Optimal. Leaf size=22 \[ \frac {2 (a c+b c x)^{5/2}}{5 b c^6} \]
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Rubi [A]
time = 0.00, antiderivative size = 22, 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 (a c+b c x)^{5/2}}{5 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)^{7/2}} \, dx &=\frac {\int (a c+b c x)^{3/2} \, dx}{c^5}\\ &=\frac {2 (a c+b c x)^{5/2}}{5 b c^6}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.14 \begin {gather*} \frac {2 (a+b x)^6}{5 b (c (a+b x))^{7/2}} \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 = 2.49, size = 47, normalized size = 2.14 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (a^2+b x \left (2 a+b x\right )\right ) \sqrt {c \left (a+b x\right )}}{5 b c^4},b\text {!=}0\right \}\right \},\frac {a^5 x}{\left (a c\right )^{\frac {7}{2}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.18, size = 19, normalized size = 0.86
method | result | size |
derivativedivides | \(\frac {2 \left (b c x +a c \right )^{\frac {5}{2}}}{5 b \,c^{6}}\) | \(19\) |
default | \(\frac {2 \left (b c x +a c \right )^{\frac {5}{2}}}{5 b \,c^{6}}\) | \(19\) |
gosper | \(\frac {2 \left (b x +a \right )^{6}}{5 b \left (b c x +a c \right )^{\frac {7}{2}}}\) | \(23\) |
trager | \(\frac {2 \left (x^{2} b^{2}+2 a b x +a^{2}\right ) \sqrt {b c x +a c}}{5 c^{4} b}\) | \(35\) |
risch | \(\frac {2 \left (x^{2} b^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )}{5 c^{3} b \sqrt {c \left (b x +a \right )}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 18, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (b c x + a c\right )}^{\frac {5}{2}}}{5 \, 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 = 34, normalized size = 1.55 \begin {gather*} \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {b c x + a c}}{5 \, b c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.89, size = 80, normalized size = 3.64 \begin {gather*} \begin {cases} \frac {2 a^{2} \sqrt {a c + b c x}}{5 b c^{4}} + \frac {4 a x \sqrt {a c + b c x}}{5 c^{4}} + \frac {2 b x^{2} \sqrt {a c + b c x}}{5 c^{4}} & \text {for}\: b \neq 0 \\\frac {a^{5} x}{\left (a c\right )^{\frac {7}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 106 vs.
\(2 (18) = 36\).
time = 0.00, size = 155, normalized size = 7.05 \begin {gather*} \frac {\frac {2 b^{2} \left (\frac {1}{5} \sqrt {a c+b c x} \left (a c+b c x\right )^{2}-\frac {2}{3} \sqrt {a c+b c x} \left (a c+b c x\right ) a c+\sqrt {a c+b c x} a^{2} c^{2}\right )}{c^{2} b^{2}}+2 a^{2} \sqrt {a c+b c x}+\frac {4 a \left (\frac {1}{3} \sqrt {a c+b c x} \left (a c+b c x\right )-a c \sqrt {a c+b c x}\right )}{c}}{c^{4} b} \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 = 0.77 \begin {gather*} \frac {2\,{\left (c\,\left (a+b\,x\right )\right )}^{5/2}}{5\,b\,c^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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