3.4.57 \(\int \frac {e^{\sinh ^{-1}(a+b x)}}{x^5} \, dx\) [357]

Optimal. Leaf size=207 \[ -\frac {a}{4 x^4}-\frac {b}{3 x^3}+\frac {\left (1-4 a^2\right ) b^2 \left (1+a^2+a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}}{8 \left (1+a^2\right )^3 x^2}-\frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 \left (1+a^2\right ) x^4}+\frac {5 a b \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{12 \left (1+a^2\right )^2 x^3}+\frac {\left (1-4 a^2\right ) b^4 \tanh ^{-1}\left (\frac {1+a^2+a b x}{\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}}\right )}{8 \left (1+a^2\right )^{7/2}} \]

[Out]

-1/4*a/x^4-1/3*b/x^3-1/4*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)/(a^2+1)/x^4+5/12*a*b*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)/(a^2
+1)^2/x^3+1/8*(-4*a^2+1)*b^4*arctanh((a*b*x+a^2+1)/(a^2+1)^(1/2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(a^2+1)^(7/2)+
1/8*(-4*a^2+1)*b^2*(a*b*x+a^2+1)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)/(a^2+1)^3/x^2

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Rubi [A]
time = 0.13, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5878, 14, 758, 820, 734, 738, 212} \begin {gather*} \frac {\left (1-4 a^2\right ) b^2 \left (a^2+a b x+1\right ) \sqrt {a^2+2 a b x+b^2 x^2+1}}{8 \left (a^2+1\right )^3 x^2}-\frac {\left (a^2+2 a b x+b^2 x^2+1\right )^{3/2}}{4 \left (a^2+1\right ) x^4}+\frac {5 a b \left (a^2+2 a b x+b^2 x^2+1\right )^{3/2}}{12 \left (a^2+1\right )^2 x^3}+\frac {\left (1-4 a^2\right ) b^4 \tanh ^{-1}\left (\frac {a^2+a b x+1}{\sqrt {a^2+1} \sqrt {a^2+2 a b x+b^2 x^2+1}}\right )}{8 \left (a^2+1\right )^{7/2}}-\frac {a}{4 x^4}-\frac {b}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcSinh[a + b*x]/x^5,x]

[Out]

-1/4*a/x^4 - b/(3*x^3) + ((1 - 4*a^2)*b^2*(1 + a^2 + a*b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])/(8*(1 + a^2)^3*
x^2) - (1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(4*(1 + a^2)*x^4) + (5*a*b*(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(12
*(1 + a^2)^2*x^3) + ((1 - 4*a^2)*b^4*ArcTanh[(1 + a^2 + a*b*x)/(Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2
])])/(8*(1 + a^2)^(7/2))

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))
*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[p*((b^2
- 4*a*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m
+ 2*p + 2, 0] && GtQ[p, 0]

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 758

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*
((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e
, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 5878

Int[E^(ArcSinh[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(u + Sqrt[1 + u^2])^n, x] /; RationalQ[m] && Intege
rQ[n] && PolyQ[u, x]

Rubi steps

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

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Mathematica [A]
time = 0.51, size = 192, normalized size = 0.93 \begin {gather*} \frac {1}{24} \left (-\frac {6 a}{x^4}-\frac {8 b}{x^3}-\frac {\sqrt {1+a^2+2 a b x+b^2 x^2} \left (6+\frac {2 a b x}{1+a^2}-\frac {\left (-3+2 a^2\right ) b^2 x^2}{\left (1+a^2\right )^2}+\frac {a \left (-13+2 a^2\right ) b^3 x^3}{\left (1+a^2\right )^3}\right )}{x^4}+\frac {3 (-1+2 a) (1+2 a) b^4 \log (x)}{\left (1+a^2\right )^{7/2}}-\frac {3 (-1+2 a) (1+2 a) b^4 \log \left (1+a^2+a b x+\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}\right )}{\left (1+a^2\right )^{7/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^ArcSinh[a + b*x]/x^5,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1002\) vs. \(2(183)=366\).
time = 0.23, size = 1003, normalized size = 4.85

method	result	size
default	\(-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{4 \left (a^{2}+1\right ) x^{4}}-\frac {5 a b \left (-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{3 \left (a^{2}+1\right ) x^{3}}-\frac {a b \left (-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{2 \left (a^{2}+1\right ) x^{2}}-\frac {a b \left (-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{\left (a^{2}+1\right ) x}+\frac {a b \left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+\frac {a b \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\sqrt {a^{2}+1}\, \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )\right )}{a^{2}+1}+\frac {2 b^{2} \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{a^{2}+1}\right )}{2 \left (a^{2}+1\right )}+\frac {b^{2} \left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+\frac {a b \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\sqrt {a^{2}+1}\, \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )\right )}{2 a^{2}+2}\right )}{a^{2}+1}\right )}{4 \left (a^{2}+1\right )}-\frac {b^{2} \left (-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{2 \left (a^{2}+1\right ) x^{2}}-\frac {a b \left (-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{\left (a^{2}+1\right ) x}+\frac {a b \left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+\frac {a b \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\sqrt {a^{2}+1}\, \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )\right )}{a^{2}+1}+\frac {2 b^{2} \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{a^{2}+1}\right )}{2 \left (a^{2}+1\right )}+\frac {b^{2} \left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+\frac {a b \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\sqrt {a^{2}+1}\, \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )\right )}{2 a^{2}+2}\right )}{4 \left (a^{2}+1\right )}-\frac {b}{3 x^{3}}-\frac {a}{4 x^{4}}\)	\(1003\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a+(1+(b*x+a)^2)^(1/2))/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)/(a^2+1)/x^4-5/4*a*b/(a^2+1)*(-1/3/(a^2+1)/x^3*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)
-a*b/(a^2+1)*(-1/2/(a^2+1)/x^2*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)-1/2*a*b/(a^2+1)*(-1/(a^2+1)/x*(b^2*x^2+2*a*b*x+a^
2+1)^(3/2)+a*b/(a^2+1)*((b^2*x^2+2*a*b*x+a^2+1)^(1/2)+a*b*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(
1/2))/(b^2)^(1/2)-(a^2+1)^(1/2)*ln((2*a^2+2+2*a*b*x+2*(a^2+1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x))+2*b^2/(
a^2+1)*(1/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*ln((b^2*x+a*b)
/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)))+1/2*b^2/(a^2+1)*((b^2*x^2+2*a*b*x+a^2+1)^(1/2)+a*b*l
n((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)-(a^2+1)^(1/2)*ln((2*a^2+2+2*a*b*x+2*(a^2+
1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x))))-1/4*b^2/(a^2+1)*(-1/2/(a^2+1)/x^2*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)-
1/2*a*b/(a^2+1)*(-1/(a^2+1)/x*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)+a*b/(a^2+1)*((b^2*x^2+2*a*b*x+a^2+1)^(1/2)+a*b*ln(
(b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)-(a^2+1)^(1/2)*ln((2*a^2+2+2*a*b*x+2*(a^2+1)
^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x))+2*b^2/(a^2+1)*(1/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)
+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)))+1/2
*b^2/(a^2+1)*((b^2*x^2+2*a*b*x+a^2+1)^(1/2)+a*b*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2
)^(1/2)-(a^2+1)^(1/2)*ln((2*a^2+2+2*a*b*x+2*(a^2+1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x)))-1/3*b/x^3-1/4*a/
x^4

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (183) = 366\).
time = 0.26, size = 594, normalized size = 2.87 \begin {gather*} \frac {5 \, a^{4} b^{4} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{8 \, {\left (a^{2} + 1\right )}^{\frac {7}{2}}} - \frac {3 \, a^{2} b^{4} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{4 \, {\left (a^{2} + 1\right )}^{\frac {5}{2}}} + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} b^{4}}{8 \, {\left (a^{2} + 1\right )}^{3}} + \frac {b^{4} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{8 \, {\left (a^{2} + 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b^{4}}{8 \, {\left (a^{2} + 1\right )}^{2}} + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3} b^{3}}{8 \, {\left (a^{2} + 1\right )}^{3} x} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a b^{3}}{8 \, {\left (a^{2} + 1\right )}^{2} x} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a^{2} b^{2}}{8 \, {\left (a^{2} + 1\right )}^{3} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} b^{2}}{8 \, {\left (a^{2} + 1\right )}^{2} x^{2}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a b}{12 \, {\left (a^{2} + 1\right )}^{2} x^{3}} - \frac {b}{3 \, x^{3}} - \frac {a}{4 \, x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{4 \, {\left (a^{2} + 1\right )} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/8*a^4*b^4*arcsinh(2*a*b*x/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)
*b^2)*abs(x)) + 2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)))/(a^2 + 1)^(7/2) - 3/4*a^2*b^4*arcsinh(2*a*b*x/(
sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2/(sqrt(-4*a^
2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)))/(a^2 + 1)^(5/2) + 5/8*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^2*b^4/(a^2 + 1)^3
+ 1/8*b^4*arcsinh(2*a*b*x/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b
^2)*abs(x)) + 2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)))/(a^2 + 1)^(3/2) - 1/8*sqrt(b^2*x^2 + 2*a*b*x + a^
2 + 1)*b^4/(a^2 + 1)^2 + 5/8*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^3*b^3/((a^2 + 1)^3*x) - 1/8*sqrt(b^2*x^2 + 2*
a*b*x + a^2 + 1)*a*b^3/((a^2 + 1)^2*x) - 5/8*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a^2*b^2/((a^2 + 1)^3*x^2) + 1
/8*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*b^2/((a^2 + 1)^2*x^2) + 5/12*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a*b/((
a^2 + 1)^2*x^3) - 1/3*b/x^3 - 1/4*a/x^4 - 1/4*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)/((a^2 + 1)*x^4)

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Fricas [A]
time = 0.36, size = 295, normalized size = 1.43 \begin {gather*} \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} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} - \sqrt {a^{2} + 1} a + 1\right )} - {\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^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 6 \, a}{24 \, {\left (a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1\right )} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b x + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1173 vs. \(2 (183) = 366\).
time = 0.45, size = 1173, normalized size = 5.67 \begin {gather*} \frac {{\left (4 \, a^{2} b^{4} - b^{4}\right )} \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{8 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} \sqrt {a^{2} + 1}} - \frac {4 \, b x + 3 \, a}{12 \, x^{4}} + \frac {32 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{5} a^{6} b^{4} + 256 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a^{8} b^{4} + 96 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{10} b^{4} + 144 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{4} a^{7} b^{3} {\left | b \right |} + 224 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{9} b^{3} {\left | b \right |} + 16 \, a^{11} b^{3} {\left | b \right |} - 12 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{7} a^{2} b^{4} + 140 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{5} a^{4} b^{4} + 716 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a^{6} b^{4} + 372 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{8} b^{4} + 432 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{4} a^{5} b^{3} {\left | b \right |} + 704 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{7} b^{3} {\left | b \right |} + 80 \, a^{9} b^{3} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{7} b^{4} + 129 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{5} a^{2} b^{4} + 685 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a^{4} b^{4} + 543 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{6} b^{4} + 432 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{4} a^{3} b^{3} {\left | b \right |} + 768 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{5} b^{3} {\left | b \right |} + 160 \, a^{7} b^{3} {\left | b \right |} + 21 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{5} b^{4} + 246 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a^{2} b^{4} + 357 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{4} b^{4} + 144 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{4} a b^{3} {\left | b \right |} + 320 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{3} b^{3} {\left | b \right |} + 160 \, a^{5} b^{3} {\left | b \right |} + 21 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} b^{4} + 93 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{2} b^{4} + 32 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a b^{3} {\left | b \right |} + 80 \, a^{3} b^{3} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} b^{4} + 16 \, a b^{3} {\left | b \right |}}{12 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} {\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} - a^{2} - 1\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*a^2*b^4 - b^4)*log(abs(-2*x*abs(b) + 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) - 2*sqrt(a^2 + 1))/abs(-2*x*ab
s(b) + 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 2*sqrt(a^2 + 1)))/((a^6 + 3*a^4 + 3*a^2 + 1)*sqrt(a^2 + 1)) - 1/1
2*(4*b*x + 3*a)/x^4 + 1/12*(32*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^5*a^6*b^4 + 256*(x*abs(b) - sqrt
(b^2*x^2 + 2*a*b*x + a^2 + 1))^3*a^8*b^4 + 96*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*a^10*b^4 + 144*(x
*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^4*a^7*b^3*abs(b) + 224*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 +
1))^2*a^9*b^3*abs(b) + 16*a^11*b^3*abs(b) - 12*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^7*a^2*b^4 + 140
*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^5*a^4*b^4 + 716*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))
^3*a^6*b^4 + 372*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*a^8*b^4 + 432*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b
*x + a^2 + 1))^4*a^5*b^3*abs(b) + 704*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2*a^7*b^3*abs(b) + 80*a^9
*b^3*abs(b) + 3*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^7*b^4 + 129*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x
+ a^2 + 1))^5*a^2*b^4 + 685*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3*a^4*b^4 + 543*(x*abs(b) - sqrt(b^
2*x^2 + 2*a*b*x + a^2 + 1))*a^6*b^4 + 432*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^4*a^3*b^3*abs(b) + 76
8*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2*a^5*b^3*abs(b) + 160*a^7*b^3*abs(b) + 21*(x*abs(b) - sqrt(b
^2*x^2 + 2*a*b*x + a^2 + 1))^5*b^4 + 246*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3*a^2*b^4 + 357*(x*abs
(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*a^4*b^4 + 144*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^4*a*b^3*
abs(b) + 320*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2*a^3*b^3*abs(b) + 160*a^5*b^3*abs(b) + 21*(x*abs(
b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3*b^4 + 93*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*a^2*b^4 + 32
*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2*a*b^3*abs(b) + 80*a^3*b^3*abs(b) + 3*(x*abs(b) - sqrt(b^2*x^
2 + 2*a*b*x + a^2 + 1))*b^4 + 16*a*b^3*abs(b))/((a^6 + 3*a^4 + 3*a^2 + 1)*((x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x
+ a^2 + 1))^2 - a^2 - 1)^4)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+\sqrt {{\left (a+b\,x\right )}^2+1}+b\,x}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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