Optimal. Leaf size=49 \[ \frac {a^2 \sqrt {c x^2} \log (x)}{x}+2 a b \sqrt {c x^2}+\frac {1}{2} b^2 x \sqrt {c x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 49, 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} \[ \frac {a^2 \sqrt {c x^2} \log (x)}{x}+2 a b \sqrt {c x^2}+\frac {1}{2} b^2 x \sqrt {c x^2} \]
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^2} \, dx &=\frac {\sqrt {c x^2} \int \frac {(a+b x)^2}{x} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx}{x}\\ &=2 a b \sqrt {c x^2}+\frac {1}{2} b^2 x \sqrt {c x^2}+\frac {a^2 \sqrt {c x^2} \log (x)}{x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.67 \[ \frac {c x \left (2 a^2 \log (x)+b x (4 a+b x)\right )}{2 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 32, normalized size = 0.65 \[ \frac {{\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \relax (x)\right )} \sqrt {c x^{2}}}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 32, normalized size = 0.65 \[ \frac {1}{2} \, {\left (b^{2} x^{2} \mathrm {sgn}\relax (x) + 4 \, a b x \mathrm {sgn}\relax (x) + 2 \, a^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\relax (x)\right )} \sqrt {c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 33, normalized size = 0.67 \[ \frac {\sqrt {c \,x^{2}}\, \left (b^{2} x^{2}+2 a^{2} \ln \relax (x )+4 a b x \right )}{2 x} \]
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 \[ \int \frac {\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {c x^{2}} \left (a + b x\right )^{2}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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