Integrand size = 13, antiderivative size = 72 \[ \int x^3 (a+b x)^{5/2} \, dx=-\frac {2 a^3 (a+b x)^{7/2}}{7 b^4}+\frac {2 a^2 (a+b x)^{9/2}}{3 b^4}-\frac {6 a (a+b x)^{11/2}}{11 b^4}+\frac {2 (a+b x)^{13/2}}{13 b^4} \]
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Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int x^3 (a+b x)^{5/2} \, dx=-\frac {2 a^3 (a+b x)^{7/2}}{7 b^4}+\frac {2 a^2 (a+b x)^{9/2}}{3 b^4}+\frac {2 (a+b x)^{13/2}}{13 b^4}-\frac {6 a (a+b x)^{11/2}}{11 b^4} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^3 (a+b x)^{5/2}}{b^3}+\frac {3 a^2 (a+b x)^{7/2}}{b^3}-\frac {3 a (a+b x)^{9/2}}{b^3}+\frac {(a+b x)^{11/2}}{b^3}\right ) \, dx \\ & = -\frac {2 a^3 (a+b x)^{7/2}}{7 b^4}+\frac {2 a^2 (a+b x)^{9/2}}{3 b^4}-\frac {6 a (a+b x)^{11/2}}{11 b^4}+\frac {2 (a+b x)^{13/2}}{13 b^4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.64 \[ \int x^3 (a+b x)^{5/2} \, dx=\frac {2 (a+b x)^{7/2} \left (-16 a^3+56 a^2 b x-126 a b^2 x^2+231 b^3 x^3\right )}{3003 b^4} \]
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Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-231 b^{3} x^{3}+126 a \,b^{2} x^{2}-56 a^{2} b x +16 a^{3}\right )}{3003 b^{4}}\) | \(43\) |
pseudoelliptic | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-231 b^{3} x^{3}+126 a \,b^{2} x^{2}-56 a^{2} b x +16 a^{3}\right )}{3003 b^{4}}\) | \(43\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {13}{2}}}{13}-\frac {6 a \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 a^{2} \left (b x +a \right )^{\frac {9}{2}}}{3}-\frac {2 a^{3} \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{4}}\) | \(50\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {13}{2}}}{13}-\frac {6 a \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 a^{2} \left (b x +a \right )^{\frac {9}{2}}}{3}-\frac {2 a^{3} \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{4}}\) | \(50\) |
trager | \(-\frac {2 \left (-231 b^{6} x^{6}-567 a \,x^{5} b^{5}-371 a^{2} x^{4} b^{4}-5 a^{3} x^{3} b^{3}+6 a^{4} x^{2} b^{2}-8 a^{5} x b +16 a^{6}\right ) \sqrt {b x +a}}{3003 b^{4}}\) | \(76\) |
risch | \(-\frac {2 \left (-231 b^{6} x^{6}-567 a \,x^{5} b^{5}-371 a^{2} x^{4} b^{4}-5 a^{3} x^{3} b^{3}+6 a^{4} x^{2} b^{2}-8 a^{5} x b +16 a^{6}\right ) \sqrt {b x +a}}{3003 b^{4}}\) | \(76\) |
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none
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.04 \[ \int x^3 (a+b x)^{5/2} \, dx=\frac {2 \, {\left (231 \, b^{6} x^{6} + 567 \, a b^{5} x^{5} + 371 \, a^{2} b^{4} x^{4} + 5 \, a^{3} b^{3} x^{3} - 6 \, a^{4} b^{2} x^{2} + 8 \, a^{5} b x - 16 \, a^{6}\right )} \sqrt {b x + a}}{3003 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (68) = 136\).
Time = 0.68 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.03 \[ \int x^3 (a+b x)^{5/2} \, dx=\begin {cases} - \frac {32 a^{6} \sqrt {a + b x}}{3003 b^{4}} + \frac {16 a^{5} x \sqrt {a + b x}}{3003 b^{3}} - \frac {4 a^{4} x^{2} \sqrt {a + b x}}{1001 b^{2}} + \frac {10 a^{3} x^{3} \sqrt {a + b x}}{3003 b} + \frac {106 a^{2} x^{4} \sqrt {a + b x}}{429} + \frac {54 a b x^{5} \sqrt {a + b x}}{143} + \frac {2 b^{2} x^{6} \sqrt {a + b x}}{13} & \text {for}\: b \neq 0 \\\frac {a^{\frac {5}{2}} x^{4}}{4} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.78 \[ \int x^3 (a+b x)^{5/2} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {13}{2}}}{13 \, b^{4}} - \frac {6 \, {\left (b x + a\right )}^{\frac {11}{2}} a}{11 \, b^{4}} + \frac {2 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2}}{3 \, b^{4}} - \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3}}{7 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (56) = 112\).
Time = 0.29 (sec) , antiderivative size = 281, normalized size of antiderivative = 3.90 \[ \int x^3 (a+b x)^{5/2} \, dx=\frac {2 \, {\left (\frac {429 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a^{3}}{b^{3}} + \frac {143 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} a^{2}}{b^{3}} + \frac {65 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} a}{b^{3}} + \frac {5 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )}}{b^{3}}\right )}}{15015 \, b} \]
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Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.78 \[ \int x^3 (a+b x)^{5/2} \, dx=\frac {2\,{\left (a+b\,x\right )}^{13/2}}{13\,b^4}-\frac {2\,a^3\,{\left (a+b\,x\right )}^{7/2}}{7\,b^4}+\frac {2\,a^2\,{\left (a+b\,x\right )}^{9/2}}{3\,b^4}-\frac {6\,a\,{\left (a+b\,x\right )}^{11/2}}{11\,b^4} \]
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