3.6.8 \(\int \frac {1+x}{\sqrt {16+18 x+13 x^2+4 x^3+x^4}} \, dx\)

Optimal. Leaf size=39 \[ \frac {1}{2} \log \left (2 x^2+2 \sqrt {x^4+4 x^3+13 x^2+18 x+16}+4 x+9\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 36, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1680, 1107, 621, 206} \begin {gather*} \frac {1}{2} \tanh ^{-1}\left (\frac {2 (x+1)^2+7}{2 \sqrt {(x+1)^4+7 (x+1)^2+8}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)/Sqrt[16 + 18*x + 13*x^2 + 4*x^3 + x^4],x]

[Out]

ArcTanh[(7 + 2*(1 + x)^2)/(2*Sqrt[8 + 7*(1 + x)^2 + (1 + x)^4])]/2

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*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+x}{\sqrt {16+18 x+13 x^2+4 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{\sqrt {8+7 x^2+x^4}} \, dx,x,1+x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {8+7 x+x^2}} \, dx,x,(1+x)^2\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {7+2 (1+x)^2}{\sqrt {8+7 (1+x)^2+(1+x)^4}}\right )\\ &=\frac {1}{2} \tanh ^{-1}\left (\frac {7+2 (1+x)^2}{2 \sqrt {8+7 (1+x)^2+(1+x)^4}}\right )\\ \end {align*}

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Mathematica [C]  time = 6.08, size = 2092, normalized size = 53.64 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(1 + x)/Sqrt[16 + 18*x + 13*x^2 + 4*x^3 + x^4],x]

[Out]

(2*((-2 - I*Sqrt[2*(7 - Sqrt[17])])/2 + (2 - I*Sqrt[2*(7 + Sqrt[17])])/2)*((2 - I*Sqrt[2*(7 - Sqrt[17])])/2 +
x)^2*Sqrt[(((2 - I*Sqrt[2*(7 - Sqrt[17])])/2 + (-2 + I*Sqrt[2*(7 + Sqrt[17])])/2)*((2 + I*Sqrt[2*(7 - Sqrt[17]
)])/2 + x))/(((2 + I*Sqrt[2*(7 - Sqrt[17])])/2 + (-2 + I*Sqrt[2*(7 + Sqrt[17])])/2)*((2 - I*Sqrt[2*(7 - Sqrt[1
7])])/2 + x))]*Sqrt[(((-2 + I*Sqrt[2*(7 - Sqrt[17])])/2 + (2 + I*Sqrt[2*(7 - Sqrt[17])])/2)*((2 - I*Sqrt[2*(7
+ Sqrt[17])])/2 + x))/(((2 + I*Sqrt[2*(7 - Sqrt[17])])/2 + (-2 + I*Sqrt[2*(7 + Sqrt[17])])/2)*((2 - I*Sqrt[2*(
7 - Sqrt[17])])/2 + x))]*Sqrt[(((-2 + I*Sqrt[2*(7 - Sqrt[17])])/2 + (2 + I*Sqrt[2*(7 - Sqrt[17])])/2)*((2 + I*
Sqrt[2*(7 + Sqrt[17])])/2 + x))/(((2 + I*Sqrt[2*(7 - Sqrt[17])])/2 + (-2 - I*Sqrt[2*(7 + Sqrt[17])])/2)*((2 -
I*Sqrt[2*(7 - Sqrt[17])])/2 + x))]*EllipticF[ArcSin[Sqrt[(((2 - I*Sqrt[2*(7 - Sqrt[17])])/2 + (-2 + I*Sqrt[2*(
7 + Sqrt[17])])/2)*((2 + I*Sqrt[2*(7 - Sqrt[17])])/2 + x))/(((2 + I*Sqrt[2*(7 - Sqrt[17])])/2 + (-2 + I*Sqrt[2
*(7 + Sqrt[17])])/2)*((2 - I*Sqrt[2*(7 - Sqrt[17])])/2 + x))]], (((-2 - I*Sqrt[2*(7 - Sqrt[17])])/2 + (2 - I*S
qrt[2*(7 + Sqrt[17])])/2)*((-2 + I*Sqrt[2*(7 - Sqrt[17])])/2 + (2 + I*Sqrt[2*(7 + Sqrt[17])])/2))/(((-2 + I*Sq
rt[2*(7 - Sqrt[17])])/2 + (2 - I*Sqrt[2*(7 + Sqrt[17])])/2)*((-2 - I*Sqrt[2*(7 - Sqrt[17])])/2 + (2 + I*Sqrt[2
*(7 + Sqrt[17])])/2))])/(((-2 + I*Sqrt[2*(7 - Sqrt[17])])/2 + (2 + I*Sqrt[2*(7 - Sqrt[17])])/2)*((2 - I*Sqrt[2
*(7 - Sqrt[17])])/2 + (-2 + I*Sqrt[2*(7 + Sqrt[17])])/2)*Sqrt[16 + 18*x + 13*x^2 + 4*x^3 + x^4]) + (2*Sqrt[(2*
(7 - Sqrt[17]))/(Sqrt[7 - Sqrt[17]] + Sqrt[7 + Sqrt[17]])]*((-2 - I*Sqrt[2*(7 - Sqrt[17])])/2 + (2 - I*Sqrt[2*
(7 + Sqrt[17])])/2)*Sqrt[((Sqrt[7 - Sqrt[17]] - Sqrt[7 + Sqrt[17]])*(-2*I + Sqrt[2*(7 - Sqrt[17])] - (2*I)*x))
/(2*I + Sqrt[2*(7 - Sqrt[17])] + (2*I)*x)]*((2 - I*Sqrt[2*(7 - Sqrt[17])])/2 + x)^2*Sqrt[(I*((2 - I*Sqrt[2*(7
+ Sqrt[17])])/2 + x))/(((2 + I*Sqrt[2*(7 - Sqrt[17])])/2 + (-2 + I*Sqrt[2*(7 + Sqrt[17])])/2)*((2 - I*Sqrt[2*(
7 - Sqrt[17])])/2 + x))]*Sqrt[(I*((2 + I*Sqrt[2*(7 + Sqrt[17])])/2 + x))/(((2 + I*Sqrt[2*(7 - Sqrt[17])])/2 +
(-2 - I*Sqrt[2*(7 + Sqrt[17])])/2)*((2 - I*Sqrt[2*(7 - Sqrt[17])])/2 + x))]*(((2 - I*Sqrt[2*(7 - Sqrt[17])])*E
llipticF[ArcSin[Sqrt[((Sqrt[7 - Sqrt[17]] - Sqrt[7 + Sqrt[17]])*(-2*I + Sqrt[2*(7 - Sqrt[17])] - (2*I)*x))/((S
qrt[7 - Sqrt[17]] + Sqrt[7 + Sqrt[17]])*(2*I + Sqrt[2*(7 - Sqrt[17])] + (2*I)*x))]], (Sqrt[7 - Sqrt[17]] + Sqr
t[7 + Sqrt[17]])^2/(Sqrt[7 - Sqrt[17]] - Sqrt[7 + Sqrt[17]])^2])/2 + I*Sqrt[2*(7 - Sqrt[17])]*EllipticPi[((2 +
I*Sqrt[2*(7 - Sqrt[17])])/2 + (-2 + I*Sqrt[2*(7 + Sqrt[17])])/2)/((2 - I*Sqrt[2*(7 - Sqrt[17])])/2 + (-2 + I*
Sqrt[2*(7 + Sqrt[17])])/2), ArcSin[Sqrt[((Sqrt[7 - Sqrt[17]] - Sqrt[7 + Sqrt[17]])*(-2*I + Sqrt[2*(7 - Sqrt[17
])] - (2*I)*x))/((Sqrt[7 - Sqrt[17]] + Sqrt[7 + Sqrt[17]])*(2*I + Sqrt[2*(7 - Sqrt[17])] + (2*I)*x))]], (Sqrt[
7 - Sqrt[17]] + Sqrt[7 + Sqrt[17]])^2/(Sqrt[7 - Sqrt[17]] - Sqrt[7 + Sqrt[17]])^2]))/(((-2 + I*Sqrt[2*(7 - Sqr
t[17])])/2 + (2 + I*Sqrt[2*(7 - Sqrt[17])])/2)*((-2 + I*Sqrt[2*(7 - Sqrt[17])])/2 + (2 - I*Sqrt[2*(7 + Sqrt[17
])])/2)*Sqrt[16 + 18*x + 13*x^2 + 4*x^3 + x^4])

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IntegrateAlgebraic [A]  time = 0.17, size = 39, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (9+4 x+2 x^2+2 \sqrt {16+18 x+13 x^2+4 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x)/Sqrt[16 + 18*x + 13*x^2 + 4*x^3 + x^4],x]

[Out]

Log[9 + 4*x + 2*x^2 + 2*Sqrt[16 + 18*x + 13*x^2 + 4*x^3 + x^4]]/2

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fricas [A]  time = 0.49, size = 35, normalized size = 0.90 \begin {gather*} \frac {1}{2} \, \log \left (2 \, x^{2} + 4 \, x + 2 \, \sqrt {x^{4} + 4 \, x^{3} + 13 \, x^{2} + 18 \, x + 16} + 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^4+4*x^3+13*x^2+18*x+16)^(1/2),x, algorithm="fricas")

[Out]

1/2*log(2*x^2 + 4*x + 2*sqrt(x^4 + 4*x^3 + 13*x^2 + 18*x + 16) + 9)

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giac [A]  time = 0.40, size = 46, normalized size = 1.18 \begin {gather*} -\frac {1}{2} \, \log \left (\sqrt {2} {\left (\sqrt {2} {\left (x^{2} + 2 \, x\right )} - 2 \, \sqrt {\frac {1}{2} \, {\left (x^{2} + 2 \, x\right )}^{2} + \frac {9}{2} \, x^{2} + 9 \, x + 8}\right )} + 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^4+4*x^3+13*x^2+18*x+16)^(1/2),x, algorithm="giac")

[Out]

-1/2*log(sqrt(2)*(sqrt(2)*(x^2 + 2*x) - 2*sqrt(1/2*(x^2 + 2*x)^2 + 9/2*x^2 + 9*x + 8)) + 9)

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maple [A]  time = 0.72, size = 36, normalized size = 0.92

method	result	size
trager	\(-\frac {\ln \left (-2 x^{2}+2 \sqrt {x^{4}+4 x^{3}+13 x^{2}+18 x +16}-4 x -9\right )}{2}\)	\(36\)
default	\(-\frac {2 i \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right ) \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )^{2} \sqrt {\frac {i \sqrt {14+2 \sqrt {17}}\, \left (x +1+\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \sqrt {\frac {i \sqrt {14+2 \sqrt {17}}\, \left (x +1-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}, \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \sqrt {14+2 \sqrt {17}}\, \sqrt {\left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}}-\frac {2 i \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right ) \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )^{2} \sqrt {\frac {i \sqrt {14+2 \sqrt {17}}\, \left (x +1+\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \sqrt {\frac {i \sqrt {14+2 \sqrt {17}}\, \left (x +1-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \left (\left (-1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}, \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\right )-i \sqrt {14+2 \sqrt {17}}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}, \frac {\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}}{\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}}, \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\right )\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \sqrt {14+2 \sqrt {17}}\, \sqrt {\left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}}\)	\(1422\)
elliptic	\(-\frac {2 i \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right ) \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )^{2} \sqrt {\frac {i \sqrt {14+2 \sqrt {17}}\, \left (x +1+\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \sqrt {\frac {i \sqrt {14+2 \sqrt {17}}\, \left (x +1-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}, \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \sqrt {14+2 \sqrt {17}}\, \sqrt {\left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}}-\frac {2 i \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right ) \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )^{2} \sqrt {\frac {i \sqrt {14+2 \sqrt {17}}\, \left (x +1+\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \sqrt {\frac {i \sqrt {14+2 \sqrt {17}}\, \left (x +1-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \left (\left (-1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}, \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\right )-i \sqrt {14+2 \sqrt {17}}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}, \frac {\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}}{\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}}, \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\right )\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \sqrt {14+2 \sqrt {17}}\, \sqrt {\left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}}\)	\(1422\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)/(x^4+4*x^3+13*x^2+18*x+16)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*ln(-2*x^2+2*(x^4+4*x^3+13*x^2+18*x+16)^(1/2)-4*x-9)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\sqrt {x^{4} + 4 \, x^{3} + 13 \, x^{2} + 18 \, x + 16}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^4+4*x^3+13*x^2+18*x+16)^(1/2),x, algorithm="maxima")

[Out]

integrate((x + 1)/sqrt(x^4 + 4*x^3 + 13*x^2 + 18*x + 16), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x+1}{\sqrt {x^4+4\,x^3+13\,x^2+18\,x+16}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)/(18*x + 13*x^2 + 4*x^3 + x^4 + 16)^(1/2),x)

[Out]

int((x + 1)/(18*x + 13*x^2 + 4*x^3 + x^4 + 16)^(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 {x^{4} + 4 x^{3} + 13 x^{2} + 18 x + 16}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x**4+4*x**3+13*x**2+18*x+16)**(1/2),x)

[Out]

Integral((x + 1)/sqrt(x**4 + 4*x**3 + 13*x**2 + 18*x + 16), x)

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