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

Optimal. Leaf size=270 \[ 2 \text {RootSum}\left [\text {$\#$1}^4 \left (-\sqrt {a}\right )+2 \text {$\#$1}^3 a+\text {$\#$1}^3 b-3 \text {$\#$1}^2 \sqrt {a} b-4 \text {$\#$1}^2 a+4 \text {$\#$1} \sqrt {a} b+2 \text {$\#$1} a c+\text {$\#$1} b^2-\text {$\#$1} b c-\sqrt {a} b c+\sqrt {a} c^2-b^2\& ,\frac {\text {$\#$1}^2 \sqrt {a} \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )-\text {$\#$1} b \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )+\sqrt {a} c \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )}{-4 \text {$\#$1}^3 \sqrt {a}+6 \text {$\#$1}^2 a+3 \text {$\#$1}^2 b-6 \text {$\#$1} \sqrt {a} b-8 \text {$\#$1} a+4 \sqrt {a} b+2 a c+b^2-b c}\& \right ] \]

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Rubi [F]  time = 0.95, 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-(1+x) \sqrt {c+b x+a x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{1-(1+x) \sqrt {c+b x+a x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.10, size = 270, normalized size = 1.00 \begin {gather*} 2 \text {RootSum}\left [-b^2-\sqrt {a} b c+\sqrt {a} c^2+4 \sqrt {a} b \text {$\#$1}+b^2 \text {$\#$1}+2 a c \text {$\#$1}-b c \text {$\#$1}-4 a \text {$\#$1}^2-3 \sqrt {a} b \text {$\#$1}^2+2 a \text {$\#$1}^3+b \text {$\#$1}^3-\sqrt {a} \text {$\#$1}^4\&,\frac {\sqrt {a} c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-b \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+\sqrt {a} \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{4 \sqrt {a} b+b^2+2 a c-b c-8 a \text {$\#$1}-6 \sqrt {a} b \text {$\#$1}+6 a \text {$\#$1}^2+3 b \text {$\#$1}^2-4 \sqrt {a} \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*RootSum[-b^2 - Sqrt[a]*b*c + Sqrt[a]*c^2 + 4*Sqrt[a]*b*#1 + b^2*#1 + 2*a*c*#1 - b*c*#1 - 4*a*#1^2 - 3*Sqrt[a
]*b*#1^2 + 2*a*#1^3 + 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^2 + 2*a*c - b*c - 8*a*#1 - 6*Sqrt[a]*b*#1 + 6*a*#1^2 + 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-(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-(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 = 0.27, size = 1540, normalized size = 5.70

method	result	size
default	\(-\left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{4}+\left (2 a +b \right ) \textit {\_Z}^{3}+\left (a +2 b +c \right ) \textit {\_Z}^{2}+\left (b +2 c \right ) \textit {\_Z} +c -1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3} a +6 \textit {\_R}^{2} a +3 \textit {\_R}^{2} b +2 \textit {\_R} a +4 \textit {\_R} b +2 \textit {\_R} c +b +2 c}\right )+\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8} a -\left (4 a^{\frac {3}{2}}+2 \sqrt {a}\, b \right ) \textit {\_Z}^{7}-\left (-4 a^{2}-10 a b -b^{2}\right ) \textit {\_Z}^{6}-\left (12 a^{\frac {3}{2}} b +4 a^{\frac {3}{2}} c +8 b^{2} \sqrt {a}-2 \sqrt {a}\, b c \right ) \textit {\_Z}^{5}-\left (-8 a^{2} c -13 a \,b^{2}-2 a b c +2 a \,c^{2}-2 b^{3}+2 b^{2} c +16 a^{2}\right ) \textit {\_Z}^{4}-\left (16 a^{\frac {3}{2}} c b -4 a^{\frac {3}{2}} c^{2}-32 a^{\frac {3}{2}} b +6 \sqrt {a}\, b^{3}-4 \sqrt {a}\, c \,b^{2}-2 \sqrt {a}\, c^{2} b \right ) \textit {\_Z}^{3}-\left (-4 a^{2} c^{2}-10 a \,b^{2} c +10 a b \,c^{2}-b^{4}+2 b^{3} c -b^{2} c^{2}+24 a \,b^{2}\right ) \textit {\_Z}^{2}-\left (4 a^{\frac {3}{2}} c^{2} b -4 a^{\frac {3}{2}} c^{3}+2 \sqrt {a}\, c \,b^{3}-4 \sqrt {a}\, c^{2} b^{2}+2 \sqrt {a}\, c^{3} b -8 \sqrt {a}\, b^{3}\right ) \textit {\_Z} +a \,b^{2} c^{2}-2 a b \,c^{3}+a \,c^{4}-b^{4}\right )}{\sum }\frac {\left (b \left (5 a \,\textit {\_R}^{4}+\left (6 a c +b^{2}\right ) \textit {\_R}^{2}+a \,c^{2}\right )-2 \textit {\_R}^{5} a^{\frac {3}{2}}+4 \textit {\_R}^{3} \left (-a^{\frac {3}{2}} c -b^{2} \sqrt {a}\right )+2 c \textit {\_R} \left (-a^{\frac {3}{2}} c -b^{2} \sqrt {a}\right )\right ) \ln \left (\sqrt {a \,x^{2}+b x +c}-\sqrt {a}\, x -\textit {\_R} \right )}{-3 \textit {\_R}^{5} b^{2}-4 \sqrt {a}\, b^{3}-12 a^{2} \textit {\_R}^{5}-2 a^{\frac {3}{2}} c^{3}+2 a^{\frac {3}{2}} c^{2} b +\sqrt {a}\, c \,b^{3}-4 \textit {\_R}^{7} a +32 \textit {\_R}^{3} a^{2}+\sqrt {a}\, c^{3} b +4 \textit {\_R}^{3} a \,c^{2}-4 \textit {\_R}^{3} b^{3}-\textit {\_R} \,b^{4}+14 a^{\frac {3}{2}} \textit {\_R}^{6}-\textit {\_R} \,b^{2} c^{2}+24 a \,b^{2} \textit {\_R} -2 \sqrt {a}\, c^{2} b^{2}-6 a^{\frac {3}{2}} \textit {\_R}^{2} c^{2}+20 \sqrt {a}\, \textit {\_R}^{4} b^{2}+9 \sqrt {a}\, \textit {\_R}^{2} b^{3}-16 a^{2} \textit {\_R}^{3} c +30 a^{\frac {3}{2}} \textit {\_R}^{4} b +10 a^{\frac {3}{2}} \textit {\_R}^{4} c +2 \textit {\_R} \,b^{3} c -30 \textit {\_R}^{5} a b -26 \textit {\_R}^{3} a \,b^{2}-4 a^{2} \textit {\_R} \,c^{2}-48 a^{\frac {3}{2}} \textit {\_R}^{2} b +7 \sqrt {a}\, \textit {\_R}^{6} b -10 \textit {\_R} a \,b^{2} c +10 \textit {\_R} a b \,c^{2}+24 a^{\frac {3}{2}} \textit {\_R}^{2} b c -5 \sqrt {a}\, \textit {\_R}^{4} b c -3 \sqrt {a}\, \textit {\_R}^{2} b \,c^{2}-6 \sqrt {a}\, \textit {\_R}^{2} b^{2} c -4 \textit {\_R}^{3} a b c +4 \textit {\_R}^{3} b^{2} c}\right )-\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8} a -\left (4 a^{\frac {3}{2}}+2 \sqrt {a}\, b \right ) \textit {\_Z}^{7}-\left (-4 a^{2}-10 a b -b^{2}\right ) \textit {\_Z}^{6}-\left (12 a^{\frac {3}{2}} b +4 a^{\frac {3}{2}} c +8 b^{2} \sqrt {a}-2 \sqrt {a}\, b c \right ) \textit {\_Z}^{5}-\left (-8 a^{2} c -13 a \,b^{2}-2 a b c +2 a \,c^{2}-2 b^{3}+2 b^{2} c +16 a^{2}\right ) \textit {\_Z}^{4}-\left (16 a^{\frac {3}{2}} c b -4 a^{\frac {3}{2}} c^{2}-32 a^{\frac {3}{2}} b +6 \sqrt {a}\, b^{3}-4 \sqrt {a}\, c \,b^{2}-2 \sqrt {a}\, c^{2} b \right ) \textit {\_Z}^{3}-\left (-4 a^{2} c^{2}-10 a \,b^{2} c +10 a b \,c^{2}-b^{4}+2 b^{3} c -b^{2} c^{2}+24 a \,b^{2}\right ) \textit {\_Z}^{2}-\left (4 a^{\frac {3}{2}} c^{2} b -4 a^{\frac {3}{2}} c^{3}+2 \sqrt {a}\, c \,b^{3}-4 \sqrt {a}\, c^{2} b^{2}+2 \sqrt {a}\, c^{3} b -8 \sqrt {a}\, b^{3}\right ) \textit {\_Z} +a \,b^{2} c^{2}-2 a b \,c^{3}+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 )}{-3 \textit {\_R}^{5} b^{2}-4 \sqrt {a}\, b^{3}-12 a^{2} \textit {\_R}^{5}-2 a^{\frac {3}{2}} c^{3}+2 a^{\frac {3}{2}} c^{2} b +\sqrt {a}\, c \,b^{3}-4 \textit {\_R}^{7} a +32 \textit {\_R}^{3} a^{2}+\sqrt {a}\, c^{3} b +4 \textit {\_R}^{3} a \,c^{2}-4 \textit {\_R}^{3} b^{3}-\textit {\_R} \,b^{4}+14 a^{\frac {3}{2}} \textit {\_R}^{6}-\textit {\_R} \,b^{2} c^{2}+24 a \,b^{2} \textit {\_R} -2 \sqrt {a}\, c^{2} b^{2}-6 a^{\frac {3}{2}} \textit {\_R}^{2} c^{2}+20 \sqrt {a}\, \textit {\_R}^{4} b^{2}+9 \sqrt {a}\, \textit {\_R}^{2} b^{3}-16 a^{2} \textit {\_R}^{3} c +30 a^{\frac {3}{2}} \textit {\_R}^{4} b +10 a^{\frac {3}{2}} \textit {\_R}^{4} c +2 \textit {\_R} \,b^{3} c -30 \textit {\_R}^{5} a b -26 \textit {\_R}^{3} a \,b^{2}-4 a^{2} \textit {\_R} \,c^{2}-48 a^{\frac {3}{2}} \textit {\_R}^{2} b +7 \sqrt {a}\, \textit {\_R}^{6} b -10 \textit {\_R} a \,b^{2} c +10 \textit {\_R} a b \,c^{2}+24 a^{\frac {3}{2}} \textit {\_R}^{2} b c -5 \sqrt {a}\, \textit {\_R}^{4} b c -3 \sqrt {a}\, \textit {\_R}^{2} b \,c^{2}-6 \sqrt {a}\, \textit {\_R}^{2} b^{2} c -4 \textit {\_R}^{3} a b c +4 \textit {\_R}^{3} b^{2} c}}{a}\)	\(1540\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sum(1/(4*_R^3*a+6*_R^2*a+3*_R^2*b+2*_R*a+4*_R*b+2*_R*c+b+2*c)*ln(x-_R),_R=RootOf(a*_Z^4+(2*a+b)*_Z^3+(a+2*b+c
)*_Z^2+(b+2*c)*_Z+c-1))+sum((b*(5*a*_R^4+(6*a*c+b^2)*_R^2+a*c^2)-2*_R^5*a^(3/2)+4*_R^3*(-a^(3/2)*c-b^2*a^(1/2)
)+2*c*_R*(-a^(3/2)*c-b^2*a^(1/2)))/(-3*_R^5*b^2-4*a^(1/2)*b^3-12*a^2*_R^5-2*a^(3/2)*c^3+2*a^(3/2)*c^2*b+a^(1/2
)*c*b^3-4*_R^7*a+32*_R^3*a^2+a^(1/2)*c^3*b+4*_R^3*a*c^2-4*_R^3*b^3-_R*b^4+14*a^(3/2)*_R^6-_R*b^2*c^2+24*a*b^2*
_R-2*a^(1/2)*c^2*b^2-6*a^(3/2)*_R^2*c^2+20*a^(1/2)*_R^4*b^2+9*a^(1/2)*_R^2*b^3-16*a^2*_R^3*c+30*a^(3/2)*_R^4*b
+10*a^(3/2)*_R^4*c+2*_R*b^3*c-30*_R^5*a*b-26*_R^3*a*b^2-4*a^2*_R*c^2-48*a^(3/2)*_R^2*b+7*a^(1/2)*_R^6*b-10*_R*
a*b^2*c+10*_R*a*b*c^2+24*a^(3/2)*_R^2*b*c-5*a^(1/2)*_R^4*b*c-3*a^(1/2)*_R^2*b*c^2-6*a^(1/2)*_R^2*b^2*c-4*_R^3*
a*b*c+4*_R^3*b^2*c)*ln((a*x^2+b*x+c)^(1/2)-a^(1/2)*x-_R),_R=RootOf(_Z^8*a-(4*a^(3/2)+2*a^(1/2)*b)*_Z^7-(-4*a^2
-10*a*b-b^2)*_Z^6-(12*a^(3/2)*b+4*a^(3/2)*c+8*b^2*a^(1/2)-2*a^(1/2)*b*c)*_Z^5-(-8*a^2*c-13*a*b^2-2*a*b*c+2*a*c
^2-2*b^3+2*b^2*c+16*a^2)*_Z^4-(16*a^(3/2)*c*b-4*a^(3/2)*c^2-32*a^(3/2)*b+6*a^(1/2)*b^3-4*a^(1/2)*c*b^2-2*a^(1/
2)*c^2*b)*_Z^3-(-4*a^2*c^2-10*a*b^2*c+10*a*b*c^2-b^4+2*b^3*c-b^2*c^2+24*a*b^2)*_Z^2-(4*a^(3/2)*c^2*b-4*a^(3/2)
*c^3+2*a^(1/2)*c*b^3-4*a^(1/2)*c^2*b^2+2*a^(1/2)*c^3*b-8*a^(1/2)*b^3)*_Z+a*b^2*c^2-2*a*b*c^3+a*c^4-b^4))-1/a*s
um((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)))/(-3*_R^5*b^2-4*a^(1/
2)*b^3-12*a^2*_R^5-2*a^(3/2)*c^3+2*a^(3/2)*c^2*b+a^(1/2)*c*b^3-4*_R^7*a+32*_R^3*a^2+a^(1/2)*c^3*b+4*_R^3*a*c^2
-4*_R^3*b^3-_R*b^4+14*a^(3/2)*_R^6-_R*b^2*c^2+24*a*b^2*_R-2*a^(1/2)*c^2*b^2-6*a^(3/2)*_R^2*c^2+20*a^(1/2)*_R^4
*b^2+9*a^(1/2)*_R^2*b^3-16*a^2*_R^3*c+30*a^(3/2)*_R^4*b+10*a^(3/2)*_R^4*c+2*_R*b^3*c-30*_R^5*a*b-26*_R^3*a*b^2
-4*a^2*_R*c^2-48*a^(3/2)*_R^2*b+7*a^(1/2)*_R^6*b-10*_R*a*b^2*c+10*_R*a*b*c^2+24*a^(3/2)*_R^2*b*c-5*a^(1/2)*_R^
4*b*c-3*a^(1/2)*_R^2*b*c^2-6*a^(1/2)*_R^2*b^2*c-4*_R^3*a*b*c+4*_R^3*b^2*c)*ln((a*x^2+b*x+c)^(1/2)-a^(1/2)*x-_R
),_R=RootOf(_Z^8*a-(4*a^(3/2)+2*a^(1/2)*b)*_Z^7-(-4*a^2-10*a*b-b^2)*_Z^6-(12*a^(3/2)*b+4*a^(3/2)*c+8*b^2*a^(1/
2)-2*a^(1/2)*b*c)*_Z^5-(-8*a^2*c-13*a*b^2-2*a*b*c+2*a*c^2-2*b^3+2*b^2*c+16*a^2)*_Z^4-(16*a^(3/2)*c*b-4*a^(3/2)
*c^2-32*a^(3/2)*b+6*a^(1/2)*b^3-4*a^(1/2)*c*b^2-2*a^(1/2)*c^2*b)*_Z^3-(-4*a^2*c^2-10*a*b^2*c+10*a*b*c^2-b^4+2*
b^3*c-b^2*c^2+24*a*b^2)*_Z^2-(4*a^(3/2)*c^2*b-4*a^(3/2)*c^3+2*a^(1/2)*c*b^3-4*a^(1/2)*c^2*b^2+2*a^(1/2)*c^3*b-
8*a^(1/2)*b^3)*_Z+a*b^2*c^2-2*a*b*c^3+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} {\left (x + 1\right )} - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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) - 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {1}{\left (x+1\right )\,\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 + 1)*(c + b*x + a*x^2)^(1/2) - 1),x)

[Out]

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

[Out]

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

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