Integrand size = 22, antiderivative size = 119 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {(4 A b-5 a B) x^3}{4 b^2 \sqrt {a+b x^2}}+\frac {B x^5}{4 b \sqrt {a+b x^2}}+\frac {3 (4 A b-5 a B) x \sqrt {a+b x^2}}{8 b^3}-\frac {3 a (4 A b-5 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{7/2}} \]
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Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {470, 294, 327, 223, 212} \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {3 a (4 A b-5 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{7/2}}+\frac {3 x \sqrt {a+b x^2} (4 A b-5 a B)}{8 b^3}-\frac {x^3 (4 A b-5 a B)}{4 b^2 \sqrt {a+b x^2}}+\frac {B x^5}{4 b \sqrt {a+b x^2}} \]
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Rule 212
Rule 223
Rule 294
Rule 327
Rule 470
Rubi steps \begin{align*} \text {integral}& = \frac {B x^5}{4 b \sqrt {a+b x^2}}-\frac {(-4 A b+5 a B) \int \frac {x^4}{\left (a+b x^2\right )^{3/2}} \, dx}{4 b} \\ & = -\frac {(4 A b-5 a B) x^3}{4 b^2 \sqrt {a+b x^2}}+\frac {B x^5}{4 b \sqrt {a+b x^2}}+\frac {(3 (4 A b-5 a B)) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{4 b^2} \\ & = -\frac {(4 A b-5 a B) x^3}{4 b^2 \sqrt {a+b x^2}}+\frac {B x^5}{4 b \sqrt {a+b x^2}}+\frac {3 (4 A b-5 a B) x \sqrt {a+b x^2}}{8 b^3}-\frac {(3 a (4 A b-5 a B)) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 b^3} \\ & = -\frac {(4 A b-5 a B) x^3}{4 b^2 \sqrt {a+b x^2}}+\frac {B x^5}{4 b \sqrt {a+b x^2}}+\frac {3 (4 A b-5 a B) x \sqrt {a+b x^2}}{8 b^3}-\frac {(3 a (4 A b-5 a B)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 b^3} \\ & = -\frac {(4 A b-5 a B) x^3}{4 b^2 \sqrt {a+b x^2}}+\frac {B x^5}{4 b \sqrt {a+b x^2}}+\frac {3 (4 A b-5 a B) x \sqrt {a+b x^2}}{8 b^3}-\frac {3 a (4 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{7/2}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.90 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {12 a A b x-15 a^2 B x+4 A b^2 x^3-5 a b B x^3+2 b^2 B x^5}{8 b^3 \sqrt {a+b x^2}}+\frac {3 a (-4 A b+5 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{4 b^{7/2}} \]
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Time = 2.90 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(\frac {-3 \sqrt {b \,x^{2}+a}\, a \left (A b -\frac {5 B a}{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+\left (3 \left (-\frac {5 x^{2} B}{12}+A \right ) a \,b^{\frac {3}{2}}+x^{2} \left (\frac {x^{2} B}{2}+A \right ) b^{\frac {5}{2}}-\frac {15 B \,a^{2} \sqrt {b}}{4}\right ) x}{2 \sqrt {b \,x^{2}+a}\, b^{\frac {7}{2}}}\) | \(94\) |
risch | \(\frac {x \left (2 b B \,x^{2}+4 A b -7 B a \right ) \sqrt {b \,x^{2}+a}}{8 b^{3}}-\frac {a \left (-\frac {7 a B x}{\sqrt {b \,x^{2}+a}}+\frac {4 b A x}{\sqrt {b \,x^{2}+a}}+\left (12 b^{2} A -15 a b B \right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )\right )}{8 b^{3}}\) | \(117\) |
default | \(B \left (\frac {x^{5}}{4 b \sqrt {b \,x^{2}+a}}-\frac {5 a \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )}{4 b}\right )+A \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )\) | \(150\) |
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Time = 0.29 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.30 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (5 \, B a^{3} - 4 \, A a^{2} b + {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (2 \, B b^{3} x^{5} - {\left (5 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 3 \, {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, {\left (b^{5} x^{2} + a b^{4}\right )}}, -\frac {3 \, {\left (5 \, B a^{3} - 4 \, A a^{2} b + {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, B b^{3} x^{5} - {\left (5 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 3 \, {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{8 \, {\left (b^{5} x^{2} + a b^{4}\right )}}\right ] \]
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Time = 6.84 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.49 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=A \left (\frac {3 \sqrt {a} x}{2 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {5}{2}}} + \frac {x^{3}}{2 \sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + B \left (- \frac {15 a^{\frac {3}{2}} x}{8 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {5 \sqrt {a} x^{3}}{8 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {15 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {7}{2}}} + \frac {x^{5}}{4 \sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.06 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {B x^{5}}{4 \, \sqrt {b x^{2} + a} b} - \frac {5 \, B a x^{3}}{8 \, \sqrt {b x^{2} + a} b^{2}} + \frac {A x^{3}}{2 \, \sqrt {b x^{2} + a} b} - \frac {15 \, B a^{2} x}{8 \, \sqrt {b x^{2} + a} b^{3}} + \frac {3 \, A a x}{2 \, \sqrt {b x^{2} + a} b^{2}} + \frac {15 \, B a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {7}{2}}} - \frac {3 \, A a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}}} \]
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Time = 0.32 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.87 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (\frac {2 \, B x^{2}}{b} - \frac {5 \, B a b^{3} - 4 \, A b^{4}}{b^{5}}\right )} x^{2} - \frac {3 \, {\left (5 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )}}{b^{5}}\right )} x}{8 \, \sqrt {b x^{2} + a}} - \frac {3 \, {\left (5 \, B a^{2} - 4 \, A a b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {7}{2}}} \]
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Timed out. \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {x^4\,\left (B\,x^2+A\right )}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
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