3.1.19 \(\int \frac {x}{\sqrt {1+x^2+(1+x^2)^{3/2}}} \, dx\) [19]

Optimal. Leaf size=32 \[ \frac {2 \sqrt {\left (1+x^2\right ) \left (1+\sqrt {1+x^2}\right )}}{\sqrt {1+x^2}} \]

[Out]

2*((x^2+1)*(1+(x^2+1)^(1/2)))^(1/2)/(x^2+1)^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6847, 1602} \begin {gather*} \frac {2 \sqrt {\left (x^2+1\right ) \left (\sqrt {x^2+1}+1\right )}}{\sqrt {x^2+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(1 + x^2)*(1 + Sqrt[1 + x^2])])/Sqrt[1 + x^2]

Rule 1602

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*x^(p - q +
1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps

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

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Mathematica [A]
time = 1.69, size = 32, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {\left (1+x^2\right ) \left (1+\sqrt {1+x^2}\right )}}{\sqrt {1+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(1 + x^2)*(1 + Sqrt[1 + x^2])])/Sqrt[1 + x^2]

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x}{\sqrt {1+x^{2}+\left (x^{2}+1\right )^{\frac {3}{2}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [A]
time = 0.67, size = 23, normalized size = 0.72 \begin {gather*} \frac {2 \, \sqrt {x^{2} + {\left (x^{2} + 1\right )}^{\frac {3}{2}} + 1}}{\sqrt {x^{2} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x^2 + (x^2 + 1)^(3/2) + 1)/sqrt(x^2 + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.70, size = 15, normalized size = 0.47 \begin {gather*} 2 \, \sqrt {\sqrt {x^{2} + 1} + 1} - 2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(sqrt(x^2 + 1) + 1) - 2

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Mupad [B]
time = 0.59, size = 47, normalized size = 1.47 \begin {gather*} \frac {2\,\left (x^2+1\right )\,\sqrt {\sqrt {x^2+1}+1}}{\left (\sqrt {\sqrt {x^2+1}+1}+1\right )\,\sqrt {{\left (x^2+1\right )}^{3/2}+x^2+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(x^2 + 1)*((x^2 + 1)^(1/2) + 1)^(1/2))/((((x^2 + 1)^(1/2) + 1)^(1/2) + 1)*((x^2 + 1)^(3/2) + x^2 + 1)^(1/2)
)

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