Integrand size = 21, antiderivative size = 241 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \, dx=\frac {a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{256 b^2}+\frac {a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {a^3 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}} \]
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Time = 0.11 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {427, 396, 201, 223, 212} \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \, dx=\frac {x \left (a+b x^2\right )^{5/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{480 b^2}+\frac {a x \left (a+b x^2\right )^{3/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{384 b^2}+\frac {a^2 x \sqrt {a+b x^2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{256 b^2}+\frac {a^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{256 b^{5/2}}+\frac {3 d x \left (a+b x^2\right )^{7/2} (4 b c-a d)}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b} \]
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Rule 201
Rule 212
Rule 223
Rule 396
Rule 427
Rubi steps \begin{align*} \text {integral}& = \frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {\int \left (a+b x^2\right )^{5/2} \left (c (10 b c-a d)+3 d (4 b c-a d) x^2\right ) \, dx}{10 b} \\ & = \frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}-\frac {(3 a d (4 b c-a d)-8 b c (10 b c-a d)) \int \left (a+b x^2\right )^{5/2} \, dx}{80 b^2} \\ & = \frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {\left (a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right )\right ) \int \left (a+b x^2\right )^{3/2} \, dx}{96 b^2} \\ & = \frac {a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {\left (a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right )\right ) \int \sqrt {a+b x^2} \, dx}{128 b^2} \\ & = \frac {a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{256 b^2}+\frac {a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {\left (a^3 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{256 b^2} \\ & = \frac {a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{256 b^2}+\frac {a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {\left (a^3 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{256 b^2} \\ & = \frac {a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{256 b^2}+\frac {a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {a^3 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.79 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (-45 a^4 d^2+30 a^3 b d \left (10 c+d x^2\right )+64 b^4 x^4 \left (10 c^2+15 c d x^2+6 d^2 x^4\right )+16 a b^3 x^2 \left (130 c^2+170 c d x^2+63 d^2 x^4\right )+8 a^2 b^2 \left (330 c^2+295 c d x^2+93 d^2 x^4\right )\right )-15 a^3 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{3840 b^{5/2}} \]
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Time = 2.44 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(\frac {\frac {3 a^{3} \left (a^{2} d^{2}-\frac {20}{3} a b c d +\frac {80}{3} b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{256}-\frac {3 x \left (-\frac {176 \left (\frac {31}{110} d^{2} x^{4}+\frac {59}{66} c d \,x^{2}+c^{2}\right ) a^{2} b^{\frac {5}{2}}}{3}-\frac {416 x^{2} \left (\frac {63}{130} d^{2} x^{4}+\frac {17}{13} c d \,x^{2}+c^{2}\right ) a \,b^{\frac {7}{2}}}{9}-\frac {128 x^{4} \left (\frac {3}{5} d^{2} x^{4}+\frac {3}{2} c d \,x^{2}+c^{2}\right ) b^{\frac {9}{2}}}{9}+\left (\frac {2 \left (-d \,x^{2}-10 c \right ) b^{\frac {3}{2}}}{3}+a d \sqrt {b}\right ) d \,a^{3}\right ) \sqrt {b \,x^{2}+a}}{256}}{b^{\frac {5}{2}}}\) | \(172\) |
risch | \(-\frac {x \left (-384 b^{4} d^{2} x^{8}-1008 a \,b^{3} d^{2} x^{6}-960 b^{4} c d \,x^{6}-744 a^{2} b^{2} d^{2} x^{4}-2720 a \,b^{3} c d \,x^{4}-640 b^{4} c^{2} x^{4}-30 a^{3} b \,d^{2} x^{2}-2360 a^{2} b^{2} c d \,x^{2}-2080 a \,b^{3} c^{2} x^{2}+45 a^{4} d^{2}-300 a^{3} b c d -2640 a^{2} b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}}{3840 b^{2}}+\frac {a^{3} \left (3 a^{2} d^{2}-20 a b c d +80 b^{2} c^{2}\right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {5}{2}}}\) | \(198\) |
default | \(c^{2} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )+d^{2} \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{10 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )+2 c d \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )\) | \(283\) |
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Time = 0.36 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.74 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \, dx=\left [\frac {15 \, {\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (384 \, b^{5} d^{2} x^{9} + 48 \, {\left (20 \, b^{5} c d + 21 \, a b^{4} d^{2}\right )} x^{7} + 8 \, {\left (80 \, b^{5} c^{2} + 340 \, a b^{4} c d + 93 \, a^{2} b^{3} d^{2}\right )} x^{5} + 10 \, {\left (208 \, a b^{4} c^{2} + 236 \, a^{2} b^{3} c d + 3 \, a^{3} b^{2} d^{2}\right )} x^{3} + 15 \, {\left (176 \, a^{2} b^{3} c^{2} + 20 \, a^{3} b^{2} c d - 3 \, a^{4} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{7680 \, b^{3}}, -\frac {15 \, {\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (384 \, b^{5} d^{2} x^{9} + 48 \, {\left (20 \, b^{5} c d + 21 \, a b^{4} d^{2}\right )} x^{7} + 8 \, {\left (80 \, b^{5} c^{2} + 340 \, a b^{4} c d + 93 \, a^{2} b^{3} d^{2}\right )} x^{5} + 10 \, {\left (208 \, a b^{4} c^{2} + 236 \, a^{2} b^{3} c d + 3 \, a^{3} b^{2} d^{2}\right )} x^{3} + 15 \, {\left (176 \, a^{2} b^{3} c^{2} + 20 \, a^{3} b^{2} c d - 3 \, a^{4} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{3840 \, b^{3}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (236) = 472\).
Time = 0.58 (sec) , antiderivative size = 520, normalized size of antiderivative = 2.16 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {b^{2} d^{2} x^{9}}{10} + \frac {x^{7} \cdot \left (\frac {21 a b^{2} d^{2}}{10} + 2 b^{3} c d\right )}{8 b} + \frac {x^{5} \cdot \left (3 a^{2} b d^{2} + 6 a b^{2} c d - \frac {7 a \left (\frac {21 a b^{2} d^{2}}{10} + 2 b^{3} c d\right )}{8 b} + b^{3} c^{2}\right )}{6 b} + \frac {x^{3} \left (a^{3} d^{2} + 6 a^{2} b c d + 3 a b^{2} c^{2} - \frac {5 a \left (3 a^{2} b d^{2} + 6 a b^{2} c d - \frac {7 a \left (\frac {21 a b^{2} d^{2}}{10} + 2 b^{3} c d\right )}{8 b} + b^{3} c^{2}\right )}{6 b}\right )}{4 b} + \frac {x \left (2 a^{3} c d + 3 a^{2} b c^{2} - \frac {3 a \left (a^{3} d^{2} + 6 a^{2} b c d + 3 a b^{2} c^{2} - \frac {5 a \left (3 a^{2} b d^{2} + 6 a b^{2} c d - \frac {7 a \left (\frac {21 a b^{2} d^{2}}{10} + 2 b^{3} c d\right )}{8 b} + b^{3} c^{2}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right ) + \left (a^{3} c^{2} - \frac {a \left (2 a^{3} c d + 3 a^{2} b c^{2} - \frac {3 a \left (a^{3} d^{2} + 6 a^{2} b c d + 3 a b^{2} c^{2} - \frac {5 a \left (3 a^{2} b d^{2} + 6 a b^{2} c d - \frac {7 a \left (\frac {21 a b^{2} d^{2}}{10} + 2 b^{3} c d\right )}{8 b} + b^{3} c^{2}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (c^{2} x + \frac {2 c d x^{3}}{3} + \frac {d^{2} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.19 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} d^{2} x^{3}}{10 \, b} + \frac {1}{6} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{2} x + \frac {5}{24} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a c^{2} x + \frac {5}{16} \, \sqrt {b x^{2} + a} a^{2} c^{2} x + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} c d x}{4 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a c d x}{24 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} c d x}{96 \, b} - \frac {5 \, \sqrt {b x^{2} + a} a^{3} c d x}{64 \, b} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a d^{2} x}{80 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} d^{2} x}{160 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} d^{2} x}{128 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} a^{4} d^{2} x}{256 \, b^{2}} + \frac {5 \, a^{3} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {b}} - \frac {5 \, a^{4} c d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{64 \, b^{\frac {3}{2}}} + \frac {3 \, a^{5} d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {5}{2}}} \]
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Time = 0.32 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.92 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \, dx=\frac {1}{3840} \, {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, b^{2} d^{2} x^{2} + \frac {20 \, b^{10} c d + 21 \, a b^{9} d^{2}}{b^{8}}\right )} x^{2} + \frac {80 \, b^{10} c^{2} + 340 \, a b^{9} c d + 93 \, a^{2} b^{8} d^{2}}{b^{8}}\right )} x^{2} + \frac {5 \, {\left (208 \, a b^{9} c^{2} + 236 \, a^{2} b^{8} c d + 3 \, a^{3} b^{7} d^{2}\right )}}{b^{8}}\right )} x^{2} + \frac {15 \, {\left (176 \, a^{2} b^{8} c^{2} + 20 \, a^{3} b^{7} c d - 3 \, a^{4} b^{6} d^{2}\right )}}{b^{8}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {5}{2}}} \]
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Timed out. \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \, dx=\int {\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^2 \,d x \]
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