Integrand size = 18, antiderivative size = 79 \[ \int \frac {(a+b x)^3 (A+B x)}{x^{3/2}} \, dx=-\frac {2 a^3 A}{\sqrt {x}}+2 a^2 (3 A b+a B) \sqrt {x}+2 a b (A b+a B) x^{3/2}+\frac {2}{5} b^2 (A b+3 a B) x^{5/2}+\frac {2}{7} b^3 B x^{7/2} \]
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Time = 0.03 (sec) , antiderivative size = 79, 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} \[ \int \frac {(a+b x)^3 (A+B x)}{x^{3/2}} \, dx=-\frac {2 a^3 A}{\sqrt {x}}+2 a^2 \sqrt {x} (a B+3 A b)+\frac {2}{5} b^2 x^{5/2} (3 a B+A b)+2 a b x^{3/2} (a B+A b)+\frac {2}{7} b^3 B x^{7/2} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^3 A}{x^{3/2}}+\frac {a^2 (3 A b+a B)}{\sqrt {x}}+3 a b (A b+a B) \sqrt {x}+b^2 (A b+3 a B) x^{3/2}+b^3 B x^{5/2}\right ) \, dx \\ & = -\frac {2 a^3 A}{\sqrt {x}}+2 a^2 (3 A b+a B) \sqrt {x}+2 a b (A b+a B) x^{3/2}+\frac {2}{5} b^2 (A b+3 a B) x^{5/2}+\frac {2}{7} b^3 B x^{7/2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^3 (A+B x)}{x^{3/2}} \, dx=\frac {2 \left (-35 a^3 (A-B x)+35 a^2 b x (3 A+B x)+7 a b^2 x^2 (5 A+3 B x)+b^3 x^3 (7 A+5 B x)\right )}{35 \sqrt {x}} \]
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Time = 1.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.96
method | result | size |
gosper | \(-\frac {2 \left (-5 b^{3} B \,x^{4}-7 A \,b^{3} x^{3}-21 B a \,b^{2} x^{3}-35 a A \,b^{2} x^{2}-35 B \,a^{2} b \,x^{2}-105 a^{2} A b x -35 a^{3} B x +35 a^{3} A \right )}{35 \sqrt {x}}\) | \(76\) |
trager | \(-\frac {2 \left (-5 b^{3} B \,x^{4}-7 A \,b^{3} x^{3}-21 B a \,b^{2} x^{3}-35 a A \,b^{2} x^{2}-35 B \,a^{2} b \,x^{2}-105 a^{2} A b x -35 a^{3} B x +35 a^{3} A \right )}{35 \sqrt {x}}\) | \(76\) |
risch | \(-\frac {2 \left (-5 b^{3} B \,x^{4}-7 A \,b^{3} x^{3}-21 B a \,b^{2} x^{3}-35 a A \,b^{2} x^{2}-35 B \,a^{2} b \,x^{2}-105 a^{2} A b x -35 a^{3} B x +35 a^{3} A \right )}{35 \sqrt {x}}\) | \(76\) |
derivativedivides | \(\frac {2 b^{3} B \,x^{\frac {7}{2}}}{7}+\frac {2 A \,b^{3} x^{\frac {5}{2}}}{5}+\frac {6 B a \,b^{2} x^{\frac {5}{2}}}{5}+2 A a \,b^{2} x^{\frac {3}{2}}+2 B \,a^{2} b \,x^{\frac {3}{2}}+6 a^{2} b A \sqrt {x}+2 a^{3} B \sqrt {x}-\frac {2 a^{3} A}{\sqrt {x}}\) | \(78\) |
default | \(\frac {2 b^{3} B \,x^{\frac {7}{2}}}{7}+\frac {2 A \,b^{3} x^{\frac {5}{2}}}{5}+\frac {6 B a \,b^{2} x^{\frac {5}{2}}}{5}+2 A a \,b^{2} x^{\frac {3}{2}}+2 B \,a^{2} b \,x^{\frac {3}{2}}+6 a^{2} b A \sqrt {x}+2 a^{3} B \sqrt {x}-\frac {2 a^{3} A}{\sqrt {x}}\) | \(78\) |
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Time = 0.22 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^3 (A+B x)}{x^{3/2}} \, dx=\frac {2 \, {\left (5 \, B b^{3} x^{4} - 35 \, A a^{3} + 7 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 35 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{35 \, \sqrt {x}} \]
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Time = 0.23 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.33 \[ \int \frac {(a+b x)^3 (A+B x)}{x^{3/2}} \, dx=- \frac {2 A a^{3}}{\sqrt {x}} + 6 A a^{2} b \sqrt {x} + 2 A a b^{2} x^{\frac {3}{2}} + \frac {2 A b^{3} x^{\frac {5}{2}}}{5} + 2 B a^{3} \sqrt {x} + 2 B a^{2} b x^{\frac {3}{2}} + \frac {6 B a b^{2} x^{\frac {5}{2}}}{5} + \frac {2 B b^{3} x^{\frac {7}{2}}}{7} \]
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Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^3 (A+B x)}{x^{3/2}} \, dx=\frac {2}{7} \, B b^{3} x^{\frac {7}{2}} - \frac {2 \, A a^{3}}{\sqrt {x}} + \frac {2}{5} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {5}{2}} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {3}{2}} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \sqrt {x} \]
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Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^3 (A+B x)}{x^{3/2}} \, dx=\frac {2}{7} \, B b^{3} x^{\frac {7}{2}} + \frac {6}{5} \, B a b^{2} x^{\frac {5}{2}} + \frac {2}{5} \, A b^{3} x^{\frac {5}{2}} + 2 \, B a^{2} b x^{\frac {3}{2}} + 2 \, A a b^{2} x^{\frac {3}{2}} + 2 \, B a^{3} \sqrt {x} + 6 \, A a^{2} b \sqrt {x} - \frac {2 \, A a^{3}}{\sqrt {x}} \]
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Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^3 (A+B x)}{x^{3/2}} \, dx=\sqrt {x}\,\left (2\,B\,a^3+6\,A\,b\,a^2\right )+x^{5/2}\,\left (\frac {2\,A\,b^3}{5}+\frac {6\,B\,a\,b^2}{5}\right )-\frac {2\,A\,a^3}{\sqrt {x}}+\frac {2\,B\,b^3\,x^{7/2}}{7}+2\,a\,b\,x^{3/2}\,\left (A\,b+B\,a\right ) \]
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