\(\int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx\) [2869]

Optimal result

Integrand size = 34, antiderivative size = 305 \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=-4 \text {RootSum}\left [b^2-2 b c^2+c^4+a^2 c \text {$\#$1}+4 b c \text {$\#$1}^2-4 c^3 \text {$\#$1}^2-a^2 \text {$\#$1}^3-2 b \text {$\#$1}^4+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-b c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+c^3 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-3 c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3+3 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^5-\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^7}{a^2 c+8 b c \text {$\#$1}-8 c^3 \text {$\#$1}-3 a^2 \text {$\#$1}^2-8 b \text {$\#$1}^3+24 c^2 \text {$\#$1}^3-24 c \text {$\#$1}^5+8 \text {$\#$1}^7}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx \]

[In]

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

[Out]

Log[b^2 - 2*b*(b + a*x) + (b + a*x)^2 - a^2*Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]]/2 + (a^2*c*Defer[Subst][Def
er[Int][(-b^2 + 2*b*(c - x^2)^2 - (c - x^2)*(c^3 + a^2*x - 3*c^2*x^2 + 3*c*x^4 - x^6))^(-1), x], x, Sqrt[c + S
qrt[b + a*x]]])/2 + (3*a^2*Defer[Subst][Defer[Int][x^2/(b^2 - 2*b*(c - x^2)^2 + (c - x^2)*(c^3 + a^2*x - 3*c^2
*x^2 + 3*c*x^4 - x^6)), x], x, Sqrt[c + Sqrt[b + a*x]]])/2

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x \left (-b+x^2\right )}{b^2-2 b x^2+x^4-a^2 x \sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right ) \\ & = 4 \text {Subst}\left (\int \frac {x \left (-c+x^2\right ) \left (-b+\left (c-x^2\right )^2\right )}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right )^4-a^2 x \left (-c+x^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ & = \frac {1}{2} \log \left (b^2-2 b (b+a x)+(b+a x)^2-a^2 \sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {-a^2 c+3 a^2 x^2}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right )^4-a^2 x \left (-c+x^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ & = \frac {1}{2} \log \left (b^2-2 b (b+a x)+(b+a x)^2-a^2 \sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}\right )+\frac {1}{2} \text {Subst}\left (\int \left (\frac {a^2 c}{-b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )-a^2 c x-4 b c \left (1-\frac {c^2}{b}\right ) x^2+a^2 x^3+2 b \left (1-\frac {3 c^2}{b}\right ) x^4+4 c x^6-x^8}+\frac {3 a^2 x^2}{b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {c^2}{b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{b}\right ) x^4-4 c x^6+x^8}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ & = \frac {1}{2} \log \left (b^2-2 b (b+a x)+(b+a x)^2-a^2 \sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}\right )+\frac {1}{2} \left (3 a^2\right ) \text {Subst}\left (\int \frac {x^2}{b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {c^2}{b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{b}\right ) x^4-4 c x^6+x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )+\frac {1}{2} \left (a^2 c\right ) \text {Subst}\left (\int \frac {1}{-b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )-a^2 c x-4 b c \left (1-\frac {c^2}{b}\right ) x^2+a^2 x^3+2 b \left (1-\frac {3 c^2}{b}\right ) x^4+4 c x^6-x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ & = \frac {1}{2} \log \left (b^2-2 b (b+a x)+(b+a x)^2-a^2 \sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}\right )+\frac {1}{2} \left (3 a^2\right ) \text {Subst}\left (\int \frac {x^2}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right ) \left (c^3+a^2 x-3 c^2 x^2+3 c x^4-x^6\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )+\frac {1}{2} \left (a^2 c\right ) \text {Subst}\left (\int \frac {1}{-b^2+2 b \left (c-x^2\right )^2-\left (c-x^2\right ) \left (c^3+a^2 x-3 c^2 x^2+3 c x^4-x^6\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00 \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=-4 \text {RootSum}\left [b^2-2 b c^2+c^4+a^2 c \text {$\#$1}+4 b c \text {$\#$1}^2-4 c^3 \text {$\#$1}^2-a^2 \text {$\#$1}^3-2 b \text {$\#$1}^4+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-b c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+c^3 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-3 c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3+3 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^5-\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^7}{a^2 c+8 b c \text {$\#$1}-8 c^3 \text {$\#$1}-3 a^2 \text {$\#$1}^2-8 b \text {$\#$1}^3+24 c^2 \text {$\#$1}^3-24 c \text {$\#$1}^5+8 \text {$\#$1}^7}\&\right ] \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 0.08 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.56

method	result	size
derivativedivides	\(4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{7}-3 c \,\textit {\_R}^{5}+\left (3 c^{2}-b \right ) \textit {\_R}^{3}+c \left (-c^{2}+b \right ) \textit {\_R} \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 c \,\textit {\_R}^{5}+24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\)	\(172\)
default	\(4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{7}-3 c \,\textit {\_R}^{5}+\left (3 c^{2}-b \right ) \textit {\_R}^{3}+c \left (-c^{2}+b \right ) \textit {\_R} \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 c \,\textit {\_R}^{5}+24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\)	\(172\)

[In]

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

[Out]

4*sum((_R^7-3*c*_R^5+(3*c^2-b)*_R^3+c*(-c^2+b)*_R)/(8*_R^7-24*_R^5*c+24*_R^3*c^2-8*_R^3*b-3*_R^2*a^2-8*_R*c^3+
8*_R*b*c+a^2*c)*ln((c+(a*x+b)^(1/2))^(1/2)-_R),_R=RootOf(_Z^8-4*c*_Z^6+(6*c^2-2*b)*_Z^4-a^2*_Z^3+(-4*c^3+4*b*c
)*_Z^2+a^2*c*_Z+c^4-2*b*c^2+b^2))

Fricas [F(-1)]

Timed out. \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {x}{x^{2} - \sqrt {a x + b} \sqrt {c + \sqrt {a x + b}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 2.37 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {x}{x^{2} - \sqrt {a x + b} \sqrt {c + \sqrt {a x + b}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 7.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.11 \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=-\int \frac {x}{\sqrt {c+\sqrt {b+a\,x}}\,\sqrt {b+a\,x}-x^2} \,d x \]

[In]

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

[Out]

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