Integrand size = 13, antiderivative size = 53 \[ \int \frac {1}{x^{5/2} (a+b x)} \, dx=-\frac {2}{3 a x^{3/2}}+\frac {2 b}{a^2 \sqrt {x}}+\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \]
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Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {53, 65, 211} \[ \int \frac {1}{x^{5/2} (a+b x)} \, dx=\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2 b}{a^2 \sqrt {x}}-\frac {2}{3 a x^{3/2}} \]
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Rule 53
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 a x^{3/2}}-\frac {b \int \frac {1}{x^{3/2} (a+b x)} \, dx}{a} \\ & = -\frac {2}{3 a x^{3/2}}+\frac {2 b}{a^2 \sqrt {x}}+\frac {b^2 \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{a^2} \\ & = -\frac {2}{3 a x^{3/2}}+\frac {2 b}{a^2 \sqrt {x}}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a^2} \\ & = -\frac {2}{3 a x^{3/2}}+\frac {2 b}{a^2 \sqrt {x}}+\frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^{5/2} (a+b x)} \, dx=-\frac {2 (a-3 b x)}{3 a^2 x^{3/2}}+\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.75
method | result | size |
risch | \(-\frac {2 \left (-3 b x +a \right )}{3 a^{2} x^{\frac {3}{2}}}+\frac {2 b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}\) | \(40\) |
derivativedivides | \(\frac {2 b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}-\frac {2}{3 a \,x^{\frac {3}{2}}}+\frac {2 b}{a^{2} \sqrt {x}}\) | \(43\) |
default | \(\frac {2 b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}-\frac {2}{3 a \,x^{\frac {3}{2}}}+\frac {2 b}{a^{2} \sqrt {x}}\) | \(43\) |
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Time = 0.24 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.23 \[ \int \frac {1}{x^{5/2} (a+b x)} \, dx=\left [\frac {3 \, b x^{2} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (3 \, b x - a\right )} \sqrt {x}}{3 \, a^{2} x^{2}}, -\frac {2 \, {\left (3 \, b x^{2} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (3 \, b x - a\right )} \sqrt {x}\right )}}{3 \, a^{2} x^{2}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (49) = 98\).
Time = 2.43 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.02 \[ \int \frac {1}{x^{5/2} (a+b x)} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{5 b x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {2}{3 a x^{\frac {3}{2}}} & \text {for}\: b = 0 \\- \frac {2}{3 a x^{\frac {3}{2}}} + \frac {b \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{a^{2} \sqrt {- \frac {a}{b}}} - \frac {b \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{a^{2} \sqrt {- \frac {a}{b}}} + \frac {2 b}{a^{2} \sqrt {x}} & \text {otherwise} \end {cases} \]
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Time = 0.31 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^{5/2} (a+b x)} \, dx=\frac {2 \, b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {2 \, {\left (3 \, b x - a\right )}}{3 \, a^{2} x^{\frac {3}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^{5/2} (a+b x)} \, dx=\frac {2 \, b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {2 \, {\left (3 \, b x - a\right )}}{3 \, a^{2} x^{\frac {3}{2}}} \]
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Time = 0.12 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^{5/2} (a+b x)} \, dx=\frac {2\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\frac {2}{3\,a}-\frac {2\,b\,x}{a^2}}{x^{3/2}} \]
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