Optimal. Leaf size=24 \[ -\frac {\sqrt {c x^2}}{b x (a+b x)} \]
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Rubi [A] time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 32} \begin {gather*} -\frac {\sqrt {c x^2}}{b x (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2}}{x (a+b x)^2} \, dx &=\frac {\sqrt {c x^2} \int \frac {1}{(a+b x)^2} \, dx}{x}\\ &=-\frac {\sqrt {c x^2}}{b x (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.96 \begin {gather*} -\frac {c x}{b \sqrt {c x^2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 24, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {c x^2}}{b x (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.25, size = 23, normalized size = 0.96 \begin {gather*} -\frac {\sqrt {c x^{2}}}{b^{2} x^{2} + a b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 29, normalized size = 1.21 \begin {gather*} -\sqrt {c} {\left (\frac {\mathrm {sgn}\relax (x)}{{\left (b x + a\right )} b} - \frac {\mathrm {sgn}\relax (x)}{a b}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 0.96 \begin {gather*} -\frac {\sqrt {c \,x^{2}}}{\left (b x +a \right ) b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 16, normalized size = 0.67 \begin {gather*} -\frac {\sqrt {c}}{b^{2} x + a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 22, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {c\,x^2}}{b\,x\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.82, size = 39, normalized size = 1.62 \begin {gather*} \begin {cases} - \frac {\sqrt {c} \sqrt {x^{2}}}{a b x + b^{2} x^{2}} & \text {for}\: b \neq 0 \\\frac {\sqrt {c} \sqrt {x^{2}}}{a^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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