Integrand size = 17, antiderivative size = 131 \[ \int x^3 \sqrt {b x+c x^2} \, dx=-\frac {7 b^3 (b+2 c x) \sqrt {b x+c x^2}}{128 c^4}+\frac {7 b^2 \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac {7 b x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac {x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {7 b^5 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{9/2}} \]
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Time = 0.04 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {684, 654, 626, 634, 212} \[ \int x^3 \sqrt {b x+c x^2} \, dx=\frac {7 b^5 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{9/2}}-\frac {7 b^3 (b+2 c x) \sqrt {b x+c x^2}}{128 c^4}+\frac {7 b^2 \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac {7 b x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac {x^2 \left (b x+c x^2\right )^{3/2}}{5 c} \]
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Rule 212
Rule 626
Rule 634
Rule 654
Rule 684
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \left (b x+c x^2\right )^{3/2}}{5 c}-\frac {(7 b) \int x^2 \sqrt {b x+c x^2} \, dx}{10 c} \\ & = -\frac {7 b x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac {x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {\left (7 b^2\right ) \int x \sqrt {b x+c x^2} \, dx}{16 c^2} \\ & = \frac {7 b^2 \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac {7 b x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac {x^2 \left (b x+c x^2\right )^{3/2}}{5 c}-\frac {\left (7 b^3\right ) \int \sqrt {b x+c x^2} \, dx}{32 c^3} \\ & = -\frac {7 b^3 (b+2 c x) \sqrt {b x+c x^2}}{128 c^4}+\frac {7 b^2 \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac {7 b x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac {x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {\left (7 b^5\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{256 c^4} \\ & = -\frac {7 b^3 (b+2 c x) \sqrt {b x+c x^2}}{128 c^4}+\frac {7 b^2 \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac {7 b x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac {x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {\left (7 b^5\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{128 c^4} \\ & = -\frac {7 b^3 (b+2 c x) \sqrt {b x+c x^2}}{128 c^4}+\frac {7 b^2 \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac {7 b x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac {x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {7 b^5 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{9/2}} \\ \end{align*}
Time = 0.64 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.90 \[ \int x^3 \sqrt {b x+c x^2} \, dx=\frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (-105 b^4+70 b^3 c x-56 b^2 c^2 x^2+48 b c^3 x^3+384 c^4 x^4\right )+\frac {210 b^5 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )}{\sqrt {x} \sqrt {b+c x}}\right )}{1920 c^{9/2}} \]
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Time = 3.47 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.64
method | result | size |
pseudoelliptic | \(\frac {\frac {7 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right ) b^{5}}{128}-\frac {7 \left (\sqrt {c}\, b^{4}-\frac {2 c^{\frac {3}{2}} b^{3} x}{3}+\frac {8 c^{\frac {5}{2}} b^{2} x^{2}}{15}-\frac {16 c^{\frac {7}{2}} b \,x^{3}}{35}-\frac {128 c^{\frac {9}{2}} x^{4}}{35}\right ) \sqrt {x \left (c x +b \right )}}{128}}{c^{\frac {9}{2}}}\) | \(84\) |
risch | \(-\frac {\left (-384 c^{4} x^{4}-48 b \,c^{3} x^{3}+56 b^{2} c^{2} x^{2}-70 b^{3} c x +105 b^{4}\right ) x \left (c x +b \right )}{1920 c^{4} \sqrt {x \left (c x +b \right )}}+\frac {7 b^{5} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{256 c^{\frac {9}{2}}}\) | \(95\) |
default | \(\frac {x^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{5 c}-\frac {7 b \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}\right )}{10 c}\) | \(129\) |
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Time = 0.27 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.47 \[ \int x^3 \sqrt {b x+c x^2} \, dx=\left [\frac {105 \, b^{5} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (384 \, c^{5} x^{4} + 48 \, b c^{4} x^{3} - 56 \, b^{2} c^{3} x^{2} + 70 \, b^{3} c^{2} x - 105 \, b^{4} c\right )} \sqrt {c x^{2} + b x}}{3840 \, c^{5}}, -\frac {105 \, b^{5} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (384 \, c^{5} x^{4} + 48 \, b c^{4} x^{3} - 56 \, b^{2} c^{3} x^{2} + 70 \, b^{3} c^{2} x - 105 \, b^{4} c\right )} \sqrt {c x^{2} + b x}}{1920 \, c^{5}}\right ] \]
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Time = 0.29 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.13 \[ \int x^3 \sqrt {b x+c x^2} \, dx=\begin {cases} \frac {7 b^{5} \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{256 c^{4}} + \sqrt {b x + c x^{2}} \left (- \frac {7 b^{4}}{128 c^{4}} + \frac {7 b^{3} x}{192 c^{3}} - \frac {7 b^{2} x^{2}}{240 c^{2}} + \frac {b x^{3}}{40 c} + \frac {x^{4}}{5}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (b x\right )^{\frac {9}{2}}}{9 b^{4}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.97 \[ \int x^3 \sqrt {b x+c x^2} \, dx=\frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} x^{2}}{5 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} b^{3} x}{64 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b x}{40 \, c^{2}} + \frac {7 \, b^{5} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {9}{2}}} - \frac {7 \, \sqrt {c x^{2} + b x} b^{4}}{128 \, c^{4}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}}{48 \, c^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.73 \[ \int x^3 \sqrt {b x+c x^2} \, dx=\frac {1}{1920} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x + \frac {b}{c}\right )} x - \frac {7 \, b^{2}}{c^{2}}\right )} x + \frac {35 \, b^{3}}{c^{3}}\right )} x - \frac {105 \, b^{4}}{c^{4}}\right )} - \frac {7 \, b^{5} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {9}{2}}} \]
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Time = 9.46 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.91 \[ \int x^3 \sqrt {b x+c x^2} \, dx=\frac {x^2\,{\left (c\,x^2+b\,x\right )}^{3/2}}{5\,c}-\frac {7\,b\,\left (\frac {x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}-\frac {5\,b\,\left (\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}\right )}{8\,c}\right )}{10\,c} \]
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