Optimal. Leaf size=28 \[ \frac {a \sqrt {c x^2} \log (x)}{x}+b \sqrt {c x^2} \]
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Rubi [A] time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 43} \begin {gather*} \frac {a \sqrt {c x^2} \log (x)}{x}+b \sqrt {c x^2} \end {gather*}
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)}{x^2} \, dx &=\frac {\sqrt {c x^2} \int \frac {a+b x}{x} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (b+\frac {a}{x}\right ) \, dx}{x}\\ &=b \sqrt {c x^2}+\frac {a \sqrt {c x^2} \log (x)}{x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 0.71 \begin {gather*} \frac {c x (a \log (x)+b x)}{\sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 19, normalized size = 0.68 \begin {gather*} \sqrt {c x^2} \left (\frac {a \log (x)}{x}+b\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.27, size = 19, normalized size = 0.68 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x + a \log \relax (x)\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 17, normalized size = 0.61 \begin {gather*} {\left (b x \mathrm {sgn}\relax (x) + a \log \left ({\left | x \right |}\right ) \mathrm {sgn}\relax (x)\right )} \sqrt {c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 20, normalized size = 0.71 \begin {gather*} \frac {\sqrt {c \,x^{2}}\, \left (a \ln \relax (x )+b x \right )}{x} \end {gather*}
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {c\,x^2}\,\left (a+b\,x\right )}{x^2} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {c x^{2}} \left (a + b x\right )}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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