Integrand size = 25, antiderivative size = 132 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {x^5 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^4 (A b+6 a C-5 b B x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {4 (A b+6 a C)}{105 b^4 \left (a+b x^2\right )^{3/2}}-\frac {4 (A b+6 a C)}{35 a b^4 \sqrt {a+b x^2}} \]
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Time = 0.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1818, 819, 272, 45} \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {4 (6 a C+A b)}{35 a b^4 \sqrt {a+b x^2}}+\frac {4 (6 a C+A b)}{105 b^4 \left (a+b x^2\right )^{3/2}}-\frac {x^4 (6 a C+A b-5 b B x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^5 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]
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Rule 45
Rule 272
Rule 819
Rule 1818
Rubi steps \begin{align*} \text {integral}& = -\frac {x^5 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x^4 (-5 a B-(A b+6 a C) x)}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b} \\ & = -\frac {x^5 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^4 (A b+6 a C-5 b B x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {(4 (A b+6 a C)) \int \frac {x^3}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a b^2} \\ & = -\frac {x^5 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^4 (A b+6 a C-5 b B x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {(2 (A b+6 a C)) \text {Subst}\left (\int \frac {x}{(a+b x)^{5/2}} \, dx,x,x^2\right )}{35 a b^2} \\ & = -\frac {x^5 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^4 (A b+6 a C-5 b B x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {(2 (A b+6 a C)) \text {Subst}\left (\int \left (-\frac {a}{b (a+b x)^{5/2}}+\frac {1}{b (a+b x)^{3/2}}\right ) \, dx,x,x^2\right )}{35 a b^2} \\ & = -\frac {x^5 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^4 (A b+6 a C-5 b B x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {4 (A b+6 a C)}{105 b^4 \left (a+b x^2\right )^{3/2}}-\frac {4 (A b+6 a C)}{35 a b^4 \sqrt {a+b x^2}} \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.67 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {-48 a^4 C+15 b^4 B x^7-35 a b^3 x^4 \left (A+3 C x^2\right )-14 a^2 b^2 x^2 \left (2 A+15 C x^2\right )-8 a^3 b \left (A+21 C x^2\right )}{105 a b^4 \left (a+b x^2\right )^{7/2}} \]
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Time = 3.56 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.72
method | result | size |
gosper | \(-\frac {-15 x^{7} B \,b^{4}+105 C \,x^{6} a \,b^{3}+35 A a \,b^{3} x^{4}+210 C \,a^{2} b^{2} x^{4}+28 A \,a^{2} b^{2} x^{2}+168 C \,a^{3} b \,x^{2}+8 A \,a^{3} b +48 C \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a \,b^{4}}\) | \(95\) |
trager | \(-\frac {-15 x^{7} B \,b^{4}+105 C \,x^{6} a \,b^{3}+35 A a \,b^{3} x^{4}+210 C \,a^{2} b^{2} x^{4}+28 A \,a^{2} b^{2} x^{2}+168 C \,a^{3} b \,x^{2}+8 A \,a^{3} b +48 C \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a \,b^{4}}\) | \(95\) |
default | \(C \left (-\frac {x^{6}}{b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {6 a \left (-\frac {x^{4}}{3 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {4 a \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )}{3 b}\right )}{b}\right )+B \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right )+A \left (-\frac {x^{4}}{3 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {4 a \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )}{3 b}\right )\) | \(289\) |
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Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.04 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (15 \, B b^{4} x^{7} - 105 \, C a b^{3} x^{6} - 48 \, C a^{4} - 8 \, A a^{3} b - 35 \, {\left (6 \, C a^{2} b^{2} + A a b^{3}\right )} x^{4} - 28 \, {\left (6 \, C a^{3} b + A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a b^{8} x^{8} + 4 \, a^{2} b^{7} x^{6} + 6 \, a^{3} b^{6} x^{4} + 4 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (121) = 242\).
Time = 20.66 (sec) , antiderivative size = 740, normalized size of antiderivative = 5.61 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=A \left (\begin {cases} - \frac {8 a^{2}}{105 a^{3} b^{3} \sqrt {a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt {a + b x^{2}} + 315 a b^{5} x^{4} \sqrt {a + b x^{2}} + 105 b^{6} x^{6} \sqrt {a + b x^{2}}} - \frac {28 a b x^{2}}{105 a^{3} b^{3} \sqrt {a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt {a + b x^{2}} + 315 a b^{5} x^{4} \sqrt {a + b x^{2}} + 105 b^{6} x^{6} \sqrt {a + b x^{2}}} - \frac {35 b^{2} x^{4}}{105 a^{3} b^{3} \sqrt {a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt {a + b x^{2}} + 315 a b^{5} x^{4} \sqrt {a + b x^{2}} + 105 b^{6} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 a^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {B x^{7}}{7 a^{\frac {9}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 21 a^{\frac {7}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 21 a^{\frac {5}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 7 a^{\frac {3}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}} + C \left (\begin {cases} - \frac {16 a^{3}}{35 a^{3} b^{4} \sqrt {a + b x^{2}} + 105 a^{2} b^{5} x^{2} \sqrt {a + b x^{2}} + 105 a b^{6} x^{4} \sqrt {a + b x^{2}} + 35 b^{7} x^{6} \sqrt {a + b x^{2}}} - \frac {56 a^{2} b x^{2}}{35 a^{3} b^{4} \sqrt {a + b x^{2}} + 105 a^{2} b^{5} x^{2} \sqrt {a + b x^{2}} + 105 a b^{6} x^{4} \sqrt {a + b x^{2}} + 35 b^{7} x^{6} \sqrt {a + b x^{2}}} - \frac {70 a b^{2} x^{4}}{35 a^{3} b^{4} \sqrt {a + b x^{2}} + 105 a^{2} b^{5} x^{2} \sqrt {a + b x^{2}} + 105 a b^{6} x^{4} \sqrt {a + b x^{2}} + 35 b^{7} x^{6} \sqrt {a + b x^{2}}} - \frac {35 b^{3} x^{6}}{35 a^{3} b^{4} \sqrt {a + b x^{2}} + 105 a^{2} b^{5} x^{2} \sqrt {a + b x^{2}} + 105 a b^{6} x^{4} \sqrt {a + b x^{2}} + 35 b^{7} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{8}}{8 a^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (116) = 232\).
Time = 0.21 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.82 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {C x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {B x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {2 \, C a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {A x^{4}}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {5 \, B a x^{3}}{8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {8 \, C a^{2} x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} - \frac {4 \, A a x^{2}}{15 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {B x}{14 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {B x}{7 \, \sqrt {b x^{2} + a} a b^{3}} + \frac {3 \, B a x}{56 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {15 \, B a^{2} x}{56 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} - \frac {16 \, C a^{3}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}} - \frac {8 \, A a^{2}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} \]
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Time = 0.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (5 \, {\left (3 \, {\left (\frac {B x}{a} - \frac {7 \, C}{b}\right )} x^{2} - \frac {7 \, {\left (6 \, C a^{4} b^{2} + A a^{3} b^{3}\right )}}{a^{3} b^{4}}\right )} x^{2} - \frac {28 \, {\left (6 \, C a^{5} b + A a^{4} b^{2}\right )}}{a^{3} b^{4}}\right )} x^{2} - \frac {8 \, {\left (6 \, C a^{6} + A a^{5} b\right )}}{a^{3} b^{4}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \]
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Time = 5.67 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.48 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {\frac {a\,\left (\frac {C}{3\,b^3}-\frac {7\,A\,b-14\,C\,a}{21\,a\,b^3}\right )}{b}-\frac {3\,B\,x}{7\,b^3}}{{\left (b\,x^2+a\right )}^{3/2}}-\frac {\frac {a^2\,\left (\frac {A}{7\,b}-\frac {C\,a}{7\,b^2}\right )}{b^2}+\frac {B\,a^2\,x}{7\,b^3}}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {\frac {C}{b^4}-\frac {B\,x}{7\,a\,b^3}}{\sqrt {b\,x^2+a}}-\frac {\frac {a\,\left (\frac {7\,C\,a^2-7\,A\,a\,b}{35\,a\,b^3}+\frac {a\,\left (\frac {C}{5\,b^2}-\frac {7\,A\,b^2-7\,C\,a\,b}{35\,a\,b^3}\right )}{b}\right )}{b}-\frac {3\,B\,a\,x}{7\,b^3}}{{\left (b\,x^2+a\right )}^{5/2}} \]
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