3.32.3 \(\int \frac {\sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x} (1+a^2 x^2)} \, dx\)

Optimal. Leaf size=586 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^8+4 \text {$\#$1}^6 b-8 \text {$\#$1}^5 b+6 \text {$\#$1}^4 b^2+2 \text {$\#$1}^4 b+16 \text {$\#$1}^4-16 \text {$\#$1}^3 b^2-32 \text {$\#$1}^3+4 \text {$\#$1}^2 b^3+20 \text {$\#$1}^2 b^2+24 \text {$\#$1}^2-8 \text {$\#$1} b^3-8 \text {$\#$1} b^2-8 \text {$\#$1}+b^4+2 b^3+b^2+1\& ,\frac {2 \text {$\#$1}^5 \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )-5 \text {$\#$1}^4 \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )+4 \text {$\#$1}^3 b \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )+4 \text {$\#$1}^3 \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )-6 \text {$\#$1}^2 b \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )-\text {$\#$1}^2 \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )+2 \text {$\#$1} b^2 \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )-b^2 \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )+2 \text {$\#$1} b \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )}{\text {$\#$1}^7+3 \text {$\#$1}^5 b-5 \text {$\#$1}^4 b+3 \text {$\#$1}^3 b^2+\text {$\#$1}^3 b+8 \text {$\#$1}^3-6 \text {$\#$1}^2 b^2-12 \text {$\#$1}^2+\text {$\#$1} b^3+5 \text {$\#$1} b^2+6 \text {$\#$1}-b^3-b^2-1}\& \right ]}{2 a} \]

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Rubi [C]  time = 1.91, antiderivative size = 590, normalized size of antiderivative = 1.01, number of steps used = 19, number of rules used = 10, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6728, 990, 621, 206, 1033, 724, 1036, 1030, 205, 208} \begin {gather*} -\frac {\sqrt {-i b+\sqrt {b-(1-i)}+(1+i)} \tan ^{-1}\left (\frac {\left (b+i \sqrt {b-(1-i)}-(1-i)\right ) \sqrt {a x-b}+\sqrt {b-(1-i)} (b+i)}{\sqrt {2} \sqrt {1-i b} \sqrt {-i b+\sqrt {b-(1-i)}+(1+i)} \sqrt [4]{b-(1-i)} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {2} a \sqrt {1-i b} \sqrt [4]{b-(1-i)}}+\frac {i \sqrt {i-\sqrt {-b+i}} \tanh ^{-1}\left (\frac {\left (1-2 \sqrt {-b+i}\right ) \left (-\sqrt {a x-b}\right )-2 b+\sqrt {-b+i}}{2 \sqrt {i-\sqrt {-b+i}} \sqrt {\sqrt {a x-b}+a x}}\right )}{2 a \sqrt {-b+i}}+\frac {i \sqrt {\sqrt {-b+i}+i} \tanh ^{-1}\left (\frac {\left (1+2 \sqrt {-b+i}\right ) \sqrt {a x-b}+2 b+\sqrt {-b+i}}{2 \sqrt {\sqrt {-b+i}+i} \sqrt {\sqrt {a x-b}+a x}}\right )}{2 a \sqrt {-b+i}}+\frac {i \sqrt {b-i \sqrt {b-(1-i)}-(1-i)} \tanh ^{-1}\left (\frac {\left (-b+i \sqrt {b-(1-i)}+(1-i)\right ) \sqrt {a x-b}+\sqrt {b-(1-i)} (b+i)}{\sqrt {2} \sqrt {-b-i} \sqrt [4]{b-(1-i)} \sqrt {b-i \sqrt {b-(1-i)}-(1-i)} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {2} a \sqrt {-b-i} \sqrt [4]{b-(1-i)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x + Sqrt[-b + a*x]]/(Sqrt[-b + a*x]*(1 + a^2*x^2)),x]

[Out]

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

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 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 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 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 1033

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

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)]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x} \left (1+a^2 x^2\right )} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {\sqrt {b+x+x^2}}{1+b^2+2 b x^2+x^4} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-\frac {i \sqrt {b+x+x^2}}{-2 i+2 b+2 x^2}+\frac {i \sqrt {b+x+x^2}}{2 i+2 b+2 x^2}\right ) \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=-\frac {(2 i) \operatorname {Subst}\left (\int \frac {\sqrt {b+x+x^2}}{-2 i+2 b+2 x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}+\frac {(2 i) \operatorname {Subst}\left (\int \frac {\sqrt {b+x+x^2}}{2 i+2 b+2 x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=\frac {i \operatorname {Subst}\left (\int \frac {-2 i-2 x}{\sqrt {b+x+x^2} \left (-2 i+2 b+2 x^2\right )} \, dx,x,\sqrt {-b+a x}\right )}{a}-\frac {i \operatorname {Subst}\left (\int \frac {2 i-2 x}{\sqrt {b+x+x^2} \left (2 i+2 b+2 x^2\right )} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=-\frac {\left (i-\frac {1}{\sqrt {i-b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2 \sqrt {i-b}+2 x\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}-\frac {\left (i+\frac {1}{\sqrt {i-b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (2 \sqrt {i-b}+2 x\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}-\frac {i \operatorname {Subst}\left (\int \frac {4 \left ((-1+i)+i \sqrt {(-1+i)+b}+b\right )-4 \sqrt {(-1+i)+b} x}{\sqrt {b+x+x^2} \left (2 i+2 b+2 x^2\right )} \, dx,x,\sqrt {-b+a x}\right )}{4 a \sqrt {(-1+i)+b}}+\frac {i \operatorname {Subst}\left (\int \frac {4 \left ((-1+i)-i \sqrt {(-1+i)+b}+b\right )+4 \sqrt {(-1+i)+b} x}{\sqrt {b+x+x^2} \left (2 i+2 b+2 x^2\right )} \, dx,x,\sqrt {-b+a x}\right )}{4 a \sqrt {(-1+i)+b}}\\ &=\frac {\left (2 \left (i-\frac {1}{\sqrt {i-b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16 \sqrt {i-b}+16 (i-b)+16 b-x^2} \, dx,x,\frac {2 \sqrt {i-b}+4 b-\left (-2-4 \sqrt {i-b}\right ) \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{a}+\frac {\left (2 \left (i+\frac {1}{\sqrt {i-b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-16 \sqrt {i-b}+16 (i-b)+16 b-x^2} \, dx,x,\frac {-2 \sqrt {i-b}+4 b-\left (-2+4 \sqrt {i-b}\right ) \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {\left (16 \left ((-1-i)+\sqrt {(-1+i)+b}+i b\right ) (i+b)\right ) \operatorname {Subst}\left (\int \frac {1}{256 \sqrt {(-1+i)+b} (i+b)^2 \left ((-1+i)-i \sqrt {(-1+i)+b}+b\right )+2 (i+b) x^2} \, dx,x,\frac {8 \sqrt {(-1+i)+b} (i+b)-8 \left ((-1+i)-i \sqrt {(-1+i)+b}+b\right ) \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {\left (16 i (i+b) \left ((-1+i)+i \sqrt {(-1+i)+b}+b\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-256 \sqrt {(-1+i)+b} (i+b)^2 \left (i \left ((1+i)+\sqrt {(-1+i)+b}\right )+b\right )+2 (i+b) x^2} \, dx,x,\frac {-8 \sqrt {(-1+i)+b} (i+b)-8 \left ((-1+i)+i \sqrt {(-1+i)+b}+b\right ) \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{a}\\ &=-\frac {\sqrt {(1+i)+\sqrt {(-1+i)+b}-i b} \tan ^{-1}\left (\frac {\sqrt {(-1+i)+b} (i+b)+\left ((-1+i)+i \sqrt {(-1+i)+b}+b\right ) \sqrt {-b+a x}}{\sqrt {2} \sqrt {1-i b} \sqrt {(1+i)+\sqrt {(-1+i)+b}-i b} \sqrt [4]{(-1+i)+b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{\sqrt {2} a \sqrt {1-i b} \sqrt [4]{(-1+i)+b}}-\frac {\left (i+\frac {1}{\sqrt {i-b}}\right ) \tanh ^{-1}\left (\frac {\sqrt {i-b}-2 b-\left (1-2 \sqrt {i-b}\right ) \sqrt {-b+a x}}{2 \sqrt {i-\sqrt {i-b}} \sqrt {a x+\sqrt {-b+a x}}}\right )}{2 a \sqrt {i-\sqrt {i-b}}}+\frac {i \sqrt {i+\sqrt {i-b}} \tanh ^{-1}\left (\frac {\sqrt {i-b}+2 b+\left (1+2 \sqrt {i-b}\right ) \sqrt {-b+a x}}{2 \sqrt {i+\sqrt {i-b}} \sqrt {a x+\sqrt {-b+a x}}}\right )}{2 a \sqrt {i-b}}+\frac {i \sqrt {(-1+i)-i \sqrt {(-1+i)+b}+b} \tanh ^{-1}\left (\frac {\sqrt {(-1+i)+b} (i+b)-\left ((-1+i)-i \sqrt {(-1+i)+b}+b\right ) \sqrt {-b+a x}}{\sqrt {2} \sqrt {-i-b} \sqrt [4]{(-1+i)+b} \sqrt {(-1+i)-i \sqrt {(-1+i)+b}+b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{\sqrt {2} a \sqrt {-i-b} \sqrt [4]{(-1+i)+b}}\\ \end {align*}

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Mathematica [C]  time = 0.85, size = 475, normalized size = 0.81 \begin {gather*} -\frac {i \left (\frac {\sqrt {\sqrt {-b-i}+i} \tan ^{-1}\left (\frac {\left (-1+2 \sqrt {-b-i}\right ) \sqrt {a x-b}-2 b+\sqrt {-b-i}}{2 \sqrt {\sqrt {-b-i}+i} \sqrt {\sqrt {a x-b}+a x}}\right )+\sqrt {i-\sqrt {-b-i}} \tan ^{-1}\left (\frac {\left (1+2 \sqrt {-b-i}\right ) \sqrt {a x-b}+2 b+\sqrt {-b-i}}{2 \sqrt {i-\sqrt {-b-i}} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {-b-i}}-\frac {\sqrt {i-\sqrt {-b+i}} \tanh ^{-1}\left (\frac {\left (-1+2 \sqrt {-b+i}\right ) \sqrt {a x-b}-2 b+\sqrt {-b+i}}{2 \sqrt {i-\sqrt {-b+i}} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {-b+i}}-\frac {\sqrt {\sqrt {-b+i}+i} \tanh ^{-1}\left (\frac {\left (1+2 \sqrt {-b+i}\right ) \sqrt {a x-b}+2 b+\sqrt {-b+i}}{2 \sqrt {\sqrt {-b+i}+i} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {-b+i}}\right )}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*x + Sqrt[-b + a*x]]/(Sqrt[-b + a*x]*(1 + a^2*x^2)),x]

[Out]

((-1/2*I)*((Sqrt[I + Sqrt[-I - b]]*ArcTan[(Sqrt[-I - b] - 2*b + (-1 + 2*Sqrt[-I - b])*Sqrt[-b + a*x])/(2*Sqrt[
I + Sqrt[-I - b]]*Sqrt[a*x + Sqrt[-b + a*x]])] + Sqrt[I - Sqrt[-I - b]]*ArcTan[(Sqrt[-I - b] + 2*b + (1 + 2*Sq
rt[-I - b])*Sqrt[-b + a*x])/(2*Sqrt[I - Sqrt[-I - b]]*Sqrt[a*x + Sqrt[-b + a*x]])])/Sqrt[-I - b] - (Sqrt[I - S
qrt[I - b]]*ArcTanh[(Sqrt[I - b] - 2*b + (-1 + 2*Sqrt[I - b])*Sqrt[-b + a*x])/(2*Sqrt[I - Sqrt[I - b]]*Sqrt[a*
x + Sqrt[-b + a*x]])])/Sqrt[I - b] - (Sqrt[I + Sqrt[I - b]]*ArcTanh[(Sqrt[I - b] + 2*b + (1 + 2*Sqrt[I - b])*S
qrt[-b + a*x])/(2*Sqrt[I + Sqrt[I - b]]*Sqrt[a*x + Sqrt[-b + a*x]])])/Sqrt[I - b]))/a

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IntegrateAlgebraic [A]  time = 0.76, size = 604, normalized size = 1.03 \begin {gather*} -\frac {\text {RootSum}\left [1+b^2+2 b^3+b^4-8 \text {$\#$1}-8 b^2 \text {$\#$1}-8 b^3 \text {$\#$1}+24 \text {$\#$1}^2+20 b^2 \text {$\#$1}^2+4 b^3 \text {$\#$1}^2-32 \text {$\#$1}^3-16 b^2 \text {$\#$1}^3+16 \text {$\#$1}^4+2 b \text {$\#$1}^4+6 b^2 \text {$\#$1}^4-8 b \text {$\#$1}^5+4 b \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-b^2 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right )+2 b \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+2 b^2 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}-\log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-6 b \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2+4 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3+4 b \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-5 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-1-b^2-b^3+6 \text {$\#$1}+5 b^2 \text {$\#$1}+b^3 \text {$\#$1}-12 \text {$\#$1}^2-6 b^2 \text {$\#$1}^2+8 \text {$\#$1}^3+b \text {$\#$1}^3+3 b^2 \text {$\#$1}^3-5 b \text {$\#$1}^4+3 b \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[a*x + Sqrt[-b + a*x]]/(Sqrt[-b + a*x]*(1 + a^2*x^2)),x]

[Out]

-1/2*RootSum[1 + b^2 + 2*b^3 + b^4 - 8*#1 - 8*b^2*#1 - 8*b^3*#1 + 24*#1^2 + 20*b^2*#1^2 + 4*b^3*#1^2 - 32*#1^3
- 16*b^2*#1^3 + 16*#1^4 + 2*b*#1^4 + 6*b^2*#1^4 - 8*b*#1^5 + 4*b*#1^6 + #1^8 & , (-(b^2*Log[-Sqrt[-b + a*x] +
Sqrt[a*x + Sqrt[-b + a*x]] - #1]) + 2*b*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1 + 2*b^2*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 - 6*b*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^2 + 4*Log[-Sqrt[-b + a*x] + Sqrt[a*x +
Sqrt[-b + a*x]] - #1]*#1^3 + 4*b*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^3 - 5*Log[-Sqrt[-b
+ a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^4 + 2*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^5
)/(-1 - b^2 - b^3 + 6*#1 + 5*b^2*#1 + b^3*#1 - 12*#1^2 - 6*b^2*#1^2 + 8*#1^3 + b*#1^3 + 3*b^2*#1^3 - 5*b*#1^4
+ 3*b*#1^5 + #1^7) & ]/a

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

[Out]

Timed out

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maple [B]  time = 0.13, size = 239, normalized size = 0.41

method	result	size
derivativedivides	\(-\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+4 b \,\textit {\_Z}^{6}-8 b \,\textit {\_Z}^{5}+\left (6 b^{2}+2 b +16\right ) \textit {\_Z}^{4}+\left (-16 b^{2}-32\right ) \textit {\_Z}^{3}+\left (4 b^{3}+20 b^{2}+24\right ) \textit {\_Z}^{2}+\left (-8 b^{3}-8 b^{2}-8\right ) \textit {\_Z} +b^{4}+2 b^{3}+b^{2}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-5 \textit {\_R}^{4}+4 \left (1+b \right ) \textit {\_R}^{3}+\left (-6 b -1\right ) \textit {\_R}^{2}+2 b \left (1+b \right ) \textit {\_R} -b^{2}\right ) \ln \left (\sqrt {a x +\sqrt {a x -b}}-\sqrt {a x -b}-\textit {\_R} \right )}{\textit {\_R}^{7}+3 \textit {\_R}^{5} b -5 \textit {\_R}^{4} b +3 \textit {\_R}^{3} b^{2}+\textit {\_R}^{3} b -6 \textit {\_R}^{2} b^{2}+\textit {\_R} \,b^{3}+8 \textit {\_R}^{3}+5 \textit {\_R} \,b^{2}-b^{3}-12 \textit {\_R}^{2}-b^{2}+6 \textit {\_R} -1}}{2 a}\)	\(239\)
default	\(-\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+4 b \,\textit {\_Z}^{6}-8 b \,\textit {\_Z}^{5}+\left (6 b^{2}+2 b +16\right ) \textit {\_Z}^{4}+\left (-16 b^{2}-32\right ) \textit {\_Z}^{3}+\left (4 b^{3}+20 b^{2}+24\right ) \textit {\_Z}^{2}+\left (-8 b^{3}-8 b^{2}-8\right ) \textit {\_Z} +b^{4}+2 b^{3}+b^{2}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-5 \textit {\_R}^{4}+4 \left (1+b \right ) \textit {\_R}^{3}+\left (-6 b -1\right ) \textit {\_R}^{2}+2 b \left (1+b \right ) \textit {\_R} -b^{2}\right ) \ln \left (\sqrt {a x +\sqrt {a x -b}}-\sqrt {a x -b}-\textit {\_R} \right )}{\textit {\_R}^{7}+3 \textit {\_R}^{5} b -5 \textit {\_R}^{4} b +3 \textit {\_R}^{3} b^{2}+\textit {\_R}^{3} b -6 \textit {\_R}^{2} b^{2}+\textit {\_R} \,b^{3}+8 \textit {\_R}^{3}+5 \textit {\_R} \,b^{2}-b^{3}-12 \textit {\_R}^{2}-b^{2}+6 \textit {\_R} -1}}{2 a}\)	\(239\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a*sum((2*_R^5-5*_R^4+4*(1+b)*_R^3+(-6*b-1)*_R^2+2*b*(1+b)*_R-b^2)/(_R^7+3*_R^5*b-5*_R^4*b+3*_R^3*b^2+_R^3
*b-6*_R^2*b^2+_R*b^3+8*_R^3+5*_R*b^2-b^3-12*_R^2-b^2+6*_R-1)*ln((a*x+(a*x-b)^(1/2))^(1/2)-(a*x-b)^(1/2)-_R),_R
=RootOf(_Z^8+4*b*_Z^6-8*b*_Z^5+(6*b^2+2*b+16)*_Z^4+(-16*b^2-32)*_Z^3+(4*b^3+20*b^2+24)*_Z^2+(-8*b^3-8*b^2-8)*_
Z+b^4+2*b^3+b^2+1))

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

[Out]

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

[Out]

int((a*x + (a*x - b)^(1/2))^(1/2)/((a^2*x^2 + 1)*(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}}}{\sqrt {a x - b} \left (a^{2} x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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