Integrand size = 19, antiderivative size = 189 \[ \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx=\frac {5 a^2 \left (8 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{128 c}+\frac {5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {5 a^3 \left (8 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {757, 655, 201, 223, 212} \[ \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx=\frac {5 a^3 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (8 c d^2-a e^2\right )}{128 c^{3/2}}+\frac {5 a^2 x \sqrt {a+c x^2} \left (8 c d^2-a e^2\right )}{128 c}+\frac {x \left (a+c x^2\right )^{5/2} \left (8 c d^2-a e^2\right )}{48 c}+\frac {5 a x \left (a+c x^2\right )^{3/2} \left (8 c d^2-a e^2\right )}{192 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e \left (a+c x^2\right )^{7/2} (d+e x)}{8 c} \]
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Rule 201
Rule 212
Rule 223
Rule 655
Rule 757
Rubi steps \begin{align*} \text {integral}& = \frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\int \left (8 c d^2-a e^2+9 c d e x\right ) \left (a+c x^2\right )^{5/2} \, dx}{8 c} \\ & = \frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\left (8 c d^2-a e^2\right ) \int \left (a+c x^2\right )^{5/2} \, dx}{8 c} \\ & = \frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 a \left (8 c d^2-a e^2\right )\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{48 c} \\ & = \frac {5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 a^2 \left (8 c d^2-a e^2\right )\right ) \int \sqrt {a+c x^2} \, dx}{64 c} \\ & = \frac {5 a^2 \left (8 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{128 c}+\frac {5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 a^3 \left (8 c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{128 c} \\ & = \frac {5 a^2 \left (8 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{128 c}+\frac {5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 a^3 \left (8 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 )}{128 c} \\ & = \frac {5 a^2 \left (8 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{128 c}+\frac {5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {5 a^3 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{3/2}} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.85 \[ \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx=\frac {\sqrt {c} \sqrt {a+c x^2} \left (3 a^3 e (256 d+35 e x)+16 c^3 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+8 a c^2 x^3 \left (182 d^2+288 d e x+119 e^2 x^2\right )+2 a^2 c x \left (924 d^2+1152 d e x+413 e^2 x^2\right )\right )+105 a^3 \left (-8 c d^2+a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{2688 c^{3/2}} \]
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Time = 1.96 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\frac {\left (336 e^{2} c^{3} x^{7}+768 d e \,c^{3} x^{6}+952 e^{2} c^{2} a \,x^{5}+448 c^{3} d^{2} x^{5}+2304 a \,c^{2} d e \,x^{4}+826 a^{2} c \,e^{2} x^{3}+1456 a \,c^{2} d^{2} x^{3}+2304 x^{2} a^{2} c d e +105 a^{3} e^{2} x +1848 c \,a^{2} d^{2} x +768 d e \,a^{3}\right ) \sqrt {c \,x^{2}+a}}{2688 c}-\frac {5 a^{3} \left (e^{2} a -8 c \,d^{2}\right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{128 c^{\frac {3}{2}}}\) | \(169\) |
default | \(d^{2} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )+e^{2} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{8 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )}{8 c}\right )+\frac {2 d e \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{7 c}\) | \(182\) |
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Time = 0.31 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.01 \[ \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx=\left [\frac {105 \, {\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (336 \, c^{4} e^{2} x^{7} + 768 \, c^{4} d e x^{6} + 2304 \, a c^{3} d e x^{4} + 2304 \, a^{2} c^{2} d e x^{2} + 768 \, a^{3} c d e + 56 \, {\left (8 \, c^{4} d^{2} + 17 \, a c^{3} e^{2}\right )} x^{5} + 14 \, {\left (104 \, a c^{3} d^{2} + 59 \, a^{2} c^{2} e^{2}\right )} x^{3} + 21 \, {\left (88 \, a^{2} c^{2} d^{2} + 5 \, a^{3} c e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{5376 \, c^{2}}, -\frac {105 \, {\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (336 \, c^{4} e^{2} x^{7} + 768 \, c^{4} d e x^{6} + 2304 \, a c^{3} d e x^{4} + 2304 \, a^{2} c^{2} d e x^{2} + 768 \, a^{3} c d e + 56 \, {\left (8 \, c^{4} d^{2} + 17 \, a c^{3} e^{2}\right )} x^{5} + 14 \, {\left (104 \, a c^{3} d^{2} + 59 \, a^{2} c^{2} e^{2}\right )} x^{3} + 21 \, {\left (88 \, a^{2} c^{2} d^{2} + 5 \, a^{3} c e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{2688 \, c^{2}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (170) = 340\).
Time = 0.61 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.96 \[ \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx=\begin {cases} \sqrt {a + c x^{2}} \cdot \left (\frac {2 a^{3} d e}{7 c} + \frac {6 a^{2} d e x^{2}}{7} + \frac {6 a c d e x^{4}}{7} + \frac {2 c^{2} d e x^{6}}{7} + \frac {c^{2} e^{2} x^{7}}{8} + \frac {x^{5} \cdot \left (\frac {17 a c^{2} e^{2}}{8} + c^{3} d^{2}\right )}{6 c} + \frac {x^{3} \cdot \left (3 a^{2} c e^{2} + 3 a c^{2} d^{2} - \frac {5 a \left (\frac {17 a c^{2} e^{2}}{8} + c^{3} d^{2}\right )}{6 c}\right )}{4 c} + \frac {x \left (a^{3} e^{2} + 3 a^{2} c d^{2} - \frac {3 a \left (3 a^{2} c e^{2} + 3 a c^{2} d^{2} - \frac {5 a \left (\frac {17 a c^{2} e^{2}}{8} + c^{3} d^{2}\right )}{6 c}\right )}{4 c}\right )}{2 c}\right ) + \left (a^{3} d^{2} - \frac {a \left (a^{3} e^{2} + 3 a^{2} c d^{2} - \frac {3 a \left (3 a^{2} c e^{2} + 3 a c^{2} d^{2} - \frac {5 a \left (\frac {17 a c^{2} e^{2}}{8} + c^{3} d^{2}\right )}{6 c}\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {a + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\a^{\frac {5}{2}} \left (\begin {cases} d^{2} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{3}}{3 e} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.98 \[ \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d^{2} x + \frac {5}{24} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a d^{2} x + \frac {5}{16} \, \sqrt {c x^{2} + a} a^{2} d^{2} x + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} e^{2} x}{8 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} a e^{2} x}{48 \, c} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} e^{2} x}{192 \, c} - \frac {5 \, \sqrt {c x^{2} + a} a^{3} e^{2} x}{128 \, c} + \frac {5 \, a^{3} d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {c}} - \frac {5 \, a^{4} e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{128 \, c^{\frac {3}{2}}} + \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} d e}{7 \, c} \]
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Time = 0.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.01 \[ \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx=\frac {1}{2688} \, {\left (\frac {768 \, a^{3} d e}{c} + {\left (2 \, {\left (1152 \, a^{2} d e + {\left (4 \, {\left (288 \, a c d e + {\left (6 \, {\left (7 \, c^{2} e^{2} x + 16 \, c^{2} d e\right )} x + \frac {7 \, {\left (8 \, c^{8} d^{2} + 17 \, a c^{7} e^{2}\right )}}{c^{6}}\right )} x\right )} x + \frac {7 \, {\left (104 \, a c^{7} d^{2} + 59 \, a^{2} c^{6} e^{2}\right )}}{c^{6}}\right )} x\right )} x + \frac {21 \, {\left (88 \, a^{2} c^{6} d^{2} + 5 \, a^{3} c^{5} e^{2}\right )}}{c^{6}}\right )} x\right )} \sqrt {c x^{2} + a} - \frac {5 \, {\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{128 \, c^{\frac {3}{2}}} \]
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Timed out. \[ \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx=\int {\left (c\,x^2+a\right )}^{5/2}\,{\left (d+e\,x\right )}^2 \,d x \]
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