Optimal. Leaf size=48 \[ \frac {c^2 \sqrt {c x^2} \log (x)}{a x}-\frac {c^2 \sqrt {c x^2} \log (a+b x)}{a x} \]
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Rubi [A]
time = 0.01, antiderivative size = 48, 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 {c^2 \sqrt {c x^2} \log (x)}{a x}-\frac {c^2 \sqrt {c x^2} \log (a+b x)}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 29
Rule 31
Rule 36
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{5/2}}{x^6 (a+b x)} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \frac {1}{x (a+b x)} \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \frac {1}{x} \, dx}{a x}-\frac {\left (b c^2 \sqrt {c x^2}\right ) \int \frac {1}{a+b x} \, dx}{a x}\\ &=\frac {c^2 \sqrt {c x^2} \log (x)}{a x}-\frac {c^2 \sqrt {c x^2} \log (a+b x)}{a x}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 0.58 \begin {gather*} \frac {c^3 x (\log (x)-\log (a+b x))}{a \sqrt {c x^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 = 0.14, size = 26, normalized size = 0.54
method | result | size |
default | \(\frac {\left (c \,x^{2}\right )^{\frac {5}{2}} \left (\ln \left (x \right )-\ln \left (b x +a \right )\right )}{a \,x^{5}}\) | \(26\) |
risch | \(\frac {c^{2} \sqrt {c \,x^{2}}\, \ln \left (-x \right )}{x a}-\frac {c^{2} \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{a x}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 24, normalized size = 0.50 \begin {gather*} -\frac {c^{\frac {5}{2}} \log \left (b x + a\right )}{a} + \frac {c^{\frac {5}{2}} \log \left (x\right )}{a} \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 = 1.46 \begin {gather*} \left [\frac {\sqrt {c x^{2}} c^{2} \log \left (\frac {x}{b x + a}\right )}{a x}, \frac {2 \, \sqrt {-c} c^{2} \arctan \left (\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )} \sqrt {-c}}{a c x}\right )}{a}\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 {\left (c x^{2}\right )^{\frac {5}{2}}}{x^{6} \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 {{\left (c\,x^2\right )}^{5/2}}{x^6\,\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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