3.5.32 \(\int \sqrt {x+x^{5/2}} \, dx\) [432]

Optimal. Leaf size=20 \[ \frac {4 \left (x+x^{5/2}\right )^{3/2}}{9 x^{3/2}} \]

[Out]

4/9*(x+x^(5/2))^(3/2)/x^(3/2)

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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2025} \begin {gather*} \frac {4 \left (x^{5/2}+x\right )^{3/2}}{9 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x + x^(5/2)],x]

[Out]

(4*(x + x^(5/2))^(3/2))/(9*x^(3/2))

Rule 2025

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

Rubi steps

\begin {align*} \int \sqrt {x+x^{5/2}} \, dx &=\frac {4 \left (x+x^{5/2}\right )^{3/2}}{9 x^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 20, normalized size = 1.00 \begin {gather*} \frac {4 \left (x+x^{5/2}\right )^{3/2}}{9 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x + x^(5/2)],x]

[Out]

(4*(x + x^(5/2))^(3/2))/(9*x^(3/2))

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Maple [A]
time = 0.37, size = 18, normalized size = 0.90

method	result	size
derivativedivides	\(\frac {4 \sqrt {x +x^{\frac {5}{2}}}\, \left (1+x^{\frac {3}{2}}\right )}{9 \sqrt {x}}\)	\(18\)
default	\(\frac {4 \sqrt {x +x^{\frac {5}{2}}}\, \left (1+x^{\frac {3}{2}}\right )}{9 \sqrt {x}}\)	\(18\)
meijerg	\(-\frac {\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2+2 x^{\frac {3}{2}}\right ) \sqrt {1+x^{\frac {3}{2}}}}{3}}{3 \sqrt {\pi }}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+x^(5/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

4/9*(x+x^(5/2))^(1/2)/x^(1/2)*(1+x^(3/2))

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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+x^(5/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x^(5/2) + x), x)

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Fricas [A]
time = 1.57, size = 19, normalized size = 0.95 \begin {gather*} \frac {4 \, \sqrt {x^{\frac {5}{2}} + x} {\left (x^{2} + \sqrt {x}\right )}}{9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+x^(5/2))^(1/2),x, algorithm="fricas")

[Out]

4/9*sqrt(x^(5/2) + x)*(x^2 + sqrt(x))/x

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{\frac {5}{2}} + x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+x**(5/2))**(1/2),x)

[Out]

Integral(sqrt(x**(5/2) + x), x)

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Giac [A]
time = 1.51, size = 11, normalized size = 0.55 \begin {gather*} \frac {4}{9} \, {\left (x^{\frac {3}{2}} + 1\right )}^{\frac {3}{2}} - \frac {4}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+x^(5/2))^(1/2),x, algorithm="giac")

[Out]

4/9*(x^(3/2) + 1)^(3/2) - 4/9

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Mupad [B]
time = 5.32, size = 27, normalized size = 1.35 \begin {gather*} \frac {2\,x\,\sqrt {x+x^{5/2}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},1;\ 2;\ -x^{3/2}\right )}{3\,\sqrt {x^{3/2}+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + x^(5/2))^(1/2),x)

[Out]

(2*x*(x + x^(5/2))^(1/2)*hypergeom([-1/2, 1], 2, -x^(3/2)))/(3*(x^(3/2) + 1)^(1/2))

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