Optimal. Leaf size=28 \[ b \sqrt {c x^2}+\frac {a \sqrt {c x^2} \log (x)}{x} \]
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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, 45}
\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 45
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 = 0.00, size = 20, normalized size = 0.71 \begin {gather*} \frac {c x (b x+a \log (x))}{\sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 20, normalized size = 0.71
method | result | size |
default | \(\frac {\sqrt {c \,x^{2}}\, \left (b x +a \ln \left (x \right )\right )}{x}\) | \(20\) |
risch | \(b \sqrt {c \,x^{2}}+\frac {a \ln \left (x \right ) \sqrt {c \,x^{2}}}{x}\) | \(25\) |
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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Fricas [A]
time = 1.09, size = 19, normalized size = 0.68 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x + a \log \left (x\right )\right )}}{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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Giac [A]
time = 1.87, size = 17, normalized size = 0.61 \begin {gather*} {\left (b x \mathrm {sgn}\left (x\right ) + a \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \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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