\(\int \frac {1}{x \sqrt {c x} \sqrt {a+b x}} \, dx\) [756]

Optimal result

Integrand size = 20, antiderivative size = 21 \[ \int \frac {1}{x \sqrt {c x} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {a+b x}}{a \sqrt {c x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {16, 37} \[ \int \frac {1}{x \sqrt {c x} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {a+b x}}{a \sqrt {c x}} \]

[In]

Int[1/(x*Sqrt[c*x]*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x])/(a*Sqrt[c*x])

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps \begin{align*} \text {integral}& = c \int \frac {1}{(c x)^{3/2} \sqrt {a+b x}} \, dx \\ & = -\frac {2 \sqrt {a+b x}}{a \sqrt {c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x \sqrt {c x} \sqrt {a+b x}} \, dx=-\frac {2 c x \sqrt {a+b x}}{a (c x)^{3/2}} \]

[In]

Integrate[1/(x*Sqrt[c*x]*Sqrt[a + b*x]),x]

[Out]

(-2*c*x*Sqrt[a + b*x])/(a*(c*x)^(3/2))

Maple [A] (verified)

Time = 1.67 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86

method	result	size
gosper	\(-\frac {2 \sqrt {b x +a}}{a \sqrt {c x}}\)	\(18\)
default	\(-\frac {2 \sqrt {b x +a}}{a \sqrt {c x}}\)	\(18\)
risch	\(-\frac {2 \sqrt {b x +a}}{a \sqrt {c x}}\)	\(18\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x \sqrt {c x} \sqrt {a+b x}} \, dx=-\frac {2 \, \sqrt {b x + a} \sqrt {c x}}{a c x} \]

[In]

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

[Out]

-2*sqrt(b*x + a)*sqrt(c*x)/(a*c*x)

Sympy [A] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x \sqrt {c x} \sqrt {a+b x}} \, dx=- \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{a \sqrt {c}} \]

[In]

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

[Out]

-2*sqrt(b)*sqrt(a/(b*x) + 1)/(a*sqrt(c))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x \sqrt {c x} \sqrt {a+b x}} \, dx=-\frac {2 \, \sqrt {b c x^{2} + a c x}}{a c x} \]

[In]

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

[Out]

-2*sqrt(b*c*x^2 + a*c*x)/(a*c*x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67 \[ \int \frac {1}{x \sqrt {c x} \sqrt {a+b x}} \, dx=-\frac {2 \, \sqrt {b x + a} b^{2}}{\sqrt {{\left (b x + a\right )} b c - a b c} a {\left | b \right |}} \]

[In]

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

[Out]

-2*sqrt(b*x + a)*b^2/(sqrt((b*x + a)*b*c - a*b*c)*a*abs(b))

Mupad [B] (verification not implemented)

Time = 1.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x \sqrt {c x} \sqrt {a+b x}} \, dx=-\frac {2\,\sqrt {a+b\,x}}{a\,\sqrt {c\,x}} \]

[In]

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

[Out]

-(2*(a + b*x)^(1/2))/(a*(c*x)^(1/2))