Optimal. Leaf size=40 \[ \frac {c \sqrt {c x^2}}{b}-\frac {a c \sqrt {c x^2} \log (a+b x)}{b^2 x} \]
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Rubi [A]
time = 0.01, antiderivative size = 40, 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 {c \sqrt {c x^2}}{b}-\frac {a c \sqrt {c x^2} \log (a+b x)}{b^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int \frac {x}{a+b x} \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx}{x}\\ &=\frac {c \sqrt {c x^2}}{b}-\frac {a c \sqrt {c x^2} \log (a+b x)}{b^2 x}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 30, normalized size = 0.75 \begin {gather*} \frac {c^2 x (b x-a \log (a+b x))}{b^2 \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 in comparison} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.13, size = 29, normalized size = 0.72
method | result | size |
default | \(-\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (a \ln \left (b x +a \right )-b x \right )}{b^{2} x^{3}}\) | \(29\) |
risch | \(\frac {c \sqrt {c \,x^{2}}}{b}-\frac {a c \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b^{2} x}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs.
\(2 (36) = 72\).
time = 0.32, size = 75, normalized size = 1.88 \begin {gather*} -\frac {\left (-1\right )^{\frac {2 \, c x}{b}} a c^{\frac {3}{2}} \log \left (\frac {2 \, c x}{b}\right )}{b^{2}} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a c^{\frac {3}{2}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2}} + \frac {\sqrt {c x^{2}} c}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 29, normalized size = 0.72 \begin {gather*} \frac {{\left (b c x - a c \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{b^{2} 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 {\left (c x^{2}\right )^{\frac {3}{2}}}{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 = 0.00, size = 41, normalized size = 1.02 \begin {gather*} \sqrt {c} c \left (\frac {x \mathrm {sign}\left (x\right )}{b}-\frac {a \mathrm {sign}\left (x\right ) \ln \left |b x+a\right |}{b^{2}}+\frac {a \ln \left |a\right |\cdot \mathrm {sign}\left (x\right )}{b^{2}}\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 = -0.02 \begin {gather*} \int \frac {{\left (c\,x^2\right )}^{3/2}}{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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