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

Optimal. Leaf size=97 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{\sqrt{a-b \sqrt{c}}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )}{\sqrt{a+b \sqrt{c}}} \]

[Out]

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

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Rubi [A]  time = 0.217547, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{\sqrt{a-b \sqrt{c}}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )}{\sqrt{a+b \sqrt{c}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.371, size = 85, normalized size = 0.88 \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c + d x}}}{\sqrt{a + b \sqrt{c}}} \right )}}{\sqrt{a + b \sqrt{c}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c + d x}}}{\sqrt{a - b \sqrt{c}}} \right )}}{\sqrt{a - b \sqrt{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0959898, size = 97, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{\sqrt{a-b \sqrt{c}}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )}{\sqrt{a+b \sqrt{c}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 92, normalized size = 1. \[ 2\,{\frac{1}{\sqrt{\sqrt{{b}^{2}c}-a}}\arctan \left ({\frac{\sqrt{a+b\sqrt{dx+c}}}{\sqrt{\sqrt{{b}^{2}c}-a}}} \right ) }+2\,{\frac{1}{\sqrt{-\sqrt{{b}^{2}c}-a}}\arctan \left ({\frac{\sqrt{a+b\sqrt{dx+c}}}{\sqrt{-\sqrt{{b}^{2}c}-a}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\sqrt{d x + c} b + a} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.348471, size = 1003, normalized size = 10.34 \[ \sqrt{-\frac{{\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a}{b^{2} c - a^{2}}} \log \left (4 \,{\left ({\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a\right )} \sqrt{-\frac{{\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a}{b^{2} c - a^{2}}} + 4 \, \sqrt{\sqrt{d x + c} b + a}\right ) - \sqrt{-\frac{{\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a}{b^{2} c - a^{2}}} \log \left (-4 \,{\left ({\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a\right )} \sqrt{-\frac{{\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a}{b^{2} c - a^{2}}} + 4 \, \sqrt{\sqrt{d x + c} b + a}\right ) - \sqrt{\frac{{\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a}{b^{2} c - a^{2}}} \log \left (4 \,{\left ({\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a\right )} \sqrt{\frac{{\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a}{b^{2} c - a^{2}}} + 4 \, \sqrt{\sqrt{d x + c} b + a}\right ) + \sqrt{\frac{{\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a}{b^{2} c - a^{2}}} \log \left (-4 \,{\left ({\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a\right )} \sqrt{\frac{{\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a}{b^{2} c - a^{2}}} + 4 \, \sqrt{\sqrt{d x + c} b + a}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) + a)/(b^2*c - a^2
))*log(4*((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) - a)*sqrt(-((b
^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) + a)/(b^2*c - a^2)) + 4*sq
rt(sqrt(d*x + c)*b + a)) - sqrt(-((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*
c + a^4)) + a)/(b^2*c - a^2))*log(-4*((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*
b^2*c + a^4)) - a)*sqrt(-((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)
) + a)/(b^2*c - a^2)) + 4*sqrt(sqrt(d*x + c)*b + a)) - sqrt(((b^2*c - a^2)*sqrt(
b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) - a)/(b^2*c - a^2))*log(4*((b^2*c - a^2)*sq
rt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) + a)*sqrt(((b^2*c - a^2)*sqrt(b^2*c/(b^4
*c^2 - 2*a^2*b^2*c + a^4)) - a)/(b^2*c - a^2)) + 4*sqrt(sqrt(d*x + c)*b + a)) +
sqrt(((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) - a)/(b^2*c - a^2)
)*log(-4*((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) + a)*sqrt(((b^
2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) - a)/(b^2*c - a^2)) + 4*sqr
t(sqrt(d*x + c)*b + a))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError