Optimal. Leaf size=24 \[ -\frac {2 (b+2 c x)}{b^2 \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {613} \[ -\frac {2 (b+2 c x)}{b^2 \sqrt {b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 613
Rubi steps
\begin {align*} \int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b+2 c x)}{b^2 \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.92 \[ -\frac {2 (b+2 c x)}{b^2 \sqrt {x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 35, normalized size = 1.46 \[ -\frac {2 \, \sqrt {c x^{2} + b x} {\left (2 \, c x + b\right )}}{b^{2} c x^{2} + b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 24, normalized size = 1.00 \[ -\frac {2 \, {\left (\frac {2 \, c x}{b^{2}} + \frac {1}{b}\right )}}{\sqrt {c x^{2} + b x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 29, normalized size = 1.21 \[ -\frac {2 \left (c x +b \right ) \left (2 c x +b \right ) x}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 35, normalized size = 1.46 \[ -\frac {4 \, c x}{\sqrt {c x^{2} + b x} b^{2}} - \frac {2}{\sqrt {c x^{2} + b x} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 24, normalized size = 1.00 \[ -\frac {2\,b+4\,c\,x}{b^2\,\sqrt {c\,x^2+b\,x}} \]
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 \[ \int \frac {1}{\left (b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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