Integrand size = 20, antiderivative size = 163 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx=-\frac {\left (2 a b+\left (b^2+8 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{16 a x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6}+\frac {b \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{32 a^{3/2}}+\frac {1}{2} c^{3/2} \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1128, 746, 824, 857, 635, 212, 738} \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx=\frac {b \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{32 a^{3/2}}+\frac {1}{2} c^{3/2} \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )-\frac {\left (x^2 \left (8 a c+b^2\right )+2 a b\right ) \sqrt {a+b x^2+c x^4}}{16 a x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6} \]
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Rule 212
Rule 635
Rule 738
Rule 746
Rule 824
Rule 857
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^4} \, dx,x,x^2\right ) \\ & = -\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6}+\frac {1}{4} \text {Subst}\left (\int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{x^3} \, dx,x,x^2\right ) \\ & = -\frac {\left (2 a b+\left (b^2+8 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{16 a x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} b \left (b^2-12 a c\right )-8 a c^2 x}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{16 a} \\ & = -\frac {\left (2 a b+\left (b^2+8 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{16 a x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6}+\frac {1}{2} c^2 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )-\frac {\left (b \left (b^2-12 a c\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{32 a} \\ & = -\frac {\left (2 a b+\left (b^2+8 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{16 a x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6}+c^2 \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^2}{\sqrt {a+b x^2+c x^4}}\right )+\frac {\left (b \left (b^2-12 a c\right )\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 )}{16 a} \\ & = -\frac {\left (2 a b+\left (b^2+8 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{16 a x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6}+\frac {b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{32 a^{3/2}}+\frac {1}{2} c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \\ \end{align*}
Time = 0.73 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx=\frac {\sqrt {a+b x^2+c x^4} \left (-8 a^2-14 a b x^2-3 b^2 x^4-32 a c x^4\right )}{48 a x^6}+\frac {\left (b^3-12 a b c\right ) \text {arctanh}\left (\frac {-\sqrt {c} x^2+\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{16 a^{3/2}}-\frac {1}{2} c^{3/2} \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right ) \]
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Time = 0.16 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.87
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (32 a c \,x^{4}+3 b^{2} x^{4}+14 a b \,x^{2}+8 a^{2}\right )}{48 x^{6} a}+\frac {8 a \,c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )-\frac {b \left (12 a c -b^{2}\right ) \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2 \sqrt {a}}}{16 a}\) | \(142\) |
pseudoelliptic | \(-\frac {3 \left (b \,x^{6} \left (a c -\frac {b^{2}}{12}\right ) \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )-\frac {4 \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right ) a^{\frac {3}{2}} c^{\frac {3}{2}} x^{6}}{3}+\left (\frac {7 x^{2} \left (\frac {16 c \,x^{2}}{7}+b \right ) a^{\frac {3}{2}}}{9}+\frac {\sqrt {a}\, b^{2} x^{4}}{6}+\frac {4 a^{\frac {5}{2}}}{9}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}+\frac {4 \ln \left (2\right ) a^{\frac {3}{2}} c^{\frac {3}{2}} x^{6}}{3}\right )}{8 a^{\frac {3}{2}} x^{6}}\) | \(161\) |
default | \(\frac {c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2}-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{6 x^{6}}-\frac {7 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{24 x^{4}}-\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 a \,x^{2}}+\frac {b^{3} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}-\frac {3 b c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{8 \sqrt {a}}-\frac {2 c \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 x^{2}}\) | \(202\) |
elliptic | \(\frac {c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2}-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{6 x^{6}}-\frac {7 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{24 x^{4}}-\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 a \,x^{2}}+\frac {b^{3} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}-\frac {3 b c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{8 \sqrt {a}}-\frac {2 c \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 x^{2}}\) | \(202\) |
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Time = 0.33 (sec) , antiderivative size = 771, normalized size of antiderivative = 4.73 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx=\left [\frac {48 \, a^{2} c^{\frac {3}{2}} x^{6} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) - 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {a} x^{6} \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 (14 \, a^{2} b x^{2} + {\left (3 \, a b^{2} + 32 \, a^{2} c\right )} x^{4} + 8 \, a^{3}\right )} \sqrt {c x^{4} + b x^{2} + a}}{192 \, a^{2} x^{6}}, -\frac {96 \, a^{2} \sqrt {-c} c x^{6} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {a} x^{6} \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 (14 \, a^{2} b x^{2} + {\left (3 \, a b^{2} + 32 \, a^{2} c\right )} x^{4} + 8 \, a^{3}\right )} \sqrt {c x^{4} + b x^{2} + a}}{192 \, a^{2} x^{6}}, \frac {24 \, a^{2} c^{\frac {3}{2}} x^{6} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) - 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {-a} x^{6} \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 (14 \, a^{2} b x^{2} + {\left (3 \, a b^{2} + 32 \, a^{2} c\right )} x^{4} + 8 \, a^{3}\right )} \sqrt {c x^{4} + b x^{2} + a}}{96 \, a^{2} x^{6}}, -\frac {48 \, a^{2} \sqrt {-c} c x^{6} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {-a} x^{6} \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 (14 \, a^{2} b x^{2} + {\left (3 \, a b^{2} + 32 \, a^{2} c\right )} x^{4} + 8 \, a^{3}\right )} \sqrt {c x^{4} + b x^{2} + a}}{96 \, a^{2} x^{6}}\right ] \]
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\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx=\int \frac {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{x^{7}}\, dx \]
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Exception generated. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (135) = 270\).
Time = 0.40 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.44 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx=-\frac {1}{2} \, c^{\frac {3}{2}} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right ) - \frac {{\left (b^{3} - 12 \, a b c\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{16 \, \sqrt {-a} a} + \frac {3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} b^{3} + 60 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a b c + 48 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a b^{2} \sqrt {c} + 96 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{2} c^{\frac {3}{2}} + 8 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a b^{3} - 96 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a^{3} c^{\frac {3}{2}} - 3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{2} b^{3} + 36 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{3} b c + 64 \, a^{4} c^{\frac {3}{2}}}{48 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{3} a} \]
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Timed out. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx=\int \frac {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{x^7} \,d x \]
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