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

Optimal. Leaf size=306 \[ \frac {2 \left (\sqrt {2} c \sqrt {a^2+4 b}+\sqrt {2} a \sqrt {a^2+4 b}+\sqrt {2} a^2+\sqrt {2} a c+2 \sqrt {2} b\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {-\sqrt {a^2+4 b}-a-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-\sqrt {a^2+4 b}-a-2 c}}+\frac {2 \left (\sqrt {2} c \sqrt {a^2+4 b}+\sqrt {2} a \sqrt {a^2+4 b}-\sqrt {2} a^2-\sqrt {2} a c-2 \sqrt {2} b\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {\sqrt {a^2+4 b}-a-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {\sqrt {a^2+4 b}-a-2 c}}+4 \sqrt {\sqrt {a x+b}+c} \]

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Rubi [A]  time = 1.74, antiderivative size = 256, normalized size of antiderivative = 0.84, number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {824, 826, 1166, 206} \begin {gather*} \frac {2 \sqrt {2} \left (-a \left (\sqrt {a^2+4 b}-c\right )-c \sqrt {a^2+4 b}+a^2+2 b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {-\sqrt {a^2+4 b}+a+2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-\sqrt {a^2+4 b}+a+2 c}}-\frac {2 \sqrt {2} \left (a \left (\sqrt {a^2+4 b}+c\right )+c \sqrt {a^2+4 b}+a^2+2 b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {\sqrt {a^2+4 b}+a+2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {\sqrt {a^2+4 b}+a+2 c}}+4 \sqrt {\sqrt {a x+b}+c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*
e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {\sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {x \sqrt {c+x}}{b+a x-x^2} \, dx,x,\sqrt {b+a x}\right )\right )\\ &=4 \sqrt {c+\sqrt {b+a x}}+2 \operatorname {Subst}\left (\int \frac {-b-(a+c) x}{\sqrt {c+x} \left (b+a x-x^2\right )} \, dx,x,\sqrt {b+a x}\right )\\ &=4 \sqrt {c+\sqrt {b+a x}}+4 \operatorname {Subst}\left (\int \frac {-b-(-a-c) c+(-a-c) x^2}{b-a c-c^2+(a+2 c) x^2-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=4 \sqrt {c+\sqrt {b+a x}}+\frac {\left (2 \left (a^2+2 b-a \left (\sqrt {a^2+4 b}-c\right )-\sqrt {a^2+4 b} c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {a^2+4 b}+\frac {1}{2} (a+2 c)-x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {a^2+4 b}}-\frac {\left (2 \left (a^2+2 b+\sqrt {a^2+4 b} c+a \left (\sqrt {a^2+4 b}+c\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {a^2+4 b}+\frac {1}{2} (a+2 c)-x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {a^2+4 b}}\\ &=4 \sqrt {c+\sqrt {b+a x}}+\frac {2 \sqrt {2} \left (a^2+2 b-a \left (\sqrt {a^2+4 b}-c\right )-\sqrt {a^2+4 b} c\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {a-\sqrt {a^2+4 b}+2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {a-\sqrt {a^2+4 b}+2 c}}-\frac {2 \sqrt {2} \left (a^2+2 b+\sqrt {a^2+4 b} c+a \left (\sqrt {a^2+4 b}+c\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {a+\sqrt {a^2+4 b}+2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {a+\sqrt {a^2+4 b}+2 c}}\\ \end {align*}

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Mathematica [A]  time = 0.46, size = 255, normalized size = 0.83 \begin {gather*} \frac {2 \sqrt {2} \left (a \left (c-\sqrt {a^2+4 b}\right )-c \sqrt {a^2+4 b}+a^2+2 b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {-\sqrt {a^2+4 b}+a+2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-\sqrt {a^2+4 b}+a+2 c}}-\frac {2 \sqrt {2} \left (a \left (\sqrt {a^2+4 b}+c\right )+c \sqrt {a^2+4 b}+a^2+2 b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {\sqrt {a^2+4 b}+a+2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {\sqrt {a^2+4 b}+a+2 c}}+4 \sqrt {\sqrt {a x+b}+c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.65, size = 306, normalized size = 1.00 \begin {gather*} 4 \sqrt {c+\sqrt {b+a x}}+\frac {2 \left (\sqrt {2} a^2+2 \sqrt {2} b+\sqrt {2} a \sqrt {a^2+4 b}+\sqrt {2} a c+\sqrt {2} \sqrt {a^2+4 b} c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {-a-\sqrt {a^2+4 b}-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-a-\sqrt {a^2+4 b}-2 c}}+\frac {2 \left (-\sqrt {2} a^2-2 \sqrt {2} b+\sqrt {2} a \sqrt {a^2+4 b}-\sqrt {2} a c+\sqrt {2} \sqrt {a^2+4 b} c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {-a+\sqrt {a^2+4 b}-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-a+\sqrt {a^2+4 b}-2 c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.81, size = 1106, normalized size = 3.61

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*sqrt((a^3 + 3*a*b + (a^2 + 2*b)*c + (a^2 + 4*b)*sqrt((a^4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3 + a*b)*c
)/(a^2 + 4*b)))/(a^2 + 4*b))*log(8*sqrt(2)*(a^4 + 5*a^2*b + 4*b^2 + (a^3 + 4*a*b)*c - (a^3 + 4*a*b)*sqrt((a^4
+ a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3 + a*b)*c)/(a^2 + 4*b)))*sqrt((a^3 + 3*a*b + (a^2 + 2*b)*c + (a^2 + 4*b)*sqr
t((a^4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3 + a*b)*c)/(a^2 + 4*b)))/(a^2 + 4*b)) + 32*(a^2*b + a*b*c + b^2)*sqrt
(c + sqrt(a*x + b))) + sqrt(2)*sqrt((a^3 + 3*a*b + (a^2 + 2*b)*c + (a^2 + 4*b)*sqrt((a^4 + a^2*c^2 + 2*a^2*b +
b^2 + 2*(a^3 + a*b)*c)/(a^2 + 4*b)))/(a^2 + 4*b))*log(-8*sqrt(2)*(a^4 + 5*a^2*b + 4*b^2 + (a^3 + 4*a*b)*c - (
a^3 + 4*a*b)*sqrt((a^4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3 + a*b)*c)/(a^2 + 4*b)))*sqrt((a^3 + 3*a*b + (a^2 + 2
*b)*c + (a^2 + 4*b)*sqrt((a^4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3 + a*b)*c)/(a^2 + 4*b)))/(a^2 + 4*b)) + 32*(a^
2*b + a*b*c + b^2)*sqrt(c + sqrt(a*x + b))) - sqrt(2)*sqrt((a^3 + 3*a*b + (a^2 + 2*b)*c - (a^2 + 4*b)*sqrt((a^
4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3 + a*b)*c)/(a^2 + 4*b)))/(a^2 + 4*b))*log(8*sqrt(2)*(a^4 + 5*a^2*b + 4*b^2
+ (a^3 + 4*a*b)*c + (a^3 + 4*a*b)*sqrt((a^4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3 + a*b)*c)/(a^2 + 4*b)))*sqrt((
a^3 + 3*a*b + (a^2 + 2*b)*c - (a^2 + 4*b)*sqrt((a^4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3 + a*b)*c)/(a^2 + 4*b)))
/(a^2 + 4*b)) + 32*(a^2*b + a*b*c + b^2)*sqrt(c + sqrt(a*x + b))) + sqrt(2)*sqrt((a^3 + 3*a*b + (a^2 + 2*b)*c
- (a^2 + 4*b)*sqrt((a^4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3 + a*b)*c)/(a^2 + 4*b)))/(a^2 + 4*b))*log(-8*sqrt(2)
*(a^4 + 5*a^2*b + 4*b^2 + (a^3 + 4*a*b)*c + (a^3 + 4*a*b)*sqrt((a^4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3 + a*b)*
c)/(a^2 + 4*b)))*sqrt((a^3 + 3*a*b + (a^2 + 2*b)*c - (a^2 + 4*b)*sqrt((a^4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3
+ a*b)*c)/(a^2 + 4*b)))/(a^2 + 4*b)) + 32*(a^2*b + a*b*c + b^2)*sqrt(c + sqrt(a*x + b))) + 4*sqrt(c + sqrt(a*x
+ b))

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giac [A]  time = 0.20, size = 210, normalized size = 0.69 \begin {gather*} -\frac {4 \, \sqrt {a^{2} + 4 \, b} \sqrt {-2 \, a - 4 \, c + 2 \, \sqrt {a^{2} + 4 \, b}} b \arctan \left (\frac {\sqrt {c + \sqrt {a x + b}}}{\sqrt {-\frac {1}{2} \, a - c + \frac {1}{2} \, \sqrt {{\left (a + 2 \, c\right )}^{2} - 4 \, a c - 4 \, c^{2} + 4 \, b}}}\right )}{a^{3} + 4 \, a b + {\left (a^{2} + 4 \, b\right )}^{\frac {3}{2}}} + \frac {4 \, \sqrt {a^{2} + 4 \, b} \sqrt {-2 \, a - 4 \, c - 2 \, \sqrt {a^{2} + 4 \, b}} b \arctan \left (\frac {\sqrt {c + \sqrt {a x + b}}}{\sqrt {-\frac {1}{2} \, a - c - \frac {1}{2} \, \sqrt {{\left (a + 2 \, c\right )}^{2} - 4 \, a c - 4 \, c^{2} + 4 \, b}}}\right )}{a^{3} + 4 \, a b - {\left (a^{2} + 4 \, b\right )}^{\frac {3}{2}}} + 4 \, \sqrt {c + \sqrt {a x + b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*sqrt(a^2 + 4*b)*sqrt(-2*a - 4*c + 2*sqrt(a^2 + 4*b))*b*arctan(sqrt(c + sqrt(a*x + b))/sqrt(-1/2*a - c + 1/2
*sqrt((a + 2*c)^2 - 4*a*c - 4*c^2 + 4*b)))/(a^3 + 4*a*b + (a^2 + 4*b)^(3/2)) + 4*sqrt(a^2 + 4*b)*sqrt(-2*a - 4
*c - 2*sqrt(a^2 + 4*b))*b*arctan(sqrt(c + sqrt(a*x + b))/sqrt(-1/2*a - c - 1/2*sqrt((a + 2*c)^2 - 4*a*c - 4*c^
2 + 4*b)))/(a^3 + 4*a*b - (a^2 + 4*b)^(3/2)) + 4*sqrt(c + sqrt(a*x + b))

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maple [A]  time = 0.25, size = 212, normalized size = 0.69

method	result	size
derivativedivides	\(4 \sqrt {c +\sqrt {a x +b}}+\frac {4 \left (a \sqrt {a^{2}+4 b}+c \sqrt {a^{2}+4 b}-a^{2}-a c -2 b \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}+\frac {4 \left (a \sqrt {a^{2}+4 b}+c \sqrt {a^{2}+4 b}+a^{2}+a c +2 b \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\)	\(212\)
default	\(4 \sqrt {c +\sqrt {a x +b}}+\frac {4 \left (a \sqrt {a^{2}+4 b}+c \sqrt {a^{2}+4 b}-a^{2}-a c -2 b \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}+\frac {4 \left (a \sqrt {a^{2}+4 b}+c \sqrt {a^{2}+4 b}+a^{2}+a c +2 b \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\)	\(212\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*(c+(a*x+b)^(1/2))^(1/2)+4*(a*(a^2+4*b)^(1/2)+c*(a^2+4*b)^(1/2)-a^2-a*c-2*b)/(a^2+4*b)^(1/2)/(2*(a^2+4*b)^(1/
2)-2*a-4*c)^(1/2)*arctan(2*(c+(a*x+b)^(1/2))^(1/2)/(2*(a^2+4*b)^(1/2)-2*a-4*c)^(1/2))+4*(a*(a^2+4*b)^(1/2)+c*(
a^2+4*b)^(1/2)+a^2+a*c+2*b)/(a^2+4*b)^(1/2)/(-2*(a^2+4*b)^(1/2)-2*a-4*c)^(1/2)*arctan(2*(c+(a*x+b)^(1/2))^(1/2
)/(-2*(a^2+4*b)^(1/2)-2*a-4*c)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(c + sqrt(a*x + b))/(x - 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 {c+\sqrt {b+a\,x}}}{x-\sqrt {b+a\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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