Integrand size = 20, antiderivative size = 53 \[ \int \frac {A+B x}{x^{5/2} \sqrt {a+b x}} \, dx=-\frac {2 A \sqrt {a+b x}}{3 a x^{3/2}}+\frac {2 (2 A b-3 a B) \sqrt {a+b x}}{3 a^2 \sqrt {x}} \]
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Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {79, 37} \[ \int \frac {A+B x}{x^{5/2} \sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} (2 A b-3 a B)}{3 a^2 \sqrt {x}}-\frac {2 A \sqrt {a+b x}}{3 a x^{3/2}} \]
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Rule 37
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A \sqrt {a+b x}}{3 a x^{3/2}}+\frac {\left (2 \left (-A b+\frac {3 a B}{2}\right )\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{3 a} \\ & = -\frac {2 A \sqrt {a+b x}}{3 a x^{3/2}}+\frac {2 (2 A b-3 a B) \sqrt {a+b x}}{3 a^2 \sqrt {x}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.66 \[ \int \frac {A+B x}{x^{5/2} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {a+b x} (a A-2 A b x+3 a B x)}{3 a^2 x^{3/2}} \]
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Time = 1.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.57
method | result | size |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (-2 A b x +3 B a x +A a \right )}{3 x^{\frac {3}{2}} a^{2}}\) | \(30\) |
default | \(-\frac {2 \sqrt {b x +a}\, \left (-2 A b x +3 B a x +A a \right )}{3 x^{\frac {3}{2}} a^{2}}\) | \(30\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-2 A b x +3 B a x +A a \right )}{3 x^{\frac {3}{2}} a^{2}}\) | \(30\) |
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none
Time = 0.22 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.57 \[ \int \frac {A+B x}{x^{5/2} \sqrt {a+b x}} \, dx=-\frac {2 \, {\left (A a + {\left (3 \, B a - 2 \, A b\right )} x\right )} \sqrt {b x + a}}{3 \, a^{2} x^{\frac {3}{2}}} \]
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Time = 1.70 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x}{x^{5/2} \sqrt {a+b x}} \, dx=- \frac {2 A \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 a x} + \frac {4 A b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a^{2}} - \frac {2 B \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{a} \]
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none
Time = 0.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.17 \[ \int \frac {A+B x}{x^{5/2} \sqrt {a+b x}} \, dx=-\frac {2 \, \sqrt {b x^{2} + a x} B}{a x} + \frac {4 \, \sqrt {b x^{2} + a x} A b}{3 \, a^{2} x} - \frac {2 \, \sqrt {b x^{2} + a x} A}{3 \, a x^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.38 \[ \int \frac {A+B x}{x^{5/2} \sqrt {a+b x}} \, dx=-\frac {2 \, \sqrt {b x + a} b {\left (\frac {{\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} {\left (b x + a\right )}}{a^{2}} - \frac {3 \, {\left (B a^{2} b^{2} - A a b^{3}\right )}}{a^{2}}\right )}}{3 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}} {\left | b \right |}} \]
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Time = 0.98 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.64 \[ \int \frac {A+B x}{x^{5/2} \sqrt {a+b x}} \, dx=-\frac {\left (\frac {2\,A}{3\,a}-\frac {x\,\left (4\,A\,b-6\,B\,a\right )}{3\,a^2}\right )\,\sqrt {a+b\,x}}{x^{3/2}} \]
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