Optimal. Leaf size=20 \[ \frac {2 \sqrt {a c+b c x}}{b c^6} \]
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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 32}
\begin {gather*} \frac {2 \sqrt {a c+b c x}}{b c^6} \end {gather*}
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*}
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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.20 \begin {gather*} \frac {2 (a+b x)}{b c^5 \sqrt {c (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 19, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {2 \sqrt {b c x +a c}}{b \,c^{6}}\) | \(19\) |
default | \(\frac {2 \sqrt {b c x +a c}}{b \,c^{6}}\) | \(19\) |
trager | \(\frac {2 \sqrt {b c x +a c}}{b \,c^{6}}\) | \(19\) |
gosper | \(\frac {2 \left (b x +a \right )^{6}}{b \left (b c x +a c \right )^{\frac {11}{2}}}\) | \(23\) |
risch | \(\frac {2 b x +2 a}{c^{5} b \sqrt {c \left (b x +a \right )}}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 18, normalized size = 0.90 \begin {gather*} \frac {2 \, \sqrt {b c x + a c}}{b c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.82, size = 18, normalized size = 0.90 \begin {gather*} \frac {2 \, \sqrt {b c x + a c}}{b c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.51, size = 29, normalized size = 1.45 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.18, size = 18, normalized size = 0.90 \begin {gather*} \frac {2 \, \sqrt {b c x + a c}}{b c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 17, normalized size = 0.85 \begin {gather*} \frac {2\,\sqrt {c\,\left (a+b\,x\right )}}{b\,c^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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