3.11.30 \(\int \frac {x}{\sqrt {-71-96 x+10 x^2+x^4}} \, dx\)

Optimal. Leaf size=78 \[ -\frac {1}{8} \log \left (-x^8-20 x^6+128 x^5-54 x^4+1408 x^3-3124 x^2+\sqrt {x^4+10 x^2-96 x-71} \left (x^6+15 x^4-80 x^3+27 x^2-528 x+781\right )-10001\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 76, normalized size of antiderivative = 0.97, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2082} \begin {gather*} \frac {1}{8} \log \left (x^8+20 x^6-128 x^5+54 x^4-1408 x^3+3124 x^2+\sqrt {x^4+10 x^2-96 x-71} \left (x^6+15 x^4-80 x^3+27 x^2-528 x+781\right )+10001\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[-71 - 96*x + 10*x^2 + x^4],x]

[Out]

Log[10001 + 3124*x^2 - 1408*x^3 + 54*x^4 - 128*x^5 + 20*x^6 + x^8 + Sqrt[-71 - 96*x + 10*x^2 + x^4]*(781 - 528
*x + 27*x^2 - 80*x^3 + 15*x^4 + x^6)]/8

Rule 2082

Int[(x_)/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (e_.)*(x_)^4], x_Symbol] :> With[{Px = (1*(33*b^2*c + 6*a*c^2
+ 40*a^2*e))/320 - (22*a*c*e*x^2)/5 + (22*b*c*e*x^3)/15 + (1*e*(5*c^2 + 4*a*e)*x^4)/4 + (4*b*e^2*x^5)/3 + 2*c
*e^2*x^6 + e^3*x^8}, Simp[(1*Log[Px + Dist[1/(8*Rt[e, 2]*x), D[Px, x], x]*Sqrt[a + b*x + c*x^2 + e*x^4]])/(8*R
t[e, 2]), x]] /; FreeQ[{a, b, c, e}, x] && EqQ[71*c^2 + 100*a*e, 0] && EqQ[1152*c^3 - 125*b^2*e, 0]

Rubi steps

\begin {align*} \int \frac {x}{\sqrt {-71-96 x+10 x^2+x^4}} \, dx &=\frac {1}{8} \log \left (10001+3124 x^2-1408 x^3+54 x^4-128 x^5+20 x^6+x^8+\sqrt {-71-96 x+10 x^2+x^4} \left (781-528 x+27 x^2-80 x^3+15 x^4+x^6\right )\right )\\ \end {align*}

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Mathematica [C]  time = 2.68, size = 1226, normalized size = 15.72

result too large to display

Warning: Unable to verify antiderivative.

[In]

Integrate[x/Sqrt[-71 - 96*x + 10*x^2 + x^4],x]

[Out]

(-2*(Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - x)*((Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])])*EllipticF[ArcSin[Sqrt[((Sqr
t[3] - 2*Sqrt[2*(-1 + Sqrt[3])] - x)*(Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 &
, 4, 0]))/((Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - x)*(Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 +
10*#1^2 + #1^4 & , 4, 0]))]], ((Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 3,
0])*(Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))/((Sqrt[3] - 2*Sqrt[2*(
-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 3, 0])*(Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 -
96*#1 + 10*#1^2 + #1^4 & , 4, 0]))] - 4*Sqrt[2*(-1 + Sqrt[3])]*EllipticPi[(Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])]
- Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0])/(Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1
^2 + #1^4 & , 4, 0]), ArcSin[Sqrt[((Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])] - x)*(Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])
] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))/((Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - x)*(Sqrt[3] - 2*Sqrt[
2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))]], ((Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - Ro
ot[-71 - 96*#1 + 10*#1^2 + #1^4 & , 3, 0])*(Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 +
#1^4 & , 4, 0]))/((Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 3, 0])*(Sqrt[3]
+ 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))])*Sqrt[(x - Root[-71 - 96*#1 + 10*#
1^2 + #1^4 & , 3, 0])/((Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - x)*(Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71
- 96*#1 + 10*#1^2 + #1^4 & , 3, 0]))]*Sqrt[((Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])] - x)*(Sqrt[3] + 2*Sqrt[2*(-1
+ Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))/((Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - x)*(Sqrt[3]
- 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))]*(x - Root[-71 - 96*#1 + 10*#1^2 +
#1^4 & , 4, 0]))/(Sqrt[-71 - 96*x + 10*x^2 + x^4]*(Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10
*#1^2 + #1^4 & , 4, 0])*Sqrt[(x - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0])/((Sqrt[3] + 2*Sqrt[2*(-1 + Sqrt
[3])] - x)*(Sqrt[3] - 2*Sqrt[2*(-1 + Sqrt[3])] - Root[-71 - 96*#1 + 10*#1^2 + #1^4 & , 4, 0]))])

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IntegrateAlgebraic [A]  time = 5.17, size = 78, normalized size = 1.00 \begin {gather*} -\frac {1}{8} \log \left (-10001-3124 x^2+1408 x^3-54 x^4+128 x^5-20 x^6-x^8+\sqrt {-71-96 x+10 x^2+x^4} \left (781-528 x+27 x^2-80 x^3+15 x^4+x^6\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/Sqrt[-71 - 96*x + 10*x^2 + x^4],x]

[Out]

-1/8*Log[-10001 - 3124*x^2 + 1408*x^3 - 54*x^4 + 128*x^5 - 20*x^6 - x^8 + Sqrt[-71 - 96*x + 10*x^2 + x^4]*(781
- 528*x + 27*x^2 - 80*x^3 + 15*x^4 + x^6)]

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fricas [A]  time = 0.55, size = 72, normalized size = 0.92 \begin {gather*} \frac {1}{8} \, \log \left (x^{8} + 20 \, x^{6} - 128 \, x^{5} + 54 \, x^{4} - 1408 \, x^{3} + 3124 \, x^{2} + {\left (x^{6} + 15 \, x^{4} - 80 \, x^{3} + 27 \, x^{2} - 528 \, x + 781\right )} \sqrt {x^{4} + 10 \, x^{2} - 96 \, x - 71} + 10001\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^4+10*x^2-96*x-71)^(1/2),x, algorithm="fricas")

[Out]

1/8*log(x^8 + 20*x^6 - 128*x^5 + 54*x^4 - 1408*x^3 + 3124*x^2 + (x^6 + 15*x^4 - 80*x^3 + 27*x^2 - 528*x + 781)
*sqrt(x^4 + 10*x^2 - 96*x - 71) + 10001)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x^{4} + 10 \, x^{2} - 96 \, x - 71}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^4+10*x^2-96*x-71)^(1/2),x, algorithm="giac")

[Out]

integrate(x/sqrt(x^4 + 10*x^2 - 96*x - 71), x)

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maple [A]  time = 1.10, size = 148, normalized size = 1.90

method	result	size
trager	\(\frac {\ln \left (x^{8}+\sqrt {x^{4}+10 x^{2}-96 x -71}\, x^{6}+20 x^{6}+15 \sqrt {x^{4}+10 x^{2}-96 x -71}\, x^{4}-128 x^{5}-80 \sqrt {x^{4}+10 x^{2}-96 x -71}\, x^{3}+54 x^{4}+27 \sqrt {x^{4}+10 x^{2}-96 x -71}\, x^{2}-1408 x^{3}-528 x \sqrt {x^{4}+10 x^{2}-96 x -71}+3124 x^{2}+781 \sqrt {x^{4}+10 x^{2}-96 x -71}+10001\right )}{8}\)	\(148\)
default	\(\text {Expression too large to display}\)	\(1290\)
elliptic	\(\text {Expression too large to display}\)	\(1290\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^4+10*x^2-96*x-71)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/8*ln(x^8+(x^4+10*x^2-96*x-71)^(1/2)*x^6+20*x^6+15*(x^4+10*x^2-96*x-71)^(1/2)*x^4-128*x^5-80*(x^4+10*x^2-96*x
-71)^(1/2)*x^3+54*x^4+27*(x^4+10*x^2-96*x-71)^(1/2)*x^2-1408*x^3-528*x*(x^4+10*x^2-96*x-71)^(1/2)+3124*x^2+781
*(x^4+10*x^2-96*x-71)^(1/2)+10001)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x^{4} + 10 \, x^{2} - 96 \, x - 71}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^4+10*x^2-96*x-71)^(1/2),x, algorithm="maxima")

[Out]

integrate(x/sqrt(x^4 + 10*x^2 - 96*x - 71), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x}{\sqrt {x^4+10\,x^2-96\,x-71}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(10*x^2 - 96*x + x^4 - 71)^(1/2),x)

[Out]

int(x/(10*x^2 - 96*x + x^4 - 71)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x^{4} + 10 x^{2} - 96 x - 71}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**4+10*x**2-96*x-71)**(1/2),x)

[Out]

Integral(x/sqrt(x**4 + 10*x**2 - 96*x - 71), x)

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