Optimal. Leaf size=26 \[ -\frac {\sqrt {c x^2} (a+b x)^2}{2 a x^3} \]
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Rubi [A] time = 0.00, antiderivative size = 26, 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, 37} \[ -\frac {\sqrt {c x^2} (a+b x)^2}{2 a x^3} \]
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2} (a+b x)}{x^4} \, dx &=\frac {\sqrt {c x^2} \int \frac {a+b x}{x^3} \, dx}{x}\\ &=-\frac {\sqrt {c x^2} (a+b x)^2}{2 a x^3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 0.85 \[ -\frac {\sqrt {c x^2} (a+2 b x)}{2 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 18, normalized size = 0.69 \[ -\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )}}{2 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 19, normalized size = 0.73 \[ -\frac {{\left (2 \, b x \mathrm {sgn}\relax (x) + a \mathrm {sgn}\relax (x)\right )} \sqrt {c}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 0.73 \[ -\frac {\left (2 b x +a \right ) \sqrt {c \,x^{2}}}{2 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 28, normalized size = 1.08 \[ -\frac {a\,\sqrt {c}\,x^2+2\,b\,\sqrt {c}\,x^3}{2\,x\,{\left (x^2\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 36, normalized size = 1.38 \[ - \frac {a \sqrt {c} \sqrt {x^{2}}}{2 x^{3}} - \frac {b \sqrt {c} \sqrt {x^{2}}}{x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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