Optimal. Leaf size=33 \[ -\frac {a}{c \sqrt {c x^2}}+\frac {b x \log (x)}{c \sqrt {c x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {15, 45}
\begin {gather*} \frac {b x \log (x)}{c \sqrt {c x^2}}-\frac {a}{c \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {x (a+b x)}{\left (c x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {a+b x}{x^2} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a}{x^2}+\frac {b}{x}\right ) \, dx}{c \sqrt {c x^2}}\\ &=-\frac {a}{c \sqrt {c x^2}}+\frac {b x \log (x)}{c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 22, normalized size = 0.67 \begin {gather*} \frac {x^2 (-a+b x \log (x))}{\left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 21, normalized size = 0.64
method | result | size |
default | \(\frac {x^{2} \left (b x \ln \left (x \right )-a \right )}{\left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(21\) |
risch | \(-\frac {a}{c \sqrt {c \,x^{2}}}+\frac {b x \ln \left (x \right )}{c \sqrt {c \,x^{2}}}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 21, normalized size = 0.64 \begin {gather*} \frac {b \log \left (x\right )}{c^{\frac {3}{2}}} - \frac {a}{\sqrt {c x^{2}} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.91, size = 23, normalized size = 0.70 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x \log \left (x\right ) - a\right )}}{c^{2} x^{2}} \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 {x \left (a + b x\right )}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.36, size = 30, normalized size = 0.91 \begin {gather*} \frac {\frac {b \log \left ({\left | x \right |}\right )}{\sqrt {c} \mathrm {sgn}\left (x\right )} - \frac {a}{\sqrt {c} x \mathrm {sgn}\left (x\right )}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 28, normalized size = 0.85 \begin {gather*} -\frac {a+b\,x-b\,\ln \left (x+\left |x\right |\right )\,\sqrt {x^2}}{c^{3/2}\,\sqrt {x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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