3.475 \(\int \frac{1}{x \left (a+b \sqrt{c+d x}\right )} \, dx\)

Optimal. Leaf size=82 \[ -\frac{2 a \log \left (a+b \sqrt{c+d x}\right )}{a^2-b^2 c}+\frac{2 b \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2-b^2 c}+\frac{a \log (x)}{a^2-b^2 c} \]

[Out]

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

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Rubi [A]  time = 0.175363, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{2 a \log \left (a+b \sqrt{c+d x}\right )}{a^2-b^2 c}+\frac{2 b \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2-b^2 c}+\frac{a \log (x)}{a^2-b^2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.1031, size = 73, normalized size = 0.89 \[ \frac{a \log{\left (- d x \right )}}{a^{2} - b^{2} c} - \frac{2 a \log{\left (a + b \sqrt{c + d x} \right )}}{a^{2} - b^{2} c} + \frac{2 b \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{a^{2} - b^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(a+b*(d*x+c)**(1/2)),x)

[Out]

a*log(-d*x)/(a**2 - b**2*c) - 2*a*log(a + b*sqrt(c + d*x))/(a**2 - b**2*c) + 2*b
*sqrt(c)*atanh(sqrt(c + d*x)/sqrt(c))/(a**2 - b**2*c)

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Mathematica [A]  time = 0.0583137, size = 61, normalized size = 0.74 \[ \frac{-2 a \log \left (a+b \sqrt{c+d x}\right )+a \log (d x)+2 b \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2-b^2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 77, normalized size = 0.9 \[{\frac{a\ln \left ( dx \right ) }{-{b}^{2}c+{a}^{2}}}+2\,{\frac{b\sqrt{c}}{-{b}^{2}c+{a}^{2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-2\,{\frac{a\ln \left ( a+b\sqrt{dx+c} \right ) }{-{b}^{2}c+{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*(d*x+c)^(1/2)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((sqrt(d*x + c)*b + a)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.287991, size = 1, normalized size = 0.01 \[ \left [\frac{b \sqrt{c} \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 2 \, a \log \left (\sqrt{d x + c} b + a\right ) - a \log \left (x\right )}{b^{2} c - a^{2}}, -\frac{2 \, b \sqrt{-c} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) - 2 \, a \log \left (\sqrt{d x + c} b + a\right ) + a \log \left (x\right )}{b^{2} c - a^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((sqrt(d*x + c)*b + a)*x),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a + b \sqrt{c + d x}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.283629, size = 155, normalized size = 1.89 \[ \frac{2 \, a b{\rm ln}\left ({\left | \sqrt{d x + c} b + a \right |}\right )}{b^{3} c - a^{2} b} + \frac{2 \, b c \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{{\left (b^{2} c - a^{2}\right )} \sqrt{-c}} - \frac{a{\rm ln}\left (d x\right )}{b^{2} c - a^{2}} + \frac{a{\rm ln}\left (-c\right ) - 2 \, a{\rm ln}\left ({\left | a \right |}\right )}{b^{2} c - a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((sqrt(d*x + c)*b + a)*x),x, algorithm="giac")

[Out]

2*a*b*ln(abs(sqrt(d*x + c)*b + a))/(b^3*c - a^2*b) + 2*b*c*arctan(sqrt(d*x + c)/
sqrt(-c))/((b^2*c - a^2)*sqrt(-c)) - a*ln(d*x)/(b^2*c - a^2) + (a*ln(-c) - 2*a*l
n(abs(a)))/(b^2*c - a^2)