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 = 0.03, 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 = {2039}
\begin {gather*} \frac {2 \left (a x^2+b x^3\right )^{5/2}}{5 b x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
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 \begin {gather*} \frac {2 \left (x^2 (a+b x)\right )^{5/2}}{5 b x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 27, normalized size = 1.08
method | result | size |
gosper | \(\frac {2 \left (b x +a \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}}}{5 b \,x^{3}}\) | \(27\) |
default | \(\frac {2 \left (b x +a \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}}}{5 b \,x^{3}}\) | \(27\) |
risch | \(\frac {2 \sqrt {x^{2} \left (b x +a \right )}\, \left (b^{2} x^{2}+2 a b x +a^{2}\right )}{5 x b}\) | \(36\) |
trager | \(\frac {2 \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \sqrt {b \,x^{3}+a \,x^{2}}}{5 b x}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 28, normalized size = 1.12 \begin {gather*} \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {b x + a}}{5 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.05, size = 37, normalized size = 1.48 \begin {gather*} \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {b x^{3} + a x^{2}}}{5 \, b 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 (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 89 vs.
\(2 (21) = 42\).
time = 1.56, size = 89, normalized size = 3.56 \begin {gather*} -\frac {2 \, a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )}{5 \, b} + \frac {2 \, {\left (15 \, \sqrt {b x + a} a^{2} \mathrm {sgn}\left (x\right ) + 10 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} a \mathrm {sgn}\left (x\right ) + {\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}\left (x\right )\right )}}{15 \, b} \end {gather*}
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 \begin {gather*} \frac {2\,\sqrt {b\,x^3+a\,x^2}\,{\left (a+b\,x\right )}^2}{5\,b\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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