Optimal. Leaf size=20 \[ -\frac {2}{b c^6 \sqrt {a c+b c x}} \]
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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}{b c^6 \sqrt {a c+b c x}} \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)^{13/2}} \, dx &=\frac {\int \frac {1}{(a c+b c x)^{3/2}} \, dx}{c^5}\\ &=-\frac {2}{b c^6 \sqrt {a c+b c x}}\\ \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 (c (a+b x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 3.64, size = 40, normalized size = 2.00 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {-2 \sqrt {c \left (a+b x\right )}}{b c^7 \left (a+b x\right )},b\text {!=}0\right \}\right \},\frac {a^5 x}{\left (a c\right )^{\frac {13}{2}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.17, size = 19, normalized size = 0.95
method | result | size |
derivativedivides | \(-\frac {2}{b \,c^{6} \sqrt {b c x +a c}}\) | \(19\) |
default | \(-\frac {2}{b \,c^{6} \sqrt {b c x +a c}}\) | \(19\) |
gosper | \(-\frac {2 \left (b x +a \right )^{6}}{b \left (b c x +a c \right )^{\frac {13}{2}}}\) | \(23\) |
trager | \(-\frac {2 \sqrt {b c x +a c}}{c^{7} b \left (b x +a \right )}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, 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 = 0.29, size = 29, normalized size = 1.45 \begin {gather*} -\frac {2 \, \sqrt {b c x + a c}}{b^{2} c^{7} x + a b c^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.00, size = 41, normalized size = 2.05 \begin {gather*} \begin {cases} - \frac {2 \sqrt {a c + b c x}}{a b c^{7} + b^{2} c^{7} x} & \text {for}\: b \neq 0 \\\frac {a^{5} x}{\left (a c\right )^{\frac {13}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 22, normalized size = 1.10 \begin {gather*} -\frac {2}{b c^{6} \sqrt {a c+b c x}} \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}{b\,c^6\,\sqrt {c\,\left (a+b\,x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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