Optimal. Leaf size=16 \[ \frac {x}{a \left (a+b \sqrt {x}\right )^2} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {190, 37} \[ \frac {x}{a \left (a+b \sqrt {x}\right )^2} \]
Antiderivative was successfully verified.
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Rule 37
Rule 190
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sqrt {x}\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {x}{(a+b x)^3} \, dx,x,\sqrt {x}\right )\\ &=\frac {x}{a \left (a+b \sqrt {x}\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 16, normalized size = 1.00 \[ \frac {x}{a \left (a+b \sqrt {x}\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 47, normalized size = 2.94 \[ -\frac {2 \, b^{3} x^{\frac {3}{2}} - 3 \, a b^{2} x + a^{3}}{b^{6} x^{2} - 2 \, a^{2} b^{4} x + a^{4} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 22, normalized size = 1.38 \[ -\frac {2 \, b \sqrt {x} + a}{{\left (b \sqrt {x} + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 131, normalized size = 8.19 \[ -3 \left (-\frac {a^{2}}{2 \left (b^{2} x -a^{2}\right )^{2} b^{4}}-\frac {1}{\left (b^{2} x -a^{2}\right ) b^{4}}\right ) a \,b^{2}+\frac {a^{3}}{2 \left (b^{2} x -a^{2}\right )^{2} b^{2}}+\frac {a}{2 \left (b \sqrt {x}+a \right )^{2} b^{2}}-\frac {a}{2 \left (b \sqrt {x}-a \right )^{2} b^{2}}-\frac {1}{\left (b \sqrt {x}+a \right ) b^{2}}-\frac {1}{\left (b \sqrt {x}-a \right ) b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.76, size = 29, normalized size = 1.81 \[ -\frac {2}{{\left (b \sqrt {x} + a\right )} b^{2}} + \frac {a}{{\left (b \sqrt {x} + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 34, normalized size = 2.12 \[ -\frac {\frac {a}{b^2}+\frac {2\,\sqrt {x}}{b}}{b^2\,x+a^2+2\,a\,b\,\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.15, size = 63, normalized size = 3.94 \[ \begin {cases} - \frac {a}{a^{2} b^{2} + 2 a b^{3} \sqrt {x} + b^{4} x} - \frac {2 b \sqrt {x}}{a^{2} b^{2} + 2 a b^{3} \sqrt {x} + b^{4} x} & \text {for}\: b \neq 0 \\\frac {x}{a^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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