Integrand size = 19, antiderivative size = 207 \[ \int (d+e x)^4 \sqrt {a+c x^2} \, dx=\frac {\left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right ) x \sqrt {a+c x^2}}{16 c^2}+\frac {3 d e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{10 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac {e \left (8 d \left (13 c d^2-8 a e^2\right )+3 e \left (16 c d^2-5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{120 c^2}+\frac {a \left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{5/2}} \]
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Time = 0.14 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {757, 847, 794, 201, 223, 212} \[ \int (d+e x)^4 \sqrt {a+c x^2} \, dx=\frac {a \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a^2 e^4-12 a c d^2 e^2+8 c^2 d^4\right )}{16 c^{5/2}}+\frac {x \sqrt {a+c x^2} \left (a^2 e^4-12 a c d^2 e^2+8 c^2 d^4\right )}{16 c^2}+\frac {e \left (a+c x^2\right )^{3/2} \left (3 e x \left (16 c d^2-5 a e^2\right )+8 d \left (13 c d^2-8 a e^2\right )\right )}{120 c^2}+\frac {e \left (a+c x^2\right )^{3/2} (d+e x)^3}{6 c}+\frac {3 d e \left (a+c x^2\right )^{3/2} (d+e x)^2}{10 c} \]
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Rule 201
Rule 212
Rule 223
Rule 757
Rule 794
Rule 847
Rubi steps \begin{align*} \text {integral}& = \frac {e (d+e x)^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac {\int (d+e x)^2 \left (3 \left (2 c d^2-a e^2\right )+9 c d e x\right ) \sqrt {a+c x^2} \, dx}{6 c} \\ & = \frac {3 d e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{10 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac {\int (d+e x) \left (3 c d \left (10 c d^2-11 a e^2\right )+3 c e \left (16 c d^2-5 a e^2\right ) x\right ) \sqrt {a+c x^2} \, dx}{30 c^2} \\ & = \frac {3 d e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{10 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac {e \left (8 d \left (13 c d^2-8 a e^2\right )+3 e \left (16 c d^2-5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{120 c^2}+\frac {\left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right ) \int \sqrt {a+c x^2} \, dx}{8 c^2} \\ & = \frac {\left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right ) x \sqrt {a+c x^2}}{16 c^2}+\frac {3 d e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{10 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac {e \left (8 d \left (13 c d^2-8 a e^2\right )+3 e \left (16 c d^2-5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{120 c^2}+\frac {\left (a \left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{16 c^2} \\ & = \frac {\left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right ) x \sqrt {a+c x^2}}{16 c^2}+\frac {3 d e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{10 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac {e \left (8 d \left (13 c d^2-8 a e^2\right )+3 e \left (16 c d^2-5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{120 c^2}+\frac {\left (a \left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{16 c^2} \\ & = \frac {\left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right ) x \sqrt {a+c x^2}}{16 c^2}+\frac {3 d e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{10 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac {e \left (8 d \left (13 c d^2-8 a e^2\right )+3 e \left (16 c d^2-5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{120 c^2}+\frac {a \left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{5/2}} \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.85 \[ \int (d+e x)^4 \sqrt {a+c x^2} \, dx=\frac {\sqrt {c} \sqrt {a+c x^2} \left (-a^2 e^3 (128 d+15 e x)+2 a c e \left (160 d^3+90 d^2 e x+32 d e^2 x^2+5 e^3 x^3\right )+8 c^2 x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )\right )-15 a \left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{240 c^{5/2}} \]
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Time = 2.21 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.87
method | result | size |
risch | \(-\frac {\left (-40 c^{2} x^{5} e^{4}-192 x^{4} c^{2} d \,e^{3}-10 x^{3} a c \,e^{4}-360 x^{3} c^{2} d^{2} e^{2}-64 x^{2} a c d \,e^{3}-320 x^{2} c^{2} d^{3} e +15 a^{2} e^{4} x -180 a c \,d^{2} e^{2} x -120 c^{2} d^{4} x +128 a^{2} d \,e^{3}-320 a c \,d^{3} e \right ) \sqrt {c \,x^{2}+a}}{240 c^{2}}+\frac {a \left (a^{2} e^{4}-12 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{16 c^{\frac {5}{2}}}\) | \(181\) |
default | \(d^{4} \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )+e^{4} \left (\frac {x^{3} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{6 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4 c}-\frac {a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4 c}\right )}{2 c}\right )+4 d \,e^{3} \left (\frac {x^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{5 c}-\frac {2 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{15 c^{2}}\right )+6 d^{2} e^{2} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4 c}-\frac {a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4 c}\right )+\frac {4 d^{3} e \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{3 c}\) | \(248\) |
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Time = 0.31 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.93 \[ \int (d+e x)^4 \sqrt {a+c x^2} \, dx=\left [\frac {15 \, {\left (8 \, a c^{2} d^{4} - 12 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (40 \, c^{3} e^{4} x^{5} + 192 \, c^{3} d e^{3} x^{4} + 320 \, a c^{2} d^{3} e - 128 \, a^{2} c d e^{3} + 10 \, {\left (36 \, c^{3} d^{2} e^{2} + a c^{2} e^{4}\right )} x^{3} + 64 \, {\left (5 \, c^{3} d^{3} e + a c^{2} d e^{3}\right )} x^{2} + 15 \, {\left (8 \, c^{3} d^{4} + 12 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{480 \, c^{3}}, -\frac {15 \, {\left (8 \, a c^{2} d^{4} - 12 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (40 \, c^{3} e^{4} x^{5} + 192 \, c^{3} d e^{3} x^{4} + 320 \, a c^{2} d^{3} e - 128 \, a^{2} c d e^{3} + 10 \, {\left (36 \, c^{3} d^{2} e^{2} + a c^{2} e^{4}\right )} x^{3} + 64 \, {\left (5 \, c^{3} d^{3} e + a c^{2} d e^{3}\right )} x^{2} + 15 \, {\left (8 \, c^{3} d^{4} + 12 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{240 \, c^{3}}\right ] \]
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Time = 0.52 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.30 \[ \int (d+e x)^4 \sqrt {a+c x^2} \, dx=\begin {cases} \sqrt {a + c x^{2}} \cdot \left (\frac {4 d e^{3} x^{4}}{5} + \frac {e^{4} x^{5}}{6} + \frac {x^{3} \left (\frac {a e^{4}}{6} + 6 c d^{2} e^{2}\right )}{4 c} + \frac {x^{2} \cdot \left (\frac {4 a d e^{3}}{5} + 4 c d^{3} e\right )}{3 c} + \frac {x \left (6 a d^{2} e^{2} - \frac {3 a \left (\frac {a e^{4}}{6} + 6 c d^{2} e^{2}\right )}{4 c} + c d^{4}\right )}{2 c} + \frac {4 a d^{3} e - \frac {2 a \left (\frac {4 a d e^{3}}{5} + 4 c d^{3} e\right )}{3 c}}{c}\right ) + \left (a d^{4} - \frac {a \left (6 a d^{2} e^{2} - \frac {3 a \left (\frac {a e^{4}}{6} + 6 c d^{2} e^{2}\right )}{4 c} + c d^{4}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {a + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\\sqrt {a} \left (\begin {cases} d^{4} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{5}}{5 e} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.15 \[ \int (d+e x)^4 \sqrt {a+c x^2} \, dx=\frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} e^{4} x^{3}}{6 \, c} + \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} d e^{3} x^{2}}{5 \, c} + \frac {1}{2} \, \sqrt {c x^{2} + a} d^{4} x + \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} d^{2} e^{2} x}{2 \, c} - \frac {3 \, \sqrt {c x^{2} + a} a d^{2} e^{2} x}{4 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} a e^{4} x}{8 \, c^{2}} + \frac {\sqrt {c x^{2} + a} a^{2} e^{4} x}{16 \, c^{2}} + \frac {a d^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {c}} - \frac {3 \, a^{2} d^{2} e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{4 \, c^{\frac {3}{2}}} + \frac {a^{3} e^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, c^{\frac {5}{2}}} + \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} d^{3} e}{3 \, c} - \frac {8 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a d e^{3}}{15 \, c^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.99 \[ \int (d+e x)^4 \sqrt {a+c x^2} \, dx=\frac {1}{240} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, e^{4} x + 24 \, d e^{3}\right )} x + \frac {5 \, {\left (36 \, c^{4} d^{2} e^{2} + a c^{3} e^{4}\right )}}{c^{4}}\right )} x + \frac {32 \, {\left (5 \, c^{4} d^{3} e + a c^{3} d e^{3}\right )}}{c^{4}}\right )} x + \frac {15 \, {\left (8 \, c^{4} d^{4} + 12 \, a c^{3} d^{2} e^{2} - a^{2} c^{2} e^{4}\right )}}{c^{4}}\right )} x + \frac {64 \, {\left (5 \, a c^{3} d^{3} e - 2 \, a^{2} c^{2} d e^{3}\right )}}{c^{4}}\right )} - \frac {{\left (8 \, a c^{2} d^{4} - 12 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{16 \, c^{\frac {5}{2}}} \]
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Timed out. \[ \int (d+e x)^4 \sqrt {a+c x^2} \, dx=\int \sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^4 \,d x \]
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