3.7.8 \(\int \frac {77-46 x+5 x^2}{(-23+82 x-23 x^2) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\)

Optimal. Leaf size=48 \[ \frac {\tan ^{-1}\left (\frac {2 \sqrt {42} \sqrt {x^4-3 x^3-21 x^2+83 x-60}}{19 x^2-86 x+103}\right )}{\sqrt {42}} \]

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Rubi [F]  time = 0.77, 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 [C]  time = 0.46, size = 196, normalized size = 4.08 \begin {gather*} \frac {\sqrt {\frac {x-4}{x-1}} \sqrt {\frac {x-3}{x-1}} (x-1)^2 \sqrt {\frac {x+5}{x-1}} \left (-7 F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {x+5}{x-1}}}{\sqrt {3}}\right )|\frac {3}{4}\right )+\left (5+4 \sqrt {2}\right ) \Pi \left (\frac {3}{14}-\frac {3 \sqrt {2}}{7};\sin ^{-1}\left (\frac {\sqrt {\frac {x+5}{x-1}}}{\sqrt {3}}\right )|\frac {3}{4}\right )+\left (5-4 \sqrt {2}\right ) \Pi \left (\frac {3}{14}+\frac {3 \sqrt {2}}{7};\sin ^{-1}\left (\frac {\sqrt {\frac {x+5}{x-1}}}{\sqrt {3}}\right )|\frac {3}{4}\right )\right )}{7 \sqrt {6} \sqrt {x^4-3 x^3-21 x^2+83 x-60}} \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[(-4 + x)/(-1 + x)]*Sqrt[(-3 + x)/(-1 + x)]*(-1 + x)^2*Sqrt[(5 + x)/(-1 + x)]*(-7*EllipticF[ArcSin[Sqrt[(
5 + x)/(-1 + x)]/Sqrt[3]], 3/4] + (5 + 4*Sqrt[2])*EllipticPi[3/14 - (3*Sqrt[2])/7, ArcSin[Sqrt[(5 + x)/(-1 + x
)]/Sqrt[3]], 3/4] + (5 - 4*Sqrt[2])*EllipticPi[3/14 + (3*Sqrt[2])/7, ArcSin[Sqrt[(5 + x)/(-1 + x)]/Sqrt[3]], 3
/4]))/(7*Sqrt[6]*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4])

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IntegrateAlgebraic [A]  time = 0.18, size = 48, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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]

ArcTan[(2*Sqrt[42]*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4])/(103 - 86*x + 19*x^2)]/Sqrt[42]

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fricas [A]  time = 0.52, 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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {5 \, x^{2} - 46 \, x + 77}{\sqrt {x^{4} - 3 \, x^{3} - 21 \, x^{2} + 83 \, x - 60} {\left (23 \, x^{2} - 82 \, x + 23\right )}}\,{d x} \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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maple [C]  time = 0.47, size = 75, normalized size = 1.56

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+42\right ) \ln \left (-\frac {19 \RootOf \left (\textit {\_Z}^{2}+42\right ) x^{2}-86 \RootOf \left (\textit {\_Z}^{2}+42\right ) x +103 \RootOf \left (\textit {\_Z}^{2}+42\right )+84 \sqrt {x^{4}-3 x^{3}-21 x^{2}+83 x -60}}{23 x^{2}-82 x +23}\right )}{42}\)	\(75\)
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]

1/42*RootOf(_Z^2+42)*ln(-(19*RootOf(_Z^2+42)*x^2-86*RootOf(_Z^2+42)*x+103*RootOf(_Z^2+42)+84*(x^4-3*x^3-21*x^2
+83*x-60)^(1/2))/(23*x^2-82*x+23))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {5 \, x^{2} - 46 \, x + 77}{\sqrt {x^{4} - 3 \, x^{3} - 21 \, x^{2} + 83 \, x - 60} {\left (23 \, x^{2} - 82 \, x + 23\right )}}\,{d x} \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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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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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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