Integrand size = 21, antiderivative size = 124 \[ \int \frac {x^7}{\sqrt {a+b x^2-c x^4}} \, dx=-\frac {x^4 \sqrt {a+b x^2-c x^4}}{6 c}-\frac {\left (15 b^2+16 a c+10 b c x^2\right ) \sqrt {a+b x^2-c x^4}}{48 c^3}-\frac {b \left (5 b^2+12 a c\right ) \arctan \left (\frac {b-2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{32 c^{7/2}} \]
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Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1128, 756, 793, 635, 210} \[ \int \frac {x^7}{\sqrt {a+b x^2-c x^4}} \, dx=-\frac {b \left (12 a c+5 b^2\right ) \arctan \left (\frac {b-2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{32 c^{7/2}}-\frac {\left (16 a c+15 b^2+10 b c x^2\right ) \sqrt {a+b x^2-c x^4}}{48 c^3}-\frac {x^4 \sqrt {a+b x^2-c x^4}}{6 c} \]
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Rule 210
Rule 635
Rule 756
Rule 793
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^3}{\sqrt {a+b x-c x^2}} \, dx,x,x^2\right ) \\ & = -\frac {x^4 \sqrt {a+b x^2-c x^4}}{6 c}-\frac {\text {Subst}\left (\int \frac {x \left (-2 a-\frac {5 b x}{2}\right )}{\sqrt {a+b x-c x^2}} \, dx,x,x^2\right )}{6 c} \\ & = -\frac {x^4 \sqrt {a+b x^2-c x^4}}{6 c}-\frac {\left (15 b^2+16 a c+10 b c x^2\right ) \sqrt {a+b x^2-c x^4}}{48 c^3}+\frac {\left (b \left (5 b^2+12 a c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x-c x^2}} \, dx,x,x^2\right )}{32 c^3} \\ & = -\frac {x^4 \sqrt {a+b x^2-c x^4}}{6 c}-\frac {\left (15 b^2+16 a c+10 b c x^2\right ) \sqrt {a+b x^2-c x^4}}{48 c^3}+\frac {\left (b \left (5 b^2+12 a c\right )\right ) \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 )}{16 c^3} \\ & = -\frac {x^4 \sqrt {a+b x^2-c x^4}}{6 c}-\frac {\left (15 b^2+16 a c+10 b c x^2\right ) \sqrt {a+b x^2-c x^4}}{48 c^3}-\frac {b \left (5 b^2+12 a c\right ) \tan ^{-1}\left (\frac {b-2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{32 c^{7/2}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.86 \[ \int \frac {x^7}{\sqrt {a+b x^2-c x^4}} \, dx=\frac {\sqrt {a+b x^2-c x^4} \left (-15 b^2-16 a c-10 b c x^2-8 c^2 x^4\right )}{48 c^3}+\frac {\left (5 b^3+12 a b c\right ) \arctan \left (\frac {\sqrt {c} x^2}{-\sqrt {a}+\sqrt {a+b x^2-c x^4}}\right )}{16 c^{7/2}} \]
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Time = 0.49 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.75
method | result | size |
risch | \(-\frac {\left (8 c^{2} x^{4}+10 b c \,x^{2}+16 a c +15 b^{2}\right ) \sqrt {-c \,x^{4}+b \,x^{2}+a}}{48 c^{3}}+\frac {b \left (12 a c +5 b^{2}\right ) \arctan \left (\frac {\sqrt {c}\, \left (x^{2}-\frac {b}{2 c}\right )}{\sqrt {-c \,x^{4}+b \,x^{2}+a}}\right )}{32 c^{\frac {7}{2}}}\) | \(93\) |
pseudoelliptic | \(\frac {-16 c^{\frac {5}{2}} x^{4} \sqrt {-c \,x^{4}+b \,x^{2}+a}-20 b \,c^{\frac {3}{2}} x^{2} \sqrt {-c \,x^{4}+b \,x^{2}+a}-32 a \,c^{\frac {3}{2}} \sqrt {-c \,x^{4}+b \,x^{2}+a}-30 b^{2} \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}-36 \arctan \left (\frac {-2 c \,x^{2}+b}{2 \sqrt {c}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}\right ) a b c -15 \arctan \left (\frac {-2 c \,x^{2}+b}{2 \sqrt {c}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}\right ) b^{3}}{96 c^{\frac {7}{2}}}\) | \(166\) |
default | \(-\frac {x^{4} \sqrt {-c \,x^{4}+b \,x^{2}+a}}{6 c}-\frac {5 b \,x^{2} \sqrt {-c \,x^{4}+b \,x^{2}+a}}{24 c^{2}}-\frac {5 b^{2} \sqrt {-c \,x^{4}+b \,x^{2}+a}}{16 c^{3}}+\frac {5 b^{3} \arctan \left (\frac {\sqrt {c}\, \left (x^{2}-\frac {b}{2 c}\right )}{\sqrt {-c \,x^{4}+b \,x^{2}+a}}\right )}{32 c^{\frac {7}{2}}}+\frac {3 b a \arctan \left (\frac {\sqrt {c}\, \left (x^{2}-\frac {b}{2 c}\right )}{\sqrt {-c \,x^{4}+b \,x^{2}+a}}\right )}{8 c^{\frac {5}{2}}}-\frac {a \sqrt {-c \,x^{4}+b \,x^{2}+a}}{3 c^{2}}\) | \(168\) |
elliptic | \(-\frac {x^{4} \sqrt {-c \,x^{4}+b \,x^{2}+a}}{6 c}-\frac {5 b \,x^{2} \sqrt {-c \,x^{4}+b \,x^{2}+a}}{24 c^{2}}-\frac {5 b^{2} \sqrt {-c \,x^{4}+b \,x^{2}+a}}{16 c^{3}}+\frac {5 b^{3} \arctan \left (\frac {\sqrt {c}\, \left (x^{2}-\frac {b}{2 c}\right )}{\sqrt {-c \,x^{4}+b \,x^{2}+a}}\right )}{32 c^{\frac {7}{2}}}+\frac {3 b a \arctan \left (\frac {\sqrt {c}\, \left (x^{2}-\frac {b}{2 c}\right )}{\sqrt {-c \,x^{4}+b \,x^{2}+a}}\right )}{8 c^{\frac {5}{2}}}-\frac {a \sqrt {-c \,x^{4}+b \,x^{2}+a}}{3 c^{2}}\) | \(168\) |
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Time = 0.28 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.01 \[ \int \frac {x^7}{\sqrt {a+b x^2-c x^4}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{3} + 12 \, a b c\right )} \sqrt {-c} \log \left (8 \, c^{2} x^{4} - 8 \, b c x^{2} + b^{2} - 4 \, \sqrt {-c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} - b\right )} \sqrt {-c} - 4 \, a c\right ) + 4 \, {\left (8 \, c^{3} x^{4} + 10 \, b c^{2} x^{2} + 15 \, b^{2} c + 16 \, a c^{2}\right )} \sqrt {-c x^{4} + b x^{2} + a}}{192 \, c^{4}}, -\frac {3 \, {\left (5 \, b^{3} + 12 \, a b c\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} - b\right )} \sqrt {c}}{2 \, {\left (c^{2} x^{4} - b c x^{2} - a c\right )}}\right ) + 2 \, {\left (8 \, c^{3} x^{4} + 10 \, b c^{2} x^{2} + 15 \, b^{2} c + 16 \, a c^{2}\right )} \sqrt {-c x^{4} + b x^{2} + a}}{96 \, c^{4}}\right ] \]
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\[ \int \frac {x^7}{\sqrt {a+b x^2-c x^4}} \, dx=\int \frac {x^{7}}{\sqrt {a + b x^{2} - c x^{4}}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.23 \[ \int \frac {x^7}{\sqrt {a+b x^2-c x^4}} \, dx=-\frac {\sqrt {-c x^{4} + b x^{2} + a} x^{4}}{6 \, c} - \frac {5 \, \sqrt {-c x^{4} + b x^{2} + a} b x^{2}}{24 \, c^{2}} - \frac {5 \, b^{3} \arcsin \left (-\frac {2 \, c x^{2} - b}{\sqrt {b^{2} + 4 \, a c}}\right )}{32 \, c^{\frac {7}{2}}} - \frac {3 \, a b \arcsin \left (-\frac {2 \, c x^{2} - b}{\sqrt {b^{2} + 4 \, a c}}\right )}{8 \, c^{\frac {5}{2}}} - \frac {5 \, \sqrt {-c x^{4} + b x^{2} + a} b^{2}}{16 \, c^{3}} - \frac {\sqrt {-c x^{4} + b x^{2} + a} a}{3 \, c^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.90 \[ \int \frac {x^7}{\sqrt {a+b x^2-c x^4}} \, dx=-\frac {1}{48} \, \sqrt {-c x^{4} + b x^{2} + a} {\left (2 \, x^{2} {\left (\frac {4 \, x^{2}}{c} + \frac {5 \, b}{c^{2}}\right )} + \frac {15 \, b^{2} + 16 \, a c}{c^{3}}\right )} - \frac {{\left (5 \, b^{3} + 12 \, a b c\right )} \log \left ({\left | 2 \, {\left (\sqrt {-c} x^{2} - \sqrt {-c x^{4} + b x^{2} + a}\right )} \sqrt {-c} + b \right |}\right )}{32 \, \sqrt {-c} c^{3}} \]
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Timed out. \[ \int \frac {x^7}{\sqrt {a+b x^2-c x^4}} \, dx=\int \frac {x^7}{\sqrt {-c\,x^4+b\,x^2+a}} \,d x \]
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