Optimal. Leaf size=22 \[ -\frac {x}{b \sqrt {c x^2} (a+b x)} \]
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Rubi [A] time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 32} \[ -\frac {x}{b \sqrt {c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {c x^2} (a+b x)^2} \, dx &=\frac {x \int \frac {1}{(a+b x)^2} \, dx}{\sqrt {c x^2}}\\ &=-\frac {x}{b \sqrt {c x^2} (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 1.00 \[ -\frac {x}{b \sqrt {c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 25, normalized size = 1.14 \[ -\frac {\sqrt {c x^{2}}}{b^{2} c x^{2} + a b c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.16, size = 38, normalized size = 1.73 \[ \frac {1}{{\left (b x + a\right )} b \sqrt {c} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.95 \[ -\frac {x}{\left (b x +a \right ) \sqrt {c \,x^{2}}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 21, normalized size = 0.95 \[ \frac {\sqrt {c x^{2}}}{a b c x + a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 25, normalized size = 1.14 \[ -\frac {\sqrt {c\,x^2}}{b\,c\,x\,\left (a+b\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.17, size = 85, normalized size = 3.86 \[ \begin {cases} \frac {\tilde {\infty }}{\sqrt {c} \sqrt {x^{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\tilde {\infty } x^{2}}{\sqrt {c} \sqrt {x^{2}}} & \text {for}\: a = - b x \\\frac {x^{2}}{a^{2} \sqrt {c} \sqrt {x^{2}}} & \text {for}\: b = 0 \\- \frac {x}{a b \sqrt {c} \sqrt {x^{2}} + b^{2} \sqrt {c} x \sqrt {x^{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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