Integrand size = 16, antiderivative size = 67 \[ \int x (a+b x)^{5/2} (A+B x) \, dx=-\frac {2 a (A b-a B) (a+b x)^{7/2}}{7 b^3}+\frac {2 (A b-2 a B) (a+b x)^{9/2}}{9 b^3}+\frac {2 B (a+b x)^{11/2}}{11 b^3} \]
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Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \[ \int x (a+b x)^{5/2} (A+B x) \, dx=\frac {2 (a+b x)^{9/2} (A b-2 a B)}{9 b^3}-\frac {2 a (a+b x)^{7/2} (A b-a B)}{7 b^3}+\frac {2 B (a+b x)^{11/2}}{11 b^3} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a (-A b+a B) (a+b x)^{5/2}}{b^2}+\frac {(A b-2 a B) (a+b x)^{7/2}}{b^2}+\frac {B (a+b x)^{9/2}}{b^2}\right ) \, dx \\ & = -\frac {2 a (A b-a B) (a+b x)^{7/2}}{7 b^3}+\frac {2 (A b-2 a B) (a+b x)^{9/2}}{9 b^3}+\frac {2 B (a+b x)^{11/2}}{11 b^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.73 \[ \int x (a+b x)^{5/2} (A+B x) \, dx=\frac {2 (a+b x)^{7/2} \left (8 a^2 B+7 b^2 x (11 A+9 B x)-2 a b (11 A+14 B x)\right )}{693 b^3} \]
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Time = 0.53 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.61
method | result | size |
pseudoelliptic | \(-\frac {4 \left (b x +a \right )^{\frac {7}{2}} \left (-\frac {7 x \left (\frac {9 B x}{11}+A \right ) b^{2}}{2}+a \left (\frac {14 B x}{11}+A \right ) b -\frac {4 a^{2} B}{11}\right )}{63 b^{3}}\) | \(41\) |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-63 b^{2} B \,x^{2}-77 A \,b^{2} x +28 B a b x +22 a b A -8 a^{2} B \right )}{693 b^{3}}\) | \(47\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{9}-\frac {2 a \left (A b -B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{3}}\) | \(52\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{9}-\frac {2 a \left (A b -B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{3}}\) | \(52\) |
trager | \(-\frac {2 \left (-63 b^{5} B \,x^{5}-77 A \,b^{5} x^{4}-161 B a \,b^{4} x^{4}-209 A a \,b^{4} x^{3}-113 B \,a^{2} b^{3} x^{3}-165 A \,a^{2} b^{3} x^{2}-3 B \,a^{3} b^{2} x^{2}-11 a^{3} b^{2} A x +4 a^{4} b B x +22 a^{4} b A -8 a^{5} B \right ) \sqrt {b x +a}}{693 b^{3}}\) | \(119\) |
risch | \(-\frac {2 \left (-63 b^{5} B \,x^{5}-77 A \,b^{5} x^{4}-161 B a \,b^{4} x^{4}-209 A a \,b^{4} x^{3}-113 B \,a^{2} b^{3} x^{3}-165 A \,a^{2} b^{3} x^{2}-3 B \,a^{3} b^{2} x^{2}-11 a^{3} b^{2} A x +4 a^{4} b B x +22 a^{4} b A -8 a^{5} B \right ) \sqrt {b x +a}}{693 b^{3}}\) | \(119\) |
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Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (56) = 112\).
Time = 0.23 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.76 \[ \int x (a+b x)^{5/2} (A+B x) \, dx=\frac {2 \, {\left (63 \, B b^{5} x^{5} + 8 \, B a^{5} - 22 \, A a^{4} b + 7 \, {\left (23 \, B a b^{4} + 11 \, A b^{5}\right )} x^{4} + {\left (113 \, B a^{2} b^{3} + 209 \, A a b^{4}\right )} x^{3} + 3 \, {\left (B a^{3} b^{2} + 55 \, A a^{2} b^{3}\right )} x^{2} - {\left (4 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b x + a}}{693 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (65) = 130\).
Time = 0.35 (sec) , antiderivative size = 245, normalized size of antiderivative = 3.66 \[ \int x (a+b x)^{5/2} (A+B x) \, dx=\begin {cases} - \frac {4 A a^{4} \sqrt {a + b x}}{63 b^{2}} + \frac {2 A a^{3} x \sqrt {a + b x}}{63 b} + \frac {10 A a^{2} x^{2} \sqrt {a + b x}}{21} + \frac {38 A a b x^{3} \sqrt {a + b x}}{63} + \frac {2 A b^{2} x^{4} \sqrt {a + b x}}{9} + \frac {16 B a^{5} \sqrt {a + b x}}{693 b^{3}} - \frac {8 B a^{4} x \sqrt {a + b x}}{693 b^{2}} + \frac {2 B a^{3} x^{2} \sqrt {a + b x}}{231 b} + \frac {226 B a^{2} x^{3} \sqrt {a + b x}}{693} + \frac {46 B a b x^{4} \sqrt {a + b x}}{99} + \frac {2 B b^{2} x^{5} \sqrt {a + b x}}{11} & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (\frac {A x^{2}}{2} + \frac {B x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.81 \[ \int x (a+b x)^{5/2} (A+B x) \, dx=\frac {2 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} B - 77 \, {\left (2 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {9}{2}} + 99 \, {\left (B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {7}{2}}\right )}}{693 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (56) = 112\).
Time = 0.28 (sec) , antiderivative size = 418, normalized size of antiderivative = 6.24 \[ \int x (a+b x)^{5/2} (A+B x) \, dx=\frac {2 \, {\left (\frac {1155 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} A a^{3}}{b} + \frac {231 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} B a^{3}}{b^{2}} + \frac {693 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} A a^{2}}{b} + \frac {297 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} B a^{2}}{b^{2}} + \frac {297 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} A a}{b} + \frac {33 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} B a}{b^{2}} + \frac {11 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} A}{b} + \frac {5 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} B}{b^{2}}\right )}}{3465 \, b} \]
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Time = 0.49 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.78 \[ \int x (a+b x)^{5/2} (A+B x) \, dx=\frac {2\,{\left (a+b\,x\right )}^{7/2}\,\left (99\,B\,a^2+63\,B\,{\left (a+b\,x\right )}^2-99\,A\,a\,b+77\,A\,b\,\left (a+b\,x\right )-154\,B\,a\,\left (a+b\,x\right )\right )}{693\,b^3} \]
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