Integrand size = 20, antiderivative size = 150 \[ \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {3 b \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^3}-\frac {b \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c}-\frac {3 b \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{7/2}} \]
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Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1128, 654, 626, 635, 212} \[ \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=-\frac {3 b \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{7/2}}+\frac {3 b \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^3}-\frac {b \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c} \]
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right ) \\ & = \frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c}-\frac {b \text {Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{4 c} \\ & = -\frac {b \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c}+\frac {\left (3 b \left (b^2-4 a c\right )\right ) \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^2\right )}{64 c^2} \\ & = \frac {3 b \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^3}-\frac {b \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c}-\frac {\left (3 b \left (b^2-4 a c\right )^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{512 c^3} \\ & = \frac {3 b \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^3}-\frac {b \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c}-\frac {\left (3 b \left (b^2-4 a c\right )^2\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 )}{256 c^3} \\ & = \frac {3 b \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^3}-\frac {b \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c}-\frac {3 b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{7/2}} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93 \[ \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {\sqrt {a+b x^2+c x^4} \left (15 b^4-10 b^3 c x^2+128 c^2 \left (a+c x^4\right )^2+4 b^2 c \left (-25 a+2 c x^4\right )+8 b c^2 x^2 \left (7 a+22 c x^4\right )\right )}{1280 c^3}+\frac {3 b \left (b^2-4 a c\right )^2 \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )}{512 c^{7/2}} \]
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Time = 0.16 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.01
method | result | size |
risch | \(\frac {\left (128 c^{4} x^{8}+176 c^{3} x^{6} b +256 a \,c^{3} x^{4}+8 b^{2} c^{2} x^{4}+56 a b \,c^{2} x^{2}-10 x^{2} c \,b^{3}+128 a^{2} c^{2}-100 a \,b^{2} c +15 b^{4}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{1280 c^{3}}-\frac {3 b \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{512 c^{\frac {7}{2}}}\) | \(152\) |
pseudoelliptic | \(\frac {-\frac {15 b \left (a c -\frac {b^{2}}{4}\right )^{2} \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right )}{16}+\left (\left (\frac {1}{16} b^{2} x^{4}+\frac {7}{16} a b \,x^{2}+a^{2}\right ) c^{\frac {5}{2}}-\frac {25 \left (\frac {b \,x^{2}}{10}+a \right ) b^{2} c^{\frac {3}{2}}}{32}+\left (\frac {11}{8} b \,x^{6}+2 a \,x^{4}\right ) c^{\frac {7}{2}}+c^{\frac {9}{2}} x^{8}+\frac {15 \sqrt {c}\, b^{4}}{128}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}+\frac {15 \ln \left (2\right ) b \left (a c -\frac {b^{2}}{4}\right )^{2}}{16}}{10 c^{\frac {7}{2}}}\) | \(156\) |
default | \(-\frac {5 b^{2} a \sqrt {c \,x^{4}+b \,x^{2}+a}}{64 c^{2}}+\frac {b^{2} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 c}-\frac {b^{3} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{128 c^{2}}-\frac {3 b^{5} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{512 c^{\frac {7}{2}}}-\frac {3 a^{2} b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{32 c^{\frac {3}{2}}}+\frac {a^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{10 c}+\frac {3 b^{3} a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{64 c^{\frac {5}{2}}}+\frac {7 b a \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 c}+\frac {c \,x^{8} \sqrt {c \,x^{4}+b \,x^{2}+a}}{10}+\frac {11 b \,x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{80}+\frac {3 b^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 c^{3}}+\frac {a \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5}\) | \(316\) |
elliptic | \(-\frac {5 b^{2} a \sqrt {c \,x^{4}+b \,x^{2}+a}}{64 c^{2}}+\frac {b^{2} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 c}-\frac {b^{3} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{128 c^{2}}-\frac {3 b^{5} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{512 c^{\frac {7}{2}}}-\frac {3 a^{2} b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{32 c^{\frac {3}{2}}}+\frac {a^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{10 c}+\frac {3 b^{3} a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{64 c^{\frac {5}{2}}}+\frac {7 b a \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 c}+\frac {c \,x^{8} \sqrt {c \,x^{4}+b \,x^{2}+a}}{10}+\frac {11 b \,x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{80}+\frac {3 b^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 c^{3}}+\frac {a \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5}\) | \(316\) |
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Time = 0.26 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.41 \[ \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\left [\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (128 \, c^{5} x^{8} + 176 \, b c^{4} x^{6} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} + 8 \, {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{4} - 2 \, {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{5120 \, c^{4}}, \frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, {\left (128 \, c^{5} x^{8} + 176 \, b c^{4} x^{6} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} + 8 \, {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{4} - 2 \, {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{2560 \, c^{4}}\right ] \]
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\[ \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int x^{3} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \]
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Exception generated. \[ \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (128) = 256\).
Time = 0.34 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.72 \[ \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {1}{96} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {3 \, b^{2} - 8 \, a c}{c^{2}}\right )} - \frac {3 \, {\left (b^{3} - 4 \, a b c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {5}{2}}}\right )} a + \frac {1}{768} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (6 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {5 \, b^{2} c - 12 \, a c^{2}}{c^{3}}\right )} x^{2} + \frac {15 \, b^{3} - 52 \, a b c}{c^{3}}\right )} + \frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {7}{2}}}\right )} b + \frac {1}{7680} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {7 \, b^{2} c^{2} - 16 \, a c^{3}}{c^{4}}\right )} x^{2} + \frac {35 \, b^{3} c - 116 \, a b c^{2}}{c^{4}}\right )} x^{2} - \frac {105 \, b^{4} - 460 \, a b^{2} c + 256 \, a^{2} c^{2}}{c^{4}}\right )} - \frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {9}{2}}}\right )} c \]
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Time = 13.94 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.49 \[ \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {{\left (c\,x^4+b\,x^2+a\right )}^{5/2}}{10\,c}-\frac {b\,\left (\frac {3\,a\,\left (\ln \left (\sqrt {c\,x^4+b\,x^2+a}+\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}\right )\,\left (\frac {a}{2\,\sqrt {c}}-\frac {b^2}{8\,c^{3/2}}\right )+\frac {\left (2\,c\,x^2+b\right )\,\sqrt {c\,x^4+b\,x^2+a}}{4\,c}\right )}{4}+\frac {x^2\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{4}-\frac {3\,b^2\,\left (\ln \left (\sqrt {c\,x^4+b\,x^2+a}+\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}\right )\,\left (\frac {a}{2\,\sqrt {c}}-\frac {b^2}{8\,c^{3/2}}\right )+\frac {\left (2\,c\,x^2+b\right )\,\sqrt {c\,x^4+b\,x^2+a}}{4\,c}\right )}{16\,c}+\frac {b\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{8\,c}\right )}{4\,c} \]
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