\(\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx\) [737]

Optimal result

Integrand size = 22, antiderivative size = 42 \[ \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} \sqrt {c}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {95, 214} \[ \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} \sqrt {c}} \]

[In]

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

[Out]

(-2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*Sqrt[c])

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{\sqrt {a} \sqrt {c}} \]

[In]

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

[Out]

(-2*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(Sqrt[a]*Sqrt[c])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(72\) vs. \(2(30)=60\).

Time = 1.38 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.74

method	result	size
default	\(-\frac {\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) \sqrt {d x +c}\, \sqrt {b x +a}}{\sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}\)	\(73\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (30) = 60\).

Time = 0.24 (sec) , antiderivative size = 187, normalized size of antiderivative = 4.45 \[ \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx=\left [\frac {\sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right )}{2 \, a c}, \frac {\sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right )}{a c}\right ] \]

[In]

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

[Out]

[1/2*sqrt(a*c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt
(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2)/(a*c), sqrt(-a*c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*s
qrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x))/(a*c)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (30) = 60\).

Time = 0.31 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.02 \[ \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {2 \, \sqrt {b d} b \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} {\left | b \right |}} \]

[In]

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

[Out]

-2*sqrt(b*d)*b*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)
/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*abs(b))

Mupad [B] (verification not implemented)

Time = 2.91 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.88 \[ \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\ln \left (\frac {\left (\sqrt {c}\,\sqrt {a+b\,x}-\sqrt {a}\,\sqrt {c+d\,x}\right )\,\left (b\,\sqrt {c}-\frac {\sqrt {a}\,d\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )-\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+d\,x}-\sqrt {c}}\right )}{\sqrt {a}\,\sqrt {c}} \]

[In]

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

[Out]

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