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

Optimal. Leaf size=229 \[ -\frac {1}{4} a \text {RootSum}\left [\text {$\#$1}^4+2 \text {$\#$1}^2 b-4 \text {$\#$1} b+b^2+b\& ,\frac {-2 \text {$\#$1}^2 \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )+2 \text {$\#$1} \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )+2 b \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )-\log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )}{\text {$\#$1}^3+\text {$\#$1} b-b}\& \right ]-\frac {\sqrt {\sqrt {a x-b}+a x}}{x} \]

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Rubi [A]  time = 0.48, antiderivative size = 171, normalized size of antiderivative = 0.75, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1014, 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^{3/4}}+\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^{3/4}}-\frac {\sqrt {\sqrt {a x-b}+a x}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x + Sqrt[-b + a*x]]/x^2,x]

[Out]

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

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 1014

Int[((g_.) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp
[((a*h - g*c*x)*(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q)/(2*a*c*(p + 1)), x] + Dist[2/(4*a*c*(p + 1)), Int[(a
+ c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[g*c*d*(2*p + 3) - a*(h*e*q) + (g*c*e*(2*p + q + 3) - a*(2*h*f*
q))*x + g*c*f*(2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ
[p, -1] && GtQ[q, 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} \, dx &=(2 a) \operatorname {Subst}\left (\int \frac {x \sqrt {b+x+x^2}}{\left (b+x^2\right )^2} \, dx,x,\sqrt {-b+a x}\right )\\ &=-\frac {\sqrt {a x+\sqrt {-b+a x}}}{x}-\frac {a \operatorname {Subst}\left (\int \frac {-\frac {b}{2}-b x}{\left (b+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{b}\\ &=-\frac {\sqrt {a x+\sqrt {-b+a x}}}{x}-\frac {a \operatorname {Subst}\left (\int \frac {-\frac {1}{2} \left (1-2 \sqrt {b}\right ) b^{3/2}+\frac {1}{2} \left (1-2 \sqrt {b}\right ) b x}{\left (b+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{2 b^{3/2}}+\frac {a \operatorname {Subst}\left (\int \frac {\frac {1}{2} \left (1+2 \sqrt {b}\right ) b^{3/2}+\frac {1}{2} \left (1+2 \sqrt {b}\right ) b x}{\left (b+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{2 b^{3/2}}\\ &=-\frac {\sqrt {a x+\sqrt {-b+a x}}}{x}-\frac {1}{4} \left (a \left (1-2 \sqrt {b}\right )^2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} \left (1-2 \sqrt {b}\right )^2 b^{9/2}+b x^2} \, dx,x,\frac {\left (1-2 \sqrt {b}\right ) b^{3/2} \left (\sqrt {b}+\sqrt {-b+a x}\right )}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )-\frac {1}{4} \left (a \left (1+2 \sqrt {b}\right )^2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2} \left (1+2 \sqrt {b}\right )^2 b^{9/2}+b x^2} \, dx,x,\frac {\left (1+2 \sqrt {b}\right ) b^{3/2} \left (\sqrt {b}-\sqrt {-b+a x}\right )}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )\\ &=-\frac {\sqrt {a x+\sqrt {-b+a x}}}{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^{3/4}}+\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^{3/4}}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 204, normalized size = 0.89 \begin {gather*} \frac {-4 (-b)^{3/4} \sqrt {\sqrt {a x-b}+a x}+a \left (2 \sqrt {-b}-1\right ) x \tan ^{-1}\left (\frac {\left (1-2 \sqrt {-b}\right ) \sqrt {a x-b}+2 b-\sqrt {-b}}{2 \sqrt [4]{-b} \sqrt {\sqrt {a x-b}+a x}}\right )-a \left (2 \sqrt {-b}+1\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)^{3/4} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(-b)^(3/4)*Sqrt[a*x + Sqrt[-b + a*x]] + a*(-1 + 2*Sqrt[-b])*x*ArcTan[(-Sqrt[-b] + 2*b + (1 - 2*Sqrt[-b])*S
qrt[-b + a*x])/(2*(-b)^(1/4)*Sqrt[a*x + Sqrt[-b + a*x]])] - a*(1 + 2*Sqrt[-b])*x*ArcTanh[(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)^(3/4)*x)

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IntegrateAlgebraic [A]  time = 0.48, size = 430, normalized size = 1.88 \begin {gather*} -\frac {\sqrt {a x+\sqrt {-b+a x}}}{x}-a \text {RootSum}\left [1-b+b^2+4 \text {$\#$1}+6 \text {$\#$1}^2+2 b \text {$\#$1}^2+4 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {b \log \left (-1-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right )-\log \left (-1-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{1+3 \text {$\#$1}+b \text {$\#$1}+3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ]+a \text {RootSum}\left [1-8 b+16 b^2+4 \text {$\#$1}-16 b \text {$\#$1}+6 \text {$\#$1}^2+8 b \text {$\#$1}^2+4 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right )+4 b \log \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right )+4 \log \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}-\log \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{1-4 b+3 \text {$\#$1}+4 b \text {$\#$1}+3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[a*x + Sqrt[-b + a*x]]/x) - a*RootSum[1 - b + b^2 + 4*#1 + 6*#1^2 + 2*b*#1^2 + 4*#1^3 + #1^4 & , (b*Log[
-1 - Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1] - Log[-1 - Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]]
- #1]*#1^2)/(1 + 3*#1 + b*#1 + 3*#1^2 + #1^3) & ] + a*RootSum[1 - 8*b + 16*b^2 + 4*#1 - 16*b*#1 + 6*#1^2 + 8*b
*#1^2 + 4*#1^3 + #1^4 & , (-Log[-1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]] - #1] + 4*b*Log[-1 - 2*Sq
rt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]] - #1] + 4*Log[-1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]]
- #1]*#1 - Log[-1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^2)/(1 - 4*b + 3*#1 + 4*b*#1 + 3*
#1^2 + #1^3) & ]

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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,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.99, size = 240, normalized size = 1.05 \begin {gather*} \frac {2 \, a^{2} {\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}^{3} + 2 \, a^{2} b {\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )} + 3 \, a^{2} {\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}^{2} + a^{2} b + a^{2} {\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}}{{\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} \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,x, algorithm="giac")

[Out]

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

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maple [B]  time = 1.06, size = 1056, normalized size = 4.61

method	result	size
derivativedivides	\(2 a \left (\frac {\sqrt {-b}\, \left (\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}}\right )}{4 b}-\frac {\sqrt {-b}\, \left (-\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}}\right )}{4 b}\right )\)	\(1056\)
default	\(2 a \left (\frac {\sqrt {-b}\, \left (\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}}\right )}{4 b}-\frac {\sqrt {-b}\, \left (-\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}}\right )}{4 b}\right )\)	\(1056\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*(1/4*(-b)^(1/2)/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)^(1/2)/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)))))

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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}}}{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,x, algorithm="maxima")

[Out]

integrate(sqrt(a*x + 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} \,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,x)

[Out]

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

[Out]

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

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