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*(2*x + x^5)^(3/2))/3

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Rubi [A]  time = 0.0083743, antiderivative size = 15, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0339676, size = 15, normalized size = 1. \[ \frac{2}{3} \left (x \left (x^4+2\right )\right )^{3/2} \]

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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Maple [A]  time = 0.003, size = 18, normalized size = 1.2 \begin{align*}{\frac{2\,x \left ({x}^{4}+2 \right ) }{3}\sqrt{{x}^{5}+2\,x}} \end{align*}

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 = 1.08896, size = 15, normalized size = 1. \begin{align*} \frac{2}{3} \,{\left (x^{5} + 2 \, x\right )}^{\frac{3}{2}} \end{align*}

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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Fricas [A]  time = 1.44811, size = 31, normalized size = 2.07 \begin{align*} \frac{2}{3} \,{\left (x^{5} + 2 \, x\right )}^{\frac{3}{2}} \end{align*}

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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Sympy [B]  time = 0.243079, size = 31, normalized size = 2.07 \begin{align*} \frac{2 x^{5} \sqrt{x^{5} + 2 x}}{3} + \frac{4 x \sqrt{x^{5} + 2 x}}{3} \end{align*}

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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Giac [A]  time = 1.14557, size = 15, normalized size = 1. \begin{align*} \frac{2}{3} \,{\left (x^{5} + 2 \, x\right )}^{\frac{3}{2}} \end{align*}

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)