Integrand size = 13, antiderivative size = 53 \[ \int x^2 (a+b x)^{9/2} \, dx=\frac {2 a^2 (a+b x)^{11/2}}{11 b^3}-\frac {4 a (a+b x)^{13/2}}{13 b^3}+\frac {2 (a+b x)^{15/2}}{15 b^3} \]
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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} \[ \int x^2 (a+b x)^{9/2} \, dx=\frac {2 a^2 (a+b x)^{11/2}}{11 b^3}+\frac {2 (a+b x)^{15/2}}{15 b^3}-\frac {4 a (a+b x)^{13/2}}{13 b^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 (a+b x)^{9/2}}{b^2}-\frac {2 a (a+b x)^{11/2}}{b^2}+\frac {(a+b x)^{13/2}}{b^2}\right ) \, dx \\ & = \frac {2 a^2 (a+b x)^{11/2}}{11 b^3}-\frac {4 a (a+b x)^{13/2}}{13 b^3}+\frac {2 (a+b x)^{15/2}}{15 b^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.66 \[ \int x^2 (a+b x)^{9/2} \, dx=\frac {2 (a+b x)^{11/2} \left (8 a^2-44 a b x+143 b^2 x^2\right )}{2145 b^3} \]
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Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (143 b^{2} x^{2}-44 a b x +8 a^{2}\right )}{2145 b^{3}}\) | \(32\) |
pseudoelliptic | \(\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (143 b^{2} x^{2}-44 a b x +8 a^{2}\right )}{2145 b^{3}}\) | \(32\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {15}{2}}}{15}-\frac {4 a \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 a^{2} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{3}}\) | \(38\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {15}{2}}}{15}-\frac {4 a \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 a^{2} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{3}}\) | \(38\) |
trager | \(\frac {2 \left (143 b^{7} x^{7}+671 a \,b^{6} x^{6}+1218 a^{2} b^{5} x^{5}+1030 a^{3} b^{4} x^{4}+355 a^{4} b^{3} x^{3}+3 a^{5} b^{2} x^{2}-4 a^{6} b x +8 a^{7}\right ) \sqrt {b x +a}}{2145 b^{3}}\) | \(87\) |
risch | \(\frac {2 \left (143 b^{7} x^{7}+671 a \,b^{6} x^{6}+1218 a^{2} b^{5} x^{5}+1030 a^{3} b^{4} x^{4}+355 a^{4} b^{3} x^{3}+3 a^{5} b^{2} x^{2}-4 a^{6} b x +8 a^{7}\right ) \sqrt {b x +a}}{2145 b^{3}}\) | \(87\) |
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Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (41) = 82\).
Time = 0.22 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.62 \[ \int x^2 (a+b x)^{9/2} \, dx=\frac {2 \, {\left (143 \, b^{7} x^{7} + 671 \, a b^{6} x^{6} + 1218 \, a^{2} b^{5} x^{5} + 1030 \, a^{3} b^{4} x^{4} + 355 \, a^{4} b^{3} x^{3} + 3 \, a^{5} b^{2} x^{2} - 4 \, a^{6} b x + 8 \, a^{7}\right )} \sqrt {b x + a}}{2145 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (49) = 98\).
Time = 1.11 (sec) , antiderivative size = 168, normalized size of antiderivative = 3.17 \[ \int x^2 (a+b x)^{9/2} \, dx=\begin {cases} \frac {16 a^{7} \sqrt {a + b x}}{2145 b^{3}} - \frac {8 a^{6} x \sqrt {a + b x}}{2145 b^{2}} + \frac {2 a^{5} x^{2} \sqrt {a + b x}}{715 b} + \frac {142 a^{4} x^{3} \sqrt {a + b x}}{429} + \frac {412 a^{3} b x^{4} \sqrt {a + b x}}{429} + \frac {812 a^{2} b^{2} x^{5} \sqrt {a + b x}}{715} + \frac {122 a b^{3} x^{6} \sqrt {a + b x}}{195} + \frac {2 b^{4} x^{7} \sqrt {a + b x}}{15} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{3}}{3} & \text {otherwise} \end {cases} \]
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none
Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int x^2 (a+b x)^{9/2} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {15}{2}}}{15 \, b^{3}} - \frac {4 \, {\left (b x + a\right )}^{\frac {13}{2}} a}{13 \, b^{3}} + \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{2}}{11 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 421 vs. \(2 (41) = 82\).
Time = 0.33 (sec) , antiderivative size = 421, normalized size of antiderivative = 7.94 \[ \int x^2 (a+b x)^{9/2} \, dx=\frac {2 \, {\left (\frac {3003 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} a^{5}}{b^{2}} + \frac {6435 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a^{4}}{b^{2}} + \frac {1430 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} a^{3}}{b^{2}} + \frac {650 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} a^{2}}{b^{2}} + \frac {75 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )} a}{b^{2}} + \frac {7 \, {\left (429 \, {\left (b x + a\right )}^{\frac {15}{2}} - 3465 \, {\left (b x + a\right )}^{\frac {13}{2}} a + 12285 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{2} - 25025 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{3} + 32175 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{4} - 27027 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{5} + 15015 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{6} - 6435 \, \sqrt {b x + a} a^{7}\right )}}{b^{2}}\right )}}{45045 \, b} \]
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Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.68 \[ \int x^2 (a+b x)^{9/2} \, dx=\frac {\frac {2\,{\left (a+b\,x\right )}^{15/2}}{15}-\frac {4\,a\,{\left (a+b\,x\right )}^{13/2}}{13}+\frac {2\,a^2\,{\left (a+b\,x\right )}^{11/2}}{11}}{b^3} \]
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