3.10.93 \(\int \frac {1}{\sqrt {c x (a+b x)}} \, dx\) [993]

Optimal. Leaf size=40 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a c x+b c x^2}}\right )}{\sqrt {b} \sqrt {c}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1976, 634, 212} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a c x+b c x^2}}\right )}{\sqrt {b} \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[c*x*(a + b*x)],x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 634

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 1976

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[u*(a*c*e + (b*c
+ a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 56, normalized size = 1.40 \begin {gather*} -\frac {2 \sqrt {x} \sqrt {a+b x} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{\sqrt {b} \sqrt {c x (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[c*x*(a + b*x)],x]

[Out]

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

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Maple [A]
time = 0.23, size = 37, normalized size = 0.92

method	result	size
default	\(\frac {\ln \left (\frac {\frac {1}{2} a c +b c x}{\sqrt {b c}}+\sqrt {b c \,x^{2}+a c x}\right )}{\sqrt {b c}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.33, size = 36, normalized size = 0.90 \begin {gather*} \frac {\log \left (2 \, b c x + a c + 2 \, \sqrt {b c x^{2} + a c x} \sqrt {b c}\right )}{\sqrt {b c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.33, size = 87, normalized size = 2.18 \begin {gather*} \left [\frac {\sqrt {b c} \log \left (2 \, b c x + a c + 2 \, \sqrt {b c x^{2} + a c x} \sqrt {b c}\right )}{b c}, -\frac {2 \, \sqrt {-b c} \arctan \left (\frac {\sqrt {b c x^{2} + a c x} \sqrt {-b c}}{b c x}\right )}{b c}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[sqrt(b*c)*log(2*b*c*x + a*c + 2*sqrt(b*c*x^2 + a*c*x)*sqrt(b*c))/(b*c), -2*sqrt(-b*c)*arctan(sqrt(b*c*x^2 + a
*c*x)*sqrt(-b*c)/(b*c*x))/(b*c)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {c x \left (a + b x\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(c*x*(a + b*x)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (30) = 60\).
time = 2.21, size = 76, normalized size = 1.90 \begin {gather*} \frac {a^{2} c \log \left ({\left | -a c - 2 \, \sqrt {b c} {\left (\sqrt {b c} x - \sqrt {b c x^{2} + a c x}\right )} \right |}\right )}{8 \, \sqrt {b c} b} + \frac {1}{4} \, \sqrt {b c x^{2} + a c x} {\left (2 \, x + \frac {a}{b}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a^2*c*log(abs(-a*c - 2*sqrt(b*c)*(sqrt(b*c)*x - sqrt(b*c*x^2 + a*c*x))))/(sqrt(b*c)*b) + 1/4*sqrt(b*c*x^2
+ a*c*x)*(2*x + a/b)

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Mupad [B]
time = 3.47, size = 33, normalized size = 0.82 \begin {gather*} \frac {\ln \left (a\,c+2\,\sqrt {b\,c}\,\sqrt {c\,x\,\left (a+b\,x\right )}+2\,b\,c\,x\right )}{\sqrt {b\,c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x*(a + b*x))^(1/2),x)

[Out]

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

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