3.7.4 \(\int (1+x^4) \sqrt {5 x+x^5} \, dx\) [604]

Optimal. Leaf size=15 \[ \frac {2}{15} \left (5 x+x^5\right )^{3/2} \]

[Out]

2/15*(x^5+5*x)^(3/2)

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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1602} \begin {gather*} \frac {2}{15} \left (x^5+5 x\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)*Sqrt[5*x + x^5],x]

[Out]

(2*(5*x + x^5)^(3/2))/15

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]

Rubi steps

\begin {align*} \int \left (1+x^4\right ) \sqrt {5 x+x^5} \, dx &=\frac {2}{15} \left (5 x+x^5\right )^{3/2}\\ \end {align*}

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Mathematica [A]
time = 10.03, size = 15, normalized size = 1.00 \begin {gather*} \frac {2}{15} \left (x \left (5+x^4\right )\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^4)*Sqrt[5*x + x^5],x]

[Out]

(2*(x*(5 + x^4))^(3/2))/15

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Maple [A]
time = 0.35, size = 12, normalized size = 0.80

method	result	size
default	\(\frac {2 \left (x^{5}+5 x \right )^{\frac {3}{2}}}{15}\)	\(12\)
gosper	\(\frac {2 x \left (x^{4}+5\right ) \sqrt {x^{5}+5 x}}{15}\)	\(18\)
trager	\(\frac {2 x \left (x^{4}+5\right ) \sqrt {x^{5}+5 x}}{15}\)	\(18\)
risch	\(\frac {2 x^{2} \left (x^{4}+5\right )^{2}}{15 \sqrt {x \left (x^{4}+5\right )}}\)	\(22\)
meijerg	\(\frac {2 \sqrt {5}\, x^{\frac {3}{2}} \hypergeom \left (\left [-\frac {1}{2}, \frac {3}{8}\right ], \left [\frac {11}{8}\right ], -\frac {x^{4}}{5}\right )}{3}+\frac {2 \sqrt {5}\, x^{\frac {11}{2}} \hypergeom \left (\left [-\frac {1}{2}, \frac {11}{8}\right ], \left [\frac {19}{8}\right ], -\frac {x^{4}}{5}\right )}{11}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+1)*(x^5+5*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/15*(x^5+5*x)^(3/2)

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Maxima [A]
time = 0.28, size = 11, normalized size = 0.73 \begin {gather*} \frac {2}{15} \, {\left (x^{5} + 5 \, x\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)*(x^5+5*x)^(1/2),x, algorithm="maxima")

[Out]

2/15*(x^5 + 5*x)^(3/2)

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Fricas [A]
time = 0.35, size = 11, normalized size = 0.73 \begin {gather*} \frac {2}{15} \, {\left (x^{5} + 5 \, x\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)*(x^5+5*x)^(1/2),x, algorithm="fricas")

[Out]

2/15*(x^5 + 5*x)^(3/2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).
time = 0.08, size = 31, normalized size = 2.07 \begin {gather*} \frac {2 x^{5} \sqrt {x^{5} + 5 x}}{15} + \frac {2 x \sqrt {x^{5} + 5 x}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+1)*(x**5+5*x)**(1/2),x)

[Out]

2*x**5*sqrt(x**5 + 5*x)/15 + 2*x*sqrt(x**5 + 5*x)/3

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Giac [A]
time = 3.70, size = 11, normalized size = 0.73 \begin {gather*} \frac {2}{15} \, {\left (x^{5} + 5 \, x\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)*(x^5+5*x)^(1/2),x, algorithm="giac")

[Out]

2/15*(x^5 + 5*x)^(3/2)

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Mupad [B]
time = 3.10, size = 11, normalized size = 0.73 \begin {gather*} \frac {2\,{\left (x^5+5\,x\right )}^{3/2}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + x^5)^(1/2)*(x^4 + 1),x)

[Out]

(2*(5*x + x^5)^(3/2))/15

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