Integrand size = 18, antiderivative size = 95 \[ \int x^2 (a+b x)^{5/2} (A+B x) \, dx=\frac {2 a^2 (A b-a B) (a+b x)^{7/2}}{7 b^4}-\frac {2 a (2 A b-3 a B) (a+b x)^{9/2}}{9 b^4}+\frac {2 (A b-3 a B) (a+b x)^{11/2}}{11 b^4}+\frac {2 B (a+b x)^{13/2}}{13 b^4} \]
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Time = 0.02 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78} \[ \int x^2 (a+b x)^{5/2} (A+B x) \, dx=\frac {2 a^2 (a+b x)^{7/2} (A b-a B)}{7 b^4}+\frac {2 (a+b x)^{11/2} (A b-3 a B)}{11 b^4}-\frac {2 a (a+b x)^{9/2} (2 A b-3 a B)}{9 b^4}+\frac {2 B (a+b x)^{13/2}}{13 b^4} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^2 (-A b+a B) (a+b x)^{5/2}}{b^3}+\frac {a (-2 A b+3 a B) (a+b x)^{7/2}}{b^3}+\frac {(A b-3 a B) (a+b x)^{9/2}}{b^3}+\frac {B (a+b x)^{11/2}}{b^3}\right ) \, dx \\ & = \frac {2 a^2 (A b-a B) (a+b x)^{7/2}}{7 b^4}-\frac {2 a (2 A b-3 a B) (a+b x)^{9/2}}{9 b^4}+\frac {2 (A b-3 a B) (a+b x)^{11/2}}{11 b^4}+\frac {2 B (a+b x)^{13/2}}{13 b^4} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.72 \[ \int x^2 (a+b x)^{5/2} (A+B x) \, dx=\frac {2 (a+b x)^{7/2} \left (-48 a^3 B+63 b^3 x^2 (13 A+11 B x)+8 a^2 b (13 A+21 B x)-14 a b^2 x (26 A+27 B x)\right )}{9009 b^4} \]
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Time = 0.53 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.61
method | result | size |
pseudoelliptic | \(\frac {16 \left (b x +a \right )^{\frac {7}{2}} \left (\frac {63 x^{2} \left (\frac {11 B x}{13}+A \right ) b^{3}}{8}-\frac {7 x a \left (\frac {27 B x}{26}+A \right ) b^{2}}{2}+a^{2} \left (\frac {21 B x}{13}+A \right ) b -\frac {6 a^{3} B}{13}\right )}{693 b^{4}}\) | \(58\) |
gosper | \(\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (693 b^{3} B \,x^{3}+819 A \,b^{3} x^{2}-378 B a \,b^{2} x^{2}-364 a \,b^{2} A x +168 a^{2} b B x +104 a^{2} b A -48 a^{3} B \right )}{9009 b^{4}}\) | \(71\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 \left (A b -3 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a^{2} B -2 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 a^{2} \left (A b -B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{4}}\) | \(80\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 \left (A b -3 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a^{2} B -2 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 a^{2} \left (A b -B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{4}}\) | \(80\) |
trager | \(\frac {2 \left (693 B \,b^{6} x^{6}+819 A \,b^{6} x^{5}+1701 B a \,b^{5} x^{5}+2093 A a \,b^{5} x^{4}+1113 B \,a^{2} b^{4} x^{4}+1469 A \,a^{2} b^{4} x^{3}+15 B \,a^{3} b^{3} x^{3}+39 A \,a^{3} b^{3} x^{2}-18 B \,a^{4} b^{2} x^{2}-52 A \,a^{4} b^{2} x +24 B \,a^{5} b x +104 A \,a^{5} b -48 B \,a^{6}\right ) \sqrt {b x +a}}{9009 b^{4}}\) | \(143\) |
risch | \(\frac {2 \left (693 B \,b^{6} x^{6}+819 A \,b^{6} x^{5}+1701 B a \,b^{5} x^{5}+2093 A a \,b^{5} x^{4}+1113 B \,a^{2} b^{4} x^{4}+1469 A \,a^{2} b^{4} x^{3}+15 B \,a^{3} b^{3} x^{3}+39 A \,a^{3} b^{3} x^{2}-18 B \,a^{4} b^{2} x^{2}-52 A \,a^{4} b^{2} x +24 B \,a^{5} b x +104 A \,a^{5} b -48 B \,a^{6}\right ) \sqrt {b x +a}}{9009 b^{4}}\) | \(143\) |
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Time = 0.22 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.51 \[ \int x^2 (a+b x)^{5/2} (A+B x) \, dx=\frac {2 \, {\left (693 \, B b^{6} x^{6} - 48 \, B a^{6} + 104 \, A a^{5} b + 63 \, {\left (27 \, B a b^{5} + 13 \, A b^{6}\right )} x^{5} + 7 \, {\left (159 \, B a^{2} b^{4} + 299 \, A a b^{5}\right )} x^{4} + {\left (15 \, B a^{3} b^{3} + 1469 \, A a^{2} b^{4}\right )} x^{3} - 3 \, {\left (6 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{2} + 4 \, {\left (6 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x\right )} \sqrt {b x + a}}{9009 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (94) = 188\).
Time = 0.39 (sec) , antiderivative size = 292, normalized size of antiderivative = 3.07 \[ \int x^2 (a+b x)^{5/2} (A+B x) \, dx=\begin {cases} \frac {16 A a^{5} \sqrt {a + b x}}{693 b^{3}} - \frac {8 A a^{4} x \sqrt {a + b x}}{693 b^{2}} + \frac {2 A a^{3} x^{2} \sqrt {a + b x}}{231 b} + \frac {226 A a^{2} x^{3} \sqrt {a + b x}}{693} + \frac {46 A a b x^{4} \sqrt {a + b x}}{99} + \frac {2 A b^{2} x^{5} \sqrt {a + b x}}{11} - \frac {32 B a^{6} \sqrt {a + b x}}{3003 b^{4}} + \frac {16 B a^{5} x \sqrt {a + b x}}{3003 b^{3}} - \frac {4 B a^{4} x^{2} \sqrt {a + b x}}{1001 b^{2}} + \frac {10 B a^{3} x^{3} \sqrt {a + b x}}{3003 b} + \frac {106 B a^{2} x^{4} \sqrt {a + b x}}{429} + \frac {54 B a b x^{5} \sqrt {a + b x}}{143} + \frac {2 B b^{2} x^{6} \sqrt {a + b x}}{13} & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (\frac {A x^{3}}{3} + \frac {B x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.81 \[ \int x^2 (a+b x)^{5/2} (A+B x) \, dx=\frac {2 \, {\left (693 \, {\left (b x + a\right )}^{\frac {13}{2}} B - 819 \, {\left (3 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {11}{2}} + 1001 \, {\left (3 \, B a^{2} - 2 \, A a b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 1287 \, {\left (B a^{3} - A a^{2} b\right )} {\left (b x + a\right )}^{\frac {7}{2}}\right )}}{9009 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (80) = 160\).
Time = 0.28 (sec) , antiderivative size = 516, normalized size of antiderivative = 5.43 \[ \int x^2 (a+b x)^{5/2} (A+B x) \, 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 a^{3}}{b^{2}} + \frac {1287 \, {\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^{3}}{b^{3}} + \frac {3861 \, {\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^{2}}{b^{2}} + \frac {429 \, {\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^{2}}{b^{3}} + \frac {429 \, {\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 a}{b^{2}} + \frac {195 \, {\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 a}{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^{2}} + \frac {15 \, {\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}{b^{3}}\right )}}{45045 \, b} \]
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Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int x^2 (a+b x)^{5/2} (A+B x) \, dx=\frac {\left (6\,B\,a^2-4\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{9/2}}{9\,b^4}+\frac {2\,B\,{\left (a+b\,x\right )}^{13/2}}{13\,b^4}+\frac {\left (2\,A\,b-6\,B\,a\right )\,{\left (a+b\,x\right )}^{11/2}}{11\,b^4}-\frac {\left (2\,B\,a^3-2\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^4} \]
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