3.8.40 \(\int \frac {\sqrt {\frac {a+b x}{c+d x}}}{a+b x} \, dx\) [740]

Optimal. Leaf size=41 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {a+b x}{c+d x}}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1983, 12, 214} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {a+b x}{c+d x}}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]

Rule 1983

Int[(u_)^(r_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> With[{q = Den
ominator[p]}, Dist[q*e*((b*c - a*d)/n), Subst[Int[SimplifyIntegrand[x^(q*(p + 1) - 1)*(((-a)*e + c*x^q)^(1/n -
1)/(b*e - d*x^q)^(1/n + 1))*(u /. x -> ((-a)*e + c*x^q)^(1/n)/(b*e - d*x^q)^(1/n))^r, x], x], x, (e*((a + b*x
^n)/(c + d*x^n)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[u, x] && FractionQ[p] && IntegerQ[1/
n] && IntegerQ[r]

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 77, normalized size = 1.88 \begin {gather*} \frac {2 \sqrt {\frac {a+b x}{c+d x}} \sqrt {c+d x} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {b} \sqrt {d} \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(31)=62\).
time = 0.40, size = 80, normalized size = 1.95

method	result	size
default	\(\frac {\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \left (d x +c \right ) \sqrt {\frac {b x +a}{d x +c}}}{\sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, size = 59, normalized size = 1.44 \begin {gather*} -\frac {\log \left (\frac {d \sqrt {\frac {b x + a}{d x + c}} - \sqrt {b d}}{d \sqrt {\frac {b x + a}{d x + c}} + \sqrt {b d}}\right )}{\sqrt {b d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log((d*sqrt((b*x + a)/(d*x + c)) - sqrt(b*d))/(d*sqrt((b*x + a)/(d*x + c)) + sqrt(b*d)))/sqrt(b*d)

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Fricas [A]
time = 0.34, size = 105, normalized size = 2.56 \begin {gather*} \left [\frac {\sqrt {b d} \log \left (2 \, b d x + b c + a d + 2 \, \sqrt {b d} {\left (d x + c\right )} \sqrt {\frac {b x + a}{d x + c}}\right )}{b d}, -\frac {2 \, \sqrt {-b d} \arctan \left (\frac {\sqrt {-b d} {\left (d x + c\right )} \sqrt {\frac {b x + a}{d x + c}}}{b d x + a d}\right )}{b d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[sqrt(b*d)*log(2*b*d*x + b*c + a*d + 2*sqrt(b*d)*(d*x + c)*sqrt((b*x + a)/(d*x + c)))/(b*d), -2*sqrt(-b*d)*arc
tan(sqrt(-b*d)*(d*x + c)*sqrt((b*x + a)/(d*x + c))/(b*d*x + a*d))/(b*d)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\frac {a + b x}{c + d x}}}{a + b x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (31) = 62\).
time = 4.22, size = 74, normalized size = 1.80 \begin {gather*} -\frac {\sqrt {b d} \log \left ({\left | -2 \, {\left (\sqrt {b d} x - \sqrt {b d x^{2} + b c x + a d x + a c}\right )} b d - \sqrt {b d} b c - \sqrt {b d} a d \right |}\right ) \mathrm {sgn}\left (d x + c\right )}{b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.20, size = 31, normalized size = 0.76 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\frac {\sqrt {d}\,\sqrt {\frac {a+b\,x}{c+d\,x}}}{\sqrt {b}}\right )}{\sqrt {b}\,\sqrt {d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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