Integrand size = 22, antiderivative size = 155 \[ \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {a^2 (8 A b-3 a B) x \sqrt {a+b x^2}}{128 b^2}+\frac {a (8 A b-3 a B) x^3 \sqrt {a+b x^2}}{64 b}+\frac {(8 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{48 b}+\frac {B x^3 \left (a+b x^2\right )^{5/2}}{8 b}-\frac {a^3 (8 A b-3 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {470, 285, 327, 223, 212} \[ \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=-\frac {a^3 (8 A b-3 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}}+\frac {a^2 x \sqrt {a+b x^2} (8 A b-3 a B)}{128 b^2}+\frac {a x^3 \sqrt {a+b x^2} (8 A b-3 a B)}{64 b}+\frac {x^3 \left (a+b x^2\right )^{3/2} (8 A b-3 a B)}{48 b}+\frac {B x^3 \left (a+b x^2\right )^{5/2}}{8 b} \]
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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 )^{5/2}}{8 b}-\frac {(-8 A b+3 a B) \int x^2 \left (a+b x^2\right )^{3/2} \, dx}{8 b} \\ & = \frac {(8 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{48 b}+\frac {B x^3 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {(a (8 A b-3 a B)) \int x^2 \sqrt {a+b x^2} \, dx}{16 b} \\ & = \frac {a (8 A b-3 a B) x^3 \sqrt {a+b x^2}}{64 b}+\frac {(8 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{48 b}+\frac {B x^3 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {\left (a^2 (8 A b-3 a B)\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{64 b} \\ & = \frac {a^2 (8 A b-3 a B) x \sqrt {a+b x^2}}{128 b^2}+\frac {a (8 A b-3 a B) x^3 \sqrt {a+b x^2}}{64 b}+\frac {(8 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{48 b}+\frac {B x^3 \left (a+b x^2\right )^{5/2}}{8 b}-\frac {\left (a^3 (8 A b-3 a B)\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{128 b^2} \\ & = \frac {a^2 (8 A b-3 a B) x \sqrt {a+b x^2}}{128 b^2}+\frac {a (8 A b-3 a B) x^3 \sqrt {a+b x^2}}{64 b}+\frac {(8 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{48 b}+\frac {B x^3 \left (a+b x^2\right )^{5/2}}{8 b}-\frac {\left (a^3 (8 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 )}{128 b^2} \\ & = \frac {a^2 (8 A b-3 a B) x \sqrt {a+b x^2}}{128 b^2}+\frac {a (8 A b-3 a B) x^3 \sqrt {a+b x^2}}{64 b}+\frac {(8 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{48 b}+\frac {B x^3 \left (a+b x^2\right )^{5/2}}{8 b}-\frac {a^3 (8 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.84 \[ \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (-9 a^3 B+6 a^2 b \left (4 A+B x^2\right )+16 b^3 x^4 \left (4 A+3 B x^2\right )+8 a b^2 x^2 \left (14 A+9 B x^2\right )\right )+6 a^3 (-8 A b+3 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{384 b^{5/2}} \]
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Time = 2.97 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(\frac {\frac {7 \left (-\frac {3}{14} A \,a^{3} b +\frac {9}{112} B \,a^{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{24}+\frac {7 x \sqrt {b \,x^{2}+a}\, \left (\frac {3 \left (\frac {x^{2} B}{4}+A \right ) a^{2} b^{\frac {3}{2}}}{14}+x^{2} a \left (\frac {9 x^{2} B}{14}+A \right ) b^{\frac {5}{2}}+\frac {4 \left (\frac {3 x^{2} B}{4}+A \right ) x^{4} b^{\frac {7}{2}}}{7}-\frac {9 B \,a^{3} \sqrt {b}}{112}\right )}{24}}{b^{\frac {5}{2}}}\) | \(108\) |
risch | \(\frac {x \left (48 b^{3} B \,x^{6}+64 A \,b^{3} x^{4}+72 B a \,b^{2} x^{4}+112 a A \,b^{2} x^{2}+6 B \,a^{2} b \,x^{2}+24 a^{2} b A -9 a^{3} B \right ) \sqrt {b \,x^{2}+a}}{384 b^{2}}-\frac {a^{3} \left (8 A b -3 B a \right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {5}{2}}}\) | \(112\) |
default | \(B \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )+A \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )\) | \(176\) |
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Time = 0.28 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.68 \[ \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\left [-\frac {3 \, {\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (48 \, B b^{4} x^{7} + 8 \, {\left (9 \, B a b^{3} + 8 \, A b^{4}\right )} x^{5} + 2 \, {\left (3 \, B a^{2} b^{2} + 56 \, A a b^{3}\right )} x^{3} - 3 \, {\left (3 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{768 \, b^{3}}, -\frac {3 \, {\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (48 \, B b^{4} x^{7} + 8 \, {\left (9 \, B a b^{3} + 8 \, A b^{4}\right )} x^{5} + 2 \, {\left (3 \, B a^{2} b^{2} + 56 \, A a b^{3}\right )} x^{3} - 3 \, {\left (3 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{384 \, b^{3}}\right ] \]
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Time = 0.42 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.52 \[ \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\begin {cases} - \frac {a \left (A a^{2} - \frac {3 a \left (2 A a b + B a^{2} - \frac {5 a \left (A b^{2} + \frac {9 B a b}{8}\right )}{6 b}\right )}{4 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2 b} + \sqrt {a + b x^{2}} \left (\frac {B b x^{7}}{8} + \frac {x^{5} \left (A b^{2} + \frac {9 B a b}{8}\right )}{6 b} + \frac {x^{3} \cdot \left (2 A a b + B a^{2} - \frac {5 a \left (A b^{2} + \frac {9 B a b}{8}\right )}{6 b}\right )}{4 b} + \frac {x \left (A a^{2} - \frac {3 a \left (2 A a b + B a^{2} - \frac {5 a \left (A b^{2} + \frac {9 B a b}{8}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right ) & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (\frac {A x^{3}}{3} + \frac {B x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.05 \[ \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B x^{3}}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B a x}{16 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} x}{64 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} B a^{3} x}{128 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A x}{6 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A a x}{24 \, b} - \frac {\sqrt {b x^{2} + a} A a^{2} x}{16 \, b} + \frac {3 \, B a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {5}{2}}} - \frac {A a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {3}{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.86 \[ \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (6 \, B b x^{2} + \frac {9 \, B a b^{6} + 8 \, A b^{7}}{b^{6}}\right )} x^{2} + \frac {3 \, B a^{2} b^{5} + 56 \, A a b^{6}}{b^{6}}\right )} x^{2} - \frac {3 \, {\left (3 \, B a^{3} b^{4} - 8 \, A a^{2} b^{5}\right )}}{b^{6}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {5}{2}}} \]
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Timed out. \[ \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\int x^2\,\left (B\,x^2+A\right )\,{\left (b\,x^2+a\right )}^{3/2} \,d x \]
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