}{x}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCosh[a + b*x]/x^3,x]

[Out]

(-(a/x^2) - (2*b)/x - (Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(-1 + a^2 + a*b*x))/((-1 + a^2)*x^2) - (I*b^2*Log[
((4*I)*Sqrt[1 - a^2]*(-1 + a^2 + a*b*x - I*Sqrt[1 - a^2]*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]))/(b^2*x)])/(1 -
a^2)^(3/2))/2

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fricas [A]  time = 0.47, size = 338, normalized size = 2.45 \[ \left [\frac {\sqrt {a^{2} - 1} b^{2} x^{2} \log \left (\frac {a^{2} b x + a^{3} + {\left (a^{2} + \sqrt {a^{2} - 1} a - 1\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) - a^{5} - {\left (a^{3} - a\right )} b^{2} x^{2} + 2 \, a^{3} - 2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} b x - {\left (a^{4} + {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} - a}{2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2}}, -\frac {2 \, \sqrt {-a^{2} + 1} b^{2} x^{2} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {-a^{2} + 1} \sqrt {b x + a + 1} \sqrt {b x + a - 1}}{a^{2} - 1}\right ) + a^{5} + {\left (a^{3} - a\right )} b^{2} x^{2} - 2 \, a^{3} + 2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} b x + {\left (a^{4} + {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + a}{2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2}}\right ] \]

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^3,x, algorithm="fricas")

[Out]

[1/2*(sqrt(a^2 - 1)*b^2*x^2*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) - a^5 - (a^3 - a)*b^2*x^2 + 2*a^3 - 2*(a^4 - 2*a^2 + 1)*b*x - (a^
4 + (a^3 - a)*b*x - 2*a^2 + 1)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) - a)/((a^4 - 2*a^2 + 1)*x^2), -1/2*(2*sqrt(
-a^2 + 1)*b^2*x^2*arctan(-(sqrt(-a^2 + 1)*b*x - sqrt(-a^2 + 1)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1))/(a^2 - 1))
+ a^5 + (a^3 - a)*b^2*x^2 - 2*a^3 + 2*(a^4 - 2*a^2 + 1)*b*x + (a^4 + (a^3 - a)*b*x - 2*a^2 + 1)*sqrt(b*x + a
+ 1)*sqrt(b*x + a - 1) + a)/((a^4 - 2*a^2 + 1)*x^2)]

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giac [B]  time = 0.33, size = 326, normalized size = 2.36 \[ \frac {\frac {2 \, b^{3} \arctan \left (\frac {{\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} - 2 \, a}{2 \, \sqrt {-a^{2} + 1}}\right )}{{\left (a^{2} - 1\right )} \sqrt {-a^{2} + 1}} + \frac {4 \, {\left (2 \, a^{2} b^{3} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} - 4 \, a^{3} b^{3} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} - b^{3} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} - 2 \, a b^{3} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} + 8 \, a^{2} b^{3} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} + 4 \, b^{3} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} - 8 \, a b^{3}\right )}}{{\left ({\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} - 4 \, a {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} + 4\right )}^{2} {\left (a^{2} - 1\right )}} - \frac {2 \, {\left (b x + a + 1\right )} b^{3} - a b^{3} - 2 \, b^{3}}{b^{2} x^{2}}}{2 \, b} \]

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^3,x, algorithm="giac")

[Out]

1/2*(2*b^3*arctan(1/2*((sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 - 2*a)/sqrt(-a^2 + 1))/((a^2 - 1)*sqrt(-a^2 +
1)) + 4*(2*a^2*b^3*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^6 - 4*a^3*b^3*(sqrt(b*x + a + 1) - sqrt(b*x + a -
1))^4 - b^3*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^6 - 2*a*b^3*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^4 + 8*
a^2*b^3*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 + 4*b^3*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 - 8*a*b^3)
/(((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)^2*(a^2 - 1))
- (2*(b*x + a + 1)*b^3 - a*b^3 - 2*b^3)/(b^2*x^2))/b

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maple [B]  time = 0.02, size = 236, normalized size = 1.71 \[ \frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (\sqrt {a^{2}-1}\, \ln \left (\frac {2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+2 a^{2}-2}{x}\right ) x^{2} b^{2}-x \,a^{3} b \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-a^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, x a b +2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{2}-\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\right )}{2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{2} x^{2}}-\frac {b}{x}-\frac {a}{2 x^{2}} \]

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^3,x)

[Out]

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

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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^3,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 = 14.15, size = 958, normalized size = 6.94 \[ \frac {b^2\,\ln \left (\frac {\sqrt {a-1}-\sqrt {a+b\,x-1}}{\sqrt {a+1}-\sqrt {a+b\,x+1}}\right )\,\sqrt {a-1}\,\sqrt {a+1}}{2\,a^4-4\,a^2+2}-\frac {\frac {a\,b^2\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^5}{8\,\left (a^2-1\right )\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^5}-\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^3\,\left (\frac {3\,a\,b^2}{8}-\frac {7\,a^3\,b^2}{8}\right )}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^3\,\left (a^4-2\,a^2+1\right )}-\frac {b^2\,\sqrt {a-1}\,\sqrt {a+1}}{32\,\left (a^2-1\right )}+\frac {a\,b^2\,\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}{4\,\left (a^2-1\right )\,\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2\,\left (\frac {b^2}{16}-\frac {11\,a^2\,b^2}{16}\right )\,\sqrt {a-1}\,\sqrt {a+1}}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2\,\left (a^4-2\,a^2+1\right )}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^4\,\sqrt {a-1}\,\sqrt {a+1}\,\left (-\frac {17\,a^4\,b^2}{32}+\frac {9\,a^2\,b^2}{16}+\frac {15\,b^2}{32}\right )}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^4\,\left (a^6-3\,a^4+3\,a^2-1\right )}}{\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^6}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^6}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^4\,\left (6\,a^2-2\right )}{\left (a^2-1\right )\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^4}-\frac {4\,a\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^3\,\sqrt {a-1}\,\sqrt {a+1}}{\left (a^2-1\right )\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^3}-\frac {4\,a\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^5\,\sqrt {a-1}\,\sqrt {a+1}}{\left (a^2-1\right )\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^5}}-\frac {\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )\,\left (\frac {a\,b^2}{2\,\left (a-1\right )\,\left (a+1\right )}-\frac {3\,a\,b^2\,{\left (a^2-1\right )}^2}{8\,{\left (a-1\right )}^3\,{\left (a+1\right )}^3}\right )}{\sqrt {a+1}-\sqrt {a+b\,x+1}}-\frac {b^2\,\ln \left (\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2}-a^2-\frac {a^2\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2}+\frac {2\,a\,\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )\,\sqrt {a-1}\,\sqrt {a+1}}{\sqrt {a+1}-\sqrt {a+b\,x+1}}+1\right )\,\sqrt {a-1}\,\sqrt {a+1}}{2\,a^4-4\,a^2+2}-\frac {\frac {a}{2}+b\,x}{x^2}+\frac {b^2\,\left (a^2-1\right )\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2}{32\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2\,{\left (a-1\right )}^{3/2}\,{\left (a+1\right )}^{3/2}} \]

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^3,x)

[Out]

(b^2*log(((a - 1)^(1/2) - (a + b*x - 1)^(1/2))/((a + 1)^(1/2) - (a + b*x + 1)^(1/2)))*(a - 1)^(1/2)*(a + 1)^(1
/2))/(2*a^4 - 4*a^2 + 2) - ((a*b^2*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^5)/(8*(a^2 - 1)*((a + 1)^(1/2) - (a +
b*x + 1)^(1/2))^5) - (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^3*((3*a*b^2)/8 - (7*a^3*b^2)/8))/(((a + 1)^(1/2)
- (a + b*x + 1)^(1/2))^3*(a^4 - 2*a^2 + 1)) - (b^2*(a - 1)^(1/2)*(a + 1)^(1/2))/(32*(a^2 - 1)) + (a*b^2*((a -
1)^(1/2) - (a + b*x - 1)^(1/2)))/(4*(a^2 - 1)*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))) + (((a - 1)^(1/2) - (a +
b*x - 1)^(1/2))^2*(b^2/16 - (11*a^2*b^2)/16)*(a - 1)^(1/2)*(a + 1)^(1/2))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2
))^2*(a^4 - 2*a^2 + 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^4*(a - 1)^(1/2)*(a + 1)^(1/2)*((15*b^2)/32 +
(9*a^2*b^2)/16 - (17*a^4*b^2)/32))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^4*(3*a^2 - 3*a^4 + a^6 - 1)))/(((a -
1)^(1/2) - (a + b*x - 1)^(1/2))^2/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^2 + ((a - 1)^(1/2) - (a + b*x - 1)^(1
/2))^6/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^6 + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^4*(6*a^2 - 2))/((a^2 -
1)*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^4) - (4*a*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^3*(a - 1)^(1/2)*(a +
1)^(1/2))/((a^2 - 1)*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^3) - (4*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)) - (((a - 1)^(1/2) - (a + b*x
- 1)^(1/2))*((a*b^2)/(2*(a - 1)*(a + 1)) - (3*a*b^2*(a^2 - 1)^2)/(8*(a - 1)^3*(a + 1)^3)))/((a + 1)^(1/2) - (
a + b*x + 1)^(1/2)) - (b^2*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)*(a - 1)^
(1/2)*(a + 1)^(1/2))/(2*a^4 - 4*a^2 + 2) - (a/2 + b*x)/x^2 + (b^2*(a^2 - 1)*((a - 1)^(1/2) - (a + b*x - 1)^(1/
2))^2)/(32*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^2*(a - 1)^(3/2)*(a + 1)^(3/2))

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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**3,x)

[Out]

Timed out

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