Optimal. Leaf size=22 \[ \frac {2 (a c+b c x)^{7/2}}{7 b c^6} \]
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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)^{7/2}}{7 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)^{5/2}} \, dx &=\frac {\int (a c+b c x)^{5/2} \, dx}{c^5}\\ &=\frac {2 (a c+b c x)^{7/2}}{7 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}{7 b (c (a+b x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 22, normalized size = 1.00 \begin {gather*} \frac {2 (a c+b c x)^{7/2}}{7 b c^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.44, size = 45, normalized size = 2.05 \begin {gather*} \frac {2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt {b c x + a c}}{7 \, b c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.96, size = 178, normalized size = 8.09 \begin {gather*} \frac {2 \, {\left (35 \, \sqrt {b c x + a c} a^{3} - \frac {35 \, {\left (3 \, \sqrt {b c x + a c} a c - {\left (b c x + a c\right )}^{\frac {3}{2}}\right )} a^{2}}{c} + \frac {7 \, {\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}{c^{2}} - \frac {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}}}{c^{3}}\right )}}{35 \, b c^{3}} \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}}{7 \left (b c x +a c \right )^{\frac {5}{2}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 18, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (b c x + a c\right )}^{\frac {7}{2}}}{7 \, 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 )}^{7/2}}{7\,b\,c^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.62, size = 73, normalized size = 3.32 \begin {gather*} \begin {cases} \frac {2 b^{\frac {5}{2}} \left (\frac {a}{b} + x\right )^{\frac {7}{2}}}{7 c^{\frac {5}{2}}} & \text {for}\: \left |{\frac {a}{b} + x}\right | > 1 \vee \left |{\frac {a}{b} + x}\right | < 1 \\\frac {b^{\frac {5}{2}} {G_{2, 2}^{1, 1}\left (\begin {matrix} 1 & \frac {9}{2} \\\frac {7}{2} & 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )}}{c^{\frac {5}{2}}} + \frac {b^{\frac {5}{2}} {G_{2, 2}^{0, 2}\left (\begin {matrix} \frac {9}{2}, 1 & \\ & \frac {7}{2}, 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )}}{c^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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