Optimal. Leaf size=39 \[ \frac {x^2}{b \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 \sqrt {c x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 39, 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, 45}
\begin {gather*} \frac {x^2}{b \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 \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^2}{\sqrt {c x^2} (a+b x)} \, dx &=\frac {x \int \frac {x}{a+b x} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {x^2}{b \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 0.69 \begin {gather*} \frac {x (b x-a \log (a+b x))}{b^2 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 27, normalized size = 0.69
method | result | size |
default | \(-\frac {x \left (a \ln \left (b x +a \right )-b x \right )}{\sqrt {c \,x^{2}}\, b^{2}}\) | \(27\) |
risch | \(\frac {x^{2}}{b \sqrt {c \,x^{2}}}-\frac {a x \ln \left (b x +a \right )}{b^{2} \sqrt {c \,x^{2}}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 64, normalized size = 1.64 \begin {gather*} -\frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2} \sqrt {c}} - \frac {a \log \left (b x\right )}{b^{2} \sqrt {c}} + \frac {\sqrt {c x^{2}}}{b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 30, normalized size = 0.77 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x - a \log \left (b x + a\right )\right )}}{b^{2} c 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 {x^{2}}{\sqrt {c x^{2}} \left (a + b x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 46, normalized size = 1.18 \begin {gather*} \frac {a \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{2} \sqrt {c}} + \frac {x}{b \sqrt {c} \mathrm {sgn}\left (x\right )} - \frac {a \log \left ({\left | b x + a \right |}\right )}{b^{2} \sqrt {c} \mathrm {sgn}\left (x\right )} \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 = -0.03 \begin {gather*} \int \frac {x^2}{\sqrt {c\,x^2}\,\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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