Integrand size = 15, antiderivative size = 42 \[ \int (a+b x)^{3/2} (A+B x) \, dx=\frac {2 (A b-a B) (a+b x)^{5/2}}{5 b^2}+\frac {2 B (a+b x)^{7/2}}{7 b^2} \]
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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} \[ \int (a+b x)^{3/2} (A+B x) \, dx=\frac {2 (a+b x)^{5/2} (A b-a B)}{5 b^2}+\frac {2 B (a+b x)^{7/2}}{7 b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(A b-a B) (a+b x)^{3/2}}{b}+\frac {B (a+b x)^{5/2}}{b}\right ) \, dx \\ & = \frac {2 (A b-a B) (a+b x)^{5/2}}{5 b^2}+\frac {2 B (a+b x)^{7/2}}{7 b^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int (a+b x)^{3/2} (A+B x) \, dx=\frac {2 (a+b x)^{5/2} (7 A b-2 a B+5 b B x)}{35 b^2} \]
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Time = 1.37 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (5 b B x +7 A b -2 B a \right )}{35 b^{2}}\) | \(27\) |
pseudoelliptic | \(\frac {2 \left (\left (5 B x +7 A \right ) b -2 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{35 b^{2}}\) | \(28\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (A b -B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{2}}\) | \(34\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (A b -B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{2}}\) | \(34\) |
trager | \(\frac {2 \left (5 b^{3} B \,x^{3}+7 A \,b^{3} x^{2}+8 B a \,b^{2} x^{2}+14 a \,b^{2} A x +a^{2} b B x +7 a^{2} b A -2 a^{3} B \right ) \sqrt {b x +a}}{35 b^{2}}\) | \(70\) |
risch | \(\frac {2 \left (5 b^{3} B \,x^{3}+7 A \,b^{3} x^{2}+8 B a \,b^{2} x^{2}+14 a \,b^{2} A x +a^{2} b B x +7 a^{2} b A -2 a^{3} B \right ) \sqrt {b x +a}}{35 b^{2}}\) | \(70\) |
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Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (34) = 68\).
Time = 0.22 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.64 \[ \int (a+b x)^{3/2} (A+B x) \, dx=\frac {2 \, {\left (5 \, B b^{3} x^{3} - 2 \, B a^{3} + 7 \, A a^{2} b + {\left (8 \, B a b^{2} + 7 \, A b^{3}\right )} x^{2} + {\left (B a^{2} b + 14 \, A a b^{2}\right )} x\right )} \sqrt {b x + a}}{35 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (41) = 82\).
Time = 0.20 (sec) , antiderivative size = 146, normalized size of antiderivative = 3.48 \[ \int (a+b x)^{3/2} (A+B x) \, dx=\begin {cases} \frac {2 A a^{2} \sqrt {a + b x}}{5 b} + \frac {4 A a x \sqrt {a + b x}}{5} + \frac {2 A b x^{2} \sqrt {a + b x}}{5} - \frac {4 B a^{3} \sqrt {a + b x}}{35 b^{2}} + \frac {2 B a^{2} x \sqrt {a + b x}}{35 b} + \frac {16 B a x^{2} \sqrt {a + b x}}{35} + \frac {2 B b x^{3} \sqrt {a + b x}}{7} & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (A x + \frac {B x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.79 \[ \int (a+b x)^{3/2} (A+B x) \, dx=\frac {2 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} B - 7 \, {\left (B a - A b\right )} {\left (b x + a\right )}^{\frac {5}{2}}\right )}}{35 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (34) = 68\).
Time = 0.29 (sec) , antiderivative size = 192, normalized size of antiderivative = 4.57 \[ \int (a+b x)^{3/2} (A+B x) \, dx=\frac {2 \, {\left (105 \, \sqrt {b x + a} A a^{2} + 70 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} A a + \frac {35 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} B a^{2}}{b} + 7 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} A + \frac {14 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} B a}{b} + \frac {3 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} B}{b}\right )}}{105 \, b} \]
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Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.69 \[ \int (a+b x)^{3/2} (A+B x) \, dx=\frac {2\,{\left (a+b\,x\right )}^{5/2}\,\left (7\,A\,b-7\,B\,a+5\,B\,\left (a+b\,x\right )\right )}{35\,b^2} \]
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