Integrand size = 15, antiderivative size = 42 \[ \int (a+b x)^{5/2} (A+B x) \, dx=\frac {2 (A b-a B) (a+b x)^{7/2}}{7 b^2}+\frac {2 B (a+b x)^{9/2}}{9 b^2} \]
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Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int (a+b x)^{5/2} (A+B x) \, dx=\frac {2 (a+b x)^{7/2} (A b-a B)}{7 b^2}+\frac {2 B (a+b x)^{9/2}}{9 b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(A b-a B) (a+b x)^{5/2}}{b}+\frac {B (a+b x)^{7/2}}{b}\right ) \, dx \\ & = \frac {2 (A b-a B) (a+b x)^{7/2}}{7 b^2}+\frac {2 B (a+b x)^{9/2}}{9 b^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int (a+b x)^{5/2} (A+B x) \, dx=\frac {2 (a+b x)^{7/2} (9 A b-2 a B+7 b B x)}{63 b^2} \]
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Time = 0.51 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (7 b B x +9 A b -2 B a \right )}{63 b^{2}}\) | \(27\) |
pseudoelliptic | \(\frac {2 \left (\left (7 B x +9 A \right ) b -2 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{63 b^{2}}\) | \(28\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (A b -B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{2}}\) | \(34\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (A b -B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{2}}\) | \(34\) |
trager | \(\frac {2 \left (7 B \,x^{4} b^{4}+9 A \,x^{3} b^{4}+19 B \,x^{3} a \,b^{3}+27 A \,x^{2} a \,b^{3}+15 B \,x^{2} a^{2} b^{2}+27 A x \,a^{2} b^{2}+B x \,a^{3} b +9 A \,a^{3} b -2 B \,a^{4}\right ) \sqrt {b x +a}}{63 b^{2}}\) | \(94\) |
risch | \(\frac {2 \left (7 B \,x^{4} b^{4}+9 A \,x^{3} b^{4}+19 B \,x^{3} a \,b^{3}+27 A \,x^{2} a \,b^{3}+15 B \,x^{2} a^{2} b^{2}+27 A x \,a^{2} b^{2}+B x \,a^{3} b +9 A \,a^{3} b -2 B \,a^{4}\right ) \sqrt {b x +a}}{63 b^{2}}\) | \(94\) |
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Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (34) = 68\).
Time = 0.22 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.21 \[ \int (a+b x)^{5/2} (A+B x) \, dx=\frac {2 \, {\left (7 \, B b^{4} x^{4} - 2 \, B a^{4} + 9 \, A a^{3} b + {\left (19 \, B a b^{3} + 9 \, A b^{4}\right )} x^{3} + 3 \, {\left (5 \, B a^{2} b^{2} + 9 \, A a b^{3}\right )} x^{2} + {\left (B a^{3} b + 27 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x + a}}{63 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (41) = 82\).
Time = 0.30 (sec) , antiderivative size = 194, normalized size of antiderivative = 4.62 \[ \int (a+b x)^{5/2} (A+B x) \, dx=\begin {cases} \frac {2 A a^{3} \sqrt {a + b x}}{7 b} + \frac {6 A a^{2} x \sqrt {a + b x}}{7} + \frac {6 A a b x^{2} \sqrt {a + b x}}{7} + \frac {2 A b^{2} x^{3} \sqrt {a + b x}}{7} - \frac {4 B a^{4} \sqrt {a + b x}}{63 b^{2}} + \frac {2 B a^{3} x \sqrt {a + b x}}{63 b} + \frac {10 B a^{2} x^{2} \sqrt {a + b x}}{21} + \frac {38 B a b x^{3} \sqrt {a + b x}}{63} + \frac {2 B b^{2} x^{4} \sqrt {a + b x}}{9} & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (A x + \frac {B x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.79 \[ \int (a+b x)^{5/2} (A+B x) \, dx=\frac {2 \, {\left (7 \, {\left (b x + a\right )}^{\frac {9}{2}} B - 9 \, {\left (B a - A b\right )} {\left (b x + a\right )}^{\frac {7}{2}}\right )}}{63 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (34) = 68\).
Time = 0.28 (sec) , antiderivative size = 306, normalized size of antiderivative = 7.29 \[ \int (a+b x)^{5/2} (A+B x) \, dx=\frac {2 \, {\left (315 \, \sqrt {b x + a} A a^{3} + 315 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} A a^{2} + \frac {105 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} B a^{3}}{b} + 63 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} A a + \frac {63 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} B a^{2}}{b} + 9 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} A + \frac {27 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} B a}{b} + \frac {{\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} B}{b}\right )}}{315 \, b} \]
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Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.69 \[ \int (a+b x)^{5/2} (A+B x) \, dx=\frac {2\,{\left (a+b\,x\right )}^{7/2}\,\left (9\,A\,b-9\,B\,a+7\,B\,\left (a+b\,x\right )\right )}{63\,b^2} \]
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