Integrand size = 13, antiderivative size = 70 \[ \int \frac {1}{\sqrt {x} (a+b x)^3} \, dx=\frac {\sqrt {x}}{2 a (a+b x)^2}+\frac {3 \sqrt {x}}{4 a^2 (a+b x)}+\frac {3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}} \]
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Time = 0.01 (sec) , antiderivative size = 70, 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 = {44, 65, 211} \[ \int \frac {1}{\sqrt {x} (a+b x)^3} \, dx=\frac {3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}}+\frac {3 \sqrt {x}}{4 a^2 (a+b x)}+\frac {\sqrt {x}}{2 a (a+b x)^2} \]
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Rule 44
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {x}}{2 a (a+b x)^2}+\frac {3 \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx}{4 a} \\ & = \frac {\sqrt {x}}{2 a (a+b x)^2}+\frac {3 \sqrt {x}}{4 a^2 (a+b x)}+\frac {3 \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 a^2} \\ & = \frac {\sqrt {x}}{2 a (a+b x)^2}+\frac {3 \sqrt {x}}{4 a^2 (a+b x)}+\frac {3 \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a^2} \\ & = \frac {\sqrt {x}}{2 a (a+b x)^2}+\frac {3 \sqrt {x}}{4 a^2 (a+b x)}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\sqrt {x} (a+b x)^3} \, dx=\frac {\sqrt {x} (5 a+3 b x)}{4 a^2 (a+b x)^2}+\frac {3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}} \]
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Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {\sqrt {x}}{2 a \left (b x +a \right )^{2}}+\frac {\frac {3 \sqrt {x}}{4 a \left (b x +a \right )}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 a \sqrt {a b}}}{a}\) | \(59\) |
default | \(\frac {\sqrt {x}}{2 a \left (b x +a \right )^{2}}+\frac {\frac {3 \sqrt {x}}{4 a \left (b x +a \right )}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 a \sqrt {a b}}}{a}\) | \(59\) |
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Time = 0.23 (sec) , antiderivative size = 186, normalized size of antiderivative = 2.66 \[ \int \frac {1}{\sqrt {x} (a+b x)^3} \, dx=\left [-\frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (3 \, a b^{2} x + 5 \, a^{2} b\right )} \sqrt {x}}{8 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}}, -\frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) - {\left (3 \, a b^{2} x + 5 \, a^{2} b\right )} \sqrt {x}}{4 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 632 vs. \(2 (61) = 122\).
Time = 12.06 (sec) , antiderivative size = 632, normalized size of antiderivative = 9.03 \[ \int \frac {1}{\sqrt {x} (a+b x)^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 \sqrt {x}}{a^{3}} & \text {for}\: b = 0 \\- \frac {2}{5 b^{3} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\\frac {3 a^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{4} b \sqrt {- \frac {a}{b}} + 16 a^{3} b^{2} x \sqrt {- \frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {- \frac {a}{b}}} - \frac {3 a^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{4} b \sqrt {- \frac {a}{b}} + 16 a^{3} b^{2} x \sqrt {- \frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {- \frac {a}{b}}} + \frac {10 a b \sqrt {x} \sqrt {- \frac {a}{b}}}{8 a^{4} b \sqrt {- \frac {a}{b}} + 16 a^{3} b^{2} x \sqrt {- \frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {- \frac {a}{b}}} + \frac {6 a b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{4} b \sqrt {- \frac {a}{b}} + 16 a^{3} b^{2} x \sqrt {- \frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {- \frac {a}{b}}} - \frac {6 a b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{4} b \sqrt {- \frac {a}{b}} + 16 a^{3} b^{2} x \sqrt {- \frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {- \frac {a}{b}}} + \frac {6 b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{8 a^{4} b \sqrt {- \frac {a}{b}} + 16 a^{3} b^{2} x \sqrt {- \frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {- \frac {a}{b}}} + \frac {3 b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{4} b \sqrt {- \frac {a}{b}} + 16 a^{3} b^{2} x \sqrt {- \frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {- \frac {a}{b}}} - \frac {3 b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{4} b \sqrt {- \frac {a}{b}} + 16 a^{3} b^{2} x \sqrt {- \frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.32 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt {x} (a+b x)^3} \, dx=\frac {3 \, b x^{\frac {3}{2}} + 5 \, a \sqrt {x}}{4 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} + \frac {3 \, \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt {x} (a+b x)^3} \, dx=\frac {3 \, \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{2}} + \frac {3 \, b x^{\frac {3}{2}} + 5 \, a \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{2}} \]
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Time = 0.14 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {x} (a+b x)^3} \, dx=\frac {\frac {5\,\sqrt {x}}{4\,a}+\frac {3\,b\,x^{3/2}}{4\,a^2}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,a^{5/2}\,\sqrt {b}} \]
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