Integrand size = 21, antiderivative size = 210 \[ \int (d+e x)^3 \sqrt {b x+c x^2} \, dx=\frac {(2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {e \left (192 c^2 d^2-150 b c d e+35 b^2 e^2+42 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{240 c^3}-\frac {b^2 (2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{9/2}} \]
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Time = 0.17 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {756, 793, 626, 634, 212} \[ \int (d+e x)^3 \sqrt {b x+c x^2} \, dx=-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right )}{128 c^{9/2}}+\frac {(b+2 c x) \sqrt {b x+c x^2} (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right )}{128 c^4}+\frac {e \left (b x+c x^2\right )^{3/2} \left (35 b^2 e^2+42 c e x (2 c d-b e)-150 b c d e+192 c^2 d^2\right )}{240 c^3}+\frac {e \left (b x+c x^2\right )^{3/2} (d+e x)^2}{5 c} \]
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Rule 212
Rule 626
Rule 634
Rule 756
Rule 793
Rubi steps \begin{align*} \text {integral}& = \frac {e (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {\int (d+e x) \left (\frac {1}{2} d (10 c d-3 b e)+\frac {7}{2} e (2 c d-b e) x\right ) \sqrt {b x+c x^2} \, dx}{5 c} \\ & = \frac {e (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {e \left (192 c^2 d^2-150 b c d e+35 b^2 e^2+42 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{240 c^3}+\frac {\left ((2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right )\right ) \int \sqrt {b x+c x^2} \, dx}{32 c^3} \\ & = \frac {(2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {e \left (192 c^2 d^2-150 b c d e+35 b^2 e^2+42 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{240 c^3}-\frac {\left (b^2 (2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{256 c^4} \\ & = \frac {(2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {e \left (192 c^2 d^2-150 b c d e+35 b^2 e^2+42 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{240 c^3}-\frac {\left (b^2 (2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right )\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 {(2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {e \left (192 c^2 d^2-150 b c d e+35 b^2 e^2+42 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{240 c^3}-\frac {b^2 (2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{9/2}} \\ \end{align*}
Time = 1.78 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.34 \[ \int (d+e x)^3 \sqrt {b x+c x^2} \, dx=\frac {\sqrt {x (b+c x)} \left (480 b c^3 d^3-720 b^2 c^2 d^2 e+450 b^3 c d e^2-105 b^4 e^3+960 c^4 d^3 x+480 b c^3 d^2 e x-300 b^2 c^2 d e^2 x+70 b^3 c e^3 x+1920 c^4 d^2 e x^2+240 b c^3 d e^2 x^2-56 b^2 c^2 e^3 x^2+1440 c^4 d e^2 x^3+48 b c^3 e^3 x^3+384 c^4 e^3 x^4\right )}{1920 c^4}+\frac {b^2 \left (-32 c^3 d^3+48 b c^2 d^2 e-30 b^2 c d e^2+7 b^3 e^3\right ) \sqrt {x (b+c x)} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )}{64 c^{9/2} \sqrt {x} \sqrt {b+c x}} \]
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Time = 1.99 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.90
method | result | size |
pseudoelliptic | \(\frac {\frac {7 \left (b^{2} e^{2}-\frac {16}{7} b c d e +\frac {16}{7} c^{2} d^{2}\right ) \left (b e -2 c d \right ) b^{2} \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )}{128}-\frac {7 \left (-\frac {32 \left (\frac {1}{10} e^{3} x^{3}+\frac {1}{2} d \,e^{2} x^{2}+d^{2} e x +d^{3}\right ) b \,c^{\frac {7}{2}}}{7}-\frac {64 x \left (\frac {2}{5} e^{3} x^{3}+\frac {3}{2} d \,e^{2} x^{2}+2 d^{2} e x +d^{3}\right ) c^{\frac {9}{2}}}{7}+\left (\left (\frac {8}{15} x^{2} e^{2}+\frac {20}{7} d e x +\frac {48}{7} d^{2}\right ) c^{\frac {5}{2}}+\left (\left (-\frac {2 e x}{3}-\frac {30 d}{7}\right ) c^{\frac {3}{2}}+\sqrt {c}\, b e \right ) e b \right ) e \,b^{2}\right ) \sqrt {x \left (c x +b \right )}}{128}}{c^{\frac {9}{2}}}\) | \(188\) |
risch | \(-\frac {\left (-384 c^{4} e^{3} x^{4}-48 b \,c^{3} e^{3} x^{3}-1440 c^{4} d \,e^{2} x^{3}+56 b^{2} c^{2} e^{3} x^{2}-240 b \,c^{3} d \,e^{2} x^{2}-1920 c^{4} d^{2} e \,x^{2}-70 b^{3} c \,e^{3} x +300 b^{2} c^{2} d \,e^{2} x -480 b \,c^{3} d^{2} e x -960 c^{4} d^{3} x +105 b^{4} e^{3}-450 b^{3} c d \,e^{2}+720 b^{2} c^{2} d^{2} e -480 c^{3} b \,d^{3}\right ) x \left (c x +b \right )}{1920 c^{4} \sqrt {x \left (c x +b \right )}}+\frac {b^{2} \left (7 b^{3} e^{3}-30 b^{2} d \,e^{2} c +48 b \,c^{2} d^{2} e -32 c^{3} d^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{256 c^{\frac {9}{2}}}\) | \(248\) |
default | \(d^{3} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )+e^{3} \left (\frac {x^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{5 c}-\frac {7 b \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}\right )}{10 c}\right )+3 d \,e^{2} \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}\right )+3 d^{2} e \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )\) | \(385\) |
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Time = 0.32 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.37 \[ \int (d+e x)^3 \sqrt {b x+c x^2} \, dx=\left [-\frac {15 \, {\left (32 \, b^{2} c^{3} d^{3} - 48 \, b^{3} c^{2} d^{2} e + 30 \, b^{4} c d e^{2} - 7 \, b^{5} e^{3}\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (384 \, c^{5} e^{3} x^{4} + 480 \, b c^{4} d^{3} - 720 \, b^{2} c^{3} d^{2} e + 450 \, b^{3} c^{2} d e^{2} - 105 \, b^{4} c e^{3} + 48 \, {\left (30 \, c^{5} d e^{2} + b c^{4} e^{3}\right )} x^{3} + 8 \, {\left (240 \, c^{5} d^{2} e + 30 \, b c^{4} d e^{2} - 7 \, b^{2} c^{3} e^{3}\right )} x^{2} + 10 \, {\left (96 \, c^{5} d^{3} + 48 \, b c^{4} d^{2} e - 30 \, b^{2} c^{3} d e^{2} + 7 \, b^{3} c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{3840 \, c^{5}}, \frac {15 \, {\left (32 \, b^{2} c^{3} d^{3} - 48 \, b^{3} c^{2} d^{2} e + 30 \, b^{4} c d e^{2} - 7 \, b^{5} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (384 \, c^{5} e^{3} x^{4} + 480 \, b c^{4} d^{3} - 720 \, b^{2} c^{3} d^{2} e + 450 \, b^{3} c^{2} d e^{2} - 105 \, b^{4} c e^{3} + 48 \, {\left (30 \, c^{5} d e^{2} + b c^{4} e^{3}\right )} x^{3} + 8 \, {\left (240 \, c^{5} d^{2} e + 30 \, b c^{4} d e^{2} - 7 \, b^{2} c^{3} e^{3}\right )} x^{2} + 10 \, {\left (96 \, c^{5} d^{3} + 48 \, b c^{4} d^{2} e - 30 \, b^{2} c^{3} d e^{2} + 7 \, b^{3} c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{1920 \, c^{5}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (204) = 408\).
Time = 0.47 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.07 \[ \int (d+e x)^3 \sqrt {b x+c x^2} \, dx=\begin {cases} - \frac {b \left (b d^{3} - \frac {3 b \left (3 b d^{2} e - \frac {5 b \left (3 b d e^{2} - \frac {7 b \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{8 c} + 3 c d^{2} e\right )}{6 c} + c d^{3}\right )}{4 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{2 c} + \sqrt {b x + c x^{2}} \left (\frac {e^{3} x^{4}}{5} + \frac {x^{3} \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{4 c} + \frac {x^{2} \cdot \left (3 b d e^{2} - \frac {7 b \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{8 c} + 3 c d^{2} e\right )}{3 c} + \frac {x \left (3 b d^{2} e - \frac {5 b \left (3 b d e^{2} - \frac {7 b \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{8 c} + 3 c d^{2} e\right )}{6 c} + c d^{3}\right )}{2 c} + \frac {b d^{3} - \frac {3 b \left (3 b d^{2} e - \frac {5 b \left (3 b d e^{2} - \frac {7 b \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{8 c} + 3 c d^{2} e\right )}{6 c} + c d^{3}\right )}{4 c}}{c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {d^{3} \left (b x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} e \left (b x\right )^{\frac {5}{2}}}{5 b} + \frac {3 d e^{2} \left (b x\right )^{\frac {7}{2}}}{7 b^{2}} + \frac {e^{3} \left (b x\right )^{\frac {9}{2}}}{9 b^{3}}\right )}{b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (190) = 380\).
Time = 0.21 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.09 \[ \int (d+e x)^3 \sqrt {b x+c x^2} \, dx=\frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} e^{3} x^{2}}{5 \, c} + \frac {1}{2} \, \sqrt {c x^{2} + b x} d^{3} x - \frac {3 \, \sqrt {c x^{2} + b x} b d^{2} e x}{4 \, c} + \frac {15 \, \sqrt {c x^{2} + b x} b^{2} d e^{2} x}{32 \, c^{2}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} d e^{2} x}{4 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} b^{3} e^{3} x}{64 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b e^{3} x}{40 \, c^{2}} - \frac {b^{2} d^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {3}{2}}} + \frac {3 \, b^{3} d^{2} e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {5}{2}}} - \frac {15 \, b^{4} d e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {7}{2}}} + \frac {7 \, b^{5} e^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {9}{2}}} + \frac {\sqrt {c x^{2} + b x} b d^{3}}{4 \, c} - \frac {3 \, \sqrt {c x^{2} + b x} b^{2} d^{2} e}{8 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} d^{2} e}{c} + \frac {15 \, \sqrt {c x^{2} + b x} b^{3} d e^{2}}{64 \, c^{3}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b d e^{2}}{8 \, c^{2}} - \frac {7 \, \sqrt {c x^{2} + b x} b^{4} e^{3}}{128 \, c^{4}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} e^{3}}{48 \, c^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.21 \[ \int (d+e x)^3 \sqrt {b x+c x^2} \, dx=\frac {1}{1920} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, e^{3} x + \frac {30 \, c^{4} d e^{2} + b c^{3} e^{3}}{c^{4}}\right )} x + \frac {240 \, c^{4} d^{2} e + 30 \, b c^{3} d e^{2} - 7 \, b^{2} c^{2} e^{3}}{c^{4}}\right )} x + \frac {5 \, {\left (96 \, c^{4} d^{3} + 48 \, b c^{3} d^{2} e - 30 \, b^{2} c^{2} d e^{2} + 7 \, b^{3} c e^{3}\right )}}{c^{4}}\right )} x + \frac {15 \, {\left (32 \, b c^{3} d^{3} - 48 \, b^{2} c^{2} d^{2} e + 30 \, b^{3} c d e^{2} - 7 \, b^{4} e^{3}\right )}}{c^{4}}\right )} + \frac {{\left (32 \, b^{2} c^{3} d^{3} - 48 \, b^{3} c^{2} d^{2} e + 30 \, b^{4} c d e^{2} - 7 \, b^{5} e^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {9}{2}}} \]
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Time = 10.24 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.72 \[ \int (d+e x)^3 \sqrt {b x+c x^2} \, dx=d^3\,\sqrt {c\,x^2+b\,x}\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )-\frac {7\,b\,e^3\,\left (\frac {x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}-\frac {5\,b\,\left (\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}\right )}{8\,c}\right )}{10\,c}+\frac {e^3\,x^2\,{\left (c\,x^2+b\,x\right )}^{3/2}}{5\,c}-\frac {b^2\,d^3\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{8\,c^{3/2}}+\frac {3\,d\,e^2\,x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}-\frac {15\,b\,d\,e^2\,\left (\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}\right )}{8\,c}+\frac {3\,b^3\,d^2\,e\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {d^2\,e\,\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{8\,c^2} \]
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