Integrand size = 23, antiderivative size = 119 \[ \int \frac {x \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {x (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {5 A b+2 a C-b B x}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {4 B x}{105 a^2 b \left (a+b x^2\right )^{3/2}}+\frac {8 B x}{105 a^3 b \sqrt {a+b x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1818, 653, 198, 197} \[ \int \frac {x \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {8 B x}{105 a^3 b \sqrt {a+b x^2}}+\frac {4 B x}{105 a^2 b \left (a+b x^2\right )^{3/2}}-\frac {2 a C+5 A b-b B x}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]
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Rule 197
Rule 198
Rule 653
Rule 1818
Rubi steps \begin{align*} \text {integral}& = -\frac {x (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {-a B-(5 A b+2 a C) x}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b} \\ & = -\frac {x (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {5 A b+2 a C-b B x}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {(4 B) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a b} \\ & = -\frac {x (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {5 A b+2 a C-b B x}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {4 B x}{105 a^2 b \left (a+b x^2\right )^{3/2}}+\frac {(8 B) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^2 b} \\ & = -\frac {x (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {5 A b+2 a C-b B x}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {4 B x}{105 a^2 b \left (a+b x^2\right )^{3/2}}+\frac {8 B x}{105 a^3 b \sqrt {a+b x^2}} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.64 \[ \int \frac {x \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {-15 a^3 A b-6 a^4 C-21 a^3 b C x^2+35 a^2 b^2 B x^3+28 a b^3 B x^5+8 b^4 B x^7}{105 a^3 b^2 \left (a+b x^2\right )^{7/2}} \]
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Time = 3.59 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.61
method | result | size |
gosper | \(-\frac {-8 x^{7} B \,b^{4}-28 x^{5} B a \,b^{3}-35 B \,a^{2} b^{2} x^{3}+21 C \,a^{3} b \,x^{2}+15 A \,a^{3} b +6 C \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{3} b^{2}}\) | \(73\) |
trager | \(-\frac {-8 x^{7} B \,b^{4}-28 x^{5} B a \,b^{3}-35 B \,a^{2} b^{2} x^{3}+21 C \,a^{3} b \,x^{2}+15 A \,a^{3} b +6 C \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{3} b^{2}}\) | \(73\) |
default | \(C \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )+B \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )-\frac {A}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\) | \(149\) |
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Time = 0.40 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (8 \, B b^{4} x^{7} + 28 \, B a b^{3} x^{5} + 35 \, B a^{2} b^{2} x^{3} - 21 \, C a^{3} b x^{2} - 6 \, C a^{4} - 15 \, A a^{3} b\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{3} b^{6} x^{8} + 4 \, a^{4} b^{5} x^{6} + 6 \, a^{5} b^{4} x^{4} + 4 \, a^{6} b^{3} x^{2} + a^{7} b^{2}\right )}} \]
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Time = 19.63 (sec) , antiderivative size = 796, normalized size of antiderivative = 6.69 \[ \int \frac {x \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=A \left (\begin {cases} - \frac {1}{7 a^{3} b \sqrt {a + b x^{2}} + 21 a^{2} b^{2} x^{2} \sqrt {a + b x^{2}} + 21 a b^{3} x^{4} \sqrt {a + b x^{2}} + 7 b^{4} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) + B \left (\frac {35 a^{5} x^{3}}{105 a^{\frac {19}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {17}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 630 a^{\frac {15}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {13}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {11}{2}} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {63 a^{4} b x^{5}}{105 a^{\frac {19}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {17}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 630 a^{\frac {15}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {13}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {11}{2}} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {36 a^{3} b^{2} x^{7}}{105 a^{\frac {19}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {17}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 630 a^{\frac {15}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {13}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {11}{2}} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {8 a^{2} b^{3} x^{9}}{105 a^{\frac {19}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {17}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 630 a^{\frac {15}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {13}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {11}{2}} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + C \left (\begin {cases} - \frac {2 a}{35 a^{3} b^{2} \sqrt {a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt {a + b x^{2}} + 105 a b^{4} x^{4} \sqrt {a + b x^{2}} + 35 b^{5} x^{6} \sqrt {a + b x^{2}}} - \frac {7 b x^{2}}{35 a^{3} b^{2} \sqrt {a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt {a + b x^{2}} + 105 a b^{4} x^{4} \sqrt {a + b x^{2}} + 35 b^{5} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.03 \[ \int \frac {x \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {C x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {B x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, B x}{105 \, \sqrt {b x^{2} + a} a^{3} b} + \frac {4 \, B x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} + \frac {B x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b} - \frac {2 \, C a}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {A}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} \]
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Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.69 \[ \int \frac {x \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left (4 \, {\left (\frac {2 \, B b^{2} x^{2}}{a^{3}} + \frac {7 \, B b}{a^{2}}\right )} x^{2} + \frac {35 \, B}{a}\right )} x - \frac {21 \, C}{b}\right )} x^{2} - \frac {3 \, {\left (2 \, C a^{4} b + 5 \, A a^{3} b^{2}\right )}}{a^{3} b^{3}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \]
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Time = 5.76 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.83 \[ \int \frac {x \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {8\,B\,x}{105\,a^3\,b\,\sqrt {b\,x^2+a}}-\frac {\frac {A}{7\,b}-\frac {C\,a}{7\,b^2}+\frac {B\,x}{7\,b}}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {\frac {C}{5\,b^2}-\frac {B\,x}{35\,a\,b}}{{\left (b\,x^2+a\right )}^{5/2}}+\frac {4\,B\,x}{105\,a^2\,b\,{\left (b\,x^2+a\right )}^{3/2}} \]
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