3.606 \(\int \frac{x+3 x^2}{\sqrt{x^2+2 x^3}} \, dx\)

Optimal. Leaf size=13 \[ \sqrt{2 x^3+x^2} \]

[Out]

Sqrt[x^2 + 2*x^3]

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Rubi [A]  time = 0.0097622, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {1588} \[ \sqrt{2 x^3+x^2} \]

Antiderivative was successfully verified.

[In]

Int[(x + 3*x^2)/Sqrt[x^2 + 2*x^3],x]

[Out]

Sqrt[x^2 + 2*x^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 \frac{x+3 x^2}{\sqrt{x^2+2 x^3}} \, dx &=\sqrt{x^2+2 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0092088, size = 13, normalized size = 1. \[ \sqrt{x^2 (2 x+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(x + 3*x^2)/Sqrt[x^2 + 2*x^3],x]

[Out]

Sqrt[x^2*(1 + 2*x)]

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Maple [A]  time = 0.002, size = 21, normalized size = 1.6 \begin{align*}{{x}^{2} \left ( 1+2\,x \right ){\frac{1}{\sqrt{2\,{x}^{3}+{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2+x)/(2*x^3+x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.089, size = 15, normalized size = 1.15 \begin{align*} \sqrt{2 \, x^{3} + x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+x)/(2*x^3+x^2)^(1/2),x, algorithm="maxima")

[Out]

sqrt(2*x^3 + x^2)

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Fricas [A]  time = 1.40801, size = 26, normalized size = 2. \begin{align*} \sqrt{2 \, x^{3} + x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+x)/(2*x^3+x^2)^(1/2),x, algorithm="fricas")

[Out]

sqrt(2*x^3 + x^2)

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Sympy [A]  time = 0.139449, size = 10, normalized size = 0.77 \begin{align*} \sqrt{2 x^{3} + x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2+x)/(2*x**3+x**2)**(1/2),x)

[Out]

sqrt(2*x**3 + x**2)

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Giac [A]  time = 1.14004, size = 15, normalized size = 1.15 \begin{align*} \sqrt{2 \, x^{3} + x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+x)/(2*x^3+x^2)^(1/2),x, algorithm="giac")

[Out]

sqrt(2*x^3 + x^2)