Optimal. Leaf size=48 \[ -\frac {2 \sqrt {b x+c x^2}}{3 b x^2}+\frac {4 c \sqrt {b x+c x^2}}{3 b^2 x} \]
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Rubi [A]
time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {672, 664}
\begin {gather*} \frac {4 c \sqrt {b x+c x^2}}{3 b^2 x}-\frac {2 \sqrt {b x+c x^2}}{3 b x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rule 672
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx &=-\frac {2 \sqrt {b x+c x^2}}{3 b x^2}-\frac {(2 c) \int \frac {1}{x \sqrt {b x+c x^2}} \, dx}{3 b}\\ &=-\frac {2 \sqrt {b x+c x^2}}{3 b x^2}+\frac {4 c \sqrt {b x+c x^2}}{3 b^2 x}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 29, normalized size = 0.60 \begin {gather*} \frac {2 \sqrt {x (b+c x)} (-b+2 c x)}{3 b^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 41, normalized size = 0.85
method | result | size |
trager | \(-\frac {2 \left (-2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{3 b^{2} x^{2}}\) | \(26\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (-2 c x +b \right )}{3 b^{2} x \sqrt {x \left (c x +b \right )}}\) | \(29\) |
gosper | \(-\frac {2 \left (c x +b \right ) \left (-2 c x +b \right )}{3 x \,b^{2} \sqrt {c \,x^{2}+b x}}\) | \(31\) |
default | \(-\frac {2 \sqrt {c \,x^{2}+b x}}{3 b \,x^{2}}+\frac {4 c \sqrt {c \,x^{2}+b x}}{3 b^{2} x}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 40, normalized size = 0.83 \begin {gather*} \frac {4 \, \sqrt {c x^{2} + b x} c}{3 \, b^{2} x} - \frac {2 \, \sqrt {c x^{2} + b x}}{3 \, b x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.55, size = 27, normalized size = 0.56 \begin {gather*} \frac {2 \, \sqrt {c x^{2} + b x} {\left (2 \, c x - b\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 {1}{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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Giac [A]
time = 1.15, size = 49, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + 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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Mupad [B]
time = 0.17, size = 25, normalized size = 0.52 \begin {gather*} -\frac {2\,\sqrt {c\,x^2+b\,x}\,\left (b-2\,c\,x\right )}{3\,b^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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