Integrand size = 13, antiderivative size = 53 \[ \int x^2 (a+b x)^{5/2} \, dx=\frac {2 a^2 (a+b x)^{7/2}}{7 b^3}-\frac {4 a (a+b x)^{9/2}}{9 b^3}+\frac {2 (a+b x)^{11/2}}{11 b^3} \]
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Time = 0.01 (sec) , antiderivative size = 53, 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} \[ \int x^2 (a+b x)^{5/2} \, dx=\frac {2 a^2 (a+b x)^{7/2}}{7 b^3}+\frac {2 (a+b x)^{11/2}}{11 b^3}-\frac {4 a (a+b x)^{9/2}}{9 b^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 (a+b x)^{5/2}}{b^2}-\frac {2 a (a+b x)^{7/2}}{b^2}+\frac {(a+b x)^{9/2}}{b^2}\right ) \, dx \\ & = \frac {2 a^2 (a+b x)^{7/2}}{7 b^3}-\frac {4 a (a+b x)^{9/2}}{9 b^3}+\frac {2 (a+b x)^{11/2}}{11 b^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.66 \[ \int x^2 (a+b x)^{5/2} \, dx=\frac {2 (a+b x)^{7/2} \left (8 a^2-28 a b x+63 b^2 x^2\right )}{693 b^3} \]
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Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (63 b^{2} x^{2}-28 a b x +8 a^{2}\right )}{693 b^{3}}\) | \(32\) |
pseudoelliptic | \(\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (63 b^{2} x^{2}-28 a b x +8 a^{2}\right )}{693 b^{3}}\) | \(32\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {11}{2}}}{11}-\frac {4 a \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 a^{2} \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{3}}\) | \(38\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {11}{2}}}{11}-\frac {4 a \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 a^{2} \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{3}}\) | \(38\) |
trager | \(\frac {2 \left (63 b^{5} x^{5}+161 a \,b^{4} x^{4}+113 a^{2} b^{3} x^{3}+3 a^{3} b^{2} x^{2}-4 a^{4} b x +8 a^{5}\right ) \sqrt {b x +a}}{693 b^{3}}\) | \(65\) |
risch | \(\frac {2 \left (63 b^{5} x^{5}+161 a \,b^{4} x^{4}+113 a^{2} b^{3} x^{3}+3 a^{3} b^{2} x^{2}-4 a^{4} b x +8 a^{5}\right ) \sqrt {b x +a}}{693 b^{3}}\) | \(65\) |
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none
Time = 0.22 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.21 \[ \int x^2 (a+b x)^{5/2} \, dx=\frac {2 \, {\left (63 \, b^{5} x^{5} + 161 \, a b^{4} x^{4} + 113 \, a^{2} b^{3} x^{3} + 3 \, a^{3} b^{2} x^{2} - 4 \, a^{4} b x + 8 \, a^{5}\right )} \sqrt {b x + a}}{693 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (49) = 98\).
Time = 0.53 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.34 \[ \int x^2 (a+b x)^{5/2} \, dx=\begin {cases} \frac {16 a^{5} \sqrt {a + b x}}{693 b^{3}} - \frac {8 a^{4} x \sqrt {a + b x}}{693 b^{2}} + \frac {2 a^{3} x^{2} \sqrt {a + b x}}{231 b} + \frac {226 a^{2} x^{3} \sqrt {a + b x}}{693} + \frac {46 a b x^{4} \sqrt {a + b x}}{99} + \frac {2 b^{2} x^{5} \sqrt {a + b x}}{11} & \text {for}\: b \neq 0 \\\frac {a^{\frac {5}{2}} x^{3}}{3} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int x^2 (a+b x)^{5/2} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}}}{11 \, b^{3}} - \frac {4 \, {\left (b x + a\right )}^{\frac {9}{2}} a}{9 \, b^{3}} + \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2}}{7 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (41) = 82\).
Time = 0.30 (sec) , antiderivative size = 233, normalized size of antiderivative = 4.40 \[ \int x^2 (a+b x)^{5/2} \, dx=\frac {2 \, {\left (\frac {231 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} a^{3}}{b^{2}} + \frac {297 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a^{2}}{b^{2}} + \frac {33 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} a}{b^{2}} + \frac {5 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )}}{b^{2}}\right )}}{3465 \, b} \]
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Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int x^2 (a+b x)^{5/2} \, dx=\frac {126\,{\left (a+b\,x\right )}^{11/2}-308\,a\,{\left (a+b\,x\right )}^{9/2}+198\,a^2\,{\left (a+b\,x\right )}^{7/2}}{693\,b^3} \]
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