\(\int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx\) [1713]

Optimal result

Integrand size = 30, antiderivative size = 115 \[ \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=4 \text {RootSum}\left [b-c^2-a c \text {$\#$1}+2 c \text {$\#$1}^2+a \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {-c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{a c-4 c \text {$\#$1}-3 a \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx \]

[In]

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

[Out]

Log[b + Sqrt[b + a*x]*(c + Sqrt[c + Sqrt[b + a*x]]*(a - Sqrt[c + Sqrt[b + a*x]]))] - 3*a*Defer[Subst][Defer[In
t][x^2/(b - c^2 - a*c*x + 2*c*x^2 + a*x^3 - x^4), x], x, Sqrt[c + Sqrt[b + a*x]]] - a*c*Defer[Subst][Defer[Int
][(-b + c^2 + a*c*x - 2*c*x^2 - a*x^3 + x^4)^(-1), x], x, Sqrt[c + Sqrt[b + a*x]]]

Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {x}{b-x^2+a x \sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right )\right ) \\ & = -\left (4 \text {Subst}\left (\int \frac {x \left (c-x^2\right )}{-b+(c+(a-x) x) \left (c-x^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\right ) \\ & = \log \left (b+\sqrt {b+a x} \left (c+\sqrt {c+\sqrt {b+a x}} \left (a-\sqrt {c+\sqrt {b+a x}}\right )\right )\right )-\text {Subst}\left (\int \frac {a c-3 a x^2}{-b+(c+(a-x) x) \left (c-x^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ & = \log \left (b+\sqrt {b+a x} \left (c+\sqrt {c+\sqrt {b+a x}} \left (a-\sqrt {c+\sqrt {b+a x}}\right )\right )\right )-\text {Subst}\left (\int \frac {-a c+3 a x^2}{b-c^2-a c x+2 c x^2+a x^3-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ & = \log \left (b+\sqrt {b+a x} \left (c+\sqrt {c+\sqrt {b+a x}} \left (a-\sqrt {c+\sqrt {b+a x}}\right )\right )\right )-\text {Subst}\left (\int \left (\frac {3 a x^2}{b-c^2-a c x+2 c x^2+a x^3-x^4}+\frac {a c}{-b+c^2+a c x-2 c x^2-a x^3+x^4}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ & = \log \left (b+\sqrt {b+a x} \left (c+\sqrt {c+\sqrt {b+a x}} \left (a-\sqrt {c+\sqrt {b+a x}}\right )\right )\right )-(3 a) \text {Subst}\left (\int \frac {x^2}{b-c^2-a c x+2 c x^2+a x^3-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-(a c) \text {Subst}\left (\int \frac {1}{-b+c^2+a c x-2 c x^2-a x^3+x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=4 \text {RootSum}\left [b-c^2-a c \text {$\#$1}+2 c \text {$\#$1}^2+a \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {-c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{a c-4 c \text {$\#$1}-3 a \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 0.38 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(-4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{3}-2 c \,\textit {\_Z}^{2}+a c \textit {\_Z} +c^{2}-b \right )}{\sum }\frac {\left (-\textit {\_R}^{3}+\textit {\_R} c \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2} a -4 \textit {\_R} c +a c}\right )\)	\(80\)
default	\(\text {Expression too large to display}\)	\(3139\)

[In]

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

[Out]

-4*sum((-_R^3+_R*c)/(4*_R^3-3*_R^2*a-4*_R*c+a*c)*ln((c+(a*x+b)^(1/2))^(1/2)-_R),_R=RootOf(_Z^4-_Z^3*a-2*_Z^2*c
+_Z*a*c+c^2-b))

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\text {Exception raised: AttributeError} \]

[In]

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

[Out]

Exception raised: AttributeError

Sympy [N/A]

Not integrable

Time = 21.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int \frac {1}{x - \sqrt {c + \sqrt {a x + b}} \sqrt {a x + b}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int { -\frac {1}{\sqrt {a x + b} \sqrt {c + \sqrt {a x + b}} - x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int { -\frac {1}{\sqrt {a x + b} \sqrt {c + \sqrt {a x + b}} - x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int \frac {1}{x-\sqrt {c+\sqrt {b+a\,x}}\,\sqrt {b+a\,x}} \,d x \]

[In]

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

[Out]

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