Integrand size = 16, antiderivative size = 63 \[ \int \frac {x (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 a (A b-a B)}{3 b^3 (a+b x)^{3/2}}-\frac {2 (A b-2 a B)}{b^3 \sqrt {a+b x}}+\frac {2 B \sqrt {a+b x}}{b^3} \]
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Time = 0.02 (sec) , antiderivative size = 63, 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} \[ \int \frac {x (A+B x)}{(a+b x)^{5/2}} \, dx=-\frac {2 (A b-2 a B)}{b^3 \sqrt {a+b x}}+\frac {2 a (A b-a B)}{3 b^3 (a+b x)^{3/2}}+\frac {2 B \sqrt {a+b x}}{b^3} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a (-A b+a B)}{b^2 (a+b x)^{5/2}}+\frac {A b-2 a B}{b^2 (a+b x)^{3/2}}+\frac {B}{b^2 \sqrt {a+b x}}\right ) \, dx \\ & = \frac {2 a (A b-a B)}{3 b^3 (a+b x)^{3/2}}-\frac {2 (A b-2 a B)}{b^3 \sqrt {a+b x}}+\frac {2 B \sqrt {a+b x}}{b^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.73 \[ \int \frac {x (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {16 a^2 B-4 a b (A-6 B x)+6 b^2 x (-A+B x)}{3 b^3 (a+b x)^{3/2}} \]
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Time = 0.54 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(-\frac {4 \left (\frac {3 x \left (-B x +A \right ) b^{2}}{2}+a \left (-6 B x +A \right ) b -4 a^{2} B \right )}{3 \left (b x +a \right )^{\frac {3}{2}} b^{3}}\) | \(41\) |
gosper | \(-\frac {2 \left (-3 b^{2} B \,x^{2}+3 A \,b^{2} x -12 B a b x +2 a b A -8 a^{2} B \right )}{3 \left (b x +a \right )^{\frac {3}{2}} b^{3}}\) | \(47\) |
trager | \(-\frac {2 \left (-3 b^{2} B \,x^{2}+3 A \,b^{2} x -12 B a b x +2 a b A -8 a^{2} B \right )}{3 \left (b x +a \right )^{\frac {3}{2}} b^{3}}\) | \(47\) |
derivativedivides | \(\frac {2 B \sqrt {b x +a}-\frac {2 \left (A b -2 B a \right )}{\sqrt {b x +a}}+\frac {2 a \left (A b -B a \right )}{3 \left (b x +a \right )^{\frac {3}{2}}}}{b^{3}}\) | \(51\) |
default | \(\frac {2 B \sqrt {b x +a}-\frac {2 \left (A b -2 B a \right )}{\sqrt {b x +a}}+\frac {2 a \left (A b -B a \right )}{3 \left (b x +a \right )^{\frac {3}{2}}}}{b^{3}}\) | \(51\) |
risch | \(\frac {2 B \sqrt {b x +a}}{b^{3}}-\frac {2 \left (3 A \,b^{2} x -6 B a b x +2 a b A -5 a^{2} B \right )}{3 b^{3} \left (b x +a \right )^{\frac {3}{2}}}\) | \(52\) |
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Time = 0.23 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.10 \[ \int \frac {x (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, B b^{2} x^{2} + 8 \, B a^{2} - 2 \, A a b + 3 \, {\left (4 \, B a b - A b^{2}\right )} x\right )} \sqrt {b x + a}}{3 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (60) = 120\).
Time = 0.34 (sec) , antiderivative size = 211, normalized size of antiderivative = 3.35 \[ \int \frac {x (A+B x)}{(a+b x)^{5/2}} \, dx=\begin {cases} - \frac {4 A a b}{3 a b^{3} \sqrt {a + b x} + 3 b^{4} x \sqrt {a + b x}} - \frac {6 A b^{2} x}{3 a b^{3} \sqrt {a + b x} + 3 b^{4} x \sqrt {a + b x}} + \frac {16 B a^{2}}{3 a b^{3} \sqrt {a + b x} + 3 b^{4} x \sqrt {a + b x}} + \frac {24 B a b x}{3 a b^{3} \sqrt {a + b x} + 3 b^{4} x \sqrt {a + b x}} + \frac {6 B b^{2} x^{2}}{3 a b^{3} \sqrt {a + b x} + 3 b^{4} x \sqrt {a + b x}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{2}}{2} + \frac {B x^{3}}{3}}{a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.92 \[ \int \frac {x (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {3 \, \sqrt {b x + a} B}{b} - \frac {B a^{2} - A a b - 3 \, {\left (2 \, B a - A b\right )} {\left (b x + a\right )}}{{\left (b x + a\right )}^{\frac {3}{2}} b}\right )}}{3 \, b^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.87 \[ \int \frac {x (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 \, \sqrt {b x + a} B}{b^{3}} + \frac {2 \, {\left (6 \, {\left (b x + a\right )} B a - B a^{2} - 3 \, {\left (b x + a\right )} A b + A a b\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{3}} \]
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Time = 0.42 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {x (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {6\,B\,{\left (a+b\,x\right )}^2-2\,B\,a^2+2\,A\,a\,b-6\,A\,b\,\left (a+b\,x\right )+12\,B\,a\,\left (a+b\,x\right )}{3\,b^3\,{\left (a+b\,x\right )}^{3/2}} \]
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