Optimal. Leaf size=52 \[ \frac {a^2 x \log (x)}{\sqrt {c x^2}}+\frac {2 a b x^2}{\sqrt {c x^2}}+\frac {b^2 x^3}{2 \sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 43} \[ \frac {a^2 x \log (x)}{\sqrt {c x^2}}+\frac {2 a b x^2}{\sqrt {c x^2}}+\frac {b^2 x^3}{2 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{\sqrt {c x^2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {2 a b x^2}{\sqrt {c x^2}}+\frac {b^2 x^3}{2 \sqrt {c x^2}}+\frac {a^2 x \log (x)}{\sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 32, normalized size = 0.62 \[ \frac {x \left (2 a^2 \log (x)+b x (4 a+b x)\right )}{2 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 35, normalized size = 0.67 \[ \frac {{\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \relax (x)\right )} \sqrt {c x^{2}}}{2 \, c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 50, normalized size = 0.96 \[ -\frac {a^{2} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2}} \right |}\right )}{\sqrt {c}} + \frac {1}{2} \, \sqrt {c x^{2}} {\left (\frac {b^{2} x}{c} + \frac {4 \, a b}{c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 0.60 \[ \frac {\left (b^{2} x^{2}+2 a^{2} \ln \relax (x )+4 a b x \right ) x}{2 \sqrt {c \,x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 35, normalized size = 0.67 \[ \frac {b^{2} x^{2}}{2 \, \sqrt {c}} + \frac {a^{2} \log \relax (x)}{\sqrt {c}} + \frac {2 \, \sqrt {c x^{2}} a b}{c} \]
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 \[ \int \frac {{\left (a+b\,x\right )}^2}{\sqrt {c\,x^2}} \,d x \]
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 \[ \int \frac {\left (a + b x\right )^{2}}{\sqrt {c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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