Optimal. Leaf size=23 \[ -\frac {2 \left (b x+c x^2\right )^{3/2}}{3 b x^3} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {650} \[ -\frac {2 \left (b x+c x^2\right )^{3/2}}{3 b x^3} \]
Antiderivative was successfully verified.
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Rule 650
Rubi steps
\begin {align*} \int \frac {\sqrt {b x+c x^2}}{x^3} \, dx &=-\frac {2 \left (b x+c x^2\right )^{3/2}}{3 b x^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 0.91 \[ -\frac {2 (x (b+c x))^{3/2}}{3 b x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 24, normalized size = 1.04 \[ -\frac {2 \, \sqrt {c x^{2} + b x} {\left (c x + b\right )}}{3 \, b x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 76, normalized size = 3.30 \[ \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} c + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b \sqrt {c} + b^{2}\right )}}{3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 25, normalized size = 1.09 \[ -\frac {2 \left (c x +b \right ) \sqrt {c \,x^{2}+b x}}{3 b \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 37, normalized size = 1.61 \[ -\frac {2 \, \sqrt {c x^{2} + b x} c}{3 \, b x} - \frac {2 \, \sqrt {c x^{2} + b x}}{3 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 24, normalized size = 1.04 \[ -\frac {2\,\sqrt {c\,x^2+b\,x}\,\left (b+c\,x\right )}{3\,b\,x^2} \]
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 \[ \int \frac {\sqrt {x \left (b + c x\right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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