Optimal. Leaf size=44 \[ \frac {x \log (x)}{a c \sqrt {c x^2}}-\frac {x \log (a+b x)}{a c \sqrt {c x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 36, 29, 31}
\begin {gather*} \frac {x \log (x)}{a c \sqrt {c x^2}}-\frac {x \log (a+b x)}{a c \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 29
Rule 31
Rule 36
Rubi steps
\begin {align*} \int \frac {x^2}{\left (c x^2\right )^{3/2} (a+b x)} \, dx &=\frac {x \int \frac {1}{x (a+b x)} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \frac {1}{x} \, dx}{a c \sqrt {c x^2}}-\frac {(b x) \int \frac {1}{a+b x} \, dx}{a c \sqrt {c x^2}}\\ &=\frac {x \log (x)}{a c \sqrt {c x^2}}-\frac {x \log (a+b x)}{a c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 27, normalized size = 0.61 \begin {gather*} \frac {x^3 (\log (x)-\log (a+b x))}{a \left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.13, size = 26, normalized size = 0.59
method | result | size |
default | \(\frac {x^{3} \left (\ln \left (x \right )-\ln \left (b x +a \right )\right )}{\left (c \,x^{2}\right )^{\frac {3}{2}} a}\) | \(26\) |
risch | \(\frac {x \ln \left (-x \right )}{c \sqrt {c \,x^{2}}\, a}-\frac {x \ln \left (b x +a \right )}{a c \sqrt {c \,x^{2}}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 35, normalized size = 0.80 \begin {gather*} -\frac {\left (-1\right )^{\frac {2 \, a c x}{b}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{a c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 70, normalized size = 1.59 \begin {gather*} \left [\frac {\sqrt {c x^{2}} \log \left (\frac {x}{b x + a}\right )}{a c^{2} x}, \frac {2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )} \sqrt {-c}}{a c x}\right )}{a c^{2}}\right ] \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}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Limit: Max order reached or unable to make series expansion Error: Bad Argument Value} \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 = -0.02 \begin {gather*} \int \frac {x^2}{{\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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