Integrand size = 13, antiderivative size = 118 \[ \int \left (a x+b x^2\right )^{5/2} \, dx=\frac {5 a^4 (a+2 b x) \sqrt {a x+b x^2}}{512 b^3}-\frac {5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac {5 a^6 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{512 b^{7/2}} \]
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Time = 0.03 (sec) , antiderivative size = 118, 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 = {626, 634, 212} \[ \int \left (a x+b x^2\right )^{5/2} \, dx=-\frac {5 a^6 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{512 b^{7/2}}+\frac {5 a^4 (a+2 b x) \sqrt {a x+b x^2}}{512 b^3}-\frac {5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b} \]
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Rule 212
Rule 626
Rule 634
Rubi steps \begin{align*} \text {integral}& = \frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac {\left (5 a^2\right ) \int \left (a x+b x^2\right )^{3/2} \, dx}{24 b} \\ & = -\frac {5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}+\frac {\left (5 a^4\right ) \int \sqrt {a x+b x^2} \, dx}{128 b^2} \\ & = \frac {5 a^4 (a+2 b x) \sqrt {a x+b x^2}}{512 b^3}-\frac {5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac {\left (5 a^6\right ) \int \frac {1}{\sqrt {a x+b x^2}} \, dx}{1024 b^3} \\ & = \frac {5 a^4 (a+2 b x) \sqrt {a x+b x^2}}{512 b^3}-\frac {5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac {\left (5 a^6\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a x+b x^2}}\right )}{512 b^3} \\ & = \frac {5 a^4 (a+2 b x) \sqrt {a x+b x^2}}{512 b^3}-\frac {5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac {5 a^6 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{512 b^{7/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.09 \[ \int \left (a x+b x^2\right )^{5/2} \, dx=\frac {\sqrt {x (a+b x)} \left (\sqrt {b} \left (15 a^5-10 a^4 b x+8 a^3 b^2 x^2+432 a^2 b^3 x^3+640 a b^4 x^4+256 b^5 x^5\right )+\frac {30 a^6 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )}{\sqrt {x} \sqrt {a+b x}}\right )}{1536 b^{7/2}} \]
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Time = 2.02 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {\left (256 b^{5} x^{5}+640 a \,b^{4} x^{4}+432 a^{2} b^{3} x^{3}+8 a^{3} b^{2} x^{2}-10 a^{4} b x +15 a^{5}\right ) x \left (b x +a \right )}{1536 b^{3} \sqrt {x \left (b x +a \right )}}-\frac {5 a^{6} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{1024 b^{\frac {7}{2}}}\) | \(106\) |
default | \(\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{12 b}-\frac {5 a^{2} \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{24 b}\) | \(118\) |
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Time = 0.28 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.81 \[ \int \left (a x+b x^2\right )^{5/2} \, dx=\left [\frac {15 \, a^{6} \sqrt {b} \log \left (2 \, b x + a - 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (256 \, b^{6} x^{5} + 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} + 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x + 15 \, a^{5} b\right )} \sqrt {b x^{2} + a x}}{3072 \, b^{4}}, \frac {15 \, a^{6} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x}\right ) + {\left (256 \, b^{6} x^{5} + 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} + 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x + 15 \, a^{5} b\right )} \sqrt {b x^{2} + a x}}{1536 \, b^{4}}\right ] \]
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Time = 0.56 (sec) , antiderivative size = 459, normalized size of antiderivative = 3.89 \[ \int \left (a x+b x^2\right )^{5/2} \, dx=a^{2} \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 ) + 2 a b \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 ) + b^{2} \left (\begin {cases} - \frac {21 a^{6} \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 )}{1024 b^{5}} + \sqrt {a x + b x^{2}} \cdot \left (\frac {21 a^{5}}{512 b^{5}} - \frac {7 a^{4} x}{256 b^{4}} + \frac {7 a^{3} x^{2}}{320 b^{3}} - \frac {3 a^{2} x^{3}}{160 b^{2}} + \frac {a x^{4}}{60 b} + \frac {x^{5}}{6}\right ) & \text {for}\: b \neq 0 \\\frac {2 \left (a x\right )^{\frac {11}{2}}}{11 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.19 \[ \int \left (a x+b x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} x + \frac {5 \, \sqrt {b x^{2} + a x} a^{4} x}{256 \, b^{2}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} x}{96 \, b} - \frac {5 \, a^{6} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{1024 \, b^{\frac {7}{2}}} + \frac {5 \, \sqrt {b x^{2} + a x} a^{5}}{512 \, b^{3}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3}}{192 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} a}{12 \, b} \]
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Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.89 \[ \int \left (a x+b x^2\right )^{5/2} \, dx=\frac {5 \, a^{6} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{1024 \, b^{\frac {7}{2}}} + \frac {1}{1536} \, \sqrt {b x^{2} + a x} {\left (\frac {15 \, a^{5}}{b^{3}} - 2 \, {\left (\frac {5 \, a^{4}}{b^{2}} - 4 \, {\left (\frac {a^{3}}{b} + 2 \, {\left (27 \, a^{2} + 8 \, {\left (2 \, b^{2} x + 5 \, a b\right )} x\right )} x\right )} x\right )} x\right )} \]
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Time = 9.47 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.01 \[ \int \left (a x+b x^2\right )^{5/2} \, dx=\frac {{\left (b\,x^2+a\,x\right )}^{5/2}\,\left (\frac {a}{2}+b\,x\right )}{6\,b}-\frac {5\,a^2\,\left (\frac {{\left (b\,x^2+a\,x\right )}^{3/2}\,\left (\frac {a}{2}+b\,x\right )}{4\,b}-\frac {3\,a^2\,\left (\sqrt {b\,x^2+a\,x}\,\left (\frac {x}{2}+\frac {a}{4\,b}\right )-\frac {a^2\,\ln \left (\frac {\frac {a}{2}+b\,x}{\sqrt {b}}+\sqrt {b\,x^2+a\,x}\right )}{8\,b^{3/2}}\right )}{16\,b}\right )}{24\,b} \]
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