Integrand size = 13, antiderivative size = 68 \[ \int \frac {x^{5/2}}{a+b x} \, dx=\frac {2 a^2 \sqrt {x}}{b^3}-\frac {2 a x^{3/2}}{3 b^2}+\frac {2 x^{5/2}}{5 b}-\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{7/2}} \]
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Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {52, 65, 211} \[ \int \frac {x^{5/2}}{a+b x} \, dx=-\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{7/2}}+\frac {2 a^2 \sqrt {x}}{b^3}-\frac {2 a x^{3/2}}{3 b^2}+\frac {2 x^{5/2}}{5 b} \]
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Rule 52
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^{5/2}}{5 b}-\frac {a \int \frac {x^{3/2}}{a+b x} \, dx}{b} \\ & = -\frac {2 a x^{3/2}}{3 b^2}+\frac {2 x^{5/2}}{5 b}+\frac {a^2 \int \frac {\sqrt {x}}{a+b x} \, dx}{b^2} \\ & = \frac {2 a^2 \sqrt {x}}{b^3}-\frac {2 a x^{3/2}}{3 b^2}+\frac {2 x^{5/2}}{5 b}-\frac {a^3 \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{b^3} \\ & = \frac {2 a^2 \sqrt {x}}{b^3}-\frac {2 a x^{3/2}}{3 b^2}+\frac {2 x^{5/2}}{5 b}-\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b^3} \\ & = \frac {2 a^2 \sqrt {x}}{b^3}-\frac {2 a x^{3/2}}{3 b^2}+\frac {2 x^{5/2}}{5 b}-\frac {2 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{7/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int \frac {x^{5/2}}{a+b x} \, dx=\frac {2 \sqrt {x} \left (15 a^2-5 a b x+3 b^2 x^2\right )}{15 b^3}-\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{7/2}} \]
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Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78
method | result | size |
risch | \(\frac {2 \left (3 b^{2} x^{2}-5 a b x +15 a^{2}\right ) \sqrt {x}}{15 b^{3}}-\frac {2 a^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b^{3} \sqrt {a b}}\) | \(53\) |
derivativedivides | \(\frac {\frac {2 b^{2} x^{\frac {5}{2}}}{5}-\frac {2 a b \,x^{\frac {3}{2}}}{3}+2 a^{2} \sqrt {x}}{b^{3}}-\frac {2 a^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b^{3} \sqrt {a b}}\) | \(54\) |
default | \(\frac {\frac {2 b^{2} x^{\frac {5}{2}}}{5}-\frac {2 a b \,x^{\frac {3}{2}}}{3}+2 a^{2} \sqrt {x}}{b^{3}}-\frac {2 a^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b^{3} \sqrt {a b}}\) | \(54\) |
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Time = 0.23 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.94 \[ \int \frac {x^{5/2}}{a+b x} \, dx=\left [\frac {15 \, a^{2} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) + 2 \, {\left (3 \, b^{2} x^{2} - 5 \, a b x + 15 \, a^{2}\right )} \sqrt {x}}{15 \, b^{3}}, -\frac {2 \, {\left (15 \, a^{2} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) - {\left (3 \, b^{2} x^{2} - 5 \, a b x + 15 \, a^{2}\right )} \sqrt {x}\right )}}{15 \, b^{3}}\right ] \]
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Time = 2.46 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.79 \[ \int \frac {x^{5/2}}{a+b x} \, dx=\begin {cases} \tilde {\infty } x^{\frac {5}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {7}{2}}}{7 a} & \text {for}\: b = 0 \\\frac {2 x^{\frac {5}{2}}}{5 b} & \text {for}\: a = 0 \\- \frac {a^{3} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{b^{4} \sqrt {- \frac {a}{b}}} + \frac {a^{3} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{b^{4} \sqrt {- \frac {a}{b}}} + \frac {2 a^{2} \sqrt {x}}{b^{3}} - \frac {2 a x^{\frac {3}{2}}}{3 b^{2}} + \frac {2 x^{\frac {5}{2}}}{5 b} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.79 \[ \int \frac {x^{5/2}}{a+b x} \, dx=-\frac {2 \, a^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {2 \, {\left (3 \, b^{2} x^{\frac {5}{2}} - 5 \, a b x^{\frac {3}{2}} + 15 \, a^{2} \sqrt {x}\right )}}{15 \, b^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.87 \[ \int \frac {x^{5/2}}{a+b x} \, dx=-\frac {2 \, a^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {2 \, {\left (3 \, b^{4} x^{\frac {5}{2}} - 5 \, a b^{3} x^{\frac {3}{2}} + 15 \, a^{2} b^{2} \sqrt {x}\right )}}{15 \, b^{5}} \]
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Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.71 \[ \int \frac {x^{5/2}}{a+b x} \, dx=\frac {2\,x^{5/2}}{5\,b}-\frac {2\,a\,x^{3/2}}{3\,b^2}+\frac {2\,a^2\,\sqrt {x}}{b^3}-\frac {2\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{b^{7/2}} \]
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