3.26.3 \(\int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x^2 \sqrt {-b+a x}} \, dx\)

Optimal. Leaf size=208 \[ \frac {\sqrt {a x-b} \sqrt {\sqrt {a x-b}+a x}}{b x}-\frac {a \text {RootSum}\left [\text {$\#$1}^4+2 \text {$\#$1}^2 b-4 \text {$\#$1} b+b^2+b\& ,\frac {\text {$\#$1}^2 \left (-\log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )\right )+4 \text {$\#$1} b \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )-b \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )}{\text {$\#$1}^3+\text {$\#$1} b-b}\& \right ]}{4 b} \]

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Rubi [A]  time = 1.04, antiderivative size = 198, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {976, 1021, 1036, 1030, 208, 205} \begin {gather*} -\frac {a \left (2 \sqrt {b}+1\right ) \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {a x-b}}{\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a x-b}+a x}}\right )}{2 \sqrt {2} b^{5/4}}-\frac {a \left (1-2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {a x-b}+\sqrt {b}}{\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a x-b}+a x}}\right )}{2 \sqrt {2} b^{5/4}}+\frac {\left (\sqrt {a x-b}+a x\right )^{3/2}}{b x}-\frac {a \sqrt {\sqrt {a x-b}+a x}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x + Sqrt[-b + a*x]]/(x^2*Sqrt[-b + a*x]),x]

[Out]

-((a*Sqrt[a*x + Sqrt[-b + a*x]])/b) + (a*x + Sqrt[-b + a*x])^(3/2)/(b*x) - (a*(1 + 2*Sqrt[b])*ArcTan[(Sqrt[b]
- Sqrt[-b + a*x])/(Sqrt[2]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a*x]])])/(2*Sqrt[2]*b^(5/4)) - (a*(1 - 2*Sqrt[b])*ArcT
anh[(Sqrt[b] + Sqrt[-b + a*x])/(Sqrt[2]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a*x]])])/(2*Sqrt[2]*b^(5/4))

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 208

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

Rule 976

Int[((a_.) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[((2*a*c^2*e + c*(2
*c^2*d - c*(2*a*f))*x)*(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1))/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p +
1)), x] - Dist[1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)), Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Si
mp[2*c*((c*d - a*f)^2 - (-(a*e))*(c*e))*(p + 1) - (2*c^2*d - c*(2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(-2*a*
c^2*e)*(p + q + 2) + (2*f*(2*a*c^2*e)*(p + q + 2) - (2*c^2*d - c*(2*a*f))*(-(c*e*(2*p + q + 4))))*x + c*f*(2*c
^2*d - c*(2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, q}, x] && NeQ[e^2 - 4*d*f, 0] && Lt
Q[p, -1] && NeQ[a*c*e^2 + (c*d - a*f)^2, 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 1021

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp
[(h*(a + b*x + c*x^2)^p*(d + f*x^2)^(q + 1))/(2*f*(p + q + 1)), x] - Dist[1/(2*f*(p + q + 1)), Int[(a + b*x +
c*x^2)^(p - 1)*(d + f*x^2)^q*Simp[h*p*(b*d) + a*(-2*g*f)*(p + q + 1) + (2*h*p*(c*d - a*f) + b*(-2*g*f)*(p + q
+ 1))*x + (h*p*(-(b*f)) + c*(-2*g*f)*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}, x] && NeQ
[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0]

Rule 1030

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*
a*g*h, Subst[Int[1/Simp[2*a^2*g*h*c + a*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; F
reeQ[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]

Rule 1036

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Dist[1/(2*q), Int[Simp[-(a*h*e) - g*(c*d - a*f - q) + (h*(c*d - a*f + q) -
g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[Simp[-(a*h*e) - g*(c*d - a*f + q
) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g,
h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[-(a*c)]

Rubi steps

\begin {align*} \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x^2 \sqrt {-b+a x}} \, dx &=(2 a) \operatorname {Subst}\left (\int \frac {\sqrt {b+x+x^2}}{\left (b+x^2\right )^2} \, dx,x,\sqrt {-b+a x}\right )\\ &=\frac {\left (a x+\sqrt {-b+a x}\right )^{3/2}}{b x}-\frac {a \operatorname {Subst}\left (\int \frac {(-b+2 b x) \sqrt {b+x+x^2}}{b+x^2} \, dx,x,\sqrt {-b+a x}\right )}{2 b^2}\\ &=-\frac {a \sqrt {a x+\sqrt {-b+a x}}}{b}+\frac {\left (a x+\sqrt {-b+a x}\right )^{3/2}}{b x}+\frac {a \operatorname {Subst}\left (\int \frac {2 b^2+b x}{\left (b+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{2 b^2}\\ &=-\frac {a \sqrt {a x+\sqrt {-b+a x}}}{b}+\frac {\left (a x+\sqrt {-b+a x}\right )^{3/2}}{b x}+\frac {a \operatorname {Subst}\left (\int \frac {-\left (\left (1-2 \sqrt {b}\right ) b^2\right )+\left (1-2 \sqrt {b}\right ) b^{3/2} x}{\left (b+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{4 b^{5/2}}-\frac {a \operatorname {Subst}\left (\int \frac {-\left (\left (1+2 \sqrt {b}\right ) b^2\right )-\left (1+2 \sqrt {b}\right ) b^{3/2} x}{\left (b+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{4 b^{5/2}}\\ &=-\frac {a \sqrt {a x+\sqrt {-b+a x}}}{b}+\frac {\left (a x+\sqrt {-b+a x}\right )^{3/2}}{b x}+\frac {1}{2} \left (a \left (1-2 \sqrt {b}\right )^2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-2 \left (1-2 \sqrt {b}\right )^2 b^{11/2}+b x^2} \, dx,x,\frac {\left (1-2 \sqrt {b}\right ) b^2 \left (\sqrt {b}+\sqrt {-b+a x}\right )}{\sqrt {a x+\sqrt {-b+a x}}}\right )+\frac {1}{2} \left (a \left (1+2 \sqrt {b}\right )^2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (1+2 \sqrt {b}\right )^2 b^{11/2}+b x^2} \, dx,x,\frac {\left (1+2 \sqrt {b}\right ) b^2 \left (-\sqrt {b}+\sqrt {-b+a x}\right )}{\sqrt {a x+\sqrt {-b+a x}}}\right )\\ &=-\frac {a \sqrt {a x+\sqrt {-b+a x}}}{b}+\frac {\left (a x+\sqrt {-b+a x}\right )^{3/2}}{b x}-\frac {a \left (1+2 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {-b+a x}}{\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{2 \sqrt {2} b^{5/4}}-\frac {a \left (1-2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {b}+\sqrt {-b+a x}}{\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{2 \sqrt {2} b^{5/4}}\\ \end {align*}

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Mathematica [A]  time = 0.55, size = 216, normalized size = 1.04 \begin {gather*} \frac {4 b \sqrt {a x-b} \sqrt {\sqrt {a x-b}+a x}+a \left (\sqrt {-b}-2 b\right ) \sqrt [4]{-b} x \tan ^{-1}\left (\frac {\left (2 \sqrt {-b}-1\right ) \sqrt {a x-b}-2 b+\sqrt {-b}}{2 \sqrt [4]{-b} \sqrt {\sqrt {a x-b}+a x}}\right )+a \sqrt [4]{-b} \left (2 b+\sqrt {-b}\right ) x \tanh ^{-1}\left (\frac {\left (2 \sqrt {-b}+1\right ) \sqrt {a x-b}+2 b+\sqrt {-b}}{2 \sqrt [4]{-b} \sqrt {\sqrt {a x-b}+a x}}\right )}{4 b^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*x + Sqrt[-b + a*x]]/(x^2*Sqrt[-b + a*x]),x]

[Out]

(4*b*Sqrt[-b + a*x]*Sqrt[a*x + Sqrt[-b + a*x]] + a*(Sqrt[-b] - 2*b)*(-b)^(1/4)*x*ArcTan[(Sqrt[-b] - 2*b + (-1
+ 2*Sqrt[-b])*Sqrt[-b + a*x])/(2*(-b)^(1/4)*Sqrt[a*x + Sqrt[-b + a*x]])] + a*(-b)^(1/4)*(Sqrt[-b] + 2*b)*x*Arc
Tanh[(Sqrt[-b] + 2*b + (1 + 2*Sqrt[-b])*Sqrt[-b + a*x])/(2*(-b)^(1/4)*Sqrt[a*x + Sqrt[-b + a*x]])])/(4*b^2*x)

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IntegrateAlgebraic [A]  time = 0.45, size = 338, normalized size = 1.62 \begin {gather*} \frac {\sqrt {-b+a x} \sqrt {a x+\sqrt {-b+a x}}}{b x}-a \text {RootSum}\left [b+b^2-4 b \text {$\#$1}+2 b \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {-5 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right )+2 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}}{-b+b \text {$\#$1}+\text {$\#$1}^3}\&\right ]+\frac {a \text {RootSum}\left [b+b^2-4 b \text {$\#$1}+2 b \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {-19 b \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right )+4 b \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-b+b \text {$\#$1}+\text {$\#$1}^3}\&\right ]}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[a*x + Sqrt[-b + a*x]]/(x^2*Sqrt[-b + a*x]),x]

[Out]

(Sqrt[-b + a*x]*Sqrt[a*x + Sqrt[-b + a*x]])/(b*x) - a*RootSum[b + b^2 - 4*b*#1 + 2*b*#1^2 + #1^4 & , (-5*Log[-
Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1] + 2*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#
1)/(-b + b*#1 + #1^3) & ] + (a*RootSum[b + b^2 - 4*b*#1 + 2*b*#1^2 + #1^4 & , (-19*b*Log[-Sqrt[-b + a*x] + Sqr
t[a*x + Sqrt[-b + a*x]] - #1] + 4*b*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1 + Log[-Sqrt[-b +
a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^2)/(-b + b*#1 + #1^3) & ])/(4*b)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [B]  time = 0.28, size = 221, normalized size = 1.06 \begin {gather*} \frac {2 \, a^{2} b {\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}^{2} - a^{2} {\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}^{3} + 2 \, a^{2} b^{2} + 5 \, a^{2} b {\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )} + a^{2} b}{{\left ({\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}^{4} + 2 \, b {\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}^{2} + b^{2} + 4 \, b {\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )} + b\right )} a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 1.22, size = 1580, normalized size = 7.60

method	result	size
derivativedivides	\(2 a \left (-\frac {-\frac {\left (\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}\right )^{\frac {3}{2}}}{\sqrt {-b}\, \left (\sqrt {a x -b}-\sqrt {-b}\right )}+\frac {\left (1+2 \sqrt {-b}\right ) \left (\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}+\frac {\left (1+2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}\right )}{2}-\left (-b \right )^{\frac {1}{4}} \ln \left (\frac {2 \sqrt {-b}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+2 \left (-b \right )^{\frac {1}{4}} \sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}}{\sqrt {a x -b}-\sqrt {-b}}\right )\right )}{2 \sqrt {-b}}+\frac {\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}}{2}+\frac {\left (4 \sqrt {-b}-\left (1+2 \sqrt {-b}\right )^{2}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}\right )}{4}}{\sqrt {-b}}}{4 b}+\frac {\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}+\frac {\left (1+2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}\right )}{2}-\left (-b \right )^{\frac {1}{4}} \ln \left (\frac {2 \sqrt {-b}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+2 \left (-b \right )^{\frac {1}{4}} \sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}}{\sqrt {a x -b}-\sqrt {-b}}\right )}{4 b \sqrt {-b}}-\frac {\frac {\left (\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}\right )^{\frac {3}{2}}}{\sqrt {-b}\, \left (\sqrt {a x -b}+\sqrt {-b}\right )}-\frac {\left (1-2 \sqrt {-b}\right ) \left (\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}+\frac {\left (1-2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}\right )}{2}+\frac {\sqrt {-b}\, \ln \left (\frac {-2 \sqrt {-b}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )+2 \sqrt {-\sqrt {-b}}\, \sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}}{\sqrt {a x -b}+\sqrt {-b}}\right )}{\sqrt {-\sqrt {-b}}}\right )}{2 \sqrt {-b}}-\frac {2 \left (\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}}{4}+\frac {\left (-4 \sqrt {-b}-\left (1-2 \sqrt {-b}\right )^{2}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}\right )}{8}\right )}{\sqrt {-b}}}{4 b}-\frac {\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}+\frac {\left (1-2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}\right )}{2}+\frac {\sqrt {-b}\, \ln \left (\frac {-2 \sqrt {-b}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )+2 \sqrt {-\sqrt {-b}}\, \sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}}{\sqrt {a x -b}+\sqrt {-b}}\right )}{\sqrt {-\sqrt {-b}}}}{4 b \sqrt {-b}}\right )\)	\(1580\)
default	\(2 a \left (-\frac {-\frac {\left (\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}\right )^{\frac {3}{2}}}{\sqrt {-b}\, \left (\sqrt {a x -b}-\sqrt {-b}\right )}+\frac {\left (1+2 \sqrt {-b}\right ) \left (\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}+\frac {\left (1+2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}\right )}{2}-\left (-b \right )^{\frac {1}{4}} \ln \left (\frac {2 \sqrt {-b}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+2 \left (-b \right )^{\frac {1}{4}} \sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}}{\sqrt {a x -b}-\sqrt {-b}}\right )\right )}{2 \sqrt {-b}}+\frac {\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}}{2}+\frac {\left (4 \sqrt {-b}-\left (1+2 \sqrt {-b}\right )^{2}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}\right )}{4}}{\sqrt {-b}}}{4 b}+\frac {\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}+\frac {\left (1+2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}\right )}{2}-\left (-b \right )^{\frac {1}{4}} \ln \left (\frac {2 \sqrt {-b}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+2 \left (-b \right )^{\frac {1}{4}} \sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}}{\sqrt {a x -b}-\sqrt {-b}}\right )}{4 b \sqrt {-b}}-\frac {\frac {\left (\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}\right )^{\frac {3}{2}}}{\sqrt {-b}\, \left (\sqrt {a x -b}+\sqrt {-b}\right )}-\frac {\left (1-2 \sqrt {-b}\right ) \left (\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}+\frac {\left (1-2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}\right )}{2}+\frac {\sqrt {-b}\, \ln \left (\frac {-2 \sqrt {-b}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )+2 \sqrt {-\sqrt {-b}}\, \sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}}{\sqrt {a x -b}+\sqrt {-b}}\right )}{\sqrt {-\sqrt {-b}}}\right )}{2 \sqrt {-b}}-\frac {2 \left (\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}}{4}+\frac {\left (-4 \sqrt {-b}-\left (1-2 \sqrt {-b}\right )^{2}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}\right )}{8}\right )}{\sqrt {-b}}}{4 b}-\frac {\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}+\frac {\left (1-2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}\right )}{2}+\frac {\sqrt {-b}\, \ln \left (\frac {-2 \sqrt {-b}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )+2 \sqrt {-\sqrt {-b}}\, \sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}}{\sqrt {a x -b}+\sqrt {-b}}\right )}{\sqrt {-\sqrt {-b}}}}{4 b \sqrt {-b}}\right )\)	\(1580\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x + \sqrt {a x - b}}}{\sqrt {a x - b} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*x + sqrt(a*x - b))/(sqrt(a*x - b)*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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