Integrand size = 25, antiderivative size = 139 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {x^3 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x (3 a B+(3 A b+4 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {2 (3 A b+4 a C)-3 b B x}{105 a b^3 \left (a+b x^2\right )^{3/2}}+\frac {2 B x}{35 a^2 b^2 \sqrt {a+b x^2}} \]
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Time = 0.12 (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, 833, 653, 197} \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {2 B x}{35 a^2 b^2 \sqrt {a+b x^2}}-\frac {2 (4 a C+3 A b)-3 b B x}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x (x (4 a C+3 A b)+3 a B)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^3 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]
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Rule 197
Rule 653
Rule 833
Rule 1818
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x^2 (-3 a B-(3 A b+4 a C) x)}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b} \\ & = -\frac {x^3 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x (3 a B+(3 A b+4 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {\int \frac {-3 a^2 B-2 a (3 A b+4 a C) x}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b^2} \\ & = -\frac {x^3 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x (3 a B+(3 A b+4 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {2 (3 A b+4 a C)-3 b B x}{105 a b^3 \left (a+b x^2\right )^{3/2}}+\frac {(2 B) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{35 a b^2} \\ & = -\frac {x^3 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x (3 a B+(3 A b+4 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {2 (3 A b+4 a C)-3 b B x}{105 a b^3 \left (a+b x^2\right )^{3/2}}+\frac {2 B x}{35 a^2 b^2 \sqrt {a+b x^2}} \\ \end{align*}
Time = 0.84 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.60 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {-8 a^4 C+21 a b^3 B x^5+6 b^4 B x^7-7 a^2 b^2 x^2 \left (3 A+5 C x^2\right )-2 a^3 b \left (3 A+14 C x^2\right )}{105 a^2 b^3 \left (a+b x^2\right )^{7/2}} \]
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Time = 3.48 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.61
method | result | size |
gosper | \(-\frac {-6 x^{7} B \,b^{4}-21 x^{5} B a \,b^{3}+35 C \,a^{2} b^{2} x^{4}+21 A \,a^{2} b^{2} x^{2}+28 C \,a^{3} b \,x^{2}+6 A \,a^{3} b +8 C \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{2} b^{3}}\) | \(85\) |
trager | \(-\frac {-6 x^{7} B \,b^{4}-21 x^{5} B a \,b^{3}+35 C \,a^{2} b^{2} x^{4}+21 A \,a^{2} b^{2} x^{2}+28 C \,a^{3} b \,x^{2}+6 A \,a^{3} b +8 C \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{2} b^{3}}\) | \(85\) |
default | \(C \left (-\frac {x^{4}}{3 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {4 a \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )}{3 b}\right )+B \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )+A \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )\) | \(217\) |
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Time = 0.32 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.94 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (6 \, B b^{4} x^{7} + 21 \, B a b^{3} x^{5} - 35 \, C a^{2} b^{2} x^{4} - 8 \, C a^{4} - 6 \, A a^{3} b - 7 \, {\left (4 \, C a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{2} b^{7} x^{8} + 4 \, a^{3} b^{6} x^{6} + 6 \, a^{4} b^{5} x^{4} + 4 \, a^{5} b^{4} x^{2} + a^{6} b^{3}\right )}} \]
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Time = 20.05 (sec) , antiderivative size = 660, normalized size of antiderivative = 4.75 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=A \left (\begin {cases} - \frac {2 a}{35 a^{3} b^{2} \sqrt {a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt {a + b x^{2}} + 105 a b^{4} x^{4} \sqrt {a + b x^{2}} + 35 b^{5} x^{6} \sqrt {a + b x^{2}}} - \frac {7 b x^{2}}{35 a^{3} b^{2} \sqrt {a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt {a + b x^{2}} + 105 a b^{4} x^{4} \sqrt {a + b x^{2}} + 35 b^{5} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) + B \left (\frac {7 a x^{5}}{35 a^{\frac {11}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {9}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {7}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 35 a^{\frac {5}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {2 b x^{7}}{35 a^{\frac {11}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {9}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {7}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 35 a^{\frac {5}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + C \left (\begin {cases} - \frac {8 a^{2}}{105 a^{3} b^{3} \sqrt {a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt {a + b x^{2}} + 315 a b^{5} x^{4} \sqrt {a + b x^{2}} + 105 b^{6} x^{6} \sqrt {a + b x^{2}}} - \frac {28 a b x^{2}}{105 a^{3} b^{3} \sqrt {a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt {a + b x^{2}} + 315 a b^{5} x^{4} \sqrt {a + b x^{2}} + 105 b^{6} x^{6} \sqrt {a + b x^{2}}} - \frac {35 b^{2} x^{4}}{105 a^{3} b^{3} \sqrt {a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt {a + b x^{2}} + 315 a b^{5} x^{4} \sqrt {a + b x^{2}} + 105 b^{6} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 a^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.29 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {C x^{4}}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {B x^{3}}{4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {4 \, C a x^{2}}{15 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {A x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {3 \, B x}{140 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {2 \, B x}{35 \, \sqrt {b x^{2} + a} a^{2} b^{2}} + \frac {B x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2}} - \frac {3 \, B a x}{28 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {8 \, C a^{2}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} - \frac {2 \, A a}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} \]
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Time = 0.31 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.68 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left (3 \, {\left (\frac {2 \, B b x^{2}}{a^{2}} + \frac {7 \, B}{a}\right )} x - \frac {35 \, C}{b}\right )} x^{2} - \frac {7 \, {\left (4 \, C a^{4} b + 3 \, A a^{3} b^{2}\right )}}{a^{3} b^{3}}\right )} x^{2} - \frac {2 \, {\left (4 \, C a^{5} + 3 \, A a^{4} b\right )}}{a^{3} b^{3}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \]
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Time = 5.50 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {\frac {a\,\left (\frac {A}{7\,b}-\frac {C\,a}{7\,b^2}\right )}{b}+\frac {B\,a\,x}{7\,b^2}}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {\frac {C}{3\,b^3}-\frac {B\,x}{35\,a\,b^2}}{{\left (b\,x^2+a\right )}^{3/2}}+\frac {\frac {a\,\left (\frac {C}{5\,b^2}-\frac {7\,A\,b-7\,C\,a}{35\,a\,b^2}\right )}{b}-\frac {8\,B\,x}{35\,b^2}}{{\left (b\,x^2+a\right )}^{5/2}}+\frac {2\,B\,x}{35\,a^2\,b^2\,\sqrt {b\,x^2+a}} \]
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