Integrand size = 20, antiderivative size = 190 \[ \int \frac {x^9}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {x^6 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b x^4 \sqrt {a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}-\frac {\left (b \left (15 b^2-52 a c\right )-2 c \left (5 b^2-12 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c^3 \left (b^2-4 a c\right )}+\frac {3 \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{7/2}} \]
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Time = 0.16 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1128, 752, 846, 793, 635, 212} \[ \int \frac {x^9}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {3 \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{7/2}}-\frac {\left (b \left (15 b^2-52 a c\right )-2 c x^2 \left (5 b^2-12 a c\right )\right ) \sqrt {a+b x^2+c x^4}}{8 c^3 \left (b^2-4 a c\right )}-\frac {b x^4 \sqrt {a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac {x^6 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
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Rule 212
Rule 635
Rule 752
Rule 793
Rule 846
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right ) \\ & = \frac {x^6 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\text {Subst}\left (\int \frac {x^2 (6 a+3 b x)}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{b^2-4 a c} \\ & = \frac {x^6 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b x^4 \sqrt {a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}-\frac {\text {Subst}\left (\int \frac {x \left (-6 a b-\frac {3}{2} \left (5 b^2-12 a c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{3 c \left (b^2-4 a c\right )} \\ & = \frac {x^6 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b x^4 \sqrt {a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}-\frac {\left (b \left (15 b^2-52 a c\right )-2 c \left (5 b^2-12 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c^3 \left (b^2-4 a c\right )}+\frac {\left (3 \left (5 b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{16 c^3} \\ & = \frac {x^6 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b x^4 \sqrt {a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}-\frac {\left (b \left (15 b^2-52 a c\right )-2 c \left (5 b^2-12 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c^3 \left (b^2-4 a c\right )}+\frac {\left (3 \left (5 b^2-4 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 )}{8 c^3} \\ & = \frac {x^6 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b x^4 \sqrt {a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}-\frac {\left (b \left (15 b^2-52 a c\right )-2 c \left (5 b^2-12 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c^3 \left (b^2-4 a c\right )}+\frac {3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{7/2}} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.87 \[ \int \frac {x^9}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {4 a^2 c \left (-13 b+6 c x^2\right )+b^2 x^2 \left (15 b^2+5 b c x^2-2 c^2 x^4\right )+a \left (15 b^3-62 b^2 c x^2-20 b c^2 x^4+8 c^3 x^6\right )}{8 c^3 \left (-b^2+4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {3 \left (-5 b^2+4 a c\right ) \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )}{16 c^{7/2}} \]
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Time = 0.22 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(-\frac {48 \left (\left (-\frac {5}{24} b^{3} x^{4}+\frac {31}{12} a \,b^{2} x^{2}+\frac {13}{6} a^{2} b \right ) c^{\frac {3}{2}}+\left (\frac {1}{12} b^{2} x^{6}+\frac {5}{6} a b \,x^{4}-a^{2} x^{2}\right ) c^{\frac {5}{2}}-\frac {a \,c^{\frac {7}{2}} x^{6}}{3}-\frac {5 \sqrt {c}\, b^{3} \left (b \,x^{2}+a \right )}{8}+\left (a c -\frac {5 b^{2}}{4}\right ) \left (a c -\frac {b^{2}}{4}\right ) \left (-\ln \left (2\right )+\ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right )\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{\frac {7}{2}} \left (64 a c -16 b^{2}\right )}\) | \(183\) |
risch | \(-\frac {\left (-2 c \,x^{2}+7 b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c^{3}}-\frac {-\frac {\left (-4 a b c -7 b^{3}\right ) \left (b \,x^{2}+2 a \right )}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}+\frac {\left (4 c \,a^{2}-7 b^{2} a \right ) \left (2 c \,x^{2}+b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+\left (12 a \,c^{2}-15 b^{2} c \right ) \left (-\frac {x^{2}}{2 c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{4 c}+\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}\right )}{8 c^{3}}\) | \(270\) |
default | \(\frac {x^{6}}{4 c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {5 b \,x^{4}}{8 c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {15 b^{2} x^{2}}{16 c^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {15 b^{3}}{32 c^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {15 b^{4} x^{2}}{16 c^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {15 b^{5}}{32 c^{4} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {15 b^{2} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 c^{\frac {7}{2}}}-\frac {13 b a}{8 c^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {13 b^{2} a \,x^{2}}{4 c^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {13 b^{3} a}{8 c^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {3 a \,x^{2}}{4 c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {3 a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 c^{\frac {5}{2}}}\) | \(354\) |
elliptic | \(\frac {x^{6}}{4 c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {5 b \,x^{4}}{8 c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {15 b^{2} x^{2}}{16 c^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {15 b^{3}}{32 c^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {15 b^{4} x^{2}}{16 c^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {15 b^{5}}{32 c^{4} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {15 b^{2} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 c^{\frac {7}{2}}}-\frac {13 b a}{8 c^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {13 b^{2} a \,x^{2}}{4 c^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {13 b^{3} a}{8 c^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {3 a \,x^{2}}{4 c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {3 a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 c^{\frac {5}{2}}}\) | \(354\) |
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Time = 0.33 (sec) , antiderivative size = 591, normalized size of antiderivative = 3.11 \[ \int \frac {x^9}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{4} + {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{6} - 15 \, a b^{3} c + 52 \, a^{2} b c^{2} - 5 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{4} - {\left (15 \, b^{4} c - 62 \, a b^{2} c^{2} + 24 \, a^{2} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{32 \, {\left (a b^{2} c^{4} - 4 \, a^{2} c^{5} + {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{4} + {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{2}\right )}}, -\frac {3 \, {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{4} + {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) - 2 \, {\left (2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{6} - 15 \, a b^{3} c + 52 \, a^{2} b c^{2} - 5 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{4} - {\left (15 \, b^{4} c - 62 \, a b^{2} c^{2} + 24 \, a^{2} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{16 \, {\left (a b^{2} c^{4} - 4 \, a^{2} c^{5} + {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{4} + {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {x^9}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {x^{9}}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^9}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.32 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.12 \[ \int \frac {x^9}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {{\left ({\left (\frac {2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2}}{b^{2} c^{3} - 4 \, a c^{4}} - \frac {5 \, {\left (b^{3} c - 4 \, a b c^{2}\right )}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x^{2} - \frac {15 \, b^{4} - 62 \, a b^{2} c + 24 \, a^{2} c^{2}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x^{2} - \frac {15 \, a b^{3} - 52 \, a^{2} b c}{b^{2} c^{3} - 4 \, a c^{4}}}{8 \, \sqrt {c x^{4} + b x^{2} + a}} - \frac {3 \, {\left (5 \, b^{2} - 4 \, a c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right )}{16 \, c^{\frac {7}{2}}} \]
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Timed out. \[ \int \frac {x^9}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {x^9}{{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \]
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