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 = {627}
\begin {gather*} -\frac {2 (b+2 c x)}{b^2 \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 627
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 = 0.06, size = 22, normalized size = 0.92 \begin {gather*} -\frac {2 (b+2 c x)}{b^2 \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 23, normalized size = 0.96
method | result | size |
default | \(-\frac {2 \left (2 c x +b \right )}{b^{2} \sqrt {c \,x^{2}+b x}}\) | \(23\) |
gosper | \(-\frac {2 x \left (c x +b \right ) \left (2 c x +b \right )}{b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}\) | \(29\) |
trager | \(-\frac {2 \left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{\left (c x +b \right ) b^{2} x}\) | \(33\) |
risch | \(-\frac {2 \left (c x +b \right )}{b^{2} \sqrt {x \left (c x +b \right )}}-\frac {2 c x}{\sqrt {x \left (c x +b \right )}\, b^{2}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 35, normalized size = 1.46 \begin {gather*} -\frac {4 \, c x}{\sqrt {c x^{2} + b x} b^{2}} - \frac {2}{\sqrt {c x^{2} + b x} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.59, size = 35, normalized size = 1.46 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x} {\left (2 \, c x + b\right )}}{b^{2} c x^{2} + b^{3} x} \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 {1}{\left (b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.81, size = 24, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (\frac {2 \, c x}{b^{2}} + \frac {1}{b}\right )}}{\sqrt {c x^{2} + b x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 24, normalized size = 1.00 \begin {gather*} -\frac {2\,b+4\,c\,x}{b^2\,\sqrt {c\,x^2+b\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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