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3.9.18 \(\int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\)

Optimal. Leaf size=62 \[ \frac {\log \left (2 \sqrt {a} \sqrt {a x^4+a+b x^3+b x+c x^2}-2 a x^2-2 a-b x\right )}{\sqrt {a}}-\frac {\log (x)}{\sqrt {a}} \]

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Rubi [F]  time = 0.68, 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-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 1.88, size = 2477, normalized size = 39.95 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])^2*((Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] -
 Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(EllipticPi[(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2]
*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))/(Root[a
+ b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1]*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*
#1^2 + b*#1^3 + a*#1^4 & , 4])), ArcSin[Sqrt[((x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1])*(Root[a +
b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))/((x - Root[a + b*#1
 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2
+ b*#1^3 + a*#1^4 & , 4]))]], -(((Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b
*#1^3 + a*#1^4 & , 3])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*
#1^4 & , 4]))/((-Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] + Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 &
, 3])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4])))] +
 EllipticPi[(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]
)/(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]), ArcSin[
Sqrt[((x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] -
 Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))/((x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(Roo
t[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))]], -(((Root[a
 + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 3])*(Root[a + b*#1 +
c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))/((-Root[a + b*#1 + c*#1^2
+ b*#1^3 + a*#1^4 & , 1] + Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 3])*(Root[a + b*#1 + c*#1^2 + b*#1^3 +
 a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4])))]*Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4
& , 1]*Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2]) + EllipticF[ArcSin[Sqrt[((x - Root[a + b*#1 + c*#1^2 +
 b*#1^3 + a*#1^4 & , 1])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 +
a*#1^4 & , 4]))/((x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1
^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))]], -(((Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4
& , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 3])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - R
oot[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))/((-Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] + Root[a +
 b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 3])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*
#1^2 + b*#1^3 + a*#1^4 & , 4])))]*Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1]*(-1 + Root[a + b*#1 + c*#1^2
 + b*#1^3 + a*#1^4 & , 2]^2))*Sqrt[((Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2
+ b*#1^3 + a*#1^4 & , 2])*(x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 3]))/((x - Root[a + b*#1 + c*#1^2
+ b*#1^3 + a*#1^4 & , 2])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 +
 a*#1^4 & , 3]))]*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4
& , 4])*Sqrt[((x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4
& , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]
)*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))/((x - R
oot[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])^2*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a +
b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4])^2)])/(Sqrt[x*(b + c*x + b*x^2) + a*(1 + x^4)]*Root[a + b*#1 + c*#1^2 +
 b*#1^3 + a*#1^4 & , 1]*Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2]*(-Root[a + b*#1 + c*#1^2 + b*#1^3 + a*
#1^4 & , 1] + Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2
] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))

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IntegrateAlgebraic [A]  time = 0.31, size = 62, normalized size = 1.00 \begin {gather*} -\frac {\log (x)}{\sqrt {a}}+\frac {\log \left (-2 a-b x-2 a x^2+2 \sqrt {a} \sqrt {a+b x+c x^2+b x^3+a x^4}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Log[x]/Sqrt[a]) + Log[-2*a - b*x - 2*a*x^2 + 2*Sqrt[a]*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]]/Sqrt[a]

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fricas [A]  time = 2.46, size = 148, normalized size = 2.39 \begin {gather*} \left [\frac {\log \left (\frac {8 \, a^{2} x^{4} + 8 \, a b x^{3} + 8 \, a b x + {\left (8 \, a^{2} + b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (2 \, a x^{2} + b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} \sqrt {-a}}{2 \, a x^{2} + b x + 2 \, a}\right )}{a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)/x/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/2*log((8*a^2*x^4 + 8*a*b*x^3 + 8*a*b*x + (8*a^2 + b^2 + 4*a*c)*x^2 - 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a
)*(2*a*x^2 + b*x + 2*a)*sqrt(a) + 8*a^2)/x^2)/sqrt(a), sqrt(-a)*arctan(2*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)
*sqrt(-a)/(2*a*x^2 + b*x + 2*a))/a]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {x^{2} - 1}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)/x/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

integrate(-(x^2 - 1)/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*x), x)

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maple [C]  time = 0.18, size = 3365, normalized size = 54.27

method result size
default \(\text {Expression too large to display}\) \(3365\)
elliptic \(\text {Expression too large to display}\) \(3365\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+1)/x/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2*(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))*((RootOf(_Z^4*a+
_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b
+a,index=1))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-Root
Of(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2)*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))^2*((RootOf(_Z^
4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+
_Z*b+a,index=3))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-
RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2)*((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*
a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+
_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^
(1/2)/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))/(RootOf(_Z^4*a
+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)*(R
ootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)*EllipticF(((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4
*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c
+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))
^(1/2),((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3))*(-RootOf(_Z^
4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_
Z*b+a,index=3)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+Root
Of(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2))+(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)+RootOf(_Z^4*a+_
Z^3*b+_Z^2*c+_Z*b+a,index=1))*EllipticPi(((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z
^2*c+_Z*b+a,index=2))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,inde
x=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2),(Root
Of(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(RootOf(_Z^4*a+_Z^3*b+_Z^
2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)),((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)
-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3))*(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3
*b+_Z^2*c+_Z*b+a,index=1))/(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,in
dex=1))/(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2)))+2*
(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))*((RootOf(_Z^4*a+_Z^
3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,
index=1))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-RootOf(
_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2)*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))^2*((RootOf(_Z^4*a
+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*
b+a,index=3))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-Roo
tOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2)*((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_
Z^3*b+_Z^2*c+_Z*b+a,index=1))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*
b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/
2)/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))/(RootOf(_Z^4*a+_Z
^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)/RootO
f(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)*(EllipticF(((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+
_Z^3*b+_Z^2*c+_Z*b+a,index=2))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z
*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1
/2),((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3))*(-RootOf(_Z^4*a
+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b
+a,index=3)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(
_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2))+(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*
b+_Z^2*c+_Z*b+a,index=1))/RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1)*EllipticPi(((RootOf(_Z^4*a+_Z^3*b+_Z^2*c
+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/
(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-RootOf(_Z^4*a+_Z^
3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2),RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)*(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z
*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1)/(RootOf
(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)),((RootOf(_Z^4*a+_Z^3*b+_Z^2
*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3))*(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+
RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3)+RootOf(_Z^4*a+_Z^3*
b+_Z^2*c+_Z*b+a,index=1))/(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,ind
ex=2)))^(1/2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {x^{2} - 1}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)/x/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

-integrate((x^2 - 1)/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int -\frac {x^2-1}{x\,\sqrt {a\,x^4+b\,x^3+c\,x^2+b\,x+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2 - 1)/(x*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)),x)

[Out]

int(-(x^2 - 1)/(x*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {1}{x \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\right )\, dx - \int \frac {x}{\sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2+1)/x/(a*x**4+b*x**3+c*x**2+b*x+a)**(1/2),x)

[Out]

-Integral(-1/(x*sqrt(a*x**4 + a + b*x**3 + b*x + c*x**2)), x) - Integral(x/sqrt(a*x**4 + a + b*x**3 + b*x + c*
x**2), x)

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