3.7.8 \(\int \frac {77-46 x+5 x^2}{(-23+82 x-23 x^2) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\) [608]
Optimal. Leaf size=48 \[ \frac {\text {ArcTan}\left (\frac {2 \sqrt {42} \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}{103-86 x+19 x^2}\right )}{\sqrt {42}} \]
[Out]
1/42*arctan(2*42^(1/2)*(x^4-3*x^3-21*x^2+83*x-60)^(1/2)/(19*x^2-86*x+103))*42^(1/2)
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Rubi [F]
time = 0.52, 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 {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(77 - 46*x + 5*x^2)/((-23 + 82*x - 23*x^2)*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4]),x]
[Out]
(-5*Defer[Int][1/Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4], x])/23 - (24*(27 + 10*Sqrt[2])*Defer[Int][1/((82 - 4
8*Sqrt[2] - 46*x)*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4]), x])/23 - (24*(27 - 10*Sqrt[2])*Defer[Int][1/((82 +
48*Sqrt[2] - 46*x)*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4]), x])/23
Rubi steps
\begin {align*} \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx &=\int \left (-\frac {5}{23 \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}+\frac {72 (23-9 x)}{23 \left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}\right ) \, dx\\ &=-\left (\frac {5}{23} \int \frac {1}{\sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\right )+\frac {72}{23} \int \frac {23-9 x}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\\ &=-\left (\frac {5}{23} \int \frac {1}{\sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\right )+\frac {72}{23} \int \left (\frac {-9-\frac {10 \sqrt {2}}{3}}{\left (82-48 \sqrt {2}-46 x\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}+\frac {-9+\frac {10 \sqrt {2}}{3}}{\left (82+48 \sqrt {2}-46 x\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}\right ) \, dx\\ &=-\left (\frac {5}{23} \int \frac {1}{\sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\right )-\frac {1}{23} \left (24 \left (27-10 \sqrt {2}\right )\right ) \int \frac {1}{\left (82+48 \sqrt {2}-46 x\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx-\frac {1}{23} \left (24 \left (27+10 \sqrt {2}\right )\right ) \int \frac {1}{\left (82-48 \sqrt {2}-46 x\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 48, normalized size = 1.00 \begin {gather*} -\sqrt {\frac {2}{21}} \text {ArcTan}\left (\frac {\sqrt {\frac {21}{2}} \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}{-20+x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(77 - 46*x + 5*x^2)/((-23 + 82*x - 23*x^2)*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4]),x]
[Out]
-(Sqrt[2/21]*ArcTan[(Sqrt[21/2]*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4])/(-20 + x + x^2)])
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.00, size = 3042, normalized size = 63.38
| | |
| method |
result |
size |
| | |
| trager |
\(-\frac {\RootOf \left (\textit {\_Z}^{2}+42\right ) \ln \left (\frac {19 \RootOf \left (\textit {\_Z}^{2}+42\right ) x^{2}-86 \RootOf \left (\textit {\_Z}^{2}+42\right ) x +103 \RootOf \left (\textit {\_Z}^{2}+42\right )-84 \sqrt {x^{4}-3 x^{3}-21 x^{2}+83 x -60}}{23 x^{2}-82 x +23}\right )}{42}\) |
\(74\) |
| default |
\(\text {Expression too large to display}\) |
\(3042\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(3260\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((5*x^2-46*x+77)/(-23*x^2+82*x-23)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2),x,method=_RETURNVERBOSE)
[Out]
5/138*((x+5)/(-1+x))^(1/2)*(-1+x)^2*((-3+x)/(-1+x))^(1/2)*6^(1/2)*((x-4)/(-1+x))^(1/2)/((x+5)*(-1+x)*(-3+x)*(x
-4))^(1/2)*EllipticF(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2),1/2*3^(1/2))+120/529*2^(1/2)*(x/(-1+x)+5/(-1+x))^(1/2)*(
-3/(-1+x)+x/(-1+x))^(1/2)*6^(1/2)*(x/(-1+x)-4/(-1+x))^(1/2)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2)/(-18/23-24/23*2^(
1/2))*x^2/(-156/23-24/23*2^(1/2))*EllipticPi(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2),3*(-18/23-24/23*2^(1/2))/(-156/2
3-24/23*2^(1/2)),1/2*3^(1/2))-240/529*2^(1/2)*(x/(-1+x)+5/(-1+x))^(1/2)*(-3/(-1+x)+x/(-1+x))^(1/2)*6^(1/2)*(x/
(-1+x)-4/(-1+x))^(1/2)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2)/(-18/23-24/23*2^(1/2))*x/(-156/23-24/23*2^(1/2))*Ellip
ticPi(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2),3*(-18/23-24/23*2^(1/2))/(-156/23-24/23*2^(1/2)),1/2*3^(1/2))+108/529*(
x/(-1+x)+5/(-1+x))^(1/2)*(-3/(-1+x)+x/(-1+x))^(1/2)*6^(1/2)*(x/(-1+x)-4/(-1+x))^(1/2)/(x^4-3*x^3-21*x^2+83*x-6
0)^(1/2)/(-18/23+24/23*2^(1/2))*x*EllipticF(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2),1/2*3^(1/2))-324/529*(x/(-1+x)+5/
(-1+x))^(1/2)*(-3/(-1+x)+x/(-1+x))^(1/2)*6^(1/2)*(x/(-1+x)-4/(-1+x))^(1/2)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2)/(-
18/23+24/23*2^(1/2))/(-156/23+24/23*2^(1/2))*EllipticPi(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2),3*(-18/23+24/23*2^(1/
2))/(-156/23+24/23*2^(1/2)),1/2*3^(1/2))-20/529*2^(1/2)*(x/(-1+x)+5/(-1+x))^(1/2)*(-3/(-1+x)+x/(-1+x))^(1/2)*6
^(1/2)*(x/(-1+x)-4/(-1+x))^(1/2)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2)/(-18/23+24/23*2^(1/2))*EllipticF(1/3*3^(1/2)
*((x+5)/(-1+x))^(1/2),1/2*3^(1/2))-54/529*(x/(-1+x)+5/(-1+x))^(1/2)*(-3/(-1+x)+x/(-1+x))^(1/2)*6^(1/2)*(x/(-1+
x)-4/(-1+x))^(1/2)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2)/(-18/23+24/23*2^(1/2))*x^2*EllipticF(1/3*3^(1/2)*((x+5)/(-
1+x))^(1/2),1/2*3^(1/2))+108/529*(x/(-1+x)+5/(-1+x))^(1/2)*(-3/(-1+x)+x/(-1+x))^(1/2)*6^(1/2)*(x/(-1+x)-4/(-1+
x))^(1/2)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2)/(-18/23-24/23*2^(1/2))*x*EllipticF(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2)
,1/2*3^(1/2))-324/529*(x/(-1+x)+5/(-1+x))^(1/2)*(-3/(-1+x)+x/(-1+x))^(1/2)*6^(1/2)*(x/(-1+x)-4/(-1+x))^(1/2)/(
x^4-3*x^3-21*x^2+83*x-60)^(1/2)/(-18/23-24/23*2^(1/2))/(-156/23-24/23*2^(1/2))*EllipticPi(1/3*3^(1/2)*((x+5)/(
-1+x))^(1/2),3*(-18/23-24/23*2^(1/2))/(-156/23-24/23*2^(1/2)),1/2*3^(1/2))-54/529*(x/(-1+x)+5/(-1+x))^(1/2)*(-
3/(-1+x)+x/(-1+x))^(1/2)*6^(1/2)*(x/(-1+x)-4/(-1+x))^(1/2)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2)/(-18/23-24/23*2^(1
/2))*x^2*EllipticF(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2),1/2*3^(1/2))+20/529*2^(1/2)*(x/(-1+x)+5/(-1+x))^(1/2)*(-3/
(-1+x)+x/(-1+x))^(1/2)*6^(1/2)*(x/(-1+x)-4/(-1+x))^(1/2)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2)/(-18/23-24/23*2^(1/2
))*EllipticF(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2),1/2*3^(1/2))-120/529*2^(1/2)*(x/(-1+x)+5/(-1+x))^(1/2)*(-3/(-1+x
)+x/(-1+x))^(1/2)*6^(1/2)*(x/(-1+x)-4/(-1+x))^(1/2)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2)/(-18/23+24/23*2^(1/2))*x^
2/(-156/23+24/23*2^(1/2))*EllipticPi(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2),3*(-18/23+24/23*2^(1/2))/(-156/23+24/23*
2^(1/2)),1/2*3^(1/2))+240/529*2^(1/2)*(x/(-1+x)+5/(-1+x))^(1/2)*(-3/(-1+x)+x/(-1+x))^(1/2)*6^(1/2)*(x/(-1+x)-4
/(-1+x))^(1/2)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2)/(-18/23+24/23*2^(1/2))*x/(-156/23+24/23*2^(1/2))*EllipticPi(1/
3*3^(1/2)*((x+5)/(-1+x))^(1/2),3*(-18/23+24/23*2^(1/2))/(-156/23+24/23*2^(1/2)),1/2*3^(1/2))-40/529*2^(1/2)*(x
/(-1+x)+5/(-1+x))^(1/2)*(-3/(-1+x)+x/(-1+x))^(1/2)*6^(1/2)*(x/(-1+x)-4/(-1+x))^(1/2)/(x^4-3*x^3-21*x^2+83*x-60
)^(1/2)/(-18/23-24/23*2^(1/2))*x*EllipticF(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2),1/2*3^(1/2))+120/529*2^(1/2)*(x/(-
1+x)+5/(-1+x))^(1/2)*(-3/(-1+x)+x/(-1+x))^(1/2)*6^(1/2)*(x/(-1+x)-4/(-1+x))^(1/2)/(x^4-3*x^3-21*x^2+83*x-60)^(
1/2)/(-18/23-24/23*2^(1/2))/(-156/23-24/23*2^(1/2))*EllipticPi(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2),3*(-18/23-24/2
3*2^(1/2))/(-156/23-24/23*2^(1/2)),1/2*3^(1/2))-324/529*(x/(-1+x)+5/(-1+x))^(1/2)*(-3/(-1+x)+x/(-1+x))^(1/2)*6
^(1/2)*(x/(-1+x)-4/(-1+x))^(1/2)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2)/(-18/23-24/23*2^(1/2))*x^2/(-156/23-24/23*2^
(1/2))*EllipticPi(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2),3*(-18/23-24/23*2^(1/2))/(-156/23-24/23*2^(1/2)),1/2*3^(1/2
))+20/529*2^(1/2)*(x/(-1+x)+5/(-1+x))^(1/2)*(-3/(-1+x)+x/(-1+x))^(1/2)*6^(1/2)*(x/(-1+x)-4/(-1+x))^(1/2)/(x^4-
3*x^3-21*x^2+83*x-60)^(1/2)/(-18/23-24/23*2^(1/2))*x^2*EllipticF(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2),1/2*3^(1/2))
-54/529*(x/(-1+x)+5/(-1+x))^(1/2)*(-3/(-1+x)+x/(-1+x))^(1/2)*6^(1/2)*(x/(-1+x)-4/(-1+x))^(1/2)/(x^4-3*x^3-21*x
^2+83*x-60)^(1/2)/(-18/23-24/23*2^(1/2))*EllipticF(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2),1/2*3^(1/2))-54/529*(x/(-1
+x)+5/(-1+x))^(1/2)*(-3/(-1+x)+x/(-1+x))^(1/2)*6^(1/2)*(x/(-1+x)-4/(-1+x))^(1/2)/(x^4-3*x^3-21*x^2+83*x-60)^(1
/2)/(-18/23+24/23*2^(1/2))*EllipticF(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2),1/2*3^(1/2))+648/529*(x/(-1+x)+5/(-1+x))
^(1/2)*(-3/(-1+x)+x/(-1+x))^(1/2)*6^(1/2)*(x/(-1+x)-4/(-1+x))^(1/2)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2)/(-18/23-2
4/23*2^(1/2))*x/(-156/23-24/23*2^(1/2))*EllipticPi(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2),3*(-18/23-24/23*2^(1/2))/(
-156/23-24/23*2^(1/2)),1/2*3^(1/2))-20/529*2^(1/2)*(x/(-1+x)+5/(-1+x))^(1/2)*(-3/(-1+x)+x/(-1+x))^(1/2)*6^(1/2
)*(x/(-1+x)-4/(-1+x))^(1/2)/(x^4-3*x^3-21*x^2+8...
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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((5*x^2-46*x+77)/(-23*x^2+82*x-23)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2),x, algorithm="maxima")
[Out]
-integrate((5*x^2 - 46*x + 77)/(sqrt(x^4 - 3*x^3 - 21*x^2 + 83*x - 60)*(23*x^2 - 82*x + 23)), x)
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Fricas [A]
time = 0.40, size = 49, normalized size = 1.02 \begin {gather*} \frac {1}{42} \, \sqrt {21} \sqrt {2} \arctan \left (\frac {2 \, \sqrt {21} \sqrt {2} \sqrt {x^{4} - 3 \, x^{3} - 21 \, x^{2} + 83 \, x - 60}}{19 \, x^{2} - 86 \, x + 103}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((5*x^2-46*x+77)/(-23*x^2+82*x-23)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2),x, algorithm="fricas")
[Out]
1/42*sqrt(21)*sqrt(2)*arctan(2*sqrt(21)*sqrt(2)*sqrt(x^4 - 3*x^3 - 21*x^2 + 83*x - 60)/(19*x^2 - 86*x + 103))
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {46 x}{23 x^{2} \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} - 82 x \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} + 23 \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60}}\right )\, dx - \int \frac {5 x^{2}}{23 x^{2} \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} - 82 x \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} + 23 \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60}}\, dx - \int \frac {77}{23 x^{2} \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} - 82 x \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} + 23 \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((5*x**2-46*x+77)/(-23*x**2+82*x-23)/(x**4-3*x**3-21*x**2+83*x-60)**(1/2),x)
[Out]
-Integral(-46*x/(23*x**2*sqrt(x**4 - 3*x**3 - 21*x**2 + 83*x - 60) - 82*x*sqrt(x**4 - 3*x**3 - 21*x**2 + 83*x
- 60) + 23*sqrt(x**4 - 3*x**3 - 21*x**2 + 83*x - 60)), x) - Integral(5*x**2/(23*x**2*sqrt(x**4 - 3*x**3 - 21*x
**2 + 83*x - 60) - 82*x*sqrt(x**4 - 3*x**3 - 21*x**2 + 83*x - 60) + 23*sqrt(x**4 - 3*x**3 - 21*x**2 + 83*x - 6
0)), x) - Integral(77/(23*x**2*sqrt(x**4 - 3*x**3 - 21*x**2 + 83*x - 60) - 82*x*sqrt(x**4 - 3*x**3 - 21*x**2 +
83*x - 60) + 23*sqrt(x**4 - 3*x**3 - 21*x**2 + 83*x - 60)), 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((5*x^2-46*x+77)/(-23*x^2+82*x-23)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2),x, algorithm="giac")
[Out]
integrate(-(5*x^2 - 46*x + 77)/(sqrt(x^4 - 3*x^3 - 21*x^2 + 83*x - 60)*(23*x^2 - 82*x + 23)), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int -\frac {5\,x^2-46\,x+77}{\left (23\,x^2-82\,x+23\right )\,\sqrt {x^4-3\,x^3-21\,x^2+83\,x-60}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(5*x^2 - 46*x + 77)/((23*x^2 - 82*x + 23)*(83*x - 21*x^2 - 3*x^3 + x^4 - 60)^(1/2)),x)
[Out]
int(-(5*x^2 - 46*x + 77)/((23*x^2 - 82*x + 23)*(83*x - 21*x^2 - 3*x^3 + x^4 - 60)^(1/2)), x)
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