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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 = 0.01, size = 13, normalized size = 1.00 \[ \sqrt {x^2 (2 x+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[x^2*(1 + 2*x)]

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 11, normalized size = 0.85 \[ \sqrt {2 \, x^{3} + x^{2}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.36, size = 11, normalized size = 0.85 \[ \sqrt {2 \, x^{3} + x^{2}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 21, normalized size = 1.62 \[ \frac {\left (2 x +1\right ) x^{2}}{\sqrt {2 x^{3}+x^{2}}} \]

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

________________________________________________________________________________________

maxima [A]  time = 0.88, size = 11, normalized size = 0.85 \[ \sqrt {2 \, x^{3} + x^{2}} \]

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)

________________________________________________________________________________________

mupad [B]  time = 3.23, size = 10, normalized size = 0.77 \[ \relax |x|\,\sqrt {2\,x+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

abs(x)*(2*x + 1)^(1/2)

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 10, normalized size = 0.77 \[ \sqrt {2 x^{3} + x^{2}} \]

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)

________________________________________________________________________________________