Integrand size = 19, antiderivative size = 137 \[ \int (d+e x) \left (b x+c x^2\right )^{3/2} \, dx=-\frac {3 b^2 (2 c d-b e) (b+2 c x) \sqrt {b x+c x^2}}{128 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{16 c^2}+\frac {e \left (b x+c x^2\right )^{5/2}}{5 c}+\frac {3 b^4 (2 c d-b e) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{7/2}} \]
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Time = 0.04 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {654, 626, 634, 212} \[ \int (d+e x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {3 b^4 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 c d-b e)}{128 c^{7/2}}-\frac {3 b^2 (b+2 c x) \sqrt {b x+c x^2} (2 c d-b e)}{128 c^3}+\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{16 c^2}+\frac {e \left (b x+c x^2\right )^{5/2}}{5 c} \]
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Rule 212
Rule 626
Rule 634
Rule 654
Rubi steps \begin{align*} \text {integral}& = \frac {e \left (b x+c x^2\right )^{5/2}}{5 c}+\frac {(2 c d-b e) \int \left (b x+c x^2\right )^{3/2} \, dx}{2 c} \\ & = \frac {(2 c d-b e) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{16 c^2}+\frac {e \left (b x+c x^2\right )^{5/2}}{5 c}-\frac {\left (3 b^2 (2 c d-b e)\right ) \int \sqrt {b x+c x^2} \, dx}{32 c^2} \\ & = -\frac {3 b^2 (2 c d-b e) (b+2 c x) \sqrt {b x+c x^2}}{128 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{16 c^2}+\frac {e \left (b x+c x^2\right )^{5/2}}{5 c}+\frac {\left (3 b^4 (2 c d-b e)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{256 c^3} \\ & = -\frac {3 b^2 (2 c d-b e) (b+2 c x) \sqrt {b x+c x^2}}{128 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{16 c^2}+\frac {e \left (b x+c x^2\right )^{5/2}}{5 c}+\frac {\left (3 b^4 (2 c d-b e)\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{128 c^3} \\ & = -\frac {3 b^2 (2 c d-b e) (b+2 c x) \sqrt {b x+c x^2}}{128 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{16 c^2}+\frac {e \left (b x+c x^2\right )^{5/2}}{5 c}+\frac {3 b^4 (2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{7/2}} \\ \end{align*}
Time = 1.14 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.22 \[ \int (d+e x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {(x (b+c x))^{3/2} \left (\frac {\sqrt {c} \sqrt {x} \left (15 b^4 e-10 b^3 c (3 d+e x)+4 b^2 c^2 x (5 d+2 e x)+32 c^4 x^3 (5 d+4 e x)+16 b c^3 x^2 (15 d+11 e x)\right )}{b+c x}+\frac {30 b^4 (-2 c d+b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )}{(b+c x)^{3/2}}\right )}{640 c^{7/2} x^{3/2}} \]
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Time = 1.87 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.85
method | result | size |
pseudoelliptic | \(-\frac {3 \left (\left (b^{5} e -2 b^{4} c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )-\sqrt {x \left (c x +b \right )}\, \left (-2 \left (\frac {e x}{3}+d \right ) b^{3} c^{\frac {3}{2}}+\frac {4 \left (\frac {2 e x}{5}+d \right ) x \,b^{2} c^{\frac {5}{2}}}{3}+16 x^{2} \left (\frac {11 e x}{15}+d \right ) b \,c^{\frac {7}{2}}+\frac {32 x^{3} \left (\frac {4 e x}{5}+d \right ) c^{\frac {9}{2}}}{3}+\sqrt {c}\, b^{4} e \right )\right )}{128 c^{\frac {7}{2}}}\) | \(116\) |
risch | \(\frac {\left (128 c^{4} e \,x^{4}+176 b \,c^{3} e \,x^{3}+160 c^{4} d \,x^{3}+8 b^{2} c^{2} e \,x^{2}+240 b \,c^{3} d \,x^{2}-10 b^{3} c e x +20 x \,b^{2} c^{2} d +15 b^{4} e -30 b^{3} d c \right ) x \left (c x +b \right )}{640 c^{3} \sqrt {x \left (c x +b \right )}}-\frac {3 b^{4} \left (b e -2 c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{256 c^{\frac {7}{2}}}\) | \(144\) |
default | \(d \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )+e \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )\) | \(201\) |
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Time = 0.30 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.20 \[ \int (d+e x) \left (b x+c x^2\right )^{3/2} \, dx=\left [-\frac {15 \, {\left (2 \, b^{4} c d - b^{5} e\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (128 \, c^{5} e x^{4} - 30 \, b^{3} c^{2} d + 15 \, b^{4} c e + 16 \, {\left (10 \, c^{5} d + 11 \, b c^{4} e\right )} x^{3} + 8 \, {\left (30 \, b c^{4} d + b^{2} c^{3} e\right )} x^{2} + 10 \, {\left (2 \, b^{2} c^{3} d - b^{3} c^{2} e\right )} x\right )} \sqrt {c x^{2} + b x}}{1280 \, c^{4}}, -\frac {15 \, {\left (2 \, b^{4} c d - b^{5} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (128 \, c^{5} e x^{4} - 30 \, b^{3} c^{2} d + 15 \, b^{4} c e + 16 \, {\left (10 \, c^{5} d + 11 \, b c^{4} e\right )} x^{3} + 8 \, {\left (30 \, b c^{4} d + b^{2} c^{3} e\right )} x^{2} + 10 \, {\left (2 \, b^{2} c^{3} d - b^{3} c^{2} e\right )} x\right )} \sqrt {c x^{2} + b x}}{640 \, c^{4}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (126) = 252\).
Time = 0.48 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.38 \[ \int (d+e x) \left (b x+c x^2\right )^{3/2} \, dx=\begin {cases} \frac {3 b^{2} \left (b^{2} d - \frac {5 b \left (b^{2} e + 2 b c d - \frac {7 b \left (\frac {11 b c e}{10} + c^{2} d\right )}{8 c}\right )}{6 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{8 c^{2}} + \sqrt {b x + c x^{2}} \left (- \frac {3 b \left (b^{2} d - \frac {5 b \left (b^{2} e + 2 b c d - \frac {7 b \left (\frac {11 b c e}{10} + c^{2} d\right )}{8 c}\right )}{6 c}\right )}{4 c^{2}} + \frac {c e x^{4}}{5} + \frac {x^{3} \cdot \left (\frac {11 b c e}{10} + c^{2} d\right )}{4 c} + \frac {x^{2} \left (b^{2} e + 2 b c d - \frac {7 b \left (\frac {11 b c e}{10} + c^{2} d\right )}{8 c}\right )}{3 c} + \frac {x \left (b^{2} d - \frac {5 b \left (b^{2} e + 2 b c d - \frac {7 b \left (\frac {11 b c e}{10} + c^{2} d\right )}{8 c}\right )}{6 c}\right )}{2 c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {d \left (b x\right )^{\frac {5}{2}}}{5} + \frac {e \left (b x\right )^{\frac {7}{2}}}{7 b}\right )}{b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (117) = 234\).
Time = 0.19 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.72 \[ \int (d+e x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {1}{4} \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} d x - \frac {3 \, \sqrt {c x^{2} + b x} b^{2} d x}{32 \, c} + \frac {3 \, \sqrt {c x^{2} + b x} b^{3} e x}{64 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b e x}{8 \, c} + \frac {3 \, b^{4} d \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {5}{2}}} - \frac {3 \, b^{5} e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {7}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} b^{3} d}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b d}{8 \, c} + \frac {3 \, \sqrt {c x^{2} + b x} b^{4} e}{128 \, c^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} e}{16 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} e}{5 \, c} \]
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Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.19 \[ \int (d+e x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {1}{640} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, c e x + \frac {10 \, c^{5} d + 11 \, b c^{4} e}{c^{4}}\right )} x + \frac {30 \, b c^{4} d + b^{2} c^{3} e}{c^{4}}\right )} x + \frac {5 \, {\left (2 \, b^{2} c^{3} d - b^{3} c^{2} e\right )}}{c^{4}}\right )} x - \frac {15 \, {\left (2 \, b^{3} c^{2} d - b^{4} c e\right )}}{c^{4}}\right )} - \frac {3 \, {\left (2 \, b^{4} c d - b^{5} e\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {7}{2}}} \]
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Time = 9.61 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.52 \[ \int (d+e x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {e\,{\left (c\,x^2+b\,x\right )}^{5/2}}{5\,c}-\frac {3\,b^2\,d\,\left (\sqrt {c\,x^2+b\,x}\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )-\frac {b^2\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{8\,c^{3/2}}\right )}{16\,c}+\frac {d\,{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (\frac {b}{2}+c\,x\right )}{4\,c}-\frac {b\,e\,\left (\frac {x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4}+\frac {b\,{\left (c\,x^2+b\,x\right )}^{3/2}}{8\,c}-\frac {3\,b^2\,\left (\frac {\sqrt {c\,x^2+b\,x}\,\left (b+2\,c\,x\right )}{4\,c}-\frac {b^2\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{8\,c^{3/2}}\right )}{16\,c}\right )}{2\,c} \]
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