3.605 \(\int (2+5 x^4) \sqrt {2 x+x^5} \, dx\)

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

[Out]

2/3*(x^5+2*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 = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {1588} \[ \frac {2}{3} \left (x^5+2 x\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 5*x^4)*Sqrt[2*x + x^5],x]

[Out]

(2*(2*x + x^5)^(3/2))/3

Rule 1588

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

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Mathematica [A]  time = 0.03, size = 15, normalized size = 1.00 \[ \frac {2}{3} \left (x \left (x^4+2\right )\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 5*x^4)*Sqrt[2*x + x^5],x]

[Out]

(2*(x*(2 + x^4))^(3/2))/3

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fricas [A]  time = 0.39, size = 11, normalized size = 0.73 \[ \frac {2}{3} \, {\left (x^{5} + 2 \, x\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x^5 + 2*x)^(3/2)

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giac [A]  time = 0.40, size = 11, normalized size = 0.73 \[ \frac {2}{3} \, {\left (x^{5} + 2 \, x\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x^5 + 2*x)^(3/2)

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maple [A]  time = 0.00, size = 18, normalized size = 1.20 \[ \frac {2 \left (x^{4}+2\right ) \sqrt {x^{5}+2 x}\, x}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.87, size = 11, normalized size = 0.73 \[ \frac {2}{3} \, {\left (x^{5} + 2 \, x\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x^5 + 2*x)^(3/2)

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mupad [B]  time = 3.11, size = 11, normalized size = 0.73 \[ \frac {2\,{\left (x^5+2\,x\right )}^{3/2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + x^5)^(1/2)*(5*x^4 + 2),x)

[Out]

(2*(2*x + x^5)^(3/2))/3

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sympy [B]  time = 0.26, size = 31, normalized size = 2.07 \[ \frac {2 x^{5} \sqrt {x^{5} + 2 x}}{3} + \frac {4 x \sqrt {x^{5} + 2 x}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**5*sqrt(x**5 + 2*x)/3 + 4*x*sqrt(x**5 + 2*x)/3

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