Optimal. Leaf size=20 \[ \frac{2 \sqrt{a c+b c x}}{b c^6} \]
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Rubi [A] time = 0.0041583, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {21, 32} \[ \frac{2 \sqrt{a c+b c x}}{b c^6} \]
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rubi steps
\begin{align*} \int \frac{(a+b x)^5}{(a c+b c x)^{11/2}} \, dx &=\frac{\int \frac{1}{\sqrt{a c+b c x}} \, dx}{c^5}\\ &=\frac{2 \sqrt{a c+b c x}}{b c^6}\\ \end{align*}
Mathematica [A] time = 0.0071777, size = 24, normalized size = 1.2 \[ \frac{2 (a+b x)}{b c^5 \sqrt{c (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 23, normalized size = 1.2 \begin{align*} 2\,{\frac{ \left ( bx+a \right ) ^{6}}{b \left ( bcx+ac \right ) ^{11/2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963576, size = 24, normalized size = 1.2 \begin{align*} \frac{2 \, \sqrt{b c x + a c}}{b c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02837, size = 39, normalized size = 1.95 \begin{align*} \frac{2 \, \sqrt{b c x + a c}}{b c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.4151, size = 29, normalized size = 1.45 \begin{align*} \begin{cases} \frac{2 \sqrt{a c + b c x}}{b c^{6}} & \text{for}\: b \neq 0 \\\frac{a^{5} x}{\left (a c\right )^{\frac{11}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05408, size = 24, normalized size = 1.2 \begin{align*} \frac{2 \, \sqrt{b c x + a c}}{b c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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