∫ 1−x4 _______________________________________x2 √ ___________

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

Optimal. Leaf size=101 \[ -\frac {b \log \left (2 \sqrt {a} \sqrt {a x^4+a+b x^3+b x+c x^2}-2 a x^2-2 a-b x\right )}{2 a^{3/2}}+\frac {b \log (x)}{2 a^{3/2}}-\frac {\sqrt {a x^4+a+b x^3+b x+c x^2}}{a x} \]

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Rubi [F] time = 0.91, 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^4}{x^2 \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^4)/(x^2*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]

[Out]

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

*x^4], x]

Rubi steps

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

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Mathematica [C] time = 6.21, size = 3897, normalized size = 38.58 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

b*x^3 + a*x^4]*((2*(x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])^2*(-(EllipticF[ArcSin[Sqrt[((x - 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 & , 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])))]*Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2]) + Elli

pticPi[(-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[Sqr

t[((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] - Ro

ot[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])))]*(-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]))*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] + Roo

t[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 & , 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]))]*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 & , 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]))])/(Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]*(-Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] + Ro

ot[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])) + (2*(x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])^2*(-(Elliptic

F[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])))]*Root[a + b*#1 + c*#1^2 + b*#1^3

+ a*#1^4 & , 1]) - 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])))]*(-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]))*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]))]*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] - Ro

ot[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))]*Sqrt[((-Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] + Roo

t[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]))/((x - Roo

t[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 & , 4]))/(Sqrt[a + b*x + c*x^2 + b*x^3 + a*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] + Ro

ot[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]))))/(2*a*Sqrt[x*(b + c*x + b*x^2) + a*(1 + x^4)])

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IntegrateAlgebraic [A] time = 0.32, size = 105, normalized size = 1.04 \begin {gather*} -\frac {\sqrt {a+b x+c x^2+b x^3+a x^4}}{a x}+\frac {b \log (x)}{2 a^{3/2}}-\frac {b \log \left (2 a^2+a b x+2 a^2 x^2-2 a^{3/2} \sqrt {a+b x+c x^2+b x^3+a x^4}\right )}{2 a^{3/2}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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fricas [A] time = 2.18, size = 216, normalized size = 2.14 \begin {gather*} \left [\frac {\sqrt {a} b x \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 ) - 4 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} a}{4 \, a^{2} x}, -\frac {\sqrt {-a} b x \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 ) + 2 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} a}{2 \, a^{2} x}\right ] \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

a*x^4 + b*x^3 + c*x^2 + b*x + a)*a)/(a^2*x)]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C] time = 0.22, size = 3402, normalized size = 33.68


method	result	size

default	\(\text {Expression too large to display}\)	\(3402\)
risch	\(\text {Expression too large to display}\)	\(3402\)
elliptic	\(\text {Expression too large to display}\)	\(3402\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

ex=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,i

ndex=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)*(RootOf(_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,ind

ex=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))*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))/(R

ootOf(_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=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)),((RootO

f(_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)))-b/a*(-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,inde

x=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,in

dex=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)/RootOf(_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,i

ndex=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,inde

x=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-RootO

f(_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))*(-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=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)))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x^4-1}{x^2\,\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^4 - 1)/(x^2*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)),x)

[Out]

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

[Out]

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

x + c*x**2), x)

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