3.6 \(\int \frac{x}{\sqrt{-71-96 x+10 x^2+x^4}} \, dx\)
Optimal. Leaf size=76 \[ \frac{1}{8} \log \left (x^8+20 x^6-128 x^5+54 x^4-1408 x^3+3124 x^2+\sqrt{x^4+10 x^2-96 x-71} \left (x^6+15 x^4-80 x^3+27 x^2-528 x+781\right )+10001\right ) \]
[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
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Rubi [A] time = 0.0275953, antiderivative size = 76, normalized size of antiderivative = 1.,
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} \[ \frac{1}{8} \log \left (x^8+20 x^6-128 x^5+54 x^4-1408 x^3+3124 x^2+\sqrt{x^4+10 x^2-96 x-71} \left (x^6+15 x^4-80 x^3+27 x^2-528 x+781\right )+10001\right ) \]
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*}
Mathematica [C] time = 1.637, size = 1226, normalized size = 16.13 \[ \text{result too large to display} \]
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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Maple [C] time = 0.58, size = 1290, normalized size = 17. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/(x^4+10*x^2-96*x-71)^(1/2),x)
[Out]
2*(-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)+RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))*((RootOf(_Z^4+10*_Z^2-96*_Z-7
1,index=4)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2))*(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/(RootOf(_Z^4+10*_Z
^2-96*_Z-71,index=4)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)))^(1/2)*(
x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2))^2*(-(RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)-RootOf(_Z^4+10*_Z^2-96*_Z-
71,index=1))*(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=3))/(-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=3)+RootOf(_Z^4+10*
_Z^2-96*_Z-71,index=1))/(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)))^(1/2)*((RootOf(_Z^4+10*_Z^2-96*_Z-71,index=
2)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))*(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4))/(RootOf(_Z^4+10*_Z^2-96*_Z
-71,index=4)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)))^(1/2)/(RootOf(_
Z^4+10*_Z^2-96*_Z-71,index=4)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2))/(RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)-Ro
otOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/((x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))*(x-RootOf(_Z^4+10*_Z^2-96*_Z-7
1,index=2))*(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=3))*(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)))^(1/2)*(RootOf
(_Z^4+10*_Z^2-96*_Z-71,index=2)*EllipticF(((RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)-RootOf(_Z^4+10*_Z^2-96*_Z-71
,index=2))*(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/(RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)-RootOf(_Z^4+10*_Z^
2-96*_Z-71,index=1))/(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)))^(1/2),((RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)-
RootOf(_Z^4+10*_Z^2-96*_Z-71,index=3))*(-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)+RootOf(_Z^4+10*_Z^2-96*_Z-71,in
dex=1))/(-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=3)+RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/(RootOf(_Z^4+10*_Z^2-96
*_Z-71,index=2)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)))^(1/2))+(-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)+RootOf(
_Z^4+10*_Z^2-96*_Z-71,index=1))*EllipticPi(((RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)-RootOf(_Z^4+10*_Z^2-96*_Z-7
1,index=2))*(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/(RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)-RootOf(_Z^4+10*_Z
^2-96*_Z-71,index=1))/(x-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)))^(1/2),(RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)-
RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/(RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)-RootOf(_Z^4+10*_Z^2-96*_Z-71,ind
ex=2)),((RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=3))*(-RootOf(_Z^4+10*_Z^2-96
*_Z-71,index=4)+RootOf(_Z^4+10*_Z^2-96*_Z-71,index=1))/(-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=3)+RootOf(_Z^4+10*
_Z^2-96*_Z-71,index=1))/(RootOf(_Z^4+10*_Z^2-96*_Z-71,index=2)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)))^(1/2)))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x^{4} + 10 \, x^{2} - 96 \, x - 71}}\,{d x} \end{align*}
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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Fricas [A] time = 2.82829, size = 205, normalized size = 2.7 \begin{align*} \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{align*}
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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x^{4} + 10 x^{2} - 96 x - 71}}\, dx \end{align*}
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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x^{4} + 10 \, x^{2} - 96 \, x - 71}}\,{d x} \end{align*}
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)