Integrand size = 18, antiderivative size = 81 \[ \int \frac {(a+b x)^3 (A+B x)}{x^{5/2}} \, dx=-\frac {2 a^3 A}{3 x^{3/2}}-\frac {2 a^2 (3 A b+a B)}{\sqrt {x}}+6 a b (A b+a B) \sqrt {x}+\frac {2}{3} b^2 (A b+3 a B) x^{3/2}+\frac {2}{5} b^3 B x^{5/2} \]
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Time = 0.03 (sec) , antiderivative size = 81, 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 = {77} \[ \int \frac {(a+b x)^3 (A+B x)}{x^{5/2}} \, dx=-\frac {2 a^3 A}{3 x^{3/2}}-\frac {2 a^2 (a B+3 A b)}{\sqrt {x}}+\frac {2}{3} b^2 x^{3/2} (3 a B+A b)+6 a b \sqrt {x} (a B+A b)+\frac {2}{5} b^3 B x^{5/2} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^3 A}{x^{5/2}}+\frac {a^2 (3 A b+a B)}{x^{3/2}}+\frac {3 a b (A b+a B)}{\sqrt {x}}+b^2 (A b+3 a B) \sqrt {x}+b^3 B x^{3/2}\right ) \, dx \\ & = -\frac {2 a^3 A}{3 x^{3/2}}-\frac {2 a^2 (3 A b+a B)}{\sqrt {x}}+6 a b (A b+a B) \sqrt {x}+\frac {2}{3} b^2 (A b+3 a B) x^{3/2}+\frac {2}{5} b^3 B x^{5/2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^3 (A+B x)}{x^{5/2}} \, dx=\frac {2 \left (45 a^2 b x (-A+B x)+15 a b^2 x^2 (3 A+B x)-5 a^3 (A+3 B x)+b^3 x^3 (5 A+3 B x)\right )}{15 x^{3/2}} \]
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Time = 0.44 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {2 b^{3} B \,x^{\frac {5}{2}}}{5}+\frac {2 A \,b^{3} x^{\frac {3}{2}}}{3}+2 B a \,b^{2} x^{\frac {3}{2}}+6 a \,b^{2} A \sqrt {x}+6 a^{2} b B \sqrt {x}-\frac {2 a^{2} \left (3 A b +B a \right )}{\sqrt {x}}-\frac {2 a^{3} A}{3 x^{\frac {3}{2}}}\) | \(75\) |
default | \(\frac {2 b^{3} B \,x^{\frac {5}{2}}}{5}+\frac {2 A \,b^{3} x^{\frac {3}{2}}}{3}+2 B a \,b^{2} x^{\frac {3}{2}}+6 a \,b^{2} A \sqrt {x}+6 a^{2} b B \sqrt {x}-\frac {2 a^{2} \left (3 A b +B a \right )}{\sqrt {x}}-\frac {2 a^{3} A}{3 x^{\frac {3}{2}}}\) | \(75\) |
gosper | \(-\frac {2 \left (-3 b^{3} B \,x^{4}-5 A \,b^{3} x^{3}-15 B a \,b^{2} x^{3}-45 a A \,b^{2} x^{2}-45 B \,a^{2} b \,x^{2}+45 a^{2} A b x +15 a^{3} B x +5 a^{3} A \right )}{15 x^{\frac {3}{2}}}\) | \(76\) |
trager | \(-\frac {2 \left (-3 b^{3} B \,x^{4}-5 A \,b^{3} x^{3}-15 B a \,b^{2} x^{3}-45 a A \,b^{2} x^{2}-45 B \,a^{2} b \,x^{2}+45 a^{2} A b x +15 a^{3} B x +5 a^{3} A \right )}{15 x^{\frac {3}{2}}}\) | \(76\) |
risch | \(-\frac {2 \left (-3 b^{3} B \,x^{4}-5 A \,b^{3} x^{3}-15 B a \,b^{2} x^{3}-45 a A \,b^{2} x^{2}-45 B \,a^{2} b \,x^{2}+45 a^{2} A b x +15 a^{3} B x +5 a^{3} A \right )}{15 x^{\frac {3}{2}}}\) | \(76\) |
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Time = 0.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^3 (A+B x)}{x^{5/2}} \, dx=\frac {2 \, {\left (3 \, B b^{3} x^{4} - 5 \, A a^{3} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 45 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} - 15 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15 \, x^{\frac {3}{2}}} \]
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Time = 0.25 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.30 \[ \int \frac {(a+b x)^3 (A+B x)}{x^{5/2}} \, dx=- \frac {2 A a^{3}}{3 x^{\frac {3}{2}}} - \frac {6 A a^{2} b}{\sqrt {x}} + 6 A a b^{2} \sqrt {x} + \frac {2 A b^{3} x^{\frac {3}{2}}}{3} - \frac {2 B a^{3}}{\sqrt {x}} + 6 B a^{2} b \sqrt {x} + 2 B a b^{2} x^{\frac {3}{2}} + \frac {2 B b^{3} x^{\frac {5}{2}}}{5} \]
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Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^3 (A+B x)}{x^{5/2}} \, dx=\frac {2}{5} \, B b^{3} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {3}{2}} + 6 \, {\left (B a^{2} b + A a b^{2}\right )} \sqrt {x} - \frac {2 \, {\left (A a^{3} + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^3 (A+B x)}{x^{5/2}} \, dx=\frac {2}{5} \, B b^{3} x^{\frac {5}{2}} + 2 \, B a b^{2} x^{\frac {3}{2}} + \frac {2}{3} \, A b^{3} x^{\frac {3}{2}} + 6 \, B a^{2} b \sqrt {x} + 6 \, A a b^{2} \sqrt {x} - \frac {2 \, {\left (3 \, B a^{3} x + 9 \, A a^{2} b x + A a^{3}\right )}}{3 \, x^{\frac {3}{2}}} \]
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Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^3 (A+B x)}{x^{5/2}} \, dx=x^{3/2}\,\left (\frac {2\,A\,b^3}{3}+2\,B\,a\,b^2\right )-\frac {x\,\left (2\,B\,a^3+6\,A\,b\,a^2\right )+\frac {2\,A\,a^3}{3}}{x^{3/2}}+\frac {2\,B\,b^3\,x^{5/2}}{5}+6\,a\,b\,\sqrt {x}\,\left (A\,b+B\,a\right ) \]
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