Integrand size = 22, antiderivative size = 188 \[ \int x^2 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\frac {a^3 (10 A b-3 a B) x \sqrt {a+b x^2}}{256 b^2}+\frac {a^2 (10 A b-3 a B) x^3 \sqrt {a+b x^2}}{128 b}+\frac {a (10 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{96 b}+\frac {(10 A b-3 a B) x^3 \left (a+b x^2\right )^{5/2}}{80 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac {a^4 (10 A b-3 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {470, 285, 327, 223, 212} \[ \int x^2 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=-\frac {a^4 (10 A b-3 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}}+\frac {a^3 x \sqrt {a+b x^2} (10 A b-3 a B)}{256 b^2}+\frac {a^2 x^3 \sqrt {a+b x^2} (10 A b-3 a B)}{128 b}+\frac {a x^3 \left (a+b x^2\right )^{3/2} (10 A b-3 a B)}{96 b}+\frac {x^3 \left (a+b x^2\right )^{5/2} (10 A b-3 a B)}{80 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b} \]
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Rule 212
Rule 223
Rule 285
Rule 327
Rule 470
Rubi steps \begin{align*} \text {integral}& = \frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac {(-10 A b+3 a B) \int x^2 \left (a+b x^2\right )^{5/2} \, dx}{10 b} \\ & = \frac {(10 A b-3 a B) x^3 \left (a+b x^2\right )^{5/2}}{80 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}+\frac {(a (10 A b-3 a B)) \int x^2 \left (a+b x^2\right )^{3/2} \, dx}{16 b} \\ & = \frac {a (10 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{96 b}+\frac {(10 A b-3 a B) x^3 \left (a+b x^2\right )^{5/2}}{80 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}+\frac {\left (a^2 (10 A b-3 a B)\right ) \int x^2 \sqrt {a+b x^2} \, dx}{32 b} \\ & = \frac {a^2 (10 A b-3 a B) x^3 \sqrt {a+b x^2}}{128 b}+\frac {a (10 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{96 b}+\frac {(10 A b-3 a B) x^3 \left (a+b x^2\right )^{5/2}}{80 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}+\frac {\left (a^3 (10 A b-3 a B)\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{128 b} \\ & = \frac {a^3 (10 A b-3 a B) x \sqrt {a+b x^2}}{256 b^2}+\frac {a^2 (10 A b-3 a B) x^3 \sqrt {a+b x^2}}{128 b}+\frac {a (10 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{96 b}+\frac {(10 A b-3 a B) x^3 \left (a+b x^2\right )^{5/2}}{80 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac {\left (a^4 (10 A b-3 a B)\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{256 b^2} \\ & = \frac {a^3 (10 A b-3 a B) x \sqrt {a+b x^2}}{256 b^2}+\frac {a^2 (10 A b-3 a B) x^3 \sqrt {a+b x^2}}{128 b}+\frac {a (10 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{96 b}+\frac {(10 A b-3 a B) x^3 \left (a+b x^2\right )^{5/2}}{80 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac {\left (a^4 (10 A b-3 a B)\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^3 (10 A b-3 a B) x \sqrt {a+b x^2}}{256 b^2}+\frac {a^2 (10 A b-3 a B) x^3 \sqrt {a+b x^2}}{128 b}+\frac {a (10 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{96 b}+\frac {(10 A b-3 a B) x^3 \left (a+b x^2\right )^{5/2}}{80 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac {a^4 (10 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.80 \[ \int x^2 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (-45 a^4 B+30 a^3 b \left (5 A+B x^2\right )+96 b^4 x^6 \left (5 A+4 B x^2\right )+16 a b^3 x^4 \left (85 A+63 B x^2\right )+4 a^2 b^2 x^2 \left (295 A+186 B x^2\right )\right )+30 a^4 (-10 A b+3 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{3840 b^{5/2}} \]
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Time = 2.91 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(\frac {\frac {59 \left (-\frac {15}{118} a^{4} b A +\frac {9}{236} a^{5} B \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{192}+\frac {59 x \left (\frac {15 \left (\frac {x^{2} B}{5}+A \right ) a^{3} b^{\frac {3}{2}}}{118}+a^{2} x^{2} \left (\frac {186 x^{2} B}{295}+A \right ) b^{\frac {5}{2}}+\frac {68 x^{4} a \left (\frac {63 x^{2} B}{85}+A \right ) b^{\frac {7}{2}}}{59}+\frac {24 x^{6} \left (\frac {4 x^{2} B}{5}+A \right ) b^{\frac {9}{2}}}{59}-\frac {9 B \,a^{4} \sqrt {b}}{236}\right ) \sqrt {b \,x^{2}+a}}{192}}{b^{\frac {5}{2}}}\) | \(127\) |
risch | \(\frac {x \left (384 B \,x^{8} b^{4}+480 A \,x^{6} b^{4}+1008 B \,x^{6} a \,b^{3}+1360 A a \,b^{3} x^{4}+744 B \,a^{2} b^{2} x^{4}+1180 A \,a^{2} b^{2} x^{2}+30 B \,a^{3} b \,x^{2}+150 A \,a^{3} b -45 B \,a^{4}\right ) \sqrt {b \,x^{2}+a}}{3840 b^{2}}-\frac {a^{4} \left (10 A b -3 B a \right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {5}{2}}}\) | \(136\) |
default | \(B \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{10 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )+A \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )\) | \(208\) |
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Time = 0.30 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.64 \[ \int x^2 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\left [-\frac {15 \, {\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (384 \, B b^{5} x^{9} + 48 \, {\left (21 \, B a b^{4} + 10 \, A b^{5}\right )} x^{7} + 8 \, {\left (93 \, B a^{2} b^{3} + 170 \, A a b^{4}\right )} x^{5} + 10 \, {\left (3 \, B a^{3} b^{2} + 118 \, A a^{2} b^{3}\right )} x^{3} - 15 \, {\left (3 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{7680 \, b^{3}}, -\frac {15 \, {\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (384 \, B b^{5} x^{9} + 48 \, {\left (21 \, B a b^{4} + 10 \, A b^{5}\right )} x^{7} + 8 \, {\left (93 \, B a^{2} b^{3} + 170 \, A a b^{4}\right )} x^{5} + 10 \, {\left (3 \, B a^{3} b^{2} + 118 \, A a^{2} b^{3}\right )} x^{3} - 15 \, {\left (3 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{3840 \, b^{3}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (173) = 346\).
Time = 0.56 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.95 \[ \int x^2 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\begin {cases} - \frac {a \left (A a^{3} - \frac {3 a \left (3 A a^{2} b + B a^{3} - \frac {5 a \left (3 A a b^{2} + 3 B a^{2} b - \frac {7 a \left (A b^{3} + \frac {21 B a b^{2}}{10}\right )}{8 b}\right )}{6 b}\right )}{4 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2 b} + \sqrt {a + b x^{2}} \left (\frac {B b^{2} x^{9}}{10} + \frac {x^{7} \left (A b^{3} + \frac {21 B a b^{2}}{10}\right )}{8 b} + \frac {x^{5} \cdot \left (3 A a b^{2} + 3 B a^{2} b - \frac {7 a \left (A b^{3} + \frac {21 B a b^{2}}{10}\right )}{8 b}\right )}{6 b} + \frac {x^{3} \cdot \left (3 A a^{2} b + B a^{3} - \frac {5 a \left (3 A a b^{2} + 3 B a^{2} b - \frac {7 a \left (A b^{3} + \frac {21 B a b^{2}}{10}\right )}{8 b}\right )}{6 b}\right )}{4 b} + \frac {x \left (A a^{3} - \frac {3 a \left (3 A a^{2} b + B a^{3} - \frac {5 a \left (3 A a b^{2} + 3 B a^{2} b - \frac {7 a \left (A b^{3} + \frac {21 B a b^{2}}{10}\right )}{8 b}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right ) & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (\frac {A x^{3}}{3} + \frac {B x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.06 \[ \int x^2 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B x^{3}}{10 \, b} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a x}{80 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{2} x}{160 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{3} x}{128 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} B a^{4} x}{256 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A x}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A a x}{48 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a^{2} x}{192 \, b} - \frac {5 \, \sqrt {b x^{2} + a} A a^{3} x}{128 \, b} + \frac {3 \, B a^{5} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {5}{2}}} - \frac {5 \, A a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {3}{2}}} \]
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Time = 0.33 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.88 \[ \int x^2 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\frac {1}{3840} \, {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, B b^{2} x^{2} + \frac {21 \, B a b^{9} + 10 \, A b^{10}}{b^{8}}\right )} x^{2} + \frac {93 \, B a^{2} b^{8} + 170 \, A a b^{9}}{b^{8}}\right )} x^{2} + \frac {5 \, {\left (3 \, B a^{3} b^{7} + 118 \, A a^{2} b^{8}\right )}}{b^{8}}\right )} x^{2} - \frac {15 \, {\left (3 \, B a^{4} b^{6} - 10 \, A a^{3} b^{7}\right )}}{b^{8}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {5}{2}}} \]
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Timed out. \[ \int x^2 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\int x^2\,\left (B\,x^2+A\right )\,{\left (b\,x^2+a\right )}^{5/2} \,d x \]
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