Optimal. Leaf size=51 \[ \frac {2 (a+2 b x)}{3 a^2 b \sqrt {a x+b x^2}}-\frac {2 x}{3 b \left (a x+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {652, 613} \[ \frac {2 (a+2 b x)}{3 a^2 b \sqrt {a x+b x^2}}-\frac {2 x}{3 b \left (a x+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 613
Rule 652
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a x+b x^2\right )^{5/2}} \, dx &=-\frac {2 x}{3 b \left (a x+b x^2\right )^{3/2}}-\frac {\int \frac {1}{\left (a x+b x^2\right )^{3/2}} \, dx}{3 b}\\ &=-\frac {2 x}{3 b \left (a x+b x^2\right )^{3/2}}+\frac {2 (a+2 b x)}{3 a^2 b \sqrt {a x+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.57 \[ \frac {2 x^2 (3 a+2 b x)}{3 a^2 (x (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 44, normalized size = 0.86 \[ \frac {2 \, \sqrt {b x^{2} + a x} {\left (2 \, b x + 3 \, a\right )}}{3 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 61, normalized size = 1.20 \[ \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} b + 2 \, a \sqrt {b}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a\right )}^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 33, normalized size = 0.65 \[ \frac {2 \left (b x +a \right ) \left (2 b x +3 a \right ) x^{3}}{3 \left (b \,x^{2}+a x \right )^{\frac {5}{2}} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 54, normalized size = 1.06 \[ \frac {4 \, x}{3 \, \sqrt {b x^{2} + a x} a^{2}} - \frac {2 \, x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {2}{3 \, \sqrt {b x^{2} + a x} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 31, normalized size = 0.61 \[ \frac {2\,\sqrt {b\,x^2+a\,x}\,\left (3\,a+2\,b\,x\right )}{3\,a^2\,{\left (a+b\,x\right )}^2} \]
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 \[ \int \frac {x^{2}}{\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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