3.11.30 \(\int \frac {x}{\sqrt {-71-96 x+10 x^2+x^4}} \, dx\) [1030]
Optimal. Leaf size=78 \[ -\frac {1}{8} \log \left (-10001-3124 x^2+1408 x^3-54 x^4+128 x^5-20 x^6-x^8+\sqrt {-71-96 x+10 x^2+x^4} \left (781-528 x+27 x^2-80 x^3+15 x^4+x^6\right )\right ) \]
[Out]
-1/8*ln(-10001-3124*x^2+1408*x^3-54*x^4+128*x^5-20*x^6-x^8+(x^4+10*x^2-96*x-71)^(1/2)*(x^6+15*x^4-80*x^3+27*x^
2-528*x+781))
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Rubi [A]
time = 0.02, 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 = {2107}
\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 2107
Int[(x_)/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (e_.)*(x_)^4], x_Symbol] :> With[{Px = (1/320)*(33*b^2*c + 6*
a*c^2 + 40*a^2*e) - (22/5)*a*c*e*x^2 + (22/15)*b*c*e*x^3 + (1/4)*e*(5*c^2 + 4*a*e)*x^4 + (4/3)*b*e^2*x^5 + 2*c
*e^2*x^6 + e^3*x^8}, Simp[(1/(8*Rt[e, 2]))*Log[Px + Dist[1/(8*Rt[e, 2]*x), D[Px, x], x]*Sqrt[a + b*x + c*x^2 +
e*x^4]], 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 [A]
time = 4.95, 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]
Integrate[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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 2.03, size = 1290, normalized size = 16.54
| | |
| 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 |
\(\text {Expression too large to display}\) |
\(1290\) |
| | |
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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]
2*(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=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-7
1,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)-Roo
tOf(_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-71
,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)-R
ootOf(_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,inde
x=4))/(-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=1)-RootOf(_Z^
4+10*_Z^2-96*_Z-71,index=2))*EllipticPi(((RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4)-RootOf(_Z^4+10*_Z^2-96*_Z-71,i
ndex=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)-Roo
tOf(_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=
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=1)-RootOf(_Z^4+10*_Z^2-96*_Z-71,index=4))/(-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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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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Fricas [A]
time = 0.41, 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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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