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