Integrand size = 20, antiderivative size = 162 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx=-\frac {3 b \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{256 a^3 x^4}+\frac {b \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 a^2 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}}+\frac {3 b \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{512 a^{7/2}} \]
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Time = 0.10 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1128, 744, 734, 738, 212} \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx=\frac {3 b \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{512 a^{7/2}}-\frac {3 b \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{256 a^3 x^4}+\frac {b \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 a^2 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}} \]
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Rule 212
Rule 734
Rule 738
Rule 744
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx,x,x^2\right ) \\ & = -\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}}-\frac {b \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^2\right )}{4 a} \\ & = \frac {b \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 a^2 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}}+\frac {\left (3 b \left (b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx,x,x^2\right )}{64 a^2} \\ & = -\frac {3 b \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{256 a^3 x^4}+\frac {b \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 a^2 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}}-\frac {\left (3 b \left (b^2-4 a c\right )^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{512 a^3} \\ & = -\frac {3 b \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{256 a^3 x^4}+\frac {b \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 a^2 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}}+\frac {\left (3 b \left (b^2-4 a c\right )^2\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^2}{\sqrt {a+b x^2+c x^4}}\right )}{256 a^3} \\ & = -\frac {3 b \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{256 a^3 x^4}+\frac {b \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 a^2 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}}+\frac {3 b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{512 a^{7/2}} \\ \end{align*}
Time = 1.04 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx=\frac {-\frac {\sqrt {a} \sqrt {a+b x^2+c x^4} \left (128 a^4+15 b^4 x^8-10 a b^2 x^6 \left (b+10 c x^2\right )+16 a^3 \left (11 b x^2+16 c x^4\right )+8 a^2 x^4 \left (b^2+7 b c x^2+16 c^2 x^4\right )\right )}{x^{10}}-15 b \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{1280 a^{7/2}} \]
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Time = 0.20 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {\frac {3 x^{10} b \left (a c -\frac {b^{2}}{4}\right )^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32}-\frac {\left (\frac {x^{4} \left (16 c^{2} x^{4}+7 b c \,x^{2}+b^{2}\right ) a^{\frac {5}{2}}}{16}-\frac {5 b^{2} x^{6} \left (10 c \,x^{2}+b \right ) a^{\frac {3}{2}}}{64}+\left (2 c \,x^{4}+\frac {11}{8} b \,x^{2}\right ) a^{\frac {7}{2}}+\frac {15 \sqrt {a}\, b^{4} x^{8}}{128}+a^{\frac {9}{2}}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{10}}{a^{\frac {7}{2}} x^{10}}\) | \(153\) |
risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (128 a^{2} c^{2} x^{8}-100 a \,b^{2} c \,x^{8}+15 b^{4} x^{8}+56 a^{2} b c \,x^{6}-10 a \,b^{3} x^{6}+256 a^{3} c \,x^{4}+8 a^{2} b^{2} x^{4}+176 a^{3} b \,x^{2}+128 a^{4}\right )}{1280 x^{10} a^{3}}+\frac {3 b \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{512 a^{\frac {7}{2}}}\) | \(165\) |
default | \(-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{10 x^{10}}-\frac {11 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{80 x^{8}}-\frac {c \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 x^{6}}-\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 a \,x^{6}}+\frac {b^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{128 a^{2} x^{4}}-\frac {3 b^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 a^{3} x^{2}}+\frac {3 b^{5} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{512 a^{\frac {7}{2}}}-\frac {c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{10 a \,x^{2}}+\frac {3 c^{2} b \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}-\frac {3 b^{3} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{64 a^{\frac {5}{2}}}+\frac {5 b^{2} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{64 a^{2} x^{2}}-\frac {7 b c \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 a \,x^{4}}\) | \(337\) |
elliptic | \(-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{10 x^{10}}-\frac {11 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{80 x^{8}}-\frac {c \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 x^{6}}-\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 a \,x^{6}}+\frac {b^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{128 a^{2} x^{4}}-\frac {3 b^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 a^{3} x^{2}}+\frac {3 b^{5} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{512 a^{\frac {7}{2}}}-\frac {c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{10 a \,x^{2}}+\frac {3 c^{2} b \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}-\frac {3 b^{3} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{64 a^{\frac {5}{2}}}+\frac {5 b^{2} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{64 a^{2} x^{2}}-\frac {7 b c \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 a \,x^{4}}\) | \(337\) |
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Time = 0.35 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.36 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx=\left [\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {a} x^{10} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \, {\left ({\left (15 \, a b^{4} - 100 \, a^{2} b^{2} c + 128 \, a^{3} c^{2}\right )} x^{8} + 176 \, a^{4} b x^{2} - 2 \, {\left (5 \, a^{2} b^{3} - 28 \, a^{3} b c\right )} x^{6} + 128 \, a^{5} + 8 \, {\left (a^{3} b^{2} + 32 \, a^{4} c\right )} x^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{5120 \, a^{4} x^{10}}, -\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {-a} x^{10} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (15 \, a b^{4} - 100 \, a^{2} b^{2} c + 128 \, a^{3} c^{2}\right )} x^{8} + 176 \, a^{4} b x^{2} - 2 \, {\left (5 \, a^{2} b^{3} - 28 \, a^{3} b c\right )} x^{6} + 128 \, a^{5} + 8 \, {\left (a^{3} b^{2} + 32 \, a^{4} c\right )} x^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{2560 \, a^{4} x^{10}}\right ] \]
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\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx=\int \frac {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{x^{11}}\, dx \]
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Exception generated. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 832 vs. \(2 (140) = 280\).
Time = 0.37 (sec) , antiderivative size = 832, normalized size of antiderivative = 5.14 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx=-\frac {3 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{256 \, \sqrt {-a} a^{3}} + \frac {15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{9} b^{5} - 120 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{9} a b^{3} c + 240 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{9} a^{2} b c^{2} + 1280 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{8} a^{3} c^{\frac {5}{2}} - 70 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a b^{5} + 560 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a^{2} b^{3} c + 2720 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a^{3} b c^{2} + 5120 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{6} a^{3} b^{2} c^{\frac {3}{2}} + 128 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{2} b^{5} + 2560 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{3} b^{3} c + 3840 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{4} b c^{2} + 1280 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{3} b^{4} \sqrt {c} + 2560 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{4} b^{2} c^{\frac {3}{2}} + 2560 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{5} c^{\frac {5}{2}} + 70 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{3} b^{5} + 2000 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{4} b^{3} c + 2400 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{5} b c^{2} + 2560 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a^{5} b^{2} c^{\frac {3}{2}} - 15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{4} b^{5} + 120 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{5} b^{3} c + 1040 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{6} b c^{2} + 256 \, a^{7} c^{\frac {5}{2}}}{1280 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{3}} \]
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Timed out. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx=\int \frac {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{x^{11}} \,d x \]
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