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

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

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Rubi [F]  time = 0.37, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{1-x \sqrt {c+b x+a x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

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

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Mathematica [F]  time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{1-x \sqrt {c+b x+a x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.53, size = 203, normalized size = 1.00 \begin {gather*} -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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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) & ]

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

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.14, size = 382, normalized size = 1.88

method	result	size
default	\(-\left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{4}+b \,\textit {\_Z}^{3}+c \,\textit {\_Z}^{2}-1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3} a +3 \textit {\_R}^{2} b +2 \textit {\_R} c}\right )-\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8} a -2 \sqrt {a}\, \textit {\_Z}^{7} b +\textit {\_Z}^{6} b^{2}+2 \sqrt {a}\, \textit {\_Z}^{5} b c +\left (-2 a \,c^{2}-2 b^{2} c -16 a^{2}\right ) \textit {\_Z}^{4}+\left (32 a^{\frac {3}{2}} b +2 \sqrt {a}\, c^{2} b \right ) \textit {\_Z}^{3}+\left (b^{2} c^{2}-24 a \,b^{2}\right ) \textit {\_Z}^{2}+\left (-2 \sqrt {a}\, c^{3} b +8 \sqrt {a}\, b^{3}\right ) \textit {\_Z} +a \,c^{4}-b^{4}\right )}{\sum }\frac {\left (a \left (a \,\textit {\_R}^{6}-\left (-a c -b^{2}\right ) \textit {\_R}^{4}-c \left (a c +b^{2}\right ) \textit {\_R}^{2}-a \,c^{3}\right )+2 b \left (-\textit {\_R}^{5} a^{\frac {3}{2}}+c^{2} \textit {\_R} \,a^{\frac {3}{2}}\right )\right ) \ln \left (\sqrt {a \,x^{2}+b x +c}-\sqrt {a}\, x -\textit {\_R} \right )}{4 \textit {\_R}^{7} a +3 \textit {\_R}^{5} b^{2}-4 \textit {\_R}^{3} a \,c^{2}-4 \textit {\_R}^{3} b^{2} c -32 \textit {\_R}^{3} a^{2}+\textit {\_R} \,b^{2} c^{2}-24 a \,b^{2} \textit {\_R} +b \left (-7 \textit {\_R}^{6} \sqrt {a}+5 c \,\textit {\_R}^{4} \sqrt {a}+48 \textit {\_R}^{2} a^{\frac {3}{2}}+3 c^{2} \textit {\_R}^{2} \sqrt {a}-\sqrt {a}\, c^{3}+4 b^{2} \sqrt {a}\right )}}{a}\)	\(382\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sum(1/(4*_R^3*a+3*_R^2*b+2*_R*c)*ln(x-_R),_R=RootOf(_Z^4*a+_Z^3*b+_Z^2*c-1))-1/a*sum((a*(a*_R^6-(-a*c-b^2)*_R
^4-c*(a*c+b^2)*_R^2-a*c^3)+2*b*(-_R^5*a^(3/2)+c^2*_R*a^(3/2)))/(4*_R^7*a+3*_R^5*b^2-4*_R^3*a*c^2-4*_R^3*b^2*c-
32*_R^3*a^2+_R*b^2*c^2-24*a*b^2*_R+b*(-7*_R^6*a^(1/2)+5*c*_R^4*a^(1/2)+48*_R^2*a^(3/2)+3*c^2*_R^2*a^(1/2)-a^(1
/2)*c^3+4*b^2*a^(1/2)))*ln((a*x^2+b*x+c)^(1/2)-a^(1/2)*x-_R),_R=RootOf(_Z^8*a-2*a^(1/2)*_Z^7*b+_Z^6*b^2+2*a^(1
/2)*_Z^5*b*c+(-2*a*c^2-2*b^2*c-16*a^2)*_Z^4+(32*a^(3/2)*b+2*a^(1/2)*c^2*b)*_Z^3+(b^2*c^2-24*a*b^2)*_Z^2+(-2*a^
(1/2)*c^3*b+8*a^(1/2)*b^3)*_Z+a*c^4-b^4))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {1}{\sqrt {a x^{2} + b x + c} x - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {1}{x\,\sqrt {a\,x^2+b\,x+c}-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{x \sqrt {a x^{2} + b x + c} - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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