Optimal. Leaf size=25 \[ \frac {2 \left (a x^2+b x^3\right )^{5/2}}{5 b x^5} \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2014} \[ \frac {2 \left (a x^2+b x^3\right )^{5/2}}{5 b x^5} \]
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin {align*} \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^3} \, dx &=\frac {2 \left (a x^2+b x^3\right )^{5/2}}{5 b x^5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.92 \[ \frac {2 \left (x^2 (a+b x)\right )^{5/2}}{5 b x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 37, normalized size = 1.48 \[ \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {b x^{3} + a x^{2}}}{5 \, b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 89, normalized size = 3.56 \[ -\frac {2 \, a^{\frac {5}{2}} \mathrm {sgn}\relax (x)}{5 \, b} + \frac {2 \, {\left (15 \, \sqrt {b x + a} a^{2} \mathrm {sgn}\relax (x) + 10 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} a \mathrm {sgn}\relax (x) + {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} \mathrm {sgn}\relax (x)\right )}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 27, normalized size = 1.08 \[ \frac {2 \left (b x +a \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}}}{5 b \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 28, normalized size = 1.12 \[ \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {b x + a}}{5 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.62, size = 28, normalized size = 1.12 \[ \frac {2\,\sqrt {b\,x^3+a\,x^2}\,{\left (a+b\,x\right )}^2}{5\,b\,x} \]
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 \[ \int \frac {\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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