Optimal. Leaf size=77 \[ \frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {b \sqrt {a+b \sqrt {x}}}{2 a \sqrt {x}}-\frac {\sqrt {a+b \sqrt {x}}}{x} \]
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Rubi [A] time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {266, 47, 51, 63, 208} \[ \frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {b \sqrt {a+b \sqrt {x}}}{2 a \sqrt {x}}-\frac {\sqrt {a+b \sqrt {x}}}{x} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {x}}}{x^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{x}+\frac {1}{2} b \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{x}-\frac {b \sqrt {a+b \sqrt {x}}}{2 a \sqrt {x}}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sqrt {x}\right )}{4 a}\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{x}-\frac {b \sqrt {a+b \sqrt {x}}}{2 a \sqrt {x}}-\frac {b \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sqrt {x}}\right )}{2 a}\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{x}-\frac {b \sqrt {a+b \sqrt {x}}}{2 a \sqrt {x}}+\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )}{2 a^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 43, normalized size = 0.56 \[ -\frac {4 b^2 \left (a+b \sqrt {x}\right )^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {\sqrt {x} b}{a}+1\right )}{3 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.42, size = 133, normalized size = 1.73 \[ \left [\frac {\sqrt {a} b^{2} x \log \left (\frac {b x + 2 \, \sqrt {b \sqrt {x} + a} \sqrt {a} \sqrt {x} + 2 \, a \sqrt {x}}{x}\right ) - 2 \, {\left (a b \sqrt {x} + 2 \, a^{2}\right )} \sqrt {b \sqrt {x} + a}}{4 \, a^{2} x}, -\frac {\sqrt {-a} b^{2} x \arctan \left (\frac {\sqrt {b \sqrt {x} + a} \sqrt {-a}}{a}\right ) + {\left (a b \sqrt {x} + 2 \, a^{2}\right )} \sqrt {b \sqrt {x} + a}}{2 \, a^{2} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 72, normalized size = 0.94 \[ -\frac {\frac {b^{3} \arctan \left (\frac {\sqrt {b \sqrt {x} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {{\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} b^{3} + \sqrt {b \sqrt {x} + a} a b^{3}}{a b^{2} x}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 59, normalized size = 0.77 \[ 4 \left (\frac {\arctanh \left (\frac {\sqrt {b \sqrt {x}+a}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}+\frac {-\frac {\left (b \sqrt {x}+a \right )^{\frac {3}{2}}}{8 a}-\frac {\sqrt {b \sqrt {x}+a}}{8}}{b^{2} x}\right ) b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.04, size = 100, normalized size = 1.30 \[ -\frac {b^{2} \log \left (\frac {\sqrt {b \sqrt {x} + a} - \sqrt {a}}{\sqrt {b \sqrt {x} + a} + \sqrt {a}}\right )}{4 \, a^{\frac {3}{2}}} - \frac {{\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} b^{2} + \sqrt {b \sqrt {x} + a} a b^{2}}{2 \, {\left ({\left (b \sqrt {x} + a\right )}^{2} a - 2 \, {\left (b \sqrt {x} + a\right )} a^{2} + a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.57, size = 54, normalized size = 0.70 \[ \frac {b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,\sqrt {x}}}{\sqrt {a}}\right )}{2\,a^{3/2}}-\frac {\sqrt {a+b\,\sqrt {x}}}{2\,x}-\frac {{\left (a+b\,\sqrt {x}\right )}^{3/2}}{2\,a\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.24, size = 105, normalized size = 1.36 \[ - \frac {a}{\sqrt {b} x^{\frac {5}{4}} \sqrt {\frac {a}{b \sqrt {x}} + 1}} - \frac {3 \sqrt {b}}{2 x^{\frac {3}{4}} \sqrt {\frac {a}{b \sqrt {x}} + 1}} - \frac {b^{\frac {3}{2}}}{2 a \sqrt [4]{x} \sqrt {\frac {a}{b \sqrt {x}} + 1}} + \frac {b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt [4]{x}} \right )}}{2 a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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