Integrand size = 20, antiderivative size = 153 \[ \int x^5 \sqrt {a+b x^2+c x^4} \, dx=\frac {\left (5 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{128 c^3}-\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac {x^2 \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{256 c^{7/2}} \]
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Time = 0.09 (sec) , antiderivative size = 153, 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, 756, 654, 626, 635, 212} \[ \int x^5 \sqrt {a+b x^2+c x^4} \, dx=-\frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{256 c^{7/2}}+\frac {\left (5 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{128 c^3}-\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac {x^2 \left (a+b x^2+c x^4\right )^{3/2}}{8 c} \]
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 756
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x^2 \sqrt {a+b x+c x^2} \, dx,x,x^2\right ) \\ & = \frac {x^2 \left (a+b x^2+c x^4\right )^{3/2}}{8 c}+\frac {\text {Subst}\left (\int \left (-a-\frac {5 b x}{2}\right ) \sqrt {a+b x+c x^2} \, dx,x,x^2\right )}{8 c} \\ & = -\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac {x^2 \left (a+b x^2+c x^4\right )^{3/2}}{8 c}+\frac {\left (5 b^2-4 a c\right ) \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^2\right )}{32 c^2} \\ & = \frac {\left (5 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{128 c^3}-\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac {x^2 \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {\left (\left (b^2-4 a c\right ) \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 )}{256 c^3} \\ & = \frac {\left (5 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{128 c^3}-\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac {x^2 \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {\left (\left (b^2-4 a c\right ) \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 )}{128 c^3} \\ & = \frac {\left (5 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{128 c^3}-\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac {x^2 \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {\left (b^2-4 a c\right ) \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 )}{256 c^{7/2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.86 \[ \int x^5 \sqrt {a+b x^2+c x^4} \, dx=\frac {\sqrt {a+b x^2+c x^4} \left (15 b^3-52 a b c-10 b^2 c x^2+24 a c^2 x^2+8 b c^2 x^4+48 c^3 x^6\right )}{384 c^3}+\frac {\left (5 b^4-24 a b^2 c+16 a^2 c^2\right ) \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )}{256 c^{7/2}} \]
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Time = 0.13 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-\frac {\left (-48 c^{3} x^{6}-8 b \,c^{2} x^{4}-24 a \,c^{2} x^{2}+10 b^{2} c \,x^{2}+52 a b c -15 b^{3}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{384 c^{3}}-\frac {\left (16 a^{2} c^{2}-24 a \,b^{2} c +5 b^{4}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{256 c^{\frac {7}{2}}}\) | \(122\) |
pseudoelliptic | \(-\frac {\left (a^{2} c^{2}-\frac {3}{2} a \,b^{2} c +\frac {5}{16} b^{4}\right ) \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right )+\left (\frac {13 \left (\frac {5 b \,x^{2}}{26}+a \right ) b \,c^{\frac {3}{2}}}{6}+\left (-\frac {1}{3} b \,x^{4}-a \,x^{2}\right ) c^{\frac {5}{2}}-2 c^{\frac {7}{2}} x^{6}-\frac {5 \sqrt {c}\, b^{3}}{8}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}-\ln \left (2\right ) \left (a c -\frac {5 b^{2}}{4}\right ) \left (a c -\frac {b^{2}}{4}\right )}{16 c^{\frac {7}{2}}}\) | \(145\) |
default | \(\frac {x^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{8 c}-\frac {5 b \left (\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{16 c}-\frac {a \left (\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{8 c^{\frac {3}{2}}}\right )}{8 c}\) | \(206\) |
elliptic | \(\frac {x^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{8 c}-\frac {5 b \left (\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{16 c}-\frac {a \left (\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{8 c^{\frac {3}{2}}}\right )}{8 c}\) | \(206\) |
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Time = 0.26 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.98 \[ \int x^5 \sqrt {a+b x^2+c x^4} \, dx=\left [\frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (48 \, c^{4} x^{6} + 8 \, b c^{3} x^{4} + 15 \, b^{3} c - 52 \, a b c^{2} - 2 \, {\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{1536 \, c^{4}}, \frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, {\left (48 \, c^{4} x^{6} + 8 \, b c^{3} x^{4} + 15 \, b^{3} c - 52 \, a b c^{2} - 2 \, {\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{768 \, c^{4}}\right ] \]
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\[ \int x^5 \sqrt {a+b x^2+c x^4} \, dx=\int x^{5} \sqrt {a + b x^{2} + c x^{4}}\, dx \]
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Exception generated. \[ \int x^5 \sqrt {a+b x^2+c x^4} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.30 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.86 \[ \int x^5 \sqrt {a+b x^2+c x^4} \, dx=\frac {1}{384} \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (6 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {5 \, b^{2} c - 12 \, a c^{2}}{c^{3}}\right )} x^{2} + \frac {15 \, b^{3} - 52 \, a b c}{c^{3}}\right )} + \frac {{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {7}{2}}} \]
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Time = 13.39 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.26 \[ \int x^5 \sqrt {a+b x^2+c x^4} \, dx=\frac {x^2\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{8\,c}-\frac {a\,\left (\left (\frac {b}{4\,c}+\frac {x^2}{2}\right )\,\sqrt {c\,x^4+b\,x^2+a}+\frac {\ln \left (\sqrt {c\,x^4+b\,x^2+a}+\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{8\,c}-\frac {5\,b\,\left (\frac {\left (8\,c\,\left (c\,x^4+a\right )-3\,b^2+2\,b\,c\,x^2\right )\,\sqrt {c\,x^4+b\,x^2+a}}{24\,c^2}+\frac {\ln \left (2\,\sqrt {c\,x^4+b\,x^2+a}+\frac {2\,c\,x^2+b}{\sqrt {c}}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}\right )}{16\,c} \]
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