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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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 = 0.29, 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 = 1.07, 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs.
\(2 (19) = 38\).
time = 0.68, 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 = 0.92, size = 106, normalized size = 4.82 \begin {gather*} \frac {2 \, {\left (15 \, \sqrt {b c x + a c} a^{2} - \frac {10 \, {\left (3 \, \sqrt {b c x + a c} a c - {\left (b c x + a c\right )}^{\frac {3}{2}}\right )} a}{c} + \frac {15 \, \sqrt {b c x + a c} a^{2} c^{2} - 10 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a c + 3 \, {\left (b c x + a c\right )}^{\frac {5}{2}}}{c^{2}}\right )}}{15 \, b c^{4}} \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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