3.7.45 \(\int \frac {-7+x}{(-11+5 x) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\) [645]

Optimal. Leaf size=51 \[ -\frac {\tanh ^{-1}\left (\frac {6 \sqrt {6} \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}{41-106 x+29 x^2}\right )}{3 \sqrt {6}} \]

[Out]

-1/18*arctanh(6*6^(1/2)*(x^4-3*x^3-21*x^2+83*x-60)^(1/2)/(29*x^2-106*x+41))*6^(1/2)

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Rubi [F]
time = 0.21, 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 {-7+x}{(-11+5 x) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-7 + x)/((-11 + 5*x)*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4]),x]

[Out]

Defer[Int][1/Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4], x]/5 - (24*Defer[Int][1/((-11 + 5*x)*Sqrt[-60 + 83*x - 2
1*x^2 - 3*x^3 + x^4]), x])/5

Rubi steps

\begin {align*} \int \frac {-7+x}{(-11+5 x) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx &=\int \left (\frac {1}{5 \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}-\frac {24}{5 (-11+5 x) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}\right ) \, dx\\ &=\frac {1}{5} \int \frac {1}{\sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx-\frac {24}{5} \int \frac {1}{(-11+5 x) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.40, size = 51, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {3 \sqrt {\frac {3}{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[(-7 + x)/((-11 + 5*x)*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4]),x]

[Out]

-1/3*(Sqrt[2/3]*ArcTanh[(3*Sqrt[3/2]*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 = 0.65, size = 188, normalized size = 3.69

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (-\frac {-29 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{2}+106 \RootOf \left (\textit {\_Z}^{2}-6\right ) x +36 \sqrt {x^{4}-3 x^{3}-21 x^{2}+83 x -60}-41 \RootOf \left (\textit {\_Z}^{2}-6\right )}{\left (-11+5 x \right )^{2}}\right )}{18}\)	\(70\)
default	\(-\frac {\sqrt {\frac {x +5}{-1+x}}\, \left (-1+x \right )^{2} \sqrt {\frac {-3+x}{-1+x}}\, \sqrt {6}\, \sqrt {\frac {x -4}{-1+x}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {x +5}{-1+x}}}{3}, \frac {\sqrt {3}}{2}\right )}{30 \sqrt {\left (x +5\right ) \left (-1+x \right ) \left (-3+x \right ) \left (x -4\right )}}-\frac {2 \sqrt {\frac {x +5}{-1+x}}\, \left (-1+x \right )^{2} \sqrt {\frac {-3+x}{-1+x}}\, \sqrt {6}\, \sqrt {\frac {x -4}{-1+x}}\, \left (\EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {x +5}{-1+x}}}{3}, \frac {\sqrt {3}}{2}\right )-\frac {5 \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {x +5}{-1+x}}}{3}, \frac {1}{2}, \frac {\sqrt {3}}{2}\right )}{6}\right )}{15 \sqrt {\left (x +5\right ) \left (-1+x \right ) \left (-3+x \right ) \left (x -4\right )}}\)	\(188\)
elliptic	\(-\frac {\sqrt {\frac {x +5}{-1+x}}\, \left (-1+x \right )^{2} \sqrt {\frac {-3+x}{-1+x}}\, \sqrt {6}\, \sqrt {\frac {x -4}{-1+x}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {x +5}{-1+x}}}{3}, \frac {\sqrt {3}}{2}\right )}{30 \sqrt {\left (x +5\right ) \left (-1+x \right ) \left (-3+x \right ) \left (x -4\right )}}-\frac {2 \sqrt {\frac {x +5}{-1+x}}\, \left (-1+x \right )^{2} \sqrt {\frac {-3+x}{-1+x}}\, \sqrt {6}\, \sqrt {\frac {x -4}{-1+x}}\, \left (\EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {x +5}{-1+x}}}{3}, \frac {\sqrt {3}}{2}\right )-\frac {5 \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {x +5}{-1+x}}}{3}, \frac {1}{2}, \frac {\sqrt {3}}{2}\right )}{6}\right )}{15 \sqrt {\left (x +5\right ) \left (-1+x \right ) \left (-3+x \right ) \left (x -4\right )}}\)	\(188\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-7+x)/(-11+5*x)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/30*((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))-2/15*((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))-5/6*EllipticPi(1/3*3^(1/2)*((x+5)/(-1+x))^(1/2),1/2,1/2*3^(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((-7+x)/(-11+5*x)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2),x, algorithm="maxima")

[Out]

integrate((x - 7)/(sqrt(x^4 - 3*x^3 - 21*x^2 + 83*x - 60)*(5*x - 11)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (43) = 86\).
time = 0.39, size = 91, normalized size = 1.78 \begin {gather*} \frac {1}{36} \, \sqrt {3} \sqrt {2} \log \left (-\frac {1057 \, x^{4} - 6796 \, x^{3} - 12 \, \sqrt {3} \sqrt {2} \sqrt {x^{4} - 3 \, x^{3} - 21 \, x^{2} + 83 \, x - 60} {\left (29 \, x^{2} - 106 \, x + 41\right )} + 9078 \, x^{2} + 9236 \, x - 11279}{625 \, x^{4} - 5500 \, x^{3} + 18150 \, x^{2} - 26620 \, x + 14641}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-7+x)/(-11+5*x)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2),x, algorithm="fricas")

[Out]

1/36*sqrt(3)*sqrt(2)*log(-(1057*x^4 - 6796*x^3 - 12*sqrt(3)*sqrt(2)*sqrt(x^4 - 3*x^3 - 21*x^2 + 83*x - 60)*(29
*x^2 - 106*x + 41) + 9078*x^2 + 9236*x - 11279)/(625*x^4 - 5500*x^3 + 18150*x^2 - 26620*x + 14641))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 7}{\sqrt {\left (x - 4\right ) \left (x - 3\right ) \left (x - 1\right ) \left (x + 5\right )} \left (5 x - 11\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-7+x)/(-11+5*x)/(x**4-3*x**3-21*x**2+83*x-60)**(1/2),x)

[Out]

Integral((x - 7)/(sqrt((x - 4)*(x - 3)*(x - 1)*(x + 5))*(5*x - 11)), 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((-7+x)/(-11+5*x)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2),x, algorithm="giac")

[Out]

integrate((x - 7)/(sqrt(x^4 - 3*x^3 - 21*x^2 + 83*x - 60)*(5*x - 11)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x-7}{\left (5\,x-11\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((x - 7)/((5*x - 11)*(83*x - 21*x^2 - 3*x^3 + x^4 - 60)^(1/2)),x)

[Out]

int((x - 7)/((5*x - 11)*(83*x - 21*x^2 - 3*x^3 + x^4 - 60)^(1/2)), x)

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