3.7.6 \(\int \frac {x+3 x^2}{\sqrt {x^2+2 x^3}} \, dx\) [606]

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

[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 = {1602} \begin {gather*} \sqrt {2 x^3+x^2} \end {gather*}

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 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 \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 \begin {gather*} \sqrt {x^2 (1+2 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[x^2*(1 + 2*x)]

________________________________________________________________________________________

Maple [A]
time = 0.32, size = 21, normalized size = 1.62

method	result	size
trager	\(\sqrt {2 x^{3}+x^{2}}\)	\(12\)
gosper	\(\frac {x^{2} \left (2 x +1\right )}{\sqrt {2 x^{3}+x^{2}}}\)	\(21\)
default	\(\frac {x^{2} \left (2 x +1\right )}{\sqrt {2 x^{3}+x^{2}}}\)	\(21\)
risch	\(\frac {x^{2} \left (2 x +1\right )}{\sqrt {x^{2} \left (2 x +1\right )}}\)	\(21\)
meijerg	\(\frac {\sqrt {\pi }-\frac {\sqrt {\pi }\, \left (-8 x +8\right ) \sqrt {2 x +1}}{8}}{\sqrt {\pi }}+\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {2 x +1}}{2 \sqrt {\pi }}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 11, normalized size = 0.85 \begin {gather*} \sqrt {2 \, x^{3} + x^{2}} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 11, normalized size = 0.85 \begin {gather*} \sqrt {2 \, x^{3} + x^{2}} \end {gather*}

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)

________________________________________________________________________________________

Sympy [A]
time = 0.05, size = 10, normalized size = 0.77 \begin {gather*} \sqrt {2 x^{3} + x^{2}} \end {gather*}

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)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (11) = 22\).
time = 3.33, size = 23, normalized size = 1.77 \begin {gather*} \frac {{\left (2 \, x + 1\right )}^{\frac {3}{2}} - \sqrt {2 \, x + 1}}{2 \, \mathrm {sgn}\left (x\right )} \end {gather*}

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]

1/2*((2*x + 1)^(3/2) - sqrt(2*x + 1))/sgn(x)

________________________________________________________________________________________

Mupad [B]
time = 3.23, size = 10, normalized size = 0.77 \begin {gather*} \left |x\right |\,\sqrt {2\,x+1} \end {gather*}

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)

________________________________________________________________________________________