Optimal. Leaf size=54 \[ -\frac {a^2 \sqrt {c x^2}}{2 x^3}-\frac {2 a b \sqrt {c x^2}}{x^2}+\frac {b^2 \sqrt {c x^2} \log (x)}{x} \]
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Rubi [A] time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \[ -\frac {a^2 \sqrt {c x^2}}{2 x^3}-\frac {2 a b \sqrt {c x^2}}{x^2}+\frac {b^2 \sqrt {c x^2} \log (x)}{x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2} (a+b x)^2}{x^4} \, dx &=\frac {\sqrt {c x^2} \int \frac {(a+b x)^2}{x^3} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (\frac {a^2}{x^3}+\frac {2 a b}{x^2}+\frac {b^2}{x}\right ) \, dx}{x}\\ &=-\frac {a^2 \sqrt {c x^2}}{2 x^3}-\frac {2 a b \sqrt {c x^2}}{x^2}+\frac {b^2 \sqrt {c x^2} \log (x)}{x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 36, normalized size = 0.67 \[ \frac {\sqrt {c x^2} \left (2 b^2 x^2 \log (x)-a (a+4 b x)\right )}{2 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 33, normalized size = 0.61 \[ \frac {{\left (2 \, b^{2} x^{2} \log \relax (x) - 4 \, a b x - a^{2}\right )} \sqrt {c x^{2}}}{2 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 35, normalized size = 0.65 \[ \frac {1}{2} \, {\left (2 \, b^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\relax (x) - \frac {4 \, a b x \mathrm {sgn}\relax (x) + a^{2} \mathrm {sgn}\relax (x)}{x^{2}}\right )} \sqrt {c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 34, normalized size = 0.63 \[ \frac {\sqrt {c \,x^{2}}\, \left (2 b^{2} x^{2} \ln \relax (x )-4 a b x -a^{2}\right )}{2 x^{3}} \]
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 [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2}{x^4} \,d x \]
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 \[ \int \frac {\sqrt {c x^{2}} \left (a + b x\right )^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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