Integrand size = 20, antiderivative size = 22 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{3/2}} \, dx=\frac {2 (a c+b c x)^{9/2}}{9 b c^6} \]
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Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 32} \[ \int \frac {(a+b x)^5}{(a c+b c x)^{3/2}} \, dx=\frac {2 (a c+b c x)^{9/2}}{9 b c^6} \]
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Rule 21
Rule 32
Rubi steps \begin{align*} \text {integral}& = \frac {\int (a c+b c x)^{7/2} \, dx}{c^5} \\ & = \frac {2 (a c+b c x)^{9/2}}{9 b c^6} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{3/2}} \, dx=\frac {2 (a+b x)^6}{9 b (c (a+b x))^{3/2}} \]
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Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {2 \left (b c x +a c \right )^{\frac {9}{2}}}{9 b \,c^{6}}\) | \(19\) |
default | \(\frac {2 \left (b c x +a c \right )^{\frac {9}{2}}}{9 b \,c^{6}}\) | \(19\) |
gosper | \(\frac {2 \left (b x +a \right )^{6}}{9 b \left (b c x +a c \right )^{\frac {3}{2}}}\) | \(23\) |
pseudoelliptic | \(\frac {2 \left (b x +a \right )^{4} \sqrt {c \left (b x +a \right )}}{9 c^{2} b}\) | \(25\) |
trager | \(\frac {2 \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right ) \sqrt {b c x +a c}}{9 c^{2} b}\) | \(57\) |
risch | \(\frac {2 \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right ) \left (b x +a \right )}{9 c b \sqrt {c \left (b x +a \right )}}\) | \(61\) |
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (18) = 36\).
Time = 0.23 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{3/2}} \, dx=\frac {2 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt {b c x + a c}}{9 \, b c^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (19) = 38\).
Time = 0.77 (sec) , antiderivative size = 88, normalized size of antiderivative = 4.00 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{3/2}} \, dx=\begin {cases} 0 & \text {for}\: \frac {1}{\left |{\frac {a}{b} + x}\right |} < 1 \wedge \left |{\frac {a}{b} + x}\right | < 1 \\\frac {2 b^{\frac {7}{2}} \left (\frac {a}{b} + x\right )^{\frac {9}{2}}}{9 c^{\frac {3}{2}}} & \text {for}\: \frac {1}{\left |{\frac {a}{b} + x}\right |} < 1 \vee \left |{\frac {a}{b} + x}\right | < 1 \\\frac {b^{\frac {7}{2}} {G_{2, 2}^{1, 1}\left (\begin {matrix} 1 & \frac {11}{2} \\\frac {9}{2} & 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )}}{c^{\frac {3}{2}}} + \frac {b^{\frac {7}{2}} {G_{2, 2}^{0, 2}\left (\begin {matrix} \frac {11}{2}, 1 & \\ & \frac {9}{2}, 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )}}{c^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{3/2}} \, dx=\frac {2 \, {\left (b c x + a c\right )}^{\frac {9}{2}}}{9 \, b c^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (18) = 36\).
Time = 0.33 (sec) , antiderivative size = 266, normalized size of antiderivative = 12.09 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{3/2}} \, dx=\frac {2 \, {\left (315 \, \sqrt {b c x + a c} a^{4} - \frac {420 \, {\left (3 \, \sqrt {b c x + a c} a c - {\left (b c x + a c\right )}^{\frac {3}{2}}\right )} a^{3}}{c} + \frac {126 \, {\left (15 \, \sqrt {b c x + a c} a^{2} c^{2} - 10 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a c + 3 \, {\left (b c x + a c\right )}^{\frac {5}{2}}\right )} a^{2}}{c^{2}} - \frac {36 \, {\left (35 \, \sqrt {b c x + a c} a^{3} c^{3} - 35 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{2} c^{2} + 21 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a c - 5 \, {\left (b c x + a c\right )}^{\frac {7}{2}}\right )} a}{c^{3}} + \frac {315 \, \sqrt {b c x + a c} a^{4} c^{4} - 420 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{3} c^{3} + 378 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{2} c^{2} - 180 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a c + 35 \, {\left (b c x + a c\right )}^{\frac {9}{2}}}{c^{4}}\right )}}{315 \, b c^{2}} \]
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Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{3/2}} \, dx=\frac {2\,{\left (c\,\left (a+b\,x\right )\right )}^{9/2}}{9\,b\,c^6} \]
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