Integrand size = 13, antiderivative size = 56 \[ \int \frac {1}{x^{3/2} (a+b x)^2} \, dx=-\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a+b x)}-\frac {3 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \]
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Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {44, 53, 65, 211} \[ \int \frac {1}{x^{3/2} (a+b x)^2} \, dx=-\frac {3 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a+b x)} \]
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Rule 44
Rule 53
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {1}{a \sqrt {x} (a+b x)}+\frac {3 \int \frac {1}{x^{3/2} (a+b x)} \, dx}{2 a} \\ & = -\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a+b x)}-\frac {(3 b) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 a^2} \\ & = -\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a+b x)}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a^2} \\ & = -\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a+b x)}-\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^{3/2} (a+b x)^2} \, dx=\frac {-2 a-3 b x}{a^2 \sqrt {x} (a+b x)}-\frac {3 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \]
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Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(-\frac {2 b \left (\frac {\sqrt {x}}{2 b x +2 a}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}-\frac {2}{a^{2} \sqrt {x}}\) | \(47\) |
default | \(-\frac {2 b \left (\frac {\sqrt {x}}{2 b x +2 a}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}-\frac {2}{a^{2} \sqrt {x}}\) | \(47\) |
risch | \(-\frac {2}{a^{2} \sqrt {x}}-\frac {b \sqrt {x}}{a^{2} \left (b x +a \right )}-\frac {3 b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}\) | \(48\) |
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Time = 0.23 (sec) , antiderivative size = 147, normalized size of antiderivative = 2.62 \[ \int \frac {1}{x^{3/2} (a+b x)^2} \, dx=\left [\frac {3 \, {\left (b x^{2} + a x\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) - 2 \, {\left (3 \, b x + 2 \, a\right )} \sqrt {x}}{2 \, {\left (a^{2} b x^{2} + a^{3} x\right )}}, \frac {3 \, {\left (b x^{2} + a x\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (3 \, b x + 2 \, a\right )} \sqrt {x}}{a^{2} b x^{2} + a^{3} x}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (51) = 102\).
Time = 8.01 (sec) , antiderivative size = 384, normalized size of antiderivative = 6.86 \[ \int \frac {1}{x^{3/2} (a+b x)^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{a^{2} \sqrt {x}} & \text {for}\: b = 0 \\- \frac {2}{5 b^{2} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {3 a \sqrt {x} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {3 a \sqrt {x} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {4 a \sqrt {- \frac {a}{b}}}{2 a^{3} \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {3 b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {3 b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {6 b x \sqrt {- \frac {a}{b}}}{2 a^{3} \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^{3/2} (a+b x)^2} \, dx=-\frac {3 \, b x + 2 \, a}{a^{2} b x^{\frac {3}{2}} + a^{3} \sqrt {x}} - \frac {3 \, b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} \]
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Time = 0.31 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^{3/2} (a+b x)^2} \, dx=-\frac {3 \, b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} - \frac {3 \, b x + 2 \, a}{{\left (b x^{\frac {3}{2}} + a \sqrt {x}\right )} a^{2}} \]
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Time = 0.13 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^{3/2} (a+b x)^2} \, dx=-\frac {\frac {2}{a}+\frac {3\,b\,x}{a^2}}{a\,\sqrt {x}+b\,x^{3/2}}-\frac {3\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \]
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