\(\int \frac {1}{1-x \sqrt {c+b x+a x^2}} \, dx\) [2476]

Optimal result

Integrand size = 21, antiderivative size = 203 \[ \int \frac {1}{1-x \sqrt {c+b x+a x^2}} \, dx=-2 \text {RootSum}\left [b^2-\sqrt {a} c^2-4 \sqrt {a} b \text {$\#$1}+b c \text {$\#$1}+4 a \text {$\#$1}^2-b \text {$\#$1}^3+\sqrt {a} \text {$\#$1}^4\&,\frac {\sqrt {a} c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-b \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+\sqrt {a} \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {a} b+b c+8 a \text {$\#$1}-3 b \text {$\#$1}^2+4 \sqrt {a} \text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {1}{1-x \sqrt {c+b x+a x^2}} \, dx=\int \frac {1}{1-x \sqrt {c+b x+a x^2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1-x \sqrt {c+b x+a x^2}} \, dx=-2 \text {RootSum}\left [b^2-\sqrt {a} c^2-4 \sqrt {a} b \text {$\#$1}+b c \text {$\#$1}+4 a \text {$\#$1}^2-b \text {$\#$1}^3+\sqrt {a} \text {$\#$1}^4\&,\frac {\sqrt {a} c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-b \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+\sqrt {a} \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {a} b+b c+8 a \text {$\#$1}-3 b \text {$\#$1}^2+4 \sqrt {a} \text {$\#$1}^3}\&\right ] \]

[In]

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

[Out]

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

Maple [F(-1)]

Timed out.

hanged

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{1-x \sqrt {c+b x+a x^2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 1.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.10 \[ \int \frac {1}{1-x \sqrt {c+b x+a x^2}} \, dx=- \int \frac {1}{x \sqrt {a x^{2} + b x + c} - 1}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.11 \[ \int \frac {1}{1-x \sqrt {c+b x+a x^2}} \, dx=\int { -\frac {1}{\sqrt {a x^{2} + b x + c} x - 1} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{1-x \sqrt {c+b x+a x^2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 6.94 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.11 \[ \int \frac {1}{1-x \sqrt {c+b x+a x^2}} \, dx=\int -\frac {1}{x\,\sqrt {a\,x^2+b\,x+c}-1} \,d x \]

[In]

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

[Out]

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