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

Optimal. Leaf size=23 \[ \log \left (x^2+\sqrt {x^4+2 x^3+x^2+3}+x\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 11, normalized size of antiderivative = 0.48, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1680, 12, 1107, 619, 215} \begin {gather*} \sinh ^{-1}\left (\frac {x (x+1)}{\sqrt {3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSinh[(x*(1 + x))/Sqrt[3]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {1+2 x}{\sqrt {3+x^2+2 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {8 x}{\sqrt {49-8 x^2+16 x^4}} \, dx,x,\frac {1}{2}+x\right )\\ &=8 \operatorname {Subst}\left (\int \frac {x}{\sqrt {49-8 x^2+16 x^4}} \, dx,x,\frac {1}{2}+x\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {49-8 x+16 x^2}} \, dx,x,\left (\frac {1}{2}+x\right )^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3072}}} \, dx,x,32 x (1+x)\right )}{32 \sqrt {3}}\\ &=\sinh ^{-1}\left (\frac {x (1+x)}{\sqrt {3}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.59, size = 305, normalized size = 13.26 \begin {gather*} -\frac {(-1)^{2/3} \sqrt {2} \sqrt [4]{3} \sqrt {\frac {2 i x+\sqrt {3}+3 i}{x+(-1)^{2/3}}} \left (x+(-1)^{2/3}\right )^2 \sqrt {\frac {2 \sqrt {3} x+3 \sqrt {3}+3 i}{\left (\sqrt {3}-2 i\right ) \left (x+(-1)^{2/3}\right )}} \sqrt {\frac {\left (4-2 i \sqrt {3}\right ) x-i \sqrt {3}-5}{x+(-1)^{2/3}}} \left (\left (\sqrt {3}+2 i\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {2 \left (2 i+\sqrt {3}\right ) x+\sqrt {3}-5 i}{-2 i x+\sqrt {3}+i}}}{\sqrt {2}}\right )|\frac {4}{7}\right )-2 \sqrt {3} \Pi \left (\frac {2 i}{2 i+\sqrt {3}};\sin ^{-1}\left (\frac {\sqrt {-\frac {2 \left (2 i+\sqrt {3}\right ) x+\sqrt {3}-5 i}{-2 i x+\sqrt {3}+i}}}{\sqrt {2}}\right )|\frac {4}{7}\right )\right )}{\left (\sqrt {3}+9 i\right ) \sqrt {x^4+2 x^3+x^2+3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(((-1)^(2/3)*Sqrt[2]*3^(1/4)*Sqrt[(3*I + Sqrt[3] + (2*I)*x)/((-1)^(2/3) + x)]*((-1)^(2/3) + x)^2*Sqrt[(3*I +
3*Sqrt[3] + 2*Sqrt[3]*x)/((-2*I + Sqrt[3])*((-1)^(2/3) + x))]*Sqrt[(-5 - I*Sqrt[3] + (4 - (2*I)*Sqrt[3])*x)/((
-1)^(2/3) + x)]*((2*I + Sqrt[3])*EllipticF[ArcSin[Sqrt[-((-5*I + Sqrt[3] + 2*(2*I + Sqrt[3])*x)/(I + Sqrt[3] -
(2*I)*x))]/Sqrt[2]], 4/7] - 2*Sqrt[3]*EllipticPi[(2*I)/(2*I + Sqrt[3]), ArcSin[Sqrt[-((-5*I + Sqrt[3] + 2*(2*
I + Sqrt[3])*x)/(I + Sqrt[3] - (2*I)*x))]/Sqrt[2]], 4/7]))/((9*I + Sqrt[3])*Sqrt[3 + x^2 + 2*x^3 + x^4]))

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IntegrateAlgebraic [A]  time = 0.07, size = 23, normalized size = 1.00 \begin {gather*} \log \left (x+x^2+\sqrt {3+x^2+2 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x + x^2 + Sqrt[3 + x^2 + 2*x^3 + x^4]]

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fricas [A]  time = 0.48, size = 21, normalized size = 0.91 \begin {gather*} \log \left (x^{2} + x + \sqrt {x^{4} + 2 \, x^{3} + x^{2} + 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 + x + sqrt(x^4 + 2*x^3 + x^2 + 3))

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giac [A]  time = 0.40, size = 23, normalized size = 1.00 \begin {gather*} -\log \left (-x^{2} - x + \sqrt {{\left (x^{2} + x\right )}^{2} + 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-x^2 - x + sqrt((x^2 + x)^2 + 3))

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maple [A]  time = 0.31, size = 28, normalized size = 1.22

method	result	size
trager	\(-\ln \left (-x^{2}+\sqrt {x^{4}+2 x^{3}+x^{2}+3}-x \right )\)	\(28\)
default	\(\frac {i \left (2-i \sqrt {3}\right ) \sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {2}\, \sqrt {-\frac {i \sqrt {3}\, \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}{x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \EllipticF \left (\sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {\sqrt {\left (2+i \sqrt {3}\right ) \left (2-i \sqrt {3}\right )}}{2}\right )}{6 \sqrt {\left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {i \left (2-i \sqrt {3}\right ) \sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {2}\, \sqrt {-\frac {i \sqrt {3}\, \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}{x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \left (\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {\sqrt {\left (2+i \sqrt {3}\right ) \left (2-i \sqrt {3}\right )}}{2}\right )-i \sqrt {3}\, \EllipticPi \left (\sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, 1-\frac {i \sqrt {3}}{2}, \frac {\sqrt {\left (2+i \sqrt {3}\right ) \left (2-i \sqrt {3}\right )}}{2}\right )\right )}{3 \sqrt {\left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(528\)
elliptic	\(\frac {i \left (2-i \sqrt {3}\right ) \sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {2}\, \sqrt {-\frac {i \sqrt {3}\, \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}{x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \EllipticF \left (\sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {\sqrt {\left (2+i \sqrt {3}\right ) \left (2-i \sqrt {3}\right )}}{2}\right )}{6 \sqrt {\left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {i \left (2-i \sqrt {3}\right ) \sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {2}\, \sqrt {-\frac {i \sqrt {3}\, \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}{x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \left (\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {\sqrt {\left (2+i \sqrt {3}\right ) \left (2-i \sqrt {3}\right )}}{2}\right )-i \sqrt {3}\, \EllipticPi \left (\sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, 1-\frac {i \sqrt {3}}{2}, \frac {\sqrt {\left (2+i \sqrt {3}\right ) \left (2-i \sqrt {3}\right )}}{2}\right )\right )}{3 \sqrt {\left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(528\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {2\,x+1}{\sqrt {x^4+2\,x^3+x^2+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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