Optimal. Leaf size=45 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}+\frac {\sqrt {x}}{a (a+b x)} \]
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Rubi [A] time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {51, 63, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}+\frac {\sqrt {x}}{a (a+b x)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx &=\frac {\sqrt {x}}{a (a+b x)}+\frac {\int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 a}\\ &=\frac {\sqrt {x}}{a (a+b x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a}\\ &=\frac {\sqrt {x}}{a (a+b x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}+\frac {\sqrt {x}}{a^2+a b x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 116, normalized size = 2.58 \[ \left [\frac {2 \, a b \sqrt {x} - \sqrt {-a b} {\left (b x + a\right )} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right )}{2 \, {\left (a^{2} b^{2} x + a^{3} b\right )}}, \frac {a b \sqrt {x} - \sqrt {a b} {\left (b x + a\right )} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right )}{a^{2} b^{2} x + a^{3} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 35, normalized size = 0.78 \[ \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a} + \frac {\sqrt {x}}{{\left (b x + a\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 0.80 \[ \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}+\frac {\sqrt {x}}{\left (b x +a \right ) a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.92, size = 35, normalized size = 0.78 \[ \frac {\sqrt {x}}{a b x + a^{2}} + \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 33, normalized size = 0.73 \[ \frac {\sqrt {x}}{a\,\left (a+b\,x\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}\,\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.53, size = 328, normalized size = 7.29 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 \sqrt {x}}{a^{2}} & \text {for}\: b = 0 \\- \frac {2}{3 b^{2} x^{\frac {3}{2}}} & \text {for}\: a = 0 \\\frac {2 i \sqrt {a} b \sqrt {x} \sqrt {\frac {1}{b}}}{2 i a^{\frac {5}{2}} b \sqrt {\frac {1}{b}} + 2 i a^{\frac {3}{2}} b^{2} x \sqrt {\frac {1}{b}}} + \frac {a \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {5}{2}} b \sqrt {\frac {1}{b}} + 2 i a^{\frac {3}{2}} b^{2} x \sqrt {\frac {1}{b}}} - \frac {a \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {5}{2}} b \sqrt {\frac {1}{b}} + 2 i a^{\frac {3}{2}} b^{2} x \sqrt {\frac {1}{b}}} + \frac {b x \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {5}{2}} b \sqrt {\frac {1}{b}} + 2 i a^{\frac {3}{2}} b^{2} x \sqrt {\frac {1}{b}}} - \frac {b x \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {5}{2}} b \sqrt {\frac {1}{b}} + 2 i a^{\frac {3}{2}} b^{2} x \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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