Integrand size = 19, antiderivative size = 180 \[ \int (d+e x)^3 \left (a+c x^2\right )^{3/2} \, dx=\frac {3 a d \left (2 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{16 c}+\frac {d \left (2 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{8 c}+\frac {e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac {3 a^2 d \left (2 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {757, 794, 201, 223, 212} \[ \int (d+e x)^3 \left (a+c x^2\right )^{3/2} \, dx=\frac {3 a^2 d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (2 c d^2-a e^2\right )}{16 c^{3/2}}+\frac {e \left (a+c x^2\right )^{5/2} \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right )}{70 c^2}+\frac {d x \left (a+c x^2\right )^{3/2} \left (2 c d^2-a e^2\right )}{8 c}+\frac {3 a d x \sqrt {a+c x^2} \left (2 c d^2-a e^2\right )}{16 c}+\frac {e \left (a+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]
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Rule 201
Rule 212
Rule 223
Rule 757
Rule 794
Rubi steps \begin{align*} \text {integral}& = \frac {e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {\int (d+e x) \left (7 c d^2-2 a e^2+9 c d e x\right ) \left (a+c x^2\right )^{3/2} \, dx}{7 c} \\ & = \frac {e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac {\left (d \left (2 c d^2-a e^2\right )\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{2 c} \\ & = \frac {d \left (2 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{8 c}+\frac {e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac {\left (3 a d \left (2 c d^2-a e^2\right )\right ) \int \sqrt {a+c x^2} \, dx}{8 c} \\ & = \frac {3 a d \left (2 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{16 c}+\frac {d \left (2 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{8 c}+\frac {e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac {\left (3 a^2 d \left (2 c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{16 c} \\ & = \frac {3 a d \left (2 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{16 c}+\frac {d \left (2 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{8 c}+\frac {e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac {\left (3 a^2 d \left (2 c d^2-a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{16 c} \\ & = \frac {3 a d \left (2 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{16 c}+\frac {d \left (2 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{8 c}+\frac {e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac {3 a^2 d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{3/2}} \\ \end{align*}
Time = 0.68 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.96 \[ \int (d+e x)^3 \left (a+c x^2\right )^{3/2} \, dx=\frac {\sqrt {a+c x^2} \left (-32 a^3 e^3+a^2 c e \left (336 d^2+105 d e x+16 e^2 x^2\right )+4 c^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+2 a c^2 x \left (175 d^3+336 d^2 e x+245 d e^2 x^2+64 e^3 x^3\right )\right )+105 a^2 \sqrt {c} d \left (-2 c d^2+a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{560 c^2} \]
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Time = 2.26 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.05
method | result | size |
risch | \(-\frac {\left (-80 e^{3} c^{3} x^{6}-280 d \,e^{2} c^{3} x^{5}-128 e^{3} c^{2} a \,x^{4}-336 d^{2} e \,c^{3} x^{4}-490 a \,c^{2} d \,e^{2} x^{3}-140 c^{3} d^{3} x^{3}-16 a^{2} c \,e^{3} x^{2}-672 c^{2} d^{2} a e \,x^{2}-105 d \,e^{2} a^{2} c x -350 d^{3} c^{2} a x +32 a^{3} e^{3}-336 d^{2} e \,a^{2} c \right ) \sqrt {c \,x^{2}+a}}{560 c^{2}}-\frac {3 a^{2} d \left (e^{2} a -2 c \,d^{2}\right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{16 c^{\frac {3}{2}}}\) | \(189\) |
default | \(d^{3} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )+e^{3} \left (\frac {x^{2} \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{7 c}-\frac {2 a \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{35 c^{2}}\right )+3 d \,e^{2} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6 c}\right )+\frac {3 d^{2} e \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{5 c}\) | \(191\) |
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Time = 0.30 (sec) , antiderivative size = 402, normalized size of antiderivative = 2.23 \[ \int (d+e x)^3 \left (a+c x^2\right )^{3/2} \, dx=\left [\frac {105 \, {\left (2 \, a^{2} c d^{3} - a^{3} d e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (80 \, c^{3} e^{3} x^{6} + 280 \, c^{3} d e^{2} x^{5} + 336 \, a^{2} c d^{2} e - 32 \, a^{3} e^{3} + 16 \, {\left (21 \, c^{3} d^{2} e + 8 \, a c^{2} e^{3}\right )} x^{4} + 70 \, {\left (2 \, c^{3} d^{3} + 7 \, a c^{2} d e^{2}\right )} x^{3} + 16 \, {\left (42 \, a c^{2} d^{2} e + a^{2} c e^{3}\right )} x^{2} + 35 \, {\left (10 \, a c^{2} d^{3} + 3 \, a^{2} c d e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{1120 \, c^{2}}, -\frac {105 \, {\left (2 \, a^{2} c d^{3} - a^{3} d e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (80 \, c^{3} e^{3} x^{6} + 280 \, c^{3} d e^{2} x^{5} + 336 \, a^{2} c d^{2} e - 32 \, a^{3} e^{3} + 16 \, {\left (21 \, c^{3} d^{2} e + 8 \, a c^{2} e^{3}\right )} x^{4} + 70 \, {\left (2 \, c^{3} d^{3} + 7 \, a c^{2} d e^{2}\right )} x^{3} + 16 \, {\left (42 \, a c^{2} d^{2} e + a^{2} c e^{3}\right )} x^{2} + 35 \, {\left (10 \, a c^{2} d^{3} + 3 \, a^{2} c d e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{560 \, c^{2}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (165) = 330\).
Time = 0.59 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.03 \[ \int (d+e x)^3 \left (a+c x^2\right )^{3/2} \, dx=\begin {cases} \sqrt {a + c x^{2}} \left (\frac {c d e^{2} x^{5}}{2} + \frac {c e^{3} x^{6}}{7} + \frac {x^{4} \cdot \left (\frac {8 a c e^{3}}{7} + 3 c^{2} d^{2} e\right )}{5 c} + \frac {x^{3} \cdot \left (\frac {7 a c d e^{2}}{2} + c^{2} d^{3}\right )}{4 c} + \frac {x^{2} \left (a^{2} e^{3} + 6 a c d^{2} e - \frac {4 a \left (\frac {8 a c e^{3}}{7} + 3 c^{2} d^{2} e\right )}{5 c}\right )}{3 c} + \frac {x \left (3 a^{2} d e^{2} + 2 a c d^{3} - \frac {3 a \left (\frac {7 a c d e^{2}}{2} + c^{2} d^{3}\right )}{4 c}\right )}{2 c} + \frac {3 a^{2} d^{2} e - \frac {2 a \left (a^{2} e^{3} + 6 a c d^{2} e - \frac {4 a \left (\frac {8 a c e^{3}}{7} + 3 c^{2} d^{2} e\right )}{5 c}\right )}{3 c}}{c}\right ) + \left (a^{2} d^{3} - \frac {a \left (3 a^{2} d e^{2} + 2 a c d^{3} - \frac {3 a \left (\frac {7 a c d e^{2}}{2} + c^{2} d^{3}\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {a + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\a^{\frac {3}{2}} \left (\begin {cases} d^{3} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{4}}{4 e} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.06 \[ \int (d+e x)^3 \left (a+c x^2\right )^{3/2} \, dx=\frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} e^{3} x^{2}}{7 \, c} + \frac {1}{4} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} d^{3} x + \frac {3}{8} \, \sqrt {c x^{2} + a} a d^{3} x + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} d e^{2} x}{2 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} a d e^{2} x}{8 \, c} - \frac {3 \, \sqrt {c x^{2} + a} a^{2} d e^{2} x}{16 \, c} + \frac {3 \, a^{2} d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c}} - \frac {3 \, a^{3} d e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, c^{\frac {3}{2}}} + \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d^{2} e}{5 \, c} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} a e^{3}}{35 \, c^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.21 \[ \int (d+e x)^3 \left (a+c x^2\right )^{3/2} \, dx=\frac {1}{560} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (2 \, c e^{3} x + 7 \, c d e^{2}\right )} x + \frac {2 \, {\left (21 \, c^{6} d^{2} e + 8 \, a c^{5} e^{3}\right )}}{c^{5}}\right )} x + \frac {35 \, {\left (2 \, c^{6} d^{3} + 7 \, a c^{5} d e^{2}\right )}}{c^{5}}\right )} x + \frac {8 \, {\left (42 \, a c^{5} d^{2} e + a^{2} c^{4} e^{3}\right )}}{c^{5}}\right )} x + \frac {35 \, {\left (10 \, a c^{5} d^{3} + 3 \, a^{2} c^{4} d e^{2}\right )}}{c^{5}}\right )} x + \frac {16 \, {\left (21 \, a^{2} c^{4} d^{2} e - 2 \, a^{3} c^{3} e^{3}\right )}}{c^{5}}\right )} - \frac {3 \, {\left (2 \, a^{2} c d^{3} - a^{3} d e^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{16 \, c^{\frac {3}{2}}} \]
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Timed out. \[ \int (d+e x)^3 \left (a+c x^2\right )^{3/2} \, dx=\int {\left (c\,x^2+a\right )}^{3/2}\,{\left (d+e\,x\right )}^3 \,d x \]
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