Integrand size = 20, antiderivative size = 223 \[ \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {3 b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{2048 c^5}-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{256 c^4}+\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\left (21 b^2-16 a c-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{560 c^3}-\frac {3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4096 c^{11/2}} \]
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Time = 0.16 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1128, 756, 793, 626, 635, 212} \[ \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx=-\frac {3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4096 c^{11/2}}+\frac {3 b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{2048 c^5}-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{256 c^4}+\frac {\left (-16 a c+21 b^2-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{560 c^3}+\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c} \]
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Rule 212
Rule 626
Rule 635
Rule 756
Rule 793
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x^3 \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right ) \\ & = \frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\text {Subst}\left (\int x \left (-2 a-\frac {9 b x}{2}\right ) \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{14 c} \\ & = \frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\left (21 b^2-16 a c-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{560 c^3}-\frac {\left (b \left (3 b^2-4 a c\right )\right ) \text {Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{32 c^3} \\ & = -\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{256 c^4}+\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\left (21 b^2-16 a c-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{560 c^3}+\frac {\left (3 b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right )\right ) \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^2\right )}{512 c^4} \\ & = \frac {3 b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{2048 c^5}-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{256 c^4}+\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\left (21 b^2-16 a c-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{560 c^3}-\frac {\left (3 b \left (b^2-4 a c\right )^2 \left (3 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 )}{4096 c^5} \\ & = \frac {3 b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{2048 c^5}-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{256 c^4}+\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\left (21 b^2-16 a c-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{560 c^3}-\frac {\left (3 b \left (b^2-4 a c\right )^2 \left (3 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 )}{2048 c^5} \\ & = \frac {3 b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{2048 c^5}-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{256 c^4}+\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\left (21 b^2-16 a c-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{560 c^3}-\frac {3 b \left (b^2-4 a c\right )^2 \left (3 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 )}{4096 c^{11/2}} \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.99 \[ \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {\sqrt {a+b x^2+c x^4} \left (315 b^6-210 b^5 c x^2+16 b^3 c^2 x^2 \left (91 a-9 c x^4\right )+168 b^4 c \left (-15 a+c x^4\right )+1024 c^3 \left (a+c x^4\right )^2 \left (-2 a+5 c x^4\right )+16 b^2 c^2 \left (343 a^2-62 a c x^4+8 c^2 x^8\right )+32 b c^3 x^2 \left (-73 a^2+22 a c x^4+200 c^2 x^8\right )\right )}{71680 c^5}+\frac {3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )}{4096 c^{11/2}} \]
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Time = 0.27 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.08
method | result | size |
pseudoelliptic | \(-\frac {-\frac {105 b \left (a c -\frac {3 b^{2}}{4}\right ) \left (a c -\frac {b^{2}}{4}\right )^{2} \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right )}{64}+\left (\left (\frac {9}{128} b^{3} x^{6}+\frac {31}{64} b^{2} x^{4} a +\frac {73}{64} a^{2} b \,x^{2}+a^{3}\right ) c^{\frac {7}{2}}-\frac {343 b^{2} \left (\frac {3}{98} b^{2} x^{4}+\frac {13}{49} a b \,x^{2}+a^{2}\right ) c^{\frac {5}{2}}}{128}+\left (-\frac {1}{16} b^{2} x^{8}-\frac {11}{32} a b \,x^{6}-\frac {1}{2} a^{2} x^{4}\right ) c^{\frac {9}{2}}+\left (-\frac {25}{8} b \,x^{10}-4 a \,x^{8}\right ) c^{\frac {11}{2}}+\frac {315 b^{4} \left (\frac {b \,x^{2}}{12}+a \right ) c^{\frac {3}{2}}}{256}-\frac {5 c^{\frac {13}{2}} x^{12}}{2}-\frac {315 \sqrt {c}\, b^{6}}{2048}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}+\frac {105 \ln \left (2\right ) b \left (a c -\frac {3 b^{2}}{4}\right ) \left (a c -\frac {b^{2}}{4}\right )^{2}}{64}}{35 c^{\frac {11}{2}}}\) | \(241\) |
risch | \(-\frac {\left (-5120 c^{6} x^{12}-6400 b \,c^{5} x^{10}-8192 a \,c^{5} x^{8}-128 b^{2} c^{4} x^{8}-704 a b \,c^{4} x^{6}+144 b^{3} c^{3} x^{6}-1024 a^{2} c^{4} x^{4}+992 a \,b^{2} c^{3} x^{4}-168 b^{4} c^{2} x^{4}+2336 a^{2} b \,c^{3} x^{2}-1456 a \,b^{3} c^{2} x^{2}+210 b^{5} c \,x^{2}+2048 c^{3} a^{3}-5488 a^{2} b^{2} c^{2}+2520 a \,b^{4} c -315 b^{6}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{71680 c^{5}}+\frac {3 b \left (64 c^{3} a^{3}-80 a^{2} b^{2} c^{2}+28 a \,b^{4} c -3 b^{6}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4096 c^{\frac {11}{2}}}\) | \(245\) |
default | \(-\frac {9 b^{4} a \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 c^{4}}+\frac {b^{2} x^{8} \sqrt {c \,x^{4}+b \,x^{2}+a}}{560 c}-\frac {9 b^{3} x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{4480 c^{2}}+\frac {3 b^{4} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1280 c^{3}}-\frac {3 b^{5} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1024 c^{4}}-\frac {9 b^{7} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4096 c^{\frac {11}{2}}}-\frac {15 a^{2} b^{3} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{256 c^{\frac {7}{2}}}+\frac {3 a^{3} b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{64 c^{\frac {5}{2}}}+\frac {a^{2} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{70 c}+\frac {49 a^{2} b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{640 c^{3}}+\frac {21 b^{5} a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{1024 c^{\frac {9}{2}}}+\frac {13 b^{3} a \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{640 c^{3}}-\frac {31 b^{2} a \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{2240 c^{2}}+\frac {11 b a \,x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1120 c}-\frac {73 a^{2} b \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{2240 c^{2}}+\frac {4 a \,x^{8} \sqrt {c \,x^{4}+b \,x^{2}+a}}{35}+\frac {5 b \,x^{10} \sqrt {c \,x^{4}+b \,x^{2}+a}}{56}+\frac {9 b^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{2048 c^{5}}+\frac {c \,x^{12} \sqrt {c \,x^{4}+b \,x^{2}+a}}{14}-\frac {a^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 c^{2}}\) | \(534\) |
elliptic | \(-\frac {9 b^{4} a \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 c^{4}}+\frac {b^{2} x^{8} \sqrt {c \,x^{4}+b \,x^{2}+a}}{560 c}-\frac {9 b^{3} x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{4480 c^{2}}+\frac {3 b^{4} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1280 c^{3}}-\frac {3 b^{5} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1024 c^{4}}-\frac {9 b^{7} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4096 c^{\frac {11}{2}}}-\frac {15 a^{2} b^{3} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{256 c^{\frac {7}{2}}}+\frac {3 a^{3} b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{64 c^{\frac {5}{2}}}+\frac {a^{2} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{70 c}+\frac {49 a^{2} b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{640 c^{3}}+\frac {21 b^{5} a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{1024 c^{\frac {9}{2}}}+\frac {13 b^{3} a \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{640 c^{3}}-\frac {31 b^{2} a \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{2240 c^{2}}+\frac {11 b a \,x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1120 c}-\frac {73 a^{2} b \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{2240 c^{2}}+\frac {4 a \,x^{8} \sqrt {c \,x^{4}+b \,x^{2}+a}}{35}+\frac {5 b \,x^{10} \sqrt {c \,x^{4}+b \,x^{2}+a}}{56}+\frac {9 b^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{2048 c^{5}}+\frac {c \,x^{12} \sqrt {c \,x^{4}+b \,x^{2}+a}}{14}-\frac {a^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 c^{2}}\) | \(534\) |
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Time = 0.27 (sec) , antiderivative size = 535, normalized size of antiderivative = 2.40 \[ \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\left [-\frac {105 \, {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (5120 \, c^{7} x^{12} + 6400 \, b c^{6} x^{10} + 128 \, {\left (b^{2} c^{5} + 64 \, a c^{6}\right )} x^{8} + 315 \, b^{6} c - 2520 \, a b^{4} c^{2} + 5488 \, a^{2} b^{2} c^{3} - 2048 \, a^{3} c^{4} - 16 \, {\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{6} + 8 \, {\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{4} - 2 \, {\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{286720 \, c^{6}}, \frac {105 \, {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, {\left (5120 \, c^{7} x^{12} + 6400 \, b c^{6} x^{10} + 128 \, {\left (b^{2} c^{5} + 64 \, a c^{6}\right )} x^{8} + 315 \, b^{6} c - 2520 \, a b^{4} c^{2} + 5488 \, a^{2} b^{2} c^{3} - 2048 \, a^{3} c^{4} - 16 \, {\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{6} + 8 \, {\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{4} - 2 \, {\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{143360 \, c^{6}}\right ] \]
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\[ \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int x^{7} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \]
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Exception generated. \[ \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 663 vs. \(2 (197) = 394\).
Time = 0.34 (sec) , antiderivative size = 663, normalized size of antiderivative = 2.97 \[ \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\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 )} a + \frac {1}{30720} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {9 \, b^{2} c^{3} - 20 \, a c^{4}}{c^{5}}\right )} x^{2} + \frac {21 \, b^{3} c^{2} - 68 \, a b c^{3}}{c^{5}}\right )} x^{2} - \frac {105 \, b^{4} c - 448 \, a b^{2} c^{2} + 240 \, a^{2} c^{3}}{c^{5}}\right )} x^{2} + \frac {315 \, b^{5} - 1680 \, a b^{3} c + 1808 \, a^{2} b c^{2}}{c^{5}}\right )} + \frac {15 \, {\left (21 \, b^{6} - 140 \, a b^{4} c + 240 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\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 {11}{2}}}\right )} b + \frac {1}{430080} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {11 \, b^{2} c^{4} - 24 \, a c^{5}}{c^{6}}\right )} x^{2} + \frac {99 \, b^{3} c^{3} - 316 \, a b c^{4}}{c^{6}}\right )} x^{2} - \frac {231 \, b^{4} c^{2} - 972 \, a b^{2} c^{3} + 512 \, a^{2} c^{4}}{c^{6}}\right )} x^{2} + \frac {1155 \, b^{5} c - 6048 \, a b^{3} c^{2} + 6352 \, a^{2} b c^{3}}{c^{6}}\right )} x^{2} - \frac {3465 \, b^{6} - 21840 \, a b^{4} c + 34608 \, a^{2} b^{2} c^{2} - 8192 \, a^{3} c^{3}}{c^{6}}\right )} - \frac {105 \, {\left (33 \, b^{7} - 252 \, a b^{5} c + 560 \, a^{2} b^{3} c^{2} - 320 \, a^{3} b c^{3}\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 {13}{2}}}\right )} c \]
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Timed out. \[ \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int x^7\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2} \,d x \]
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