3.3.55 \(\int \frac {x}{\sqrt {a x^2+b x^3}} \, dx\) [255]

Optimal. Leaf size=23 \[ \frac {2 \sqrt {a x^2+b x^3}}{b x} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1602} \begin {gather*} \frac {2 \sqrt {a x^2+b x^3}}{b x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a*x^2 + b*x^3],x]

[Out]

(2*Sqrt[a*x^2 + b*x^3])/(b*x)

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

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Mathematica [A]
time = 0.01, size = 21, normalized size = 0.91 \begin {gather*} \frac {2 \sqrt {x^2 (a+b x)}}{b x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[a*x^2 + b*x^3],x]

[Out]

(2*Sqrt[x^2*(a + b*x)])/(b*x)

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Maple [A]
time = 0.36, size = 25, normalized size = 1.09

method	result	size
trager	\(\frac {2 \sqrt {b \,x^{3}+a \,x^{2}}}{b x}\)	\(22\)
risch	\(\frac {2 x \left (b x +a \right )}{\sqrt {x^{2} \left (b x +a \right )}\, b}\)	\(23\)
gosper	\(\frac {2 x \left (b x +a \right )}{b \sqrt {b \,x^{3}+a \,x^{2}}}\)	\(25\)
default	\(\frac {2 x \left (b x +a \right )}{b \sqrt {b \,x^{3}+a \,x^{2}}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 12, normalized size = 0.52 \begin {gather*} \frac {2 \, \sqrt {b x + a}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x + a)/b

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Fricas [A]
time = 1.35, size = 21, normalized size = 0.91 \begin {gather*} \frac {2 \, \sqrt {b x^{3} + a x^{2}}}{b x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x^3 + a*x^2)/(b*x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x^{2} \left (a + b x\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/sqrt(x**2*(a + b*x)), x)

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Giac [A]
time = 1.37, size = 27, normalized size = 1.17 \begin {gather*} -\frac {2 \, \sqrt {a} \mathrm {sgn}\left (x\right )}{b} + \frac {2 \, \sqrt {b x + a}}{b \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(a)*sgn(x)/b + 2*sqrt(b*x + a)/(b*sgn(x))

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Mupad [B]
time = 5.14, size = 17, normalized size = 0.74 \begin {gather*} \frac {2\,\left |x\right |\,\sqrt {a+b\,x}}{b\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a*x^2 + b*x^3)^(1/2),x)

[Out]

(2*abs(x)*(a + b*x)^(1/2))/(b*x)

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