Integrand size = 17, antiderivative size = 85 \[ \int \frac {x^{14}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {35 b x}{8 c^4}+\frac {35 x^3}{24 c^3}-\frac {x^7}{4 c \left (b+c x^2\right )^2}-\frac {7 x^5}{8 c^2 \left (b+c x^2\right )}+\frac {35 b^{3/2} \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 c^{9/2}} \]
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Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1598, 294, 308, 211} \[ \int \frac {x^{14}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {35 b^{3/2} \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 c^{9/2}}-\frac {35 b x}{8 c^4}-\frac {7 x^5}{8 c^2 \left (b+c x^2\right )}-\frac {x^7}{4 c \left (b+c x^2\right )^2}+\frac {35 x^3}{24 c^3} \]
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Rule 211
Rule 294
Rule 308
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^8}{\left (b+c x^2\right )^3} \, dx \\ & = -\frac {x^7}{4 c \left (b+c x^2\right )^2}+\frac {7 \int \frac {x^6}{\left (b+c x^2\right )^2} \, dx}{4 c} \\ & = -\frac {x^7}{4 c \left (b+c x^2\right )^2}-\frac {7 x^5}{8 c^2 \left (b+c x^2\right )}+\frac {35 \int \frac {x^4}{b+c x^2} \, dx}{8 c^2} \\ & = -\frac {x^7}{4 c \left (b+c x^2\right )^2}-\frac {7 x^5}{8 c^2 \left (b+c x^2\right )}+\frac {35 \int \left (-\frac {b}{c^2}+\frac {x^2}{c}+\frac {b^2}{c^2 \left (b+c x^2\right )}\right ) \, dx}{8 c^2} \\ & = -\frac {35 b x}{8 c^4}+\frac {35 x^3}{24 c^3}-\frac {x^7}{4 c \left (b+c x^2\right )^2}-\frac {7 x^5}{8 c^2 \left (b+c x^2\right )}+\frac {\left (35 b^2\right ) \int \frac {1}{b+c x^2} \, dx}{8 c^4} \\ & = -\frac {35 b x}{8 c^4}+\frac {35 x^3}{24 c^3}-\frac {x^7}{4 c \left (b+c x^2\right )^2}-\frac {7 x^5}{8 c^2 \left (b+c x^2\right )}+\frac {35 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 c^{9/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \frac {x^{14}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {105 b^3 x+175 b^2 c x^3+56 b c^2 x^5-8 c^3 x^7}{24 c^4 \left (b+c x^2\right )^2}+\frac {35 b^{3/2} \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 c^{9/2}} \]
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Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.74
method | result | size |
default | \(-\frac {-\frac {1}{3} c \,x^{3}+3 b x}{c^{4}}+\frac {b^{2} \left (\frac {-\frac {13}{8} c \,x^{3}-\frac {11}{8} b x}{\left (c \,x^{2}+b \right )^{2}}+\frac {35 \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}}\right )}{c^{4}}\) | \(63\) |
risch | \(\frac {x^{3}}{3 c^{3}}-\frac {3 b x}{c^{4}}+\frac {-\frac {13}{8} b^{2} c \,x^{3}-\frac {11}{8} b^{3} x}{c^{4} \left (c \,x^{2}+b \right )^{2}}+\frac {35 \sqrt {-b c}\, b \ln \left (-\sqrt {-b c}\, x +b \right )}{16 c^{5}}-\frac {35 \sqrt {-b c}\, b \ln \left (\sqrt {-b c}\, x +b \right )}{16 c^{5}}\) | \(93\) |
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Time = 0.27 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.71 \[ \int \frac {x^{14}}{\left (b x^2+c x^4\right )^3} \, dx=\left [\frac {16 \, c^{3} x^{7} - 112 \, b c^{2} x^{5} - 350 \, b^{2} c x^{3} - 210 \, b^{3} x + 105 \, {\left (b c^{2} x^{4} + 2 \, b^{2} c x^{2} + b^{3}\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} + 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right )}{48 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}}, \frac {8 \, c^{3} x^{7} - 56 \, b c^{2} x^{5} - 175 \, b^{2} c x^{3} - 105 \, b^{3} x + 105 \, {\left (b c^{2} x^{4} + 2 \, b^{2} c x^{2} + b^{3}\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right )}{24 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}}\right ] \]
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Time = 0.22 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.56 \[ \int \frac {x^{14}}{\left (b x^2+c x^4\right )^3} \, dx=- \frac {3 b x}{c^{4}} - \frac {35 \sqrt {- \frac {b^{3}}{c^{9}}} \log {\left (x - \frac {c^{4} \sqrt {- \frac {b^{3}}{c^{9}}}}{b} \right )}}{16} + \frac {35 \sqrt {- \frac {b^{3}}{c^{9}}} \log {\left (x + \frac {c^{4} \sqrt {- \frac {b^{3}}{c^{9}}}}{b} \right )}}{16} + \frac {- 11 b^{3} x - 13 b^{2} c x^{3}}{8 b^{2} c^{4} + 16 b c^{5} x^{2} + 8 c^{6} x^{4}} + \frac {x^{3}}{3 c^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.96 \[ \int \frac {x^{14}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {13 \, b^{2} c x^{3} + 11 \, b^{3} x}{8 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}} + \frac {35 \, b^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} c^{4}} + \frac {c x^{3} - 9 \, b x}{3 \, c^{4}} \]
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Time = 0.30 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86 \[ \int \frac {x^{14}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {35 \, b^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} c^{4}} - \frac {13 \, b^{2} c x^{3} + 11 \, b^{3} x}{8 \, {\left (c x^{2} + b\right )}^{2} c^{4}} + \frac {c^{6} x^{3} - 9 \, b c^{5} x}{3 \, c^{9}} \]
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Time = 12.83 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \frac {x^{14}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {x^3}{3\,c^3}-\frac {\frac {11\,b^3\,x}{8}+\frac {13\,c\,b^2\,x^3}{8}}{b^2\,c^4+2\,b\,c^5\,x^2+c^6\,x^4}+\frac {35\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{8\,c^{9/2}}-\frac {3\,b\,x}{c^4} \]
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