Integrand size = 24, antiderivative size = 126 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{9/2}} \, dx=\frac {a x}{d \left (d+e x^2\right )^{7/2}}+\frac {(b d+6 a e) x^3}{3 d^2 \left (d+e x^2\right )^{7/2}}+\frac {\left (3 c d^2+4 e (b d+6 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{7/2}}+\frac {2 e \left (3 c d^2+4 e (b d+6 a e)\right ) x^7}{105 d^4 \left (d+e x^2\right )^{7/2}} \]
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Time = 0.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1169, 1817, 12, 277, 270} \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{9/2}} \, dx=\frac {2 e x^7 \left (4 e (6 a e+b d)+3 c d^2\right )}{105 d^4 \left (d+e x^2\right )^{7/2}}+\frac {x^5 \left (4 e (6 a e+b d)+3 c d^2\right )}{15 d^3 \left (d+e x^2\right )^{7/2}}+\frac {x^3 (6 a e+b d)}{3 d^2 \left (d+e x^2\right )^{7/2}}+\frac {a x}{d \left (d+e x^2\right )^{7/2}} \]
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Rule 12
Rule 270
Rule 277
Rule 1169
Rule 1817
Rubi steps \begin{align*} \text {integral}& = \frac {a x}{d \left (d+e x^2\right )^{7/2}}+\frac {\int \frac {x^2 \left (6 a e+d \left (b+c x^2\right )\right )}{\left (d+e x^2\right )^{9/2}} \, dx}{d} \\ & = \frac {a x}{d \left (d+e x^2\right )^{7/2}}+\frac {(b d+6 a e) x^3}{3 d^2 \left (d+e x^2\right )^{7/2}}+\frac {\int \frac {\left (3 c d^2+4 e (b d+6 a e)\right ) x^4}{\left (d+e x^2\right )^{9/2}} \, dx}{3 d^2} \\ & = \frac {a x}{d \left (d+e x^2\right )^{7/2}}+\frac {(b d+6 a e) x^3}{3 d^2 \left (d+e x^2\right )^{7/2}}+\frac {1}{3} \left (3 c+\frac {4 e (b d+6 a e)}{d^2}\right ) \int \frac {x^4}{\left (d+e x^2\right )^{9/2}} \, dx \\ & = \frac {a x}{d \left (d+e x^2\right )^{7/2}}+\frac {(b d+6 a e) x^3}{3 d^2 \left (d+e x^2\right )^{7/2}}+\frac {\left (3 c d^2+4 e (b d+6 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{7/2}}+\frac {\left (2 e \left (3 c d^2+4 e (b d+6 a e)\right )\right ) \int \frac {x^6}{\left (d+e x^2\right )^{9/2}} \, dx}{15 d^3} \\ & = \frac {a x}{d \left (d+e x^2\right )^{7/2}}+\frac {(b d+6 a e) x^3}{3 d^2 \left (d+e x^2\right )^{7/2}}+\frac {\left (3 c d^2+4 e (b d+6 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{7/2}}+\frac {2 e \left (3 c d^2+4 e (b d+6 a e)\right ) x^7}{105 d^4 \left (d+e x^2\right )^{7/2}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.82 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{9/2}} \, dx=\frac {105 a d^3 x+35 b d^3 x^3+210 a d^2 e x^3+21 c d^3 x^5+28 b d^2 e x^5+168 a d e^2 x^5+6 c d^2 e x^7+8 b d e^2 x^7+48 a e^3 x^7}{105 d^4 \left (d+e x^2\right )^{7/2}} \]
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Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.66
method | result | size |
pseudoelliptic | \(\frac {\left (\left (\frac {1}{5} c \,x^{4}+\frac {1}{3} b \,x^{2}+a \right ) d^{3}+2 e \left (\frac {1}{35} c \,x^{4}+\frac {2}{15} b \,x^{2}+a \right ) x^{2} d^{2}+\frac {8 \left (\frac {b \,x^{2}}{21}+a \right ) e^{2} x^{4} d}{5}+\frac {16 a \,e^{3} x^{6}}{35}\right ) x}{\left (e \,x^{2}+d \right )^{\frac {7}{2}} d^{4}}\) | \(83\) |
gosper | \(\frac {x \left (48 a \,e^{3} x^{6}+8 b d \,e^{2} x^{6}+6 c \,d^{2} e \,x^{6}+168 a d \,e^{2} x^{4}+28 b \,d^{2} e \,x^{4}+21 c \,d^{3} x^{4}+210 a \,d^{2} e \,x^{2}+35 b \,d^{3} x^{2}+105 d^{3} a \right )}{105 \left (e \,x^{2}+d \right )^{\frac {7}{2}} d^{4}}\) | \(100\) |
trager | \(\frac {x \left (48 a \,e^{3} x^{6}+8 b d \,e^{2} x^{6}+6 c \,d^{2} e \,x^{6}+168 a d \,e^{2} x^{4}+28 b \,d^{2} e \,x^{4}+21 c \,d^{3} x^{4}+210 a \,d^{2} e \,x^{2}+35 b \,d^{3} x^{2}+105 d^{3} a \right )}{105 \left (e \,x^{2}+d \right )^{\frac {7}{2}} d^{4}}\) | \(100\) |
default | \(a \left (\frac {x}{7 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}}{d}\right )+c \left (-\frac {x^{3}}{4 e \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {3 d \left (-\frac {x}{6 e \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {d \left (\frac {x}{7 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}}{d}\right )}{6 e}\right )}{4 e}\right )+b \left (-\frac {x}{6 e \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {d \left (\frac {x}{7 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}}{d}\right )}{6 e}\right )\) | \(295\) |
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Time = 0.34 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.08 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{9/2}} \, dx=\frac {{\left (2 \, {\left (3 \, c d^{2} e + 4 \, b d e^{2} + 24 \, a e^{3}\right )} x^{7} + 7 \, {\left (3 \, c d^{3} + 4 \, b d^{2} e + 24 \, a d e^{2}\right )} x^{5} + 105 \, a d^{3} x + 35 \, {\left (b d^{3} + 6 \, a d^{2} e\right )} x^{3}\right )} \sqrt {e x^{2} + d}}{105 \, {\left (d^{4} e^{4} x^{8} + 4 \, d^{5} e^{3} x^{6} + 6 \, d^{6} e^{2} x^{4} + 4 \, d^{7} e x^{2} + d^{8}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1989 vs. \(2 (119) = 238\).
Time = 35.55 (sec) , antiderivative size = 1989, normalized size of antiderivative = 15.79 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{9/2}} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (112) = 224\).
Time = 0.19 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.80 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{9/2}} \, dx=-\frac {c x^{3}}{4 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} e} + \frac {16 \, a x}{35 \, \sqrt {e x^{2} + d} d^{4}} + \frac {8 \, a x}{35 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3}} + \frac {6 \, a x}{35 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2}} + \frac {a x}{7 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d} + \frac {3 \, c x}{140 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} e^{2}} + \frac {2 \, c x}{35 \, \sqrt {e x^{2} + d} d^{2} e^{2}} + \frac {c x}{35 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d e^{2}} - \frac {3 \, c d x}{28 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} e^{2}} - \frac {b x}{7 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} e} + \frac {8 \, b x}{105 \, \sqrt {e x^{2} + d} d^{3} e} + \frac {4 \, b x}{105 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{2} e} + \frac {b x}{35 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d e} \]
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Time = 0.32 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.98 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{9/2}} \, dx=\frac {{\left ({\left (x^{2} {\left (\frac {2 \, {\left (3 \, c d^{2} e^{4} + 4 \, b d e^{5} + 24 \, a e^{6}\right )} x^{2}}{d^{4} e^{3}} + \frac {7 \, {\left (3 \, c d^{3} e^{3} + 4 \, b d^{2} e^{4} + 24 \, a d e^{5}\right )}}{d^{4} e^{3}}\right )} + \frac {35 \, {\left (b d^{3} e^{3} + 6 \, a d^{2} e^{4}\right )}}{d^{4} e^{3}}\right )} x^{2} + \frac {105 \, a}{d}\right )} x}{105 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}}} \]
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Time = 7.88 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.22 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{9/2}} \, dx=\frac {x\,\left (\frac {a}{7\,d}-\frac {d\,\left (\frac {b}{7\,d}-\frac {c}{7\,e}\right )}{e}\right )}{{\left (e\,x^2+d\right )}^{7/2}}-\frac {x\,\left (\frac {c}{5\,e^2}-\frac {-c\,d^2+b\,d\,e+6\,a\,e^2}{35\,d^2\,e^2}\right )}{{\left (e\,x^2+d\right )}^{5/2}}+\frac {x\,\left (3\,c\,d^2+4\,b\,d\,e+24\,a\,e^2\right )}{105\,d^3\,e^2\,{\left (e\,x^2+d\right )}^{3/2}}+\frac {x\,\left (6\,c\,d^2+8\,b\,d\,e+48\,a\,e^2\right )}{105\,d^4\,e^2\,\sqrt {e\,x^2+d}} \]
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