Optimal. Leaf size=57 \[ -\frac {2 \sqrt {b x+c x^2} (3 b B-2 A c)}{3 b^2 x}-\frac {2 A \sqrt {b x+c x^2}}{3 b x^2} \]
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Rubi [A] time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {792, 650} \begin {gather*} -\frac {2 \sqrt {b x+c x^2} (3 b B-2 A c)}{3 b^2 x}-\frac {2 A \sqrt {b x+c x^2}}{3 b x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 650
Rule 792
Rubi steps
\begin {align*} \int \frac {A+B x}{x^2 \sqrt {b x+c x^2}} \, dx &=-\frac {2 A \sqrt {b x+c x^2}}{3 b x^2}+\frac {\left (2 \left (-2 (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right )\right ) \int \frac {1}{x \sqrt {b x+c x^2}} \, dx}{3 b}\\ &=-\frac {2 A \sqrt {b x+c x^2}}{3 b x^2}-\frac {2 (3 b B-2 A c) \sqrt {b x+c x^2}}{3 b^2 x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 0.61 \begin {gather*} -\frac {2 \sqrt {x (b+c x)} (A (b-2 c x)+3 b B x)}{3 b^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.30, size = 38, normalized size = 0.67 \begin {gather*} \frac {2 \sqrt {b x+c x^2} (-A b+2 A c x-3 b B x)}{3 b^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 34, normalized size = 0.60 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x} {\left (A b + {\left (3 \, B b - 2 \, A c\right )} x\right )}}{3 \, b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 76, normalized size = 1.33 \begin {gather*} \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A \sqrt {c} + A b\right )}}{3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 39, normalized size = 0.68 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-2 A c x +3 B b x +A b \right )}{3 \sqrt {c \,x^{2}+b x}\, b^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 62, normalized size = 1.09 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x} B}{b x} + \frac {4 \, \sqrt {c x^{2} + b x} A c}{3 \, b^{2} x} - \frac {2 \, \sqrt {c x^{2} + b x} A}{3 \, b x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 33, normalized size = 0.58 \begin {gather*} -\frac {2\,\sqrt {c\,x^2+b\,x}\,\left (A\,b-2\,A\,c\,x+3\,B\,b\,x\right )}{3\,b^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{x^{2} \sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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