3.5.5 \(\int \sqrt {x+x^4} \, dx\)

Optimal. Leaf size=33 \[ \frac {1}{3} \sqrt {x^4+x} x+\frac {1}{3} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2004, 2029, 206} \begin {gather*} \frac {1}{3} \sqrt {x^4+x} x+\frac {1}{3} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x + x^4],x]

[Out]

(x*Sqrt[x + x^4])/3 + ArcTanh[x^2/Sqrt[x + x^4]]/3

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 2004

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(x*(a*x^j + b*x^n)^p)/(n*p + 1), x] + Dist[(
a*(n - j)*p)/(n*p + 1), Int[x^j*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] &&  !IntegerQ[p] && LtQ[0,
j, n] && GtQ[p, 0] && NeQ[n*p + 1, 0]

Rule 2029

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 40, normalized size = 1.21 \begin {gather*} \frac {\sqrt {x^4+x} \left (x^{3/2}+\frac {\sinh ^{-1}\left (x^{3/2}\right )}{\sqrt {x^3+1}}\right )}{3 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x + x^4],x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.34, size = 33, normalized size = 1.00 \begin {gather*} \frac {1}{3} x \sqrt {x+x^4}+\frac {1}{3} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[x + x^4],x]

[Out]

(x*Sqrt[x + x^4])/3 + ArcTanh[x^2/Sqrt[x + x^4]]/3

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fricas [A]  time = 0.52, size = 31, normalized size = 0.94 \begin {gather*} \frac {1}{3} \, \sqrt {x^{4} + x} x + \frac {1}{6} \, \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(x^4 + x)*x + 1/6*log(-2*x^3 - 2*sqrt(x^4 + x)*x - 1)

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giac [A]  time = 0.50, size = 36, normalized size = 1.09 \begin {gather*} \frac {1}{3} \, \sqrt {x^{4} + x} x + \frac {1}{6} \, \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right ) - \frac {1}{6} \, \log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(x^4 + x)*x + 1/6*log(sqrt(1/x^3 + 1) + 1) - 1/6*log(abs(sqrt(1/x^3 + 1) - 1))

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maple [A]  time = 0.28, size = 31, normalized size = 0.94

method	result	size
meijerg	\(-\frac {-2 \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {x^{3}+1}-2 \sqrt {\pi }\, \arcsinh \left (x^{\frac {3}{2}}\right )}{6 \sqrt {\pi }}\)	\(31\)
trager	\(\frac {x \sqrt {x^{4}+x}}{3}-\frac {\ln \left (2 x^{3}-2 x \sqrt {x^{4}+x}+1\right )}{6}\)	\(32\)
default	\(\frac {x \sqrt {x^{4}+x}}{3}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(301\)
elliptic	\(\frac {x \sqrt {x^{4}+x}}{3}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(301\)
risch	\(\frac {x^{2} \left (x^{3}+1\right )}{3 \sqrt {x \left (x^{3}+1\right )}}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(310\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/Pi^(1/2)*(-2*Pi^(1/2)*x^(3/2)*(x^3+1)^(1/2)-2*Pi^(1/2)*arcsinh(x^(3/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^4 + x), x)

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mupad [B]  time = 0.30, size = 27, normalized size = 0.82 \begin {gather*} \frac {2\,x\,\sqrt {x^4+x}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {1}{2};\ \frac {3}{2};\ -x^3\right )}{3\,\sqrt {x^3+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + x^4)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x**4 + x), x)

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