Integrand size = 25, antiderivative size = 139 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {x^2 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {2 a B+(4 A b+3 a C) x}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {(4 A b+3 a C) x}{105 a^2 b^2 \left (a+b x^2\right )^{3/2}}+\frac {2 (4 A b+3 a C) x}{105 a^3 b^2 \sqrt {a+b x^2}} \]
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Time = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1818, 792, 198, 197} \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {2 x (3 a C+4 A b)}{105 a^3 b^2 \sqrt {a+b x^2}}+\frac {x (3 a C+4 A b)}{105 a^2 b^2 \left (a+b x^2\right )^{3/2}}-\frac {x (3 a C+4 A b)+2 a B}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^2 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]
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Rule 197
Rule 198
Rule 792
Rule 1818
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x (-2 a B-(4 A b+3 a C) x)}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b} \\ & = -\frac {x^2 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {2 a B+(4 A b+3 a C) x}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {(4 A b+3 a C) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a b^2} \\ & = -\frac {x^2 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {2 a B+(4 A b+3 a C) x}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {(4 A b+3 a C) x}{105 a^2 b^2 \left (a+b x^2\right )^{3/2}}+\frac {(2 (4 A b+3 a C)) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^2 b^2} \\ & = -\frac {x^2 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {2 a B+(4 A b+3 a C) x}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {(4 A b+3 a C) x}{105 a^2 b^2 \left (a+b x^2\right )^{3/2}}+\frac {2 (4 A b+3 a C) x}{105 a^3 b^2 \sqrt {a+b x^2}} \\ \end{align*}
Time = 0.67 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.63 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {-6 a^4 B-21 a^3 b B x^2+8 A b^4 x^7+7 a^2 b^2 x^3 \left (5 A+3 C x^2\right )+2 a b^3 x^5 \left (14 A+3 C x^2\right )}{105 a^3 b^2 \left (a+b x^2\right )^{7/2}} \]
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Time = 3.43 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.63
method | result | size |
gosper | \(\frac {8 A \,b^{4} x^{7}+6 C a \,x^{7} b^{3}+28 A a \,b^{3} x^{5}+21 C \,a^{2} x^{5} b^{2}+35 A \,a^{2} b^{2} x^{3}-21 B \,a^{3} b \,x^{2}-6 B \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{3} b^{2}}\) | \(88\) |
trager | \(\frac {8 A \,b^{4} x^{7}+6 C a \,x^{7} b^{3}+28 A a \,b^{3} x^{5}+21 C \,a^{2} x^{5} b^{2}+35 A \,a^{2} b^{2} x^{3}-21 B \,a^{3} b \,x^{2}-6 B \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{3} b^{2}}\) | \(88\) |
default | \(C \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )+B \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )+A \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )\) | \(255\) |
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Time = 0.36 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (35 \, A a^{2} b^{2} x^{3} + 2 \, {\left (3 \, C a b^{3} + 4 \, A b^{4}\right )} x^{7} - 21 \, B a^{3} b x^{2} + 7 \, {\left (3 \, C a^{2} b^{2} + 4 \, A a b^{3}\right )} x^{5} - 6 \, B a^{4}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{3} b^{6} x^{8} + 4 \, a^{4} b^{5} x^{6} + 6 \, a^{5} b^{4} x^{4} + 4 \, a^{6} b^{3} x^{2} + a^{7} b^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (129) = 258\).
Time = 30.65 (sec) , antiderivative size = 904, normalized size of antiderivative = 6.50 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=A \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 ) + B \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 ) + C \left (\frac {7 a x^{5}}{35 a^{\frac {11}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {9}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {7}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 35 a^{\frac {5}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {2 b x^{7}}{35 a^{\frac {11}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {9}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {7}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 35 a^{\frac {5}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.42 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {C x^{3}}{4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {B x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {3 \, C x}{140 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {2 \, C x}{35 \, \sqrt {b x^{2} + a} a^{2} b^{2}} + \frac {C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2}} - \frac {3 \, C a x}{28 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {A x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, A x}{105 \, \sqrt {b x^{2} + a} a^{3} b} + \frac {4 \, A x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} + \frac {A x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b} - \frac {2 \, B a}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} \]
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Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.68 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left (x^{2} {\left (\frac {2 \, {\left (3 \, C a b^{4} + 4 \, A b^{5}\right )} x^{2}}{a^{3} b^{3}} + \frac {7 \, {\left (3 \, C a^{2} b^{3} + 4 \, A a b^{4}\right )}}{a^{3} b^{3}}\right )} + \frac {35 \, A}{a}\right )} x - \frac {21 \, B}{b}\right )} x^{2} - \frac {6 \, B a}{b^{2}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \]
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Time = 5.48 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {x\,\left (4\,A\,b+3\,C\,a\right )}{105\,a^2\,b^2\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {\frac {B}{5\,b^2}+x\,\left (\frac {C}{5\,b^2}-\frac {A\,b-C\,a}{35\,a\,b^2}\right )}{{\left (b\,x^2+a\right )}^{5/2}}-\frac {x\,\left (\frac {A}{7\,b}-\frac {C\,a}{7\,b^2}\right )-\frac {B\,a}{7\,b^2}}{{\left (b\,x^2+a\right )}^{7/2}}+\frac {x\,\left (8\,A\,b+6\,C\,a\right )}{105\,a^3\,b^2\,\sqrt {b\,x^2+a}} \]
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