Integrand size = 21, antiderivative size = 174 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=-\frac {d x \left (a+b x^2\right )^3}{7 c (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac {(7 b c-6 a d) x \left (a+b x^2\right )^2}{35 c^2 (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac {4 a (7 b c-6 a d) x \left (a+b x^2\right )}{105 c^3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {8 a^2 (7 b c-6 a d) x}{105 c^4 (b c-a d) \sqrt {c+d x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {390, 386, 197} \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {8 a^2 x (7 b c-6 a d)}{105 c^4 \sqrt {c+d x^2} (b c-a d)}+\frac {4 a x \left (a+b x^2\right ) (7 b c-6 a d)}{105 c^3 \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {x \left (a+b x^2\right )^2 (7 b c-6 a d)}{35 c^2 \left (c+d x^2\right )^{5/2} (b c-a d)}-\frac {d x \left (a+b x^2\right )^3}{7 c \left (c+d x^2\right )^{7/2} (b c-a d)} \]
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Rule 197
Rule 386
Rule 390
Rubi steps \begin{align*} \text {integral}& = -\frac {d x \left (a+b x^2\right )^3}{7 c (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac {(7 b c-6 a d) \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^{7/2}} \, dx}{7 c (b c-a d)} \\ & = -\frac {d x \left (a+b x^2\right )^3}{7 c (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac {(7 b c-6 a d) x \left (a+b x^2\right )^2}{35 c^2 (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac {(4 a (7 b c-6 a d)) \int \frac {a+b x^2}{\left (c+d x^2\right )^{5/2}} \, dx}{35 c^2 (b c-a d)} \\ & = -\frac {d x \left (a+b x^2\right )^3}{7 c (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac {(7 b c-6 a d) x \left (a+b x^2\right )^2}{35 c^2 (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac {4 a (7 b c-6 a d) x \left (a+b x^2\right )}{105 c^3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {\left (8 a^2 (7 b c-6 a d)\right ) \int \frac {1}{\left (c+d x^2\right )^{3/2}} \, dx}{105 c^3 (b c-a d)} \\ & = -\frac {d x \left (a+b x^2\right )^3}{7 c (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac {(7 b c-6 a d) x \left (a+b x^2\right )^2}{35 c^2 (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac {4 a (7 b c-6 a d) x \left (a+b x^2\right )}{105 c^3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {8 a^2 (7 b c-6 a d) x}{105 c^4 (b c-a d) \sqrt {c+d x^2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.61 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {3 b^2 c^2 x^5 \left (7 c+2 d x^2\right )+2 a b c x^3 \left (35 c^2+28 c d x^2+8 d^2 x^4\right )+3 a^2 \left (35 c^3 x+70 c^2 d x^3+56 c d^2 x^5+16 d^3 x^7\right )}{105 c^4 \left (c+d x^2\right )^{7/2}} \]
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Time = 2.38 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.55
method | result | size |
pseudoelliptic | \(\frac {x \left (\left (\frac {1}{5} b^{2} x^{4}+\frac {2}{3} a b \,x^{2}+a^{2}\right ) c^{3}+2 x^{2} d \left (\frac {1}{35} b^{2} x^{4}+\frac {4}{15} a b \,x^{2}+a^{2}\right ) c^{2}+\frac {8 x^{4} d^{2} a \left (\frac {2 b \,x^{2}}{21}+a \right ) c}{5}+\frac {16 a^{2} d^{3} x^{6}}{35}\right )}{\left (d \,x^{2}+c \right )^{\frac {7}{2}} c^{4}}\) | \(96\) |
gosper | \(\frac {x \left (48 a^{2} d^{3} x^{6}+16 a b c \,d^{2} x^{6}+6 b^{2} c^{2} d \,x^{6}+168 a^{2} c \,d^{2} x^{4}+56 a b \,c^{2} d \,x^{4}+21 b^{2} c^{3} x^{4}+210 a^{2} c^{2} d \,x^{2}+70 a b \,c^{3} x^{2}+105 a^{2} c^{3}\right )}{105 \left (d \,x^{2}+c \right )^{\frac {7}{2}} c^{4}}\) | \(115\) |
trager | \(\frac {x \left (48 a^{2} d^{3} x^{6}+16 a b c \,d^{2} x^{6}+6 b^{2} c^{2} d \,x^{6}+168 a^{2} c \,d^{2} x^{4}+56 a b \,c^{2} d \,x^{4}+21 b^{2} c^{3} x^{4}+210 a^{2} c^{2} d \,x^{2}+70 a b \,c^{3} x^{2}+105 a^{2} c^{3}\right )}{105 \left (d \,x^{2}+c \right )^{\frac {7}{2}} c^{4}}\) | \(115\) |
default | \(a^{2} \left (\frac {x}{7 c \left (d \,x^{2}+c \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 c \left (d \,x^{2}+c \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 c \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {8 x}{15 c^{2} \sqrt {d \,x^{2}+c}}\right )}{7 c}}{c}\right )+b^{2} \left (-\frac {x^{3}}{4 d \left (d \,x^{2}+c \right )^{\frac {7}{2}}}+\frac {3 c \left (-\frac {x}{6 d \left (d \,x^{2}+c \right )^{\frac {7}{2}}}+\frac {c \left (\frac {x}{7 c \left (d \,x^{2}+c \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 c \left (d \,x^{2}+c \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 c \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {8 x}{15 c^{2} \sqrt {d \,x^{2}+c}}\right )}{7 c}}{c}\right )}{6 d}\right )}{4 d}\right )+2 a b \left (-\frac {x}{6 d \left (d \,x^{2}+c \right )^{\frac {7}{2}}}+\frac {c \left (\frac {x}{7 c \left (d \,x^{2}+c \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 c \left (d \,x^{2}+c \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 c \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {8 x}{15 c^{2} \sqrt {d \,x^{2}+c}}\right )}{7 c}}{c}\right )}{6 d}\right )\) | \(301\) |
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Time = 0.29 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {{\left (2 \, {\left (3 \, b^{2} c^{2} d + 8 \, a b c d^{2} + 24 \, a^{2} d^{3}\right )} x^{7} + 105 \, a^{2} c^{3} x + 7 \, {\left (3 \, b^{2} c^{3} + 8 \, a b c^{2} d + 24 \, a^{2} c d^{2}\right )} x^{5} + 70 \, {\left (a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c}}{105 \, {\left (c^{4} d^{4} x^{8} + 4 \, c^{5} d^{3} x^{6} + 6 \, c^{6} d^{2} x^{4} + 4 \, c^{7} d x^{2} + c^{8}\right )}} \]
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\[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {9}{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=-\frac {b^{2} x^{3}}{4 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} d} + \frac {16 \, a^{2} x}{35 \, \sqrt {d x^{2} + c} c^{4}} + \frac {8 \, a^{2} x}{35 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{3}} + \frac {6 \, a^{2} x}{35 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} c^{2}} + \frac {a^{2} x}{7 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} c} + \frac {3 \, b^{2} x}{140 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} d^{2}} + \frac {2 \, b^{2} x}{35 \, \sqrt {d x^{2} + c} c^{2} d^{2}} + \frac {b^{2} x}{35 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c d^{2}} - \frac {3 \, b^{2} c x}{28 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} d^{2}} - \frac {2 \, a b x}{7 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} d} + \frac {16 \, a b x}{105 \, \sqrt {d x^{2} + c} c^{3} d} + \frac {8 \, a b x}{105 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{2} d} + \frac {2 \, a b x}{35 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} c d} \]
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Time = 0.30 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {{\left ({\left (x^{2} {\left (\frac {2 \, {\left (3 \, b^{2} c^{2} d^{4} + 8 \, a b c d^{5} + 24 \, a^{2} d^{6}\right )} x^{2}}{c^{4} d^{3}} + \frac {7 \, {\left (3 \, b^{2} c^{3} d^{3} + 8 \, a b c^{2} d^{4} + 24 \, a^{2} c d^{5}\right )}}{c^{4} d^{3}}\right )} + \frac {70 \, {\left (a b c^{3} d^{3} + 3 \, a^{2} c^{2} d^{4}\right )}}{c^{4} d^{3}}\right )} x^{2} + \frac {105 \, a^{2}}{c}\right )} x}{105 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}}} \]
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Time = 5.04 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {x\,\left (\frac {a^2}{7\,c}+\frac {c\,\left (\frac {b^2}{7\,d}-\frac {2\,a\,b}{7\,c}\right )}{d}\right )}{{\left (d\,x^2+c\right )}^{7/2}}-\frac {x\,\left (\frac {b^2}{5\,d^2}-\frac {6\,a^2\,d^2+2\,a\,b\,c\,d-b^2\,c^2}{35\,c^2\,d^2}\right )}{{\left (d\,x^2+c\right )}^{5/2}}+\frac {x\,\left (24\,a^2\,d^2+8\,a\,b\,c\,d+3\,b^2\,c^2\right )}{105\,c^3\,d^2\,{\left (d\,x^2+c\right )}^{3/2}}+\frac {x\,\left (48\,a^2\,d^2+16\,a\,b\,c\,d+6\,b^2\,c^2\right )}{105\,c^4\,d^2\,\sqrt {d\,x^2+c}} \]
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