Optimal. Leaf size=48 \[ \frac {2 x}{3 a \left (a x+b x^2\right )^{3/2}}-\frac {8 (a+2 b x)}{3 a^3 \sqrt {a x+b x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {638, 613} \[ \frac {2 x}{3 a \left (a x+b x^2\right )^{3/2}}-\frac {8 (a+2 b x)}{3 a^3 \sqrt {a x+b x^2}} \]
Antiderivative was successfully verified.
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Rule 613
Rule 638
Rubi steps
\begin {align*} \int \frac {x}{\left (a x+b x^2\right )^{5/2}} \, dx &=\frac {2 x}{3 a \left (a x+b x^2\right )^{3/2}}+\frac {4 \int \frac {1}{\left (a x+b x^2\right )^{3/2}} \, dx}{3 a}\\ &=\frac {2 x}{3 a \left (a x+b x^2\right )^{3/2}}-\frac {8 (a+2 b x)}{3 a^3 \sqrt {a x+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 0.79 \[ -\frac {2 x \left (3 a^2+12 a b x+8 b^2 x^2\right )}{3 a^3 (x (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 59, normalized size = 1.23 \[ -\frac {2 \, {\left (8 \, b^{2} x^{2} + 12 \, a b x + 3 \, a^{2}\right )} \sqrt {b x^{2} + a x}}{3 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 44, normalized size = 0.92 \[ -\frac {2 \left (b x +a \right ) \left (8 b^{2} x^{2}+12 a b x +3 a^{2}\right ) x^{2}}{3 \left (b \,x^{2}+a x \right )^{\frac {5}{2}} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 52, normalized size = 1.08 \[ \frac {2 \, x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a} - \frac {16 \, b x}{3 \, \sqrt {b x^{2} + a x} a^{3}} - \frac {8}{3 \, \sqrt {b x^{2} + a x} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 45, normalized size = 0.94 \[ -\frac {2\,\sqrt {b\,x^2+a\,x}\,\left (3\,a^2+12\,a\,b\,x+8\,b^2\,x^2\right )}{3\,a^3\,x\,{\left (a+b\,x\right )}^2} \]
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 \[ \int \frac {x}{\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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