3.29.27 \(\int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x \sqrt {-b+a x}} \, dx\)

Optimal. Leaf size=286 \[ -2 \text {RootSum}\left [\text {$\#$1}^4+4 \text {$\#$1}^3+8 \text {$\#$1}^2 b+6 \text {$\#$1}^2-16 \text {$\#$1} b+4 \text {$\#$1}+16 b^2-8 b+1\& ,\frac {\text {$\#$1}^2 \left (-\log \left (\text {$\#$1}+2 \sqrt {a x-b}-2 \sqrt {\sqrt {a x-b}+a x}+1\right )\right )-2 \text {$\#$1} \log \left (\text {$\#$1}+2 \sqrt {a x-b}-2 \sqrt {\sqrt {a x-b}+a x}+1\right )+4 b \log \left (\text {$\#$1}+2 \sqrt {a x-b}-2 \sqrt {\sqrt {a x-b}+a x}+1\right )-\log \left (\text {$\#$1}+2 \sqrt {a x-b}-2 \sqrt {\sqrt {a x-b}+a x}+1\right )}{\text {$\#$1}^3+3 \text {$\#$1}^2+4 \text {$\#$1} b+3 \text {$\#$1}-4 b+1}\& \right ]-2 \log \left (2 \sqrt {a x-b}-2 \sqrt {\sqrt {a x-b}+a x}+1\right ) \]

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Rubi [A]  time = 0.78, antiderivative size = 164, normalized size of antiderivative = 0.57, number of steps used = 9, number of rules used = 7, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {990, 621, 206, 1036, 1030, 208, 205} \begin {gather*} -\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {a x-b}}{\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt [4]{b}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a x-b}+\sqrt {b}}{\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt [4]{b}}+2 \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 621

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

Rule 990

Int[Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]/((d_) + (f_.)*(x_)^2), x_Symbol] :> Dist[c/f, Int[1/Sqrt[a + b*x +
c*x^2], x], x] - Dist[1/f, Int[(c*d - a*f - b*f*x)/(Sqrt[a + b*x + c*x^2]*(d + f*x^2)), x], x] /; FreeQ[{a, b,
c, d, f}, x] && NeQ[b^2 - 4*a*c, 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 \sqrt {-b+a x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {\sqrt {b+x+x^2}}{b+x^2} \, dx,x,\sqrt {-b+a x}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )+2 \operatorname {Subst}\left (\int \frac {x}{\left (b+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )-\frac {\operatorname {Subst}\left (\int \frac {-b-\sqrt {b} x}{\left (b+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{\sqrt {b}}+\frac {\operatorname {Subst}\left (\int \frac {-b+\sqrt {b} x}{\left (b+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{\sqrt {b}}\\ &=2 \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )+\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-2 b^{7/2}+b x^2} \, dx,x,\frac {b^{3/2}+b \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )+\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{2 b^{7/2}+b x^2} \, dx,x,\frac {-b^{3/2}+b \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )\\ &=-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {-b+a x}}{\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{\sqrt [4]{b}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {b}+\sqrt {-b+a x}}{\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{\sqrt [4]{b}}+2 \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 182, normalized size = 0.64 \begin {gather*} 2 \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )-\frac {\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 )+\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 )}{\sqrt [4]{-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.23, size = 166, normalized size = 0.58 \begin {gather*} -2 \log \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}\right )-\text {RootSum}\left [b+b^2-4 b \text {$\#$1}+2 b \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {b \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right )-\log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-b+b \text {$\#$1}+\text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Log[-1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]]] - RootSum[b + b^2 - 4*b*#1 + 2*b*#1^2 + #1^4 & ,
(b*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1] - Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] -
#1]*#1^2)/(-b + b*#1 + #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/(a*x-b)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out

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maple [B]  time = 1.26, size = 529, normalized size = 1.85

method	result	size
derivativedivides	\(-\frac {\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}+\frac {\left (1-2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}\right )}{2}+\frac {\sqrt {-b}\, \ln \left (\frac {-2 \sqrt {-b}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )+2 \sqrt {-\sqrt {-b}}\, \sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}}{\sqrt {a x -b}+\sqrt {-b}}\right )}{\sqrt {-\sqrt {-b}}}}{\sqrt {-b}}+\frac {\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}+\frac {\left (1+2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}\right )}{2}-\left (-b \right )^{\frac {1}{4}} \ln \left (\frac {2 \sqrt {-b}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+2 \left (-b \right )^{\frac {1}{4}} \sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}}{\sqrt {a x -b}-\sqrt {-b}}\right )}{\sqrt {-b}}\)	\(529\)
default	\(-\frac {\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}+\frac {\left (1-2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}\right )}{2}+\frac {\sqrt {-b}\, \ln \left (\frac {-2 \sqrt {-b}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )+2 \sqrt {-\sqrt {-b}}\, \sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}}{\sqrt {a x -b}+\sqrt {-b}}\right )}{\sqrt {-\sqrt {-b}}}}{\sqrt {-b}}+\frac {\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}+\frac {\left (1+2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}\right )}{2}-\left (-b \right )^{\frac {1}{4}} \ln \left (\frac {2 \sqrt {-b}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+2 \left (-b \right )^{\frac {1}{4}} \sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}}{\sqrt {a x -b}-\sqrt {-b}}\right )}{\sqrt {-b}}\)	\(529\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(-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))))+1/(-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))))

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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}\,{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/(a*x-b)^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

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

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