Integrand size = 17, antiderivative size = 107 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x} \, dx=-\frac {3 a^3 (a+2 b x) \sqrt {a x+b x^2}}{128 b^2}+\frac {a (a+2 b x) \left (a x+b x^2\right )^{3/2}}{16 b}+\frac {1}{5} \left (a x+b x^2\right )^{5/2}+\frac {3 a^5 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{128 b^{5/2}} \]
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Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {678, 626, 634, 212} \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x} \, dx=\frac {3 a^5 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{128 b^{5/2}}-\frac {3 a^3 (a+2 b x) \sqrt {a x+b x^2}}{128 b^2}+\frac {a (a+2 b x) \left (a x+b x^2\right )^{3/2}}{16 b}+\frac {1}{5} \left (a x+b x^2\right )^{5/2} \]
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Rule 212
Rule 626
Rule 634
Rule 678
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \left (a x+b x^2\right )^{5/2}+\frac {1}{2} a \int \left (a x+b x^2\right )^{3/2} \, dx \\ & = \frac {a (a+2 b x) \left (a x+b x^2\right )^{3/2}}{16 b}+\frac {1}{5} \left (a x+b x^2\right )^{5/2}-\frac {\left (3 a^3\right ) \int \sqrt {a x+b x^2} \, dx}{32 b} \\ & = -\frac {3 a^3 (a+2 b x) \sqrt {a x+b x^2}}{128 b^2}+\frac {a (a+2 b x) \left (a x+b x^2\right )^{3/2}}{16 b}+\frac {1}{5} \left (a x+b x^2\right )^{5/2}+\frac {\left (3 a^5\right ) \int \frac {1}{\sqrt {a x+b x^2}} \, dx}{256 b^2} \\ & = -\frac {3 a^3 (a+2 b x) \sqrt {a x+b x^2}}{128 b^2}+\frac {a (a+2 b x) \left (a x+b x^2\right )^{3/2}}{16 b}+\frac {1}{5} \left (a x+b x^2\right )^{5/2}+\frac {\left (3 a^5\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a x+b x^2}}\right )}{128 b^2} \\ & = -\frac {3 a^3 (a+2 b x) \sqrt {a x+b x^2}}{128 b^2}+\frac {a (a+2 b x) \left (a x+b x^2\right )^{3/2}}{16 b}+\frac {1}{5} \left (a x+b x^2\right )^{5/2}+\frac {3 a^5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{128 b^{5/2}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x} \, dx=\frac {\sqrt {x (a+b x)} \left (\sqrt {b} \left (-15 a^4+10 a^3 b x+248 a^2 b^2 x^2+336 a b^3 x^3+128 b^4 x^4\right )+\frac {30 a^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{\sqrt {x} \sqrt {a+b x}}\right )}{640 b^{5/2}} \]
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Time = 2.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(\frac {\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right ) a^{5}}{128}-\frac {3 \sqrt {x \left (b x +a \right )}\, \left (a^{4} \sqrt {b}-\frac {2 a^{3} b^{\frac {3}{2}} x}{3}-\frac {248 b^{\frac {5}{2}} a^{2} x^{2}}{15}-\frac {112 b^{\frac {7}{2}} a \,x^{3}}{5}-\frac {128 x^{4} b^{\frac {9}{2}}}{15}\right )}{128}}{b^{\frac {5}{2}}}\) | \(84\) |
risch | \(-\frac {\left (-128 b^{4} x^{4}-336 a \,b^{3} x^{3}-248 a^{2} b^{2} x^{2}-10 a^{3} b x +15 a^{4}\right ) x \left (b x +a \right )}{640 b^{2} \sqrt {x \left (b x +a \right )}}+\frac {3 a^{5} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{256 b^{\frac {5}{2}}}\) | \(95\) |
default | \(\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5}+\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{2}\) | \(104\) |
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Time = 0.28 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.79 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x} \, dx=\left [\frac {15 \, a^{5} \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (128 \, b^{5} x^{4} + 336 \, a b^{4} x^{3} + 248 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x - 15 \, a^{4} b\right )} \sqrt {b x^{2} + a x}}{1280 \, b^{3}}, -\frac {15 \, a^{5} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x}\right ) - {\left (128 \, b^{5} x^{4} + 336 \, a b^{4} x^{3} + 248 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x - 15 \, a^{4} b\right )} \sqrt {b x^{2} + a x}}{640 \, b^{3}}\right ] \]
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Time = 1.63 (sec) , antiderivative size = 415, normalized size of antiderivative = 3.88 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x} \, dx=a^{2} \left (\begin {cases} \frac {a^{3} \left (\begin {cases} \frac {\log {\left (a + 2 \sqrt {b} \sqrt {a x + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: \frac {a^{2}}{b} \neq 0 \\\frac {\left (\frac {a}{2 b} + x\right ) \log {\left (\frac {a}{2 b} + x \right )}}{\sqrt {b \left (\frac {a}{2 b} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{16 b^{2}} + \sqrt {a x + b x^{2}} \left (- \frac {a^{2}}{8 b^{2}} + \frac {a x}{12 b} + \frac {x^{2}}{3}\right ) & \text {for}\: b \neq 0 \\\frac {2 \left (a x\right )^{\frac {5}{2}}}{5 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases}\right ) + 2 a b \left (\begin {cases} - \frac {5 a^{4} \left (\begin {cases} \frac {\log {\left (a + 2 \sqrt {b} \sqrt {a x + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: \frac {a^{2}}{b} \neq 0 \\\frac {\left (\frac {a}{2 b} + x\right ) \log {\left (\frac {a}{2 b} + x \right )}}{\sqrt {b \left (\frac {a}{2 b} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{128 b^{3}} + \sqrt {a x + b x^{2}} \cdot \left (\frac {5 a^{3}}{64 b^{3}} - \frac {5 a^{2} x}{96 b^{2}} + \frac {a x^{2}}{24 b} + \frac {x^{3}}{4}\right ) & \text {for}\: b \neq 0 \\\frac {2 \left (a x\right )^{\frac {7}{2}}}{7 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases}\right ) + b^{2} \left (\begin {cases} \frac {7 a^{5} \left (\begin {cases} \frac {\log {\left (a + 2 \sqrt {b} \sqrt {a x + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: \frac {a^{2}}{b} \neq 0 \\\frac {\left (\frac {a}{2 b} + x\right ) \log {\left (\frac {a}{2 b} + x \right )}}{\sqrt {b \left (\frac {a}{2 b} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{256 b^{4}} + \sqrt {a x + b x^{2}} \left (- \frac {7 a^{4}}{128 b^{4}} + \frac {7 a^{3} x}{192 b^{3}} - \frac {7 a^{2} x^{2}}{240 b^{2}} + \frac {a x^{3}}{40 b} + \frac {x^{4}}{5}\right ) & \text {for}\: b \neq 0 \\\frac {2 \left (a x\right )^{\frac {9}{2}}}{9 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x} \, dx=\frac {1}{8} \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a x - \frac {3 \, \sqrt {b x^{2} + a x} a^{3} x}{64 \, b} + \frac {3 \, a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {5}{2}}} + \frac {1}{5} \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} - \frac {3 \, \sqrt {b x^{2} + a x} a^{4}}{128 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2}}{16 \, b} \]
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Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x} \, dx=-\frac {3 \, a^{5} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{256 \, b^{\frac {5}{2}}} - \frac {1}{640} \, \sqrt {b x^{2} + a x} {\left (\frac {15 \, a^{4}}{b^{2}} - 2 \, {\left (\frac {5 \, a^{3}}{b} + 4 \, {\left (31 \, a^{2} + 2 \, {\left (8 \, b^{2} x + 21 \, a b\right )} x\right )} x\right )} x\right )} \]
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Timed out. \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x} \, dx=\int \frac {{\left (b\,x^2+a\,x\right )}^{5/2}}{x} \,d x \]
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