\(\int \frac {1}{\sqrt {x} \sqrt {a x^2+b x^3}} \, dx\) [273]

Optimal result

Integrand size = 21, antiderivative size = 25 \[ \int \frac {1}{\sqrt {x} \sqrt {a x^2+b x^3}} \, dx=-\frac {2 \sqrt {a x^2+b x^3}}{a x^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2039} \[ \int \frac {1}{\sqrt {x} \sqrt {a x^2+b x^3}} \, dx=-\frac {2 \sqrt {a x^2+b x^3}}{a x^{3/2}} \]

[In]

Int[1/(Sqrt[x]*Sqrt[a*x^2 + b*x^3]),x]

[Out]

(-2*Sqrt[a*x^2 + b*x^3])/(a*x^(3/2))

Rule 2039

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-c^(j - 1))*(c*x)^(m - j
+ 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j)*(p + 1))), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] &&
NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {a x^2+b x^3}}{a x^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt {x} \sqrt {a x^2+b x^3}} \, dx=-\frac {2 \sqrt {x^2 (a+b x)}}{a x^{3/2}} \]

[In]

Integrate[1/(Sqrt[x]*Sqrt[a*x^2 + b*x^3]),x]

[Out]

(-2*Sqrt[x^2*(a + b*x)])/(a*x^(3/2))

Maple [A] (verified)

Time = 1.87 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

method	result	size
risch	\(-\frac {2 \sqrt {x}\, \left (b x +a \right )}{\sqrt {x^{2} \left (b x +a \right )}\, a}\)	\(25\)
gosper	\(-\frac {2 \sqrt {x}\, \left (b x +a \right )}{a \sqrt {b \,x^{3}+a \,x^{2}}}\)	\(27\)
default	\(-\frac {2 \sqrt {x}\, \left (b x +a \right )}{a \sqrt {b \,x^{3}+a \,x^{2}}}\)	\(27\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\sqrt {x} \sqrt {a x^2+b x^3}} \, dx=-\frac {2 \, \sqrt {b x^{3} + a x^{2}}}{a x^{\frac {3}{2}}} \]

[In]

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

[Out]

-2*sqrt(b*x^3 + a*x^2)/(a*x^(3/2))

Sympy [F]

\[ \int \frac {1}{\sqrt {x} \sqrt {a x^2+b x^3}} \, dx=\int \frac {1}{\sqrt {x} \sqrt {x^{2} \left (a + b x\right )}}\, dx \]

[In]

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

[Out]

Integral(1/(sqrt(x)*sqrt(x**2*(a + b*x))), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {x} \sqrt {a x^2+b x^3}} \, dx=\int { \frac {1}{\sqrt {b x^{3} + a x^{2}} \sqrt {x}} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\sqrt {x} \sqrt {a x^2+b x^3}} \, dx=\frac {4 \, \sqrt {b}}{{\left ({\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{2} - a\right )} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {x} \sqrt {a x^2+b x^3}} \, dx=\int \frac {1}{\sqrt {x}\,\sqrt {b\,x^3+a\,x^2}} \,d x \]

[In]

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

[Out]

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