3.7.73 \(\int \frac {1+4 x}{\sqrt {1-2 x+3 x^2+2 x^3+x^4}} \, dx\)

Optimal. Leaf size=53 \[ -\log \left (-x^4-3 x^3-5 x^2+\left (x^2+2 x+2\right ) \sqrt {x^4+2 x^3+3 x^2-2 x+1}-2 x\right ) \]

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Rubi [F]  time = 0.14, 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+4 x}{\sqrt {1-2 x+3 x^2+2 x^3+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 1.39, size = 484, normalized size = 9.13 \begin {gather*} \frac {2 \left (-4 i-\sqrt {1-4 i}-\sqrt {1+4 i}\right ) \left (x+\frac {1}{2} \left ((1-2 i)+\sqrt {1-4 i}\right )\right )^2 \sqrt {\frac {\left (-4 i+\sqrt {1-4 i}-\sqrt {1+4 i}\right ) \left (-2 x+\sqrt {1+4 i}-(1+2 i)\right )}{\sqrt {1+4 i} \left (2 x+\sqrt {1-4 i}+(1-2 i)\right )}} \sqrt {\frac {\sqrt {1-4 i} \left (-4 i+\sqrt {1-4 i}-\sqrt {1+4 i}\right ) \left (-2 x+\sqrt {1-4 i}-(1-2 i)\right ) \left (2 x+\sqrt {1+4 i}+(1+2 i)\right )}{\left (4 i+\sqrt {1-4 i}+\sqrt {1+4 i}\right )^2 \left (2 x+\sqrt {1-4 i}+(1-2 i)\right )^2}} \left (\left ((1-4 i)+2 \sqrt {1-4 i}\right ) F\left (\sin ^{-1}\left (\sqrt {2} \sqrt {\frac {\sqrt {1-4 i} \left (2 x+\sqrt {1+4 i}+(1+2 i)\right )}{\left (4 i+\sqrt {1-4 i}+\sqrt {1+4 i}\right ) \left (2 x+\sqrt {1-4 i}+(1-2 i)\right )}}\right )|\frac {1}{2}+\frac {9}{2 \sqrt {17}}\right )+2 \left (4 i-\sqrt {1-4 i}+\sqrt {1+4 i}\right ) \Pi \left (\frac {4 i+\sqrt {1-4 i}+\sqrt {1+4 i}}{2 \sqrt {1-4 i}};\sin ^{-1}\left (\sqrt {2} \sqrt {\frac {\sqrt {1-4 i} \left (2 x+\sqrt {1+4 i}+(1+2 i)\right )}{\left (4 i+\sqrt {1-4 i}+\sqrt {1+4 i}\right ) \left (2 x+\sqrt {1-4 i}+(1-2 i)\right )}}\right )|\frac {1}{2}+\frac {9}{2 \sqrt {17}}\right )\right )}{\sqrt {1-4 i} \left (-4 i+\sqrt {1-4 i}-\sqrt {1+4 i}\right ) \sqrt {x^4+2 x^3+3 x^2-2 x+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*(-4*I - Sqrt[1 - 4*I] - Sqrt[1 + 4*I])*(((1 - 2*I) + Sqrt[1 - 4*I])/2 + x)^2*Sqrt[((-4*I + Sqrt[1 - 4*I] -
Sqrt[1 + 4*I])*((-1 - 2*I) + Sqrt[1 + 4*I] - 2*x))/(Sqrt[1 + 4*I]*((1 - 2*I) + Sqrt[1 - 4*I] + 2*x))]*Sqrt[(Sq
rt[1 - 4*I]*(-4*I + Sqrt[1 - 4*I] - Sqrt[1 + 4*I])*((-1 + 2*I) + Sqrt[1 - 4*I] - 2*x)*((1 + 2*I) + Sqrt[1 + 4*
I] + 2*x))/((4*I + Sqrt[1 - 4*I] + Sqrt[1 + 4*I])^2*((1 - 2*I) + Sqrt[1 - 4*I] + 2*x)^2)]*(((1 - 4*I) + 2*Sqrt
[1 - 4*I])*EllipticF[ArcSin[Sqrt[2]*Sqrt[(Sqrt[1 - 4*I]*((1 + 2*I) + Sqrt[1 + 4*I] + 2*x))/((4*I + Sqrt[1 - 4*
I] + Sqrt[1 + 4*I])*((1 - 2*I) + Sqrt[1 - 4*I] + 2*x))]], 1/2 + 9/(2*Sqrt[17])] + 2*(4*I - Sqrt[1 - 4*I] + Sqr
t[1 + 4*I])*EllipticPi[(4*I + Sqrt[1 - 4*I] + Sqrt[1 + 4*I])/(2*Sqrt[1 - 4*I]), ArcSin[Sqrt[2]*Sqrt[(Sqrt[1 -
4*I]*((1 + 2*I) + Sqrt[1 + 4*I] + 2*x))/((4*I + Sqrt[1 - 4*I] + Sqrt[1 + 4*I])*((1 - 2*I) + Sqrt[1 - 4*I] + 2*
x))]], 1/2 + 9/(2*Sqrt[17])]))/(Sqrt[1 - 4*I]*(-4*I + Sqrt[1 - 4*I] - Sqrt[1 + 4*I])*Sqrt[1 - 2*x + 3*x^2 + 2*
x^3 + x^4])

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IntegrateAlgebraic [A]  time = 4.82, size = 53, normalized size = 1.00 \begin {gather*} -\log \left (-2 x-5 x^2-3 x^3-x^4+\left (2+2 x+x^2\right ) \sqrt {1-2 x+3 x^2+2 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[-2*x - 5*x^2 - 3*x^3 - x^4 + (2 + 2*x + x^2)*Sqrt[1 - 2*x + 3*x^2 + 2*x^3 + x^4]]

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fricas [A]  time = 0.51, size = 47, normalized size = 0.89 \begin {gather*} \log \left (x^{4} + 3 \, x^{3} + 5 \, x^{2} + \sqrt {x^{4} + 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1} {\left (x^{2} + 2 \, x + 2\right )} + 2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^4 + 3*x^3 + 5*x^2 + sqrt(x^4 + 2*x^3 + 3*x^2 - 2*x + 1)*(x^2 + 2*x + 2) + 2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.29, size = 92, normalized size = 1.74

method	result	size
trager	\(-\ln \left (-x^{4}+\sqrt {x^{4}+2 x^{3}+3 x^{2}-2 x +1}\, x^{2}-3 x^{3}+2 \sqrt {x^{4}+2 x^{3}+3 x^{2}-2 x +1}\, x -5 x^{2}+2 \sqrt {x^{4}+2 x^{3}+3 x^{2}-2 x +1}-2 x \right )\)	\(92\)
default	\(\text {Expression too large to display}\)	\(2700\)
elliptic	\(\text {Expression too large to display}\)	\(2700\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(-x^4+(x^4+2*x^3+3*x^2-2*x+1)^(1/2)*x^2-3*x^3+2*(x^4+2*x^3+3*x^2-2*x+1)^(1/2)*x-5*x^2+2*(x^4+2*x^3+3*x^2-2*
x+1)^(1/2)-2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {4\,x+1}{\sqrt {x^4+2\,x^3+3\,x^2-2\,x+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((4*x + 1)/sqrt(x**4 + 2*x**3 + 3*x**2 - 2*x + 1), x)

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