Integrand size = 19, antiderivative size = 149 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \, dx=\frac {5 a^2 (8 b c-a d) x \sqrt {a+b x^2}}{128 b}+\frac {5 a (8 b c-a d) x \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(8 b c-a d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {d x \left (a+b x^2\right )^{7/2}}{8 b}+\frac {5 a^3 (8 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {396, 201, 223, 212} \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \, dx=\frac {5 a^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (8 b c-a d)}{128 b^{3/2}}+\frac {5 a^2 x \sqrt {a+b x^2} (8 b c-a d)}{128 b}+\frac {x \left (a+b x^2\right )^{5/2} (8 b c-a d)}{48 b}+\frac {5 a x \left (a+b x^2\right )^{3/2} (8 b c-a d)}{192 b}+\frac {d x \left (a+b x^2\right )^{7/2}}{8 b} \]
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Rule 201
Rule 212
Rule 223
Rule 396
Rubi steps \begin{align*} \text {integral}& = \frac {d x \left (a+b x^2\right )^{7/2}}{8 b}-\frac {(-8 b c+a d) \int \left (a+b x^2\right )^{5/2} \, dx}{8 b} \\ & = \frac {(8 b c-a d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {d x \left (a+b x^2\right )^{7/2}}{8 b}+\frac {(5 a (8 b c-a d)) \int \left (a+b x^2\right )^{3/2} \, dx}{48 b} \\ & = \frac {5 a (8 b c-a d) x \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(8 b c-a d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {d x \left (a+b x^2\right )^{7/2}}{8 b}+\frac {\left (5 a^2 (8 b c-a d)\right ) \int \sqrt {a+b x^2} \, dx}{64 b} \\ & = \frac {5 a^2 (8 b c-a d) x \sqrt {a+b x^2}}{128 b}+\frac {5 a (8 b c-a d) x \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(8 b c-a d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {d x \left (a+b x^2\right )^{7/2}}{8 b}+\frac {\left (5 a^3 (8 b c-a d)\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{128 b} \\ & = \frac {5 a^2 (8 b c-a d) x \sqrt {a+b x^2}}{128 b}+\frac {5 a (8 b c-a d) x \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(8 b c-a d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {d x \left (a+b x^2\right )^{7/2}}{8 b}+\frac {\left (5 a^3 (8 b c-a d)\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{128 b} \\ & = \frac {5 a^2 (8 b c-a d) x \sqrt {a+b x^2}}{128 b}+\frac {5 a (8 b c-a d) x \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(8 b c-a d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {d x \left (a+b x^2\right )^{7/2}}{8 b}+\frac {5 a^3 (8 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{3/2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.83 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \, dx=\frac {x \sqrt {a+b x^2} \left (264 a^2 b c+15 a^3 d+208 a b^2 c x^2+118 a^2 b d x^2+64 b^3 c x^4+136 a b^2 d x^4+48 b^3 d x^6\right )}{384 b}+\frac {5 a^3 (-8 b c+a d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{128 b^{3/2}} \]
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Time = 2.39 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.72
method | result | size |
pseudoelliptic | \(-\frac {5 \left (\left (a^{4} d -8 a^{3} b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-x \sqrt {b \,x^{2}+a}\, \left (\frac {88 \left (\frac {59 d \,x^{2}}{132}+c \right ) a^{2} b^{\frac {3}{2}}}{5}+\frac {208 x^{2} \left (\frac {17 d \,x^{2}}{26}+c \right ) a \,b^{\frac {5}{2}}}{15}+\frac {64 x^{4} \left (\frac {3 d \,x^{2}}{4}+c \right ) b^{\frac {7}{2}}}{15}+a^{3} d \sqrt {b}\right )\right )}{128 b^{\frac {3}{2}}}\) | \(108\) |
risch | \(\frac {x \left (48 b^{3} d \,x^{6}+136 a \,b^{2} d \,x^{4}+64 b^{3} c \,x^{4}+118 a^{2} b d \,x^{2}+208 a \,b^{2} c \,x^{2}+15 a^{3} d +264 a^{2} b c \right ) \sqrt {b \,x^{2}+a}}{384 b}-\frac {5 a^{3} \left (a d -8 b c \right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {3}{2}}}\) | \(111\) |
default | \(c \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )+d \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )\) | \(162\) |
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Time = 0.32 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.74 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \, dx=\left [-\frac {15 \, {\left (8 \, a^{3} b c - a^{4} d\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (48 \, b^{4} d x^{7} + 8 \, {\left (8 \, b^{4} c + 17 \, a b^{3} d\right )} x^{5} + 2 \, {\left (104 \, a b^{3} c + 59 \, a^{2} b^{2} d\right )} x^{3} + 3 \, {\left (88 \, a^{2} b^{2} c + 5 \, a^{3} b d\right )} x\right )} \sqrt {b x^{2} + a}}{768 \, b^{2}}, -\frac {15 \, {\left (8 \, a^{3} b c - a^{4} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (48 \, b^{4} d x^{7} + 8 \, {\left (8 \, b^{4} c + 17 \, a b^{3} d\right )} x^{5} + 2 \, {\left (104 \, a b^{3} c + 59 \, a^{2} b^{2} d\right )} x^{3} + 3 \, {\left (88 \, a^{2} b^{2} c + 5 \, a^{3} b d\right )} x\right )} \sqrt {b x^{2} + a}}{384 \, b^{2}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (134) = 268\).
Time = 0.42 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.87 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {b^{2} d x^{7}}{8} + \frac {x^{5} \cdot \left (\frac {17 a b^{2} d}{8} + b^{3} c\right )}{6 b} + \frac {x^{3} \cdot \left (3 a^{2} b d + 3 a b^{2} c - \frac {5 a \left (\frac {17 a b^{2} d}{8} + b^{3} c\right )}{6 b}\right )}{4 b} + \frac {x \left (a^{3} d + 3 a^{2} b c - \frac {3 a \left (3 a^{2} b d + 3 a b^{2} c - \frac {5 a \left (\frac {17 a b^{2} d}{8} + b^{3} c\right )}{6 b}\right )}{4 b}\right )}{2 b}\right ) + \left (a^{3} c - \frac {a \left (a^{3} d + 3 a^{2} b c - \frac {3 a \left (3 a^{2} b d + 3 a b^{2} c - \frac {5 a \left (\frac {17 a b^{2} d}{8} + b^{3} c\right )}{6 b}\right )}{4 b}\right )}{2 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (c x + \frac {d x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.01 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \, dx=\frac {1}{6} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} c x + \frac {5}{24} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a c x + \frac {5}{16} \, \sqrt {b x^{2} + a} a^{2} c x + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} d x}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a d x}{48 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} d x}{192 \, b} - \frac {5 \, \sqrt {b x^{2} + a} a^{3} d x}{128 \, b} + \frac {5 \, a^{3} c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {b}} - \frac {5 \, a^{4} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {3}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.91 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \, dx=\frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (6 \, b^{2} d x^{2} + \frac {8 \, b^{8} c + 17 \, a b^{7} d}{b^{6}}\right )} x^{2} + \frac {104 \, a b^{7} c + 59 \, a^{2} b^{6} d}{b^{6}}\right )} x^{2} + \frac {3 \, {\left (88 \, a^{2} b^{6} c + 5 \, a^{3} b^{5} d\right )}}{b^{6}}\right )} \sqrt {b x^{2} + a} x - \frac {5 \, {\left (8 \, a^{3} b c - a^{4} d\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {3}{2}}} \]
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Timed out. \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \, dx=\int {\left (b\,x^2+a\right )}^{5/2}\,\left (d\,x^2+c\right ) \,d x \]
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