Integrand size = 19, antiderivative size = 255 \[ \int (d+e x)^4 \left (a+c x^2\right )^{3/2} \, dx=\frac {3 a \left (16 c^2 d^4-16 a c d^2 e^2+a^2 e^4\right ) x \sqrt {a+c x^2}}{128 c^2}+\frac {\left (16 c^2 d^4-16 a c d^2 e^2+a^2 e^4\right ) x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac {11 d e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{56 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {e \left (4 d \left (67 c d^2-32 a e^2\right )+5 e \left (26 c d^2-7 a e^2\right ) x\right ) \left (a+c x^2\right )^{5/2}}{560 c^2}+\frac {3 a^2 \left (16 c^2 d^4-16 a c d^2 e^2+a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{5/2}} \]
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Time = 0.16 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {757, 847, 794, 201, 223, 212} \[ \int (d+e x)^4 \left (a+c x^2\right )^{3/2} \, dx=\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a^2 e^4-16 a c d^2 e^2+16 c^2 d^4\right )}{128 c^{5/2}}+\frac {x \left (a+c x^2\right )^{3/2} \left (a^2 e^4-16 a c d^2 e^2+16 c^2 d^4\right )}{64 c^2}+\frac {3 a x \sqrt {a+c x^2} \left (a^2 e^4-16 a c d^2 e^2+16 c^2 d^4\right )}{128 c^2}+\frac {e \left (a+c x^2\right )^{5/2} \left (5 e x \left (26 c d^2-7 a e^2\right )+4 d \left (67 c d^2-32 a e^2\right )\right )}{560 c^2}+\frac {e \left (a+c x^2\right )^{5/2} (d+e x)^3}{8 c}+\frac {11 d e \left (a+c x^2\right )^{5/2} (d+e x)^2}{56 c} \]
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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 )^{5/2}}{8 c}+\frac {\int (d+e x)^2 \left (8 c d^2-3 a e^2+11 c d e x\right ) \left (a+c x^2\right )^{3/2} \, dx}{8 c} \\ & = \frac {11 d e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{56 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {\int (d+e x) \left (c d \left (56 c d^2-43 a e^2\right )+3 c e \left (26 c d^2-7 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2} \, dx}{56 c^2} \\ & = \frac {11 d e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{56 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {e \left (4 d \left (67 c d^2-32 a e^2\right )+5 e \left (26 c d^2-7 a e^2\right ) x\right ) \left (a+c x^2\right )^{5/2}}{560 c^2}+\frac {\left (16 c^2 d^4-16 a c d^2 e^2+a^2 e^4\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{16 c^2} \\ & = \frac {\left (16 c^2 d^4-16 a c d^2 e^2+a^2 e^4\right ) x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac {11 d e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{56 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {e \left (4 d \left (67 c d^2-32 a e^2\right )+5 e \left (26 c d^2-7 a e^2\right ) x\right ) \left (a+c x^2\right )^{5/2}}{560 c^2}+\frac {\left (3 a \left (16 c^2 d^4-16 a c d^2 e^2+a^2 e^4\right )\right ) \int \sqrt {a+c x^2} \, dx}{64 c^2} \\ & = \frac {3 a \left (16 c^2 d^4-16 a c d^2 e^2+a^2 e^4\right ) x \sqrt {a+c x^2}}{128 c^2}+\frac {\left (16 c^2 d^4-16 a c d^2 e^2+a^2 e^4\right ) x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac {11 d e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{56 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {e \left (4 d \left (67 c d^2-32 a e^2\right )+5 e \left (26 c d^2-7 a e^2\right ) x\right ) \left (a+c x^2\right )^{5/2}}{560 c^2}+\frac {\left (3 a^2 \left (16 c^2 d^4-16 a c d^2 e^2+a^2 e^4\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{128 c^2} \\ & = \frac {3 a \left (16 c^2 d^4-16 a c d^2 e^2+a^2 e^4\right ) x \sqrt {a+c x^2}}{128 c^2}+\frac {\left (16 c^2 d^4-16 a c d^2 e^2+a^2 e^4\right ) x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac {11 d e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{56 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {e \left (4 d \left (67 c d^2-32 a e^2\right )+5 e \left (26 c d^2-7 a e^2\right ) x\right ) \left (a+c x^2\right )^{5/2}}{560 c^2}+\frac {\left (3 a^2 \left (16 c^2 d^4-16 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 )}{128 c^2} \\ & = \frac {3 a \left (16 c^2 d^4-16 a c d^2 e^2+a^2 e^4\right ) x \sqrt {a+c x^2}}{128 c^2}+\frac {\left (16 c^2 d^4-16 a c d^2 e^2+a^2 e^4\right ) x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac {11 d e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{56 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {e \left (4 d \left (67 c d^2-32 a e^2\right )+5 e \left (26 c d^2-7 a e^2\right ) x\right ) \left (a+c x^2\right )^{5/2}}{560 c^2}+\frac {3 a^2 \left (16 c^2 d^4-16 a c d^2 e^2+a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{5/2}} \\ \end{align*}
Time = 0.76 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.90 \[ \int (d+e x)^4 \left (a+c x^2\right )^{3/2} \, dx=\frac {\sqrt {c} \sqrt {a+c x^2} \left (-a^3 e^3 (1024 d+105 e x)+2 a^2 c e \left (1792 d^3+840 d^2 e x+256 d e^2 x^2+35 e^3 x^3\right )+16 c^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )+8 a c^2 x \left (350 d^4+896 d^3 e x+980 d^2 e^2 x^2+512 d e^3 x^3+105 e^4 x^4\right )\right )-105 a^2 \left (16 c^2 d^4-16 a c d^2 e^2+a^2 e^4\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{4480 c^{5/2}} \]
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Time = 2.21 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\frac {\left (-560 e^{4} c^{3} x^{7}-2560 d \,e^{3} c^{3} x^{6}-840 e^{4} c^{2} a \,x^{5}-4480 d^{2} e^{2} c^{3} x^{5}-4096 d \,e^{3} c^{2} a \,x^{4}-3584 d^{3} e \,c^{3} x^{4}-70 e^{4} a^{2} c \,x^{3}-7840 d^{2} e^{2} c^{2} a \,x^{3}-1120 d^{4} c^{3} x^{3}-512 a^{2} c d \,e^{3} x^{2}-7168 d^{3} e \,c^{2} a \,x^{2}+105 e^{4} a^{3} x -1680 d^{2} e^{2} a^{2} c x -2800 a \,c^{2} d^{4} x +1024 a^{3} d \,e^{3}-3584 a^{2} c \,d^{3} e \right ) \sqrt {c \,x^{2}+a}}{4480 c^{2}}+\frac {3 a^{2} \left (a^{2} e^{4}-16 a c \,d^{2} e^{2}+16 c^{2} d^{4}\right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{128 c^{\frac {5}{2}}}\) | \(256\) |
default | \(d^{4} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )+e^{4} \left (\frac {x^{3} \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{8 c}-\frac {3 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6 c}\right )}{8 c}\right )+4 d \,e^{3} \left (\frac {x^{2} \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{7 c}-\frac {2 a \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{35 c^{2}}\right )+6 d^{2} e^{2} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6 c}\right )+\frac {4 d^{3} e \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{5 c}\) | \(296\) |
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Time = 0.32 (sec) , antiderivative size = 544, normalized size of antiderivative = 2.13 \[ \int (d+e x)^4 \left (a+c x^2\right )^{3/2} \, dx=\left [\frac {105 \, {\left (16 \, a^{2} c^{2} d^{4} - 16 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (560 \, c^{4} e^{4} x^{7} + 2560 \, c^{4} d e^{3} x^{6} + 3584 \, a^{2} c^{2} d^{3} e - 1024 \, a^{3} c d e^{3} + 280 \, {\left (16 \, c^{4} d^{2} e^{2} + 3 \, a c^{3} e^{4}\right )} x^{5} + 512 \, {\left (7 \, c^{4} d^{3} e + 8 \, a c^{3} d e^{3}\right )} x^{4} + 70 \, {\left (16 \, c^{4} d^{4} + 112 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{3} + 512 \, {\left (14 \, a c^{3} d^{3} e + a^{2} c^{2} d e^{3}\right )} x^{2} + 35 \, {\left (80 \, a c^{3} d^{4} + 48 \, a^{2} c^{2} d^{2} e^{2} - 3 \, a^{3} c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{8960 \, c^{3}}, -\frac {105 \, {\left (16 \, a^{2} c^{2} d^{4} - 16 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (560 \, c^{4} e^{4} x^{7} + 2560 \, c^{4} d e^{3} x^{6} + 3584 \, a^{2} c^{2} d^{3} e - 1024 \, a^{3} c d e^{3} + 280 \, {\left (16 \, c^{4} d^{2} e^{2} + 3 \, a c^{3} e^{4}\right )} x^{5} + 512 \, {\left (7 \, c^{4} d^{3} e + 8 \, a c^{3} d e^{3}\right )} x^{4} + 70 \, {\left (16 \, c^{4} d^{4} + 112 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{3} + 512 \, {\left (14 \, a c^{3} d^{3} e + a^{2} c^{2} d e^{3}\right )} x^{2} + 35 \, {\left (80 \, a c^{3} d^{4} + 48 \, a^{2} c^{2} d^{2} e^{2} - 3 \, a^{3} c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{4480 \, c^{3}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (248) = 496\).
Time = 0.62 (sec) , antiderivative size = 518, normalized size of antiderivative = 2.03 \[ \int (d+e x)^4 \left (a+c x^2\right )^{3/2} \, dx=\begin {cases} \sqrt {a + c x^{2}} \cdot \left (\frac {4 c d e^{3} x^{6}}{7} + \frac {c e^{4} x^{7}}{8} + \frac {x^{5} \cdot \left (\frac {9 a c e^{4}}{8} + 6 c^{2} d^{2} e^{2}\right )}{6 c} + \frac {x^{4} \cdot \left (\frac {32 a c d e^{3}}{7} + 4 c^{2} d^{3} e\right )}{5 c} + \frac {x^{3} \left (a^{2} e^{4} + 12 a c d^{2} e^{2} - \frac {5 a \left (\frac {9 a c e^{4}}{8} + 6 c^{2} d^{2} e^{2}\right )}{6 c} + c^{2} d^{4}\right )}{4 c} + \frac {x^{2} \cdot \left (4 a^{2} d e^{3} + 8 a c d^{3} e - \frac {4 a \left (\frac {32 a c d e^{3}}{7} + 4 c^{2} d^{3} e\right )}{5 c}\right )}{3 c} + \frac {x \left (6 a^{2} d^{2} e^{2} + 2 a c d^{4} - \frac {3 a \left (a^{2} e^{4} + 12 a c d^{2} e^{2} - \frac {5 a \left (\frac {9 a c e^{4}}{8} + 6 c^{2} d^{2} e^{2}\right )}{6 c} + c^{2} d^{4}\right )}{4 c}\right )}{2 c} + \frac {4 a^{2} d^{3} e - \frac {2 a \left (4 a^{2} d e^{3} + 8 a c d^{3} e - \frac {4 a \left (\frac {32 a c d e^{3}}{7} + 4 c^{2} d^{3} e\right )}{5 c}\right )}{3 c}}{c}\right ) + \left (a^{2} d^{4} - \frac {a \left (6 a^{2} d^{2} e^{2} + 2 a c d^{4} - \frac {3 a \left (a^{2} e^{4} + 12 a c d^{2} e^{2} - \frac {5 a \left (\frac {9 a c e^{4}}{8} + 6 c^{2} d^{2} e^{2}\right )}{6 c} + c^{2} d^{4}\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {a + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\a^{\frac {3}{2}} \left (\begin {cases} d^{4} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{5}}{5 e} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.18 \[ \int (d+e x)^4 \left (a+c x^2\right )^{3/2} \, dx=\frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} e^{4} x^{3}}{8 \, c} + \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d e^{3} x^{2}}{7 \, c} + \frac {1}{4} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} d^{4} x + \frac {3}{8} \, \sqrt {c x^{2} + a} a d^{4} x + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} d^{2} e^{2} x}{c} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} a d^{2} e^{2} x}{4 \, c} - \frac {3 \, \sqrt {c x^{2} + a} a^{2} d^{2} e^{2} x}{8 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} a e^{4} x}{16 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} e^{4} x}{64 \, c^{2}} + \frac {3 \, \sqrt {c x^{2} + a} a^{3} e^{4} x}{128 \, c^{2}} + \frac {3 \, a^{2} d^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c}} - \frac {3 \, a^{3} d^{2} e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, c^{\frac {3}{2}}} + \frac {3 \, a^{4} e^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{128 \, c^{\frac {5}{2}}} + \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d^{3} e}{5 \, c} - \frac {8 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} a d e^{3}}{35 \, c^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.13 \[ \int (d+e x)^4 \left (a+c x^2\right )^{3/2} \, dx=\frac {1}{4480} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (2 \, {\left (7 \, c e^{4} x + 32 \, c d e^{3}\right )} x + \frac {7 \, {\left (16 \, c^{7} d^{2} e^{2} + 3 \, a c^{6} e^{4}\right )}}{c^{6}}\right )} x + \frac {64 \, {\left (7 \, c^{7} d^{3} e + 8 \, a c^{6} d e^{3}\right )}}{c^{6}}\right )} x + \frac {35 \, {\left (16 \, c^{7} d^{4} + 112 \, a c^{6} d^{2} e^{2} + a^{2} c^{5} e^{4}\right )}}{c^{6}}\right )} x + \frac {256 \, {\left (14 \, a c^{6} d^{3} e + a^{2} c^{5} d e^{3}\right )}}{c^{6}}\right )} x + \frac {35 \, {\left (80 \, a c^{6} d^{4} + 48 \, a^{2} c^{5} d^{2} e^{2} - 3 \, a^{3} c^{4} e^{4}\right )}}{c^{6}}\right )} x + \frac {512 \, {\left (7 \, a^{2} c^{5} d^{3} e - 2 \, a^{3} c^{4} d e^{3}\right )}}{c^{6}}\right )} - \frac {3 \, {\left (16 \, a^{2} c^{2} d^{4} - 16 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{128 \, c^{\frac {5}{2}}} \]
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Timed out. \[ \int (d+e x)^4 \left (a+c x^2\right )^{3/2} \, dx=\int {\left (c\,x^2+a\right )}^{3/2}\,{\left (d+e\,x\right )}^4 \,d x \]
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