Integrand size = 22, antiderivative size = 99 \[ \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {a^2 (A b-a B)}{b^4 \sqrt {a+b x^2}}-\frac {a (2 A b-3 a B) \sqrt {a+b x^2}}{b^4}+\frac {(A b-3 a B) \left (a+b x^2\right )^{3/2}}{3 b^4}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b^4} \]
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Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 78} \[ \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {a^2 (A b-a B)}{b^4 \sqrt {a+b x^2}}+\frac {\left (a+b x^2\right )^{3/2} (A b-3 a B)}{3 b^4}-\frac {a \sqrt {a+b x^2} (2 A b-3 a B)}{b^4}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b^4} \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2 (A+B x)}{(a+b x)^{3/2}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {a^2 (-A b+a B)}{b^3 (a+b x)^{3/2}}+\frac {a (-2 A b+3 a B)}{b^3 \sqrt {a+b x}}+\frac {(A b-3 a B) \sqrt {a+b x}}{b^3}+\frac {B (a+b x)^{3/2}}{b^3}\right ) \, dx,x,x^2\right ) \\ & = -\frac {a^2 (A b-a B)}{b^4 \sqrt {a+b x^2}}-\frac {a (2 A b-3 a B) \sqrt {a+b x^2}}{b^4}+\frac {(A b-3 a B) \left (a+b x^2\right )^{3/2}}{3 b^4}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b^4} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.78 \[ \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {48 a^3 B-8 a^2 b \left (5 A-3 B x^2\right )+b^3 x^4 \left (5 A+3 B x^2\right )-2 a b^2 x^2 \left (10 A+3 B x^2\right )}{15 b^4 \sqrt {a+b x^2}} \]
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Time = 2.80 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(-\frac {8 \left (-\frac {x^{4} \left (\frac {3 x^{2} B}{5}+A \right ) b^{3}}{8}+\frac {x^{2} \left (\frac {3 x^{2} B}{10}+A \right ) a \,b^{2}}{2}+a^{2} \left (-\frac {3 x^{2} B}{5}+A \right ) b -\frac {6 a^{3} B}{5}\right )}{3 \sqrt {b \,x^{2}+a}\, b^{4}}\) | \(68\) |
gosper | \(-\frac {-3 b^{3} B \,x^{6}-5 A \,b^{3} x^{4}+6 B a \,b^{2} x^{4}+20 a A \,b^{2} x^{2}-24 B \,a^{2} b \,x^{2}+40 a^{2} b A -48 a^{3} B}{15 \sqrt {b \,x^{2}+a}\, b^{4}}\) | \(77\) |
trager | \(-\frac {-3 b^{3} B \,x^{6}-5 A \,b^{3} x^{4}+6 B a \,b^{2} x^{4}+20 a A \,b^{2} x^{2}-24 B \,a^{2} b \,x^{2}+40 a^{2} b A -48 a^{3} B}{15 \sqrt {b \,x^{2}+a}\, b^{4}}\) | \(77\) |
risch | \(-\frac {\left (-3 b^{2} B \,x^{4}-5 A \,b^{2} x^{2}+9 B a b \,x^{2}+25 a b A -33 a^{2} B \right ) \sqrt {b \,x^{2}+a}}{15 b^{4}}-\frac {a^{2} \left (A b -B a \right )}{b^{4} \sqrt {b \,x^{2}+a}}\) | \(79\) |
default | \(B \left (\frac {x^{6}}{5 b \sqrt {b \,x^{2}+a}}-\frac {6 a \left (\frac {x^{4}}{3 b \sqrt {b \,x^{2}+a}}-\frac {4 a \left (\frac {x^{2}}{\sqrt {b \,x^{2}+a}\, b}+\frac {2 a}{b^{2} \sqrt {b \,x^{2}+a}}\right )}{3 b}\right )}{5 b}\right )+A \left (\frac {x^{4}}{3 b \sqrt {b \,x^{2}+a}}-\frac {4 a \left (\frac {x^{2}}{\sqrt {b \,x^{2}+a}\, b}+\frac {2 a}{b^{2} \sqrt {b \,x^{2}+a}}\right )}{3 b}\right )\) | \(142\) |
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Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.89 \[ \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left (3 \, B b^{3} x^{6} - {\left (6 \, B a b^{2} - 5 \, A b^{3}\right )} x^{4} + 48 \, B a^{3} - 40 \, A a^{2} b + 4 \, {\left (6 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{15 \, {\left (b^{5} x^{2} + a b^{4}\right )}} \]
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Time = 0.32 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.74 \[ \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\begin {cases} - \frac {8 A a^{2}}{3 b^{3} \sqrt {a + b x^{2}}} - \frac {4 A a x^{2}}{3 b^{2} \sqrt {a + b x^{2}}} + \frac {A x^{4}}{3 b \sqrt {a + b x^{2}}} + \frac {16 B a^{3}}{5 b^{4} \sqrt {a + b x^{2}}} + \frac {8 B a^{2} x^{2}}{5 b^{3} \sqrt {a + b x^{2}}} - \frac {2 B a x^{4}}{5 b^{2} \sqrt {a + b x^{2}}} + \frac {B x^{6}}{5 b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{6}}{6} + \frac {B x^{8}}{8}}{a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.33 \[ \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {B x^{6}}{5 \, \sqrt {b x^{2} + a} b} - \frac {2 \, B a x^{4}}{5 \, \sqrt {b x^{2} + a} b^{2}} + \frac {A x^{4}}{3 \, \sqrt {b x^{2} + a} b} + \frac {8 \, B a^{2} x^{2}}{5 \, \sqrt {b x^{2} + a} b^{3}} - \frac {4 \, A a x^{2}}{3 \, \sqrt {b x^{2} + a} b^{2}} + \frac {16 \, B a^{3}}{5 \, \sqrt {b x^{2} + a} b^{4}} - \frac {8 \, A a^{2}}{3 \, \sqrt {b x^{2} + a} b^{3}} \]
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Time = 0.32 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.14 \[ \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {B a^{3} - A a^{2} b}{\sqrt {b x^{2} + a} b^{4}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{16} - 15 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a b^{16} + 45 \, \sqrt {b x^{2} + a} B a^{2} b^{16} + 5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{17} - 30 \, \sqrt {b x^{2} + a} A a b^{17}}{15 \, b^{20}} \]
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Time = 5.24 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.90 \[ \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {B\,{\left (b\,x^2+a\right )}^3}{5}+B\,a^3+\frac {A\,b\,{\left (b\,x^2+a\right )}^2}{3}-B\,a\,{\left (b\,x^2+a\right )}^2+3\,B\,a^2\,\left (b\,x^2+a\right )-A\,a^2\,b-2\,A\,a\,b\,\left (b\,x^2+a\right )}{b^4\,\sqrt {b\,x^2+a}} \]
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