Optimal. Leaf size=22 \[ \frac {2 (a c+b c x)^{13/2}}{13 b c^6} \]
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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)^{13/2}}{13 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 \sqrt {a c+b c x} \, dx &=\frac {\int (a c+b c x)^{11/2} \, dx}{c^5}\\ &=\frac {2 (a c+b c x)^{13/2}}{13 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 \sqrt {c (a+b x)}}{13 b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 22, normalized size = 1.00 \begin {gather*} \frac {2 (a c+b c x)^{13/2}}{13 b c^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.16, size = 75, normalized size = 3.41 \begin {gather*} \frac {2 \, {\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )} \sqrt {b c x + a c}}{13 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.27, size = 495, normalized size = 22.50 \begin {gather*} \frac {2 \, {\left (3003 \, \sqrt {b c x + a c} a^{6} - \frac {6006 \, {\left (3 \, \sqrt {b c x + a c} a c - {\left (b c x + a c\right )}^{\frac {3}{2}}\right )} a^{5}}{c} + \frac {3003 \, {\left (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}}\right )} a^{4}}{c^{2}} - \frac {1716 \, {\left (35 \, \sqrt {b c x + a c} a^{3} c^{3} - 35 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{2} c^{2} + 21 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a c - 5 \, {\left (b c x + a c\right )}^{\frac {7}{2}}\right )} a^{3}}{c^{3}} + \frac {143 \, {\left (315 \, \sqrt {b c x + a c} a^{4} c^{4} - 420 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{3} c^{3} + 378 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{2} c^{2} - 180 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a c + 35 \, {\left (b c x + a c\right )}^{\frac {9}{2}}\right )} a^{2}}{c^{4}} - \frac {26 \, {\left (693 \, \sqrt {b c x + a c} a^{5} c^{5} - 1155 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{4} c^{4} + 1386 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{3} c^{3} - 990 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{2} c^{2} + 385 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a c - 63 \, {\left (b c x + a c\right )}^{\frac {11}{2}}\right )} a}{c^{5}} + \frac {3003 \, \sqrt {b c x + a c} a^{6} c^{6} - 6006 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{5} c^{5} + 9009 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{4} c^{4} - 8580 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{3} c^{3} + 5005 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a^{2} c^{2} - 1638 \, {\left (b c x + a c\right )}^{\frac {11}{2}} a c + 231 \, {\left (b c x + a c\right )}^{\frac {13}{2}}}{c^{6}}\right )}}{3003 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 1.05 \begin {gather*} \frac {2 \left (b x +a \right )^{6} \sqrt {b c x +a c}}{13 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 18, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (b c x + a c\right )}^{\frac {13}{2}}}{13 \, 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 = 0.77 \begin {gather*} \frac {2\,{\left (c\,\left (a+b\,x\right )\right )}^{13/2}}{13\,b\,c^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.06, size = 66, normalized size = 3.00 \begin {gather*} \begin {cases} \frac {2 b^{\frac {11}{2}} \sqrt {c} \left (\frac {a}{b} + x\right )^{\frac {13}{2}}}{13} & \text {for}\: \left |{\frac {a}{b} + x}\right | < 1 \\b^{\frac {11}{2}} \sqrt {c} {G_{2, 2}^{1, 1}\left (\begin {matrix} 1 & \frac {15}{2} \\\frac {13}{2} & 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )} + b^{\frac {11}{2}} \sqrt {c} {G_{2, 2}^{0, 2}\left (\begin {matrix} \frac {15}{2}, 1 & \\ & \frac {13}{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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