Optimal. Leaf size=22 \[ \frac {2 (a c+b c x)^{15/2}}{15 b c^6} \]
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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)^{15/2}}{15 b c^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rubi steps
\begin {align*} \int (a+b x)^5 (a c+b c x)^{3/2} \, dx &=\frac {\int (a c+b c x)^{13/2} \, dx}{c^5}\\ &=\frac {2 (a c+b c x)^{15/2}}{15 b c^6}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 25, normalized size = 1.14 \begin {gather*} \frac {2 (a+b x)^6 (c (a+b x))^{3/2}}{15 b} \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 = 2.28, size = 81, normalized size = 3.68 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 b^{\frac {13}{2}} c^{\frac {3}{2}} \left (\frac {a}{b}+x\right )^{\frac {15}{2}}}{15},\text {Abs}\left [\frac {a}{b}+x\right ]<1\right \}\right \},b^{\frac {13}{2}} c^{\frac {3}{2}} \text {meijerg}\left [\left \{\left \{1\right \},\left \{\frac {17}{2}\right \}\right \},\left \{\left \{\frac {15}{2}\right \},\left \{0\right \}\right \},\frac {a}{b}+x\right ]+b^{\frac {13}{2}} c^{\frac {3}{2}} \text {meijerg}\left [\left \{\left \{\frac {17}{2},1\right \},\left \{\right \}\right \},\left \{\left \{\right \},\left \{\frac {15}{2},0\right \}\right \},\frac {a}{b}+x\right ]\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.16, size = 19, normalized size = 0.86
method | result | size |
derivativedivides | \(\frac {2 \left (b c x +a c \right )^{\frac {15}{2}}}{15 b \,c^{6}}\) | \(19\) |
default | \(\frac {2 \left (b c x +a c \right )^{\frac {15}{2}}}{15 b \,c^{6}}\) | \(19\) |
gosper | \(\frac {2 \left (b x +a \right )^{6} \left (b c x +a c \right )^{\frac {3}{2}}}{15 b}\) | \(23\) |
trager | \(\frac {2 c \left (b^{7} x^{7}+7 a \,b^{6} x^{6}+21 a^{2} b^{5} x^{5}+35 a^{3} b^{4} x^{4}+35 a^{4} b^{3} x^{3}+21 a^{5} b^{2} x^{2}+7 a^{6} b x +a^{7}\right ) \sqrt {b c x +a c}}{15 b}\) | \(88\) |
risch | \(\frac {2 c^{2} \left (b^{7} x^{7}+7 a \,b^{6} x^{6}+21 a^{2} b^{5} x^{5}+35 a^{3} b^{4} x^{4}+35 a^{4} b^{3} x^{3}+21 a^{5} b^{2} x^{2}+7 a^{6} b x +a^{7}\right ) \left (b x +a \right )}{15 b \sqrt {c \left (b x +a \right )}}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 18, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (b c x + a c\right )}^{\frac {15}{2}}}{15 \, b c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs.
\(2 (18) = 36\).
time = 0.30, size = 95, normalized size = 4.32 \begin {gather*} \frac {2 \, {\left (b^{7} c x^{7} + 7 \, a b^{6} c x^{6} + 21 \, a^{2} b^{5} c x^{5} + 35 \, a^{3} b^{4} c x^{4} + 35 \, a^{4} b^{3} c x^{3} + 21 \, a^{5} b^{2} c x^{2} + 7 \, a^{6} b c x + a^{7} c\right )} \sqrt {b c x + a c}}{15 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.72, size = 66, normalized size = 3.00 \begin {gather*} \begin {cases} \frac {2 b^{\frac {13}{2}} c^{\frac {3}{2}} \left (\frac {a}{b} + x\right )^{\frac {15}{2}}}{15} & \text {for}\: \left |{\frac {a}{b} + x}\right | < 1 \\b^{\frac {13}{2}} c^{\frac {3}{2}} {G_{2, 2}^{1, 1}\left (\begin {matrix} 1 & \frac {17}{2} \\\frac {15}{2} & 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )} + b^{\frac {13}{2}} c^{\frac {3}{2}} {G_{2, 2}^{0, 2}\left (\begin {matrix} \frac {17}{2}, 1 & \\ & \frac {15}{2}, 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )} & \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. 637 vs.
\(2 (18) = 36\).
time = 0.01, size = 1080, normalized size = 49.09 \begin {gather*} \frac {\frac {2 b^{7} c \left (\frac {1}{15} \sqrt {a c+b c x} \left (a c+b c x\right )^{7}-\frac {7}{13} \sqrt {a c+b c x} \left (a c+b c x\right )^{6} a c+\frac {21}{11} \sqrt {a c+b c x} \left (a c+b c x\right )^{5} a^{2} c^{2}-\frac {35}{9} \sqrt {a c+b c x} \left (a c+b c x\right )^{4} a^{3} c^{3}+5 \sqrt {a c+b c x} \left (a c+b c x\right )^{3} a^{4} c^{4}-\frac {21}{5} \sqrt {a c+b c x} \left (a c+b c x\right )^{2} a^{5} c^{5}+\frac {7}{3} \sqrt {a c+b c x} \left (a c+b c x\right ) a^{6} c^{6}-\sqrt {a c+b c x} a^{7} c^{7}\right )}{c^{7} b^{7}}+\frac {14 a b^{6} c \left (\frac {1}{13} \sqrt {a c+b c x} \left (a c+b c x\right )^{6}-\frac {6}{11} \sqrt {a c+b c x} \left (a c+b c x\right )^{5} a c+\frac {5}{3} \sqrt {a c+b c x} \left (a c+b c x\right )^{4} a^{2} c^{2}-\frac {20}{7} \sqrt {a c+b c x} \left (a c+b c x\right )^{3} a^{3} c^{3}+3 \sqrt {a c+b c x} \left (a c+b c x\right )^{2} a^{4} c^{4}-2 \sqrt {a c+b c x} \left (a c+b c x\right ) a^{5} c^{5}+\sqrt {a c+b c x} a^{6} c^{6}\right )}{c^{6} b^{6}}+\frac {42 a^{2} b^{5} c \left (\frac {1}{11} \sqrt {a c+b c x} \left (a c+b c x\right )^{5}-\frac {5}{9} \sqrt {a c+b c x} \left (a c+b c x\right )^{4} a c+\frac {10}{7} \sqrt {a c+b c x} \left (a c+b c x\right )^{3} a^{2} c^{2}-2 \sqrt {a c+b c x} \left (a c+b c x\right )^{2} a^{3} c^{3}+\frac {5}{3} \sqrt {a c+b c x} \left (a c+b c x\right ) a^{4} c^{4}-\sqrt {a c+b c x} a^{5} c^{5}\right )}{c^{5} b^{5}}+\frac {70 a^{3} b^{4} c \left (\frac {1}{9} \sqrt {a c+b c x} \left (a c+b c x\right )^{4}-\frac {4}{7} \sqrt {a c+b c x} \left (a c+b c x\right )^{3} a c+\frac {6}{5} \sqrt {a c+b c x} \left (a c+b c x\right )^{2} a^{2} c^{2}-\frac {4}{3} \sqrt {a c+b c x} \left (a c+b c x\right ) a^{3} c^{3}+\sqrt {a c+b c x} a^{4} c^{4}\right )}{c^{4} b^{4}}+\frac {70 a^{4} b^{3} c \left (\frac {1}{7} \sqrt {a c+b c x} \left (a c+b c x\right )^{3}-\frac {3}{5} \sqrt {a c+b c x} \left (a c+b c x\right )^{2} a c+\sqrt {a c+b c x} \left (a c+b c x\right ) a^{2} c^{2}-\sqrt {a c+b c x} a^{3} c^{3}\right )}{c^{3} b^{3}}+\frac {42 a^{5} b^{2} c \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^{7} c \sqrt {a c+b c x}+\frac {14 a^{6} b c \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 )}{b c}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 17, normalized size = 0.77 \begin {gather*} \frac {2\,{\left (c\,\left (a+b\,x\right )\right )}^{15/2}}{15\,b\,c^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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