3.356 \(\int \frac {e^{\sinh ^{-1}(a+b x)}}{x^4} \, dx\)

Optimal. Leaf size=156 \[ \frac {a b \left (a^2+a b x+1\right ) \sqrt {a^2+2 a b x+b^2 x^2+1}}{2 \left (a^2+1\right )^2 x^2}-\frac {\left (a^2+2 a b x+b^2 x^2+1\right )^{3/2}}{3 \left (a^2+1\right ) x^3}+\frac {a b^3 \tanh ^{-1}\left (\frac {a^2+a b x+1}{\sqrt {a^2+1} \sqrt {a^2+2 a b x+b^2 x^2+1}}\right )}{2 \left (a^2+1\right )^{5/2}}-\frac {a}{3 x^3}-\frac {b}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5907, 14, 730, 720, 724, 206} \[ \frac {a b \left (a^2+a b x+1\right ) \sqrt {a^2+2 a b x+b^2 x^2+1}}{2 \left (a^2+1\right )^2 x^2}-\frac {\left (a^2+2 a b x+b^2 x^2+1\right )^{3/2}}{3 \left (a^2+1\right ) x^3}+\frac {a b^3 \tanh ^{-1}\left (\frac {a^2+a b x+1}{\sqrt {a^2+1} \sqrt {a^2+2 a b x+b^2 x^2+1}}\right )}{2 \left (a^2+1\right )^{5/2}}-\frac {a}{3 x^3}-\frac {b}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcSinh[a + b*x]/x^4,x]

[Out]

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

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 206

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

Rule 720

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] /; FreeQ[
{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 724

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 730

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[(2*c*d - b*e)/(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, 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] && EqQ[m + 2*p + 3, 0]

Rule 5907

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

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 162, normalized size = 1.04 \[ \frac {1}{6} \left (-\frac {3 a b^3 \log (x)}{\left (a^2+1\right )^{5/2}}+\frac {3 a b^3 \log \left (\sqrt {a^2+1} \sqrt {a^2+2 a b x+b^2 x^2+1}+a^2+a b x+1\right )}{\left (a^2+1\right )^{5/2}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2+1} \left (2 a^4+a^3 b x+a^2 \left (4-b^2 x^2\right )+a b x+2 b^2 x^2+2\right )}{\left (a^2+1\right )^2 x^3}-\frac {2 a}{x^3}-\frac {3 b}{x^2}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSinh[a + b*x]/x^4,x]

[Out]

((-2*a)/x^3 - (3*b)/x^2 - (Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(2 + 2*a^4 + a*b*x + a^3*b*x + 2*b^2*x^2 + a^2*(4
- b^2*x^2)))/((1 + a^2)^2*x^3) - (3*a*b^3*Log[x])/(1 + a^2)^(5/2) + (3*a*b^3*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)^(5/2))/6

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fricas [A]  time = 0.51, size = 230, normalized size = 1.47 \[ \frac {3 \, \sqrt {a^{2} + 1} a b^{3} x^{3} \log \left (-\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + \sqrt {a^{2} + 1} a + 1\right )} + {\left (a b x + a^{2} + 1\right )} \sqrt {a^{2} + 1} + a}{x}\right ) - 2 \, a^{7} + {\left (a^{4} - a^{2} - 2\right )} b^{3} x^{3} - 6 \, a^{5} - 6 \, a^{3} - 3 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} b x - {\left (2 \, a^{6} - {\left (a^{4} - a^{2} - 2\right )} b^{2} x^{2} + 6 \, a^{4} + {\left (a^{5} + 2 \, a^{3} + a\right )} b x + 6 \, a^{2} + 2\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 2 \, a}{6 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a+(1+(b*x+a)^2)^(1/2))/x^4,x, algorithm="fricas")

[Out]

1/6*(3*sqrt(a^2 + 1)*a*b^3*x^3*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) - 2*a^7 + (a^4 - a^2 - 2)*b^3*x^3 - 6*a^5 - 6*a^3 - 3*(a^6 + 3*
a^4 + 3*a^2 + 1)*b*x - (2*a^6 - (a^4 - a^2 - 2)*b^2*x^2 + 6*a^4 + (a^5 + 2*a^3 + a)*b*x + 6*a^2 + 2)*sqrt(b^2*
x^2 + 2*a*b*x + a^2 + 1) - 2*a)/((a^6 + 3*a^4 + 3*a^2 + 1)*x^3)

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giac [B]  time = 0.53, size = 715, normalized size = 4.58 \[ -\frac {a b^{3} \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 )}{2 \, {\left (a^{4} + 2 \, a^{2} + 1\right )} \sqrt {a^{2} + 1}} - \frac {3 \, b x + 2 \, a}{6 \, x^{3}} + \frac {20 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a^{5} b^{3} + 12 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{7} b^{3} + 6 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{4} a^{4} b^{2} {\left | b \right |} + 24 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{6} b^{2} {\left | b \right |} + 2 \, a^{8} b^{2} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{5} a b^{3} + 32 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a^{3} b^{3} + 33 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{5} b^{3} + 12 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{4} a^{2} b^{2} {\left | b \right |} + 48 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{4} b^{2} {\left | b \right |} + 8 \, a^{6} b^{2} {\left | b \right |} + 12 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a b^{3} + 30 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{3} b^{3} + 6 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{4} b^{2} {\left | b \right |} + 24 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{2} b^{2} {\left | b \right |} + 12 \, a^{4} b^{2} {\left | b \right |} + 9 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a b^{3} + 8 \, a^{2} b^{2} {\left | b \right |} + 2 \, b^{2} {\left | b \right |}}{3 \, {\left (a^{4} + 2 \, 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 )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a+(1+(b*x+a)^2)^(1/2))/x^4,x, algorithm="giac")

[Out]

-1/2*a*b^3*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*abs(b) + 2*sq
rt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 2*sqrt(a^2 + 1)))/((a^4 + 2*a^2 + 1)*sqrt(a^2 + 1)) - 1/6*(3*b*x + 2*a)/x^3
+ 1/3*(20*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3*a^5*b^3 + 12*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a
^2 + 1))*a^7*b^3 + 6*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^4*a^4*b^2*abs(b) + 24*(x*abs(b) - sqrt(b^2
*x^2 + 2*a*b*x + a^2 + 1))^2*a^6*b^2*abs(b) + 2*a^8*b^2*abs(b) + 3*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 +
1))^5*a*b^3 + 32*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3*a^3*b^3 + 33*(x*abs(b) - sqrt(b^2*x^2 + 2*a*
b*x + a^2 + 1))*a^5*b^3 + 12*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^4*a^2*b^2*abs(b) + 48*(x*abs(b) -
sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2*a^4*b^2*abs(b) + 8*a^6*b^2*abs(b) + 12*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x
+ a^2 + 1))^3*a*b^3 + 30*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*a^3*b^3 + 6*(x*abs(b) - sqrt(b^2*x^2
+ 2*a*b*x + a^2 + 1))^4*b^2*abs(b) + 24*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2*a^2*b^2*abs(b) + 12*a
^4*b^2*abs(b) + 9*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*a*b^3 + 8*a^2*b^2*abs(b) + 2*b^2*abs(b))/((a^
4 + 2*a^2 + 1)*((x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2 - a^2 - 1)^3)

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maple [B]  time = 0.01, size = 501, normalized size = 3.21 \[ -\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{3 \left (a^{2}+1\right ) x^{3}}+\frac {a b \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{2 \left (a^{2}+1\right )^{2} x^{2}}-\frac {a^{2} b^{2} \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{2 \left (a^{2}+1\right )^{3} x}+\frac {a^{3} b^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right )^{3}}+\frac {a^{4} b^{4} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 \left (a^{2}+1\right )^{3} \sqrt {b^{2}}}-\frac {a^{3} b^{3} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {a^{2} b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x}{2 \left (a^{2}+1\right )^{3}}+\frac {a^{2} b^{4} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 \left (a^{2}+1\right )^{3} \sqrt {b^{2}}}-\frac {a \,b^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right )^{2}}-\frac {a^{2} b^{4} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 \left (a^{2}+1\right )^{2} \sqrt {b^{2}}}+\frac {a \,b^{3} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {a}{3 x^{3}}-\frac {b}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.56, size = 352, normalized size = 2.26 \[ -\frac {a^{3} b^{3} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{2 \, {\left (a^{2} + 1\right )}^{\frac {5}{2}}} + \frac {a b^{3} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{2 \, {\left (a^{2} + 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a b^{3}}{2 \, {\left (a^{2} + 1\right )}^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} b^{2}}{2 \, {\left (a^{2} + 1\right )}^{2} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a b}{2 \, {\left (a^{2} + 1\right )}^{2} x^{2}} - \frac {b}{2 \, x^{2}} - \frac {a}{3 \, x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (a^{2} + 1\right )} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a+(1+(b*x+a)^2)^(1/2))/x^4,x, algorithm="maxima")

[Out]

-1/2*a^3*b^3*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) + 1/2*a*b^3*arcsinh(2*a*b*x/(s
qrt(-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) - 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a*b^3/(a^2 + 1)^2 - 1
/2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^2*b^2/((a^2 + 1)^2*x) + 1/2*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a*b/((a
^2 + 1)^2*x^2) - 1/2*b/x^2 - 1/3*a/x^3 - 1/3*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)/((a^2 + 1)*x^3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+\sqrt {{\left (a+b\,x\right )}^2+1}+b\,x}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b x + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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