3.11.81 \(\int \frac {1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx\)

Optimal. Leaf size=81 \[ -\log \left (-x^2+\sqrt {x^4-12 x^3+14 x^2+4 x-7}+6 x-5\right )+\tan ^{-1}\left (\frac {4 x-4}{x^2-\sqrt {x^4-12 x^3+14 x^2+4 x-7}-2 x+1}\right )+\log (x-1) \]

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Rubi [F]  time = 0.18, 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 {1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + x)/Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4],x]

[Out]

Defer[Int][1/Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4], x] + Defer[Int][x/Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4],
x]

Rubi steps

\begin {align*} \int \frac {1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx &=\int \left (\frac {1}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}}+\frac {x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}}\right ) \, dx\\ &=\int \frac {1}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 76, normalized size = 0.94 \begin {gather*} \frac {(x-1) \sqrt {x^2-10 x-7} \left (\tan ^{-1}\left (\frac {-x-3}{\sqrt {x^2-10 x-7}}\right )+2 \tanh ^{-1}\left (\frac {x-5}{\sqrt {x^2-10 x-7}}\right )\right )}{2 \sqrt {(x-1)^2 \left (x^2-10 x-7\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)/Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4],x]

[Out]

((-1 + x)*Sqrt[-7 - 10*x + x^2]*(ArcTan[(-3 - x)/Sqrt[-7 - 10*x + x^2]] + 2*ArcTanh[(-5 + x)/Sqrt[-7 - 10*x +
x^2]]))/(2*Sqrt[(-1 + x)^2*(-7 - 10*x + x^2)])

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IntegrateAlgebraic [A]  time = 0.28, size = 81, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {-4+4 x}{1-2 x+x^2-\sqrt {-7+4 x+14 x^2-12 x^3+x^4}}\right )+\log (-1+x)-\log \left (-5+6 x-x^2+\sqrt {-7+4 x+14 x^2-12 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x)/Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4],x]

[Out]

ArcTan[(-4 + 4*x)/(1 - 2*x + x^2 - Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4])] + Log[-1 + x] - Log[-5 + 6*x - x^2
+ Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4]]

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fricas [A]  time = 0.50, size = 79, normalized size = 0.98 \begin {gather*} \arctan \left (-\frac {x^{2} - 2 \, x - \sqrt {x^{4} - 12 \, x^{3} + 14 \, x^{2} + 4 \, x - 7} + 1}{4 \, {\left (x - 1\right )}}\right ) - \log \left (-\frac {x^{2} - 6 \, x - \sqrt {x^{4} - 12 \, x^{3} + 14 \, x^{2} + 4 \, x - 7} + 5}{x - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^4-12*x^3+14*x^2+4*x-7)^(1/2),x, algorithm="fricas")

[Out]

arctan(-1/4*(x^2 - 2*x - sqrt(x^4 - 12*x^3 + 14*x^2 + 4*x - 7) + 1)/(x - 1)) - log(-(x^2 - 6*x - sqrt(x^4 - 12
*x^3 + 14*x^2 + 4*x - 7) + 5)/(x - 1))

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giac [B]  time = 0.46, size = 183, normalized size = 2.26 \begin {gather*} -\frac {\arctan \left (\frac {1}{7} \, \sqrt {7} {\left (\sqrt {2} + \frac {3 \, {\left (\sqrt {7} \sqrt {-\frac {10}{x} - \frac {7}{x^{2}} + 1} - 4 \, \sqrt {2}\right )}}{\frac {7}{x} + 5}\right )}\right )}{\mathrm {sgn}\left (-\frac {1}{x^{2}} + \frac {1}{x^{3}}\right )} - \frac {\log \left ({\left | 10 \, \sqrt {7} + 40 \, \sqrt {2} + \frac {50 \, {\left (\sqrt {7} \sqrt {-\frac {10}{x} - \frac {7}{x^{2}} + 1} - 4 \, \sqrt {2}\right )}}{\frac {7}{x} + 5} \right |}\right )}{\mathrm {sgn}\left (-\frac {1}{x^{2}} + \frac {1}{x^{3}}\right )} + \frac {\log \left ({\left | -2 \, \sqrt {7} + 8 \, \sqrt {2} + \frac {10 \, {\left (\sqrt {7} \sqrt {-\frac {10}{x} - \frac {7}{x^{2}} + 1} - 4 \, \sqrt {2}\right )}}{\frac {7}{x} + 5} \right |}\right )}{\mathrm {sgn}\left (-\frac {1}{x^{2}} + \frac {1}{x^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^4-12*x^3+14*x^2+4*x-7)^(1/2),x, algorithm="giac")

[Out]

-arctan(1/7*sqrt(7)*(sqrt(2) + 3*(sqrt(7)*sqrt(-10/x - 7/x^2 + 1) - 4*sqrt(2))/(7/x + 5)))/sgn(-1/x^2 + 1/x^3)
- log(abs(10*sqrt(7) + 40*sqrt(2) + 50*(sqrt(7)*sqrt(-10/x - 7/x^2 + 1) - 4*sqrt(2))/(7/x + 5)))/sgn(-1/x^2 +
1/x^3) + log(abs(-2*sqrt(7) + 8*sqrt(2) + 10*(sqrt(7)*sqrt(-10/x - 7/x^2 + 1) - 4*sqrt(2))/(7/x + 5)))/sgn(-1
/x^2 + 1/x^3)

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maple [A]  time = 0.22, size = 70, normalized size = 0.86

method	result	size
default	\(\frac {\left (-1+x \right ) \sqrt {x^{2}-10 x -7}\, \left (2 \ln \left (-5+x +\sqrt {x^{2}-10 x -7}\right )-\arctan \left (\frac {3+x}{\sqrt {x^{2}-10 x -7}}\right )\right )}{2 \sqrt {x^{4}-12 x^{3}+14 x^{2}+4 x -7}}\)	\(70\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x +\sqrt {x^{4}-12 x^{3}+14 x^{2}+4 x -7}+3 \RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right )^{2}}\right )}{2}+\ln \left (-\frac {x^{2}+\sqrt {x^{4}-12 x^{3}+14 x^{2}+4 x -7}-6 x +5}{-1+x}\right )\)	\(102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)/(x^4-12*x^3+14*x^2+4*x-7)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*(-1+x)*(x^2-10*x-7)^(1/2)*(2*ln(-5+x+(x^2-10*x-7)^(1/2))-arctan((3+x)/(x^2-10*x-7)^(1/2)))/(x^4-12*x^3+14*
x^2+4*x-7)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\sqrt {x^{4} - 12 \, x^{3} + 14 \, x^{2} + 4 \, x - 7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^4-12*x^3+14*x^2+4*x-7)^(1/2),x, algorithm="maxima")

[Out]

integrate((x + 1)/sqrt(x^4 - 12*x^3 + 14*x^2 + 4*x - 7), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x+1}{\sqrt {x^4-12\,x^3+14\,x^2+4\,x-7}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)/(4*x + 14*x^2 - 12*x^3 + x^4 - 7)^(1/2),x)

[Out]

int((x + 1)/(4*x + 14*x^2 - 12*x^3 + x^4 - 7)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\sqrt {\left (x - 1\right )^{2} \left (x^{2} - 10 x - 7\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x**4-12*x**3+14*x**2+4*x-7)**(1/2),x)

[Out]

Integral((x + 1)/sqrt((x - 1)**2*(x**2 - 10*x - 7)), x)

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