Integrand size = 18, antiderivative size = 130 \[ \int \frac {x^5}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 a b+\left (b^2+2 a c\right ) x^2}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (b^2+2 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1128, 752, 652, 632, 212} \[ \int \frac {x^5}{\left (a+b x^2+c x^4\right )^3} \, dx=-\frac {\left (2 a c+b^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^2 \left (2 a c+b^2\right )+3 a b}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )} \]
[In]
[Out]
Rule 212
Rule 632
Rule 652
Rule 752
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right ) \\ & = \frac {x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\text {Subst}\left (\int \frac {2 a-2 b x}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 \left (b^2-4 a c\right )} \\ & = \frac {x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 a b+\left (b^2+2 a c\right ) x^2}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (b^2+2 a c\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )^2} \\ & = \frac {x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 a b+\left (b^2+2 a c\right ) x^2}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (b^2+2 a c\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{\left (b^2-4 a c\right )^2} \\ & = \frac {x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 a b+\left (b^2+2 a c\right ) x^2}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (b^2+2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.12 \[ \int \frac {x^5}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {1}{4} \left (\frac {\left (b^2+2 a c\right ) \left (b+2 c x^2\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {b^2 x^2+a \left (b-2 c x^2\right )}{c \left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {4 \left (b^2+2 a c\right ) \arctan \left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}\right ) \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.66
method | result | size |
default | \(\frac {\frac {c \left (2 a c +b^{2}\right ) x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {3 b \left (2 a c +b^{2}\right ) x^{4}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (2 a c -5 b^{2}\right ) x^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {3 a^{2} b}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\left (2 a c +b^{2}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}\) | \(216\) |
risch | \(\frac {\frac {c \left (2 a c +b^{2}\right ) x^{6}}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}+\frac {3 b \left (2 a c +b^{2}\right ) x^{4}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (2 a c -5 b^{2}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {3 a^{2} b}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {\ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) x^{2}-32 a^{3} c^{2}+16 a^{2} b^{2} c -2 b^{4} a \right ) a c}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}-\frac {\ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) x^{2}-32 a^{3} c^{2}+16 a^{2} b^{2} c -2 b^{4} a \right ) b^{2}}{2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {\ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) x^{2}+32 a^{3} c^{2}-16 a^{2} b^{2} c +2 b^{4} a \right ) a c}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {\ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) x^{2}+32 a^{3} c^{2}-16 a^{2} b^{2} c +2 b^{4} a \right ) b^{2}}{2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}}}\) | \(449\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (122) = 244\).
Time = 0.26 (sec) , antiderivative size = 907, normalized size of antiderivative = 6.98 \[ \int \frac {x^5}{\left (a+b x^2+c x^4\right )^3} \, dx=\left [\frac {2 \, {\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} x^{6} + 6 \, a^{2} b^{3} - 24 \, a^{3} b c + 3 \, {\left (b^{5} - 2 \, a b^{3} c - 8 \, a^{2} b c^{2}\right )} x^{4} + 2 \, {\left (5 \, a b^{4} - 22 \, a^{2} b^{2} c + 8 \, a^{3} c^{2}\right )} x^{2} + 2 \, {\left ({\left (b^{2} c^{2} + 2 \, a c^{3}\right )} x^{8} + 2 \, {\left (b^{3} c + 2 \, a b c^{2}\right )} x^{6} + {\left (b^{4} + 4 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{4} + a^{2} b^{2} + 2 \, a^{3} c + 2 \, {\left (a b^{3} + 2 \, a^{2} b c\right )} x^{2}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c - {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right )}{4 \, {\left ({\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} x^{8} + a^{2} b^{6} - 12 \, a^{3} b^{4} c + 48 \, a^{4} b^{2} c^{2} - 64 \, a^{5} c^{3} + 2 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} x^{6} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} x^{4} + 2 \, {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} x^{2}\right )}}, \frac {2 \, {\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} x^{6} + 6 \, a^{2} b^{3} - 24 \, a^{3} b c + 3 \, {\left (b^{5} - 2 \, a b^{3} c - 8 \, a^{2} b c^{2}\right )} x^{4} + 2 \, {\left (5 \, a b^{4} - 22 \, a^{2} b^{2} c + 8 \, a^{3} c^{2}\right )} x^{2} - 4 \, {\left ({\left (b^{2} c^{2} + 2 \, a c^{3}\right )} x^{8} + 2 \, {\left (b^{3} c + 2 \, a b c^{2}\right )} x^{6} + {\left (b^{4} + 4 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{4} + a^{2} b^{2} + 2 \, a^{3} c + 2 \, {\left (a b^{3} + 2 \, a^{2} b c\right )} x^{2}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{4 \, {\left ({\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} x^{8} + a^{2} b^{6} - 12 \, a^{3} b^{4} c + 48 \, a^{4} b^{2} c^{2} - 64 \, a^{5} c^{3} + 2 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} x^{6} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} x^{4} + 2 \, {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} x^{2}\right )}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (119) = 238\).
Time = 2.28 (sec) , antiderivative size = 580, normalized size of antiderivative = 4.46 \[ \int \frac {x^5}{\left (a+b x^2+c x^4\right )^3} \, dx=- \frac {\sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) \log {\left (x^{2} + \frac {- 64 a^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) + 48 a^{2} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) - 12 a b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) + 2 a b c + b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) + b^{3}}{4 a c^{2} + 2 b^{2} c} \right )}}{2} + \frac {\sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) \log {\left (x^{2} + \frac {64 a^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) - 48 a^{2} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) + 12 a b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) + 2 a b c - b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) + b^{3}}{4 a c^{2} + 2 b^{2} c} \right )}}{2} + \frac {6 a^{2} b + x^{6} \cdot \left (4 a c^{2} + 2 b^{2} c\right ) + x^{4} \cdot \left (6 a b c + 3 b^{3}\right ) + x^{2} \left (- 4 a^{2} c + 10 a b^{2}\right )}{64 a^{4} c^{2} - 32 a^{3} b^{2} c + 4 a^{2} b^{4} + x^{8} \cdot \left (64 a^{2} c^{4} - 32 a b^{2} c^{3} + 4 b^{4} c^{2}\right ) + x^{6} \cdot \left (128 a^{2} b c^{3} - 64 a b^{3} c^{2} + 8 b^{5} c\right ) + x^{4} \cdot \left (128 a^{3} c^{3} - 24 a b^{4} c + 4 b^{6}\right ) + x^{2} \cdot \left (128 a^{3} b c^{2} - 64 a^{2} b^{3} c + 8 a b^{5}\right )} \]
[In]
[Out]
Exception generated. \[ \int \frac {x^5}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 1.40 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.24 \[ \int \frac {x^5}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {{\left (b^{2} + 2 \, a c\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {2 \, b^{2} c x^{6} + 4 \, a c^{2} x^{6} + 3 \, b^{3} x^{4} + 6 \, a b c x^{4} + 10 \, a b^{2} x^{2} - 4 \, a^{2} c x^{2} + 6 \, a^{2} b}{4 \, {\left (c x^{4} + b x^{2} + a\right )}^{2} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}} \]
[In]
[Out]
Time = 13.19 (sec) , antiderivative size = 460, normalized size of antiderivative = 3.54 \[ \int \frac {x^5}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {\frac {3\,a^2\,b}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^2\,\left (5\,a\,b^2-2\,a^2\,c\right )}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {3\,b\,x^4\,\left (b^2+2\,a\,c\right )}{4\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {c\,x^6\,\left (b^2+2\,a\,c\right )}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^4\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^8+2\,a\,b\,x^2+2\,b\,c\,x^6}+\frac {\mathrm {atan}\left (\frac {\left (x^2\,\left (\frac {\left (b^2+2\,a\,c\right )\,\left (b^2\,c^2+2\,a\,c^3\right )}{a\,{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {b\,{\left (b^2+2\,a\,c\right )}^2\,\left (32\,a^2\,b\,c^4-16\,a\,b^3\,c^3+2\,b^5\,c^2\right )}{2\,a\,{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )+\frac {2\,b\,c^2\,{\left (b^2+2\,a\,c\right )}^2}{{\left (4\,a\,c-b^2\right )}^{15/2}}\right )\,\left (b^4\,{\left (4\,a\,c-b^2\right )}^5+16\,a^2\,c^2\,{\left (4\,a\,c-b^2\right )}^5-8\,a\,b^2\,c\,{\left (4\,a\,c-b^2\right )}^5\right )}{8\,a^2\,c^4+8\,a\,b^2\,c^3+2\,b^4\,c^2}\right )\,\left (b^2+2\,a\,c\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}} \]
[In]
[Out]