Integrand size = 24, antiderivative size = 244 \[ \int x^{3/2} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x+b x^3+c x^5}}{1280 c^3 \sqrt {x}}-\frac {x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac {3 b \left (b^2-4 a c\right )^2 \sqrt {x} \sqrt {a+b x^2+c x^4} \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{7/2} \sqrt {a x+b x^3+c x^5}} \]
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Time = 0.22 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1933, 1959, 1963, 12, 1928, 1121, 635, 212} \[ \int x^{3/2} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\frac {\left (128 a^2 c^2-100 a b^2 c+15 b^4\right ) \sqrt {a x+b x^3+c x^5}}{1280 c^3 \sqrt {x}}-\frac {3 b \sqrt {x} \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4} \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{7/2} \sqrt {a x+b x^3+c x^5}}-\frac {x^{3/2} \left (4 c x^2 \left (5 b^2-16 a c\right )+b \left (5 b^2-4 a c\right )\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c} \]
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Rule 12
Rule 212
Rule 635
Rule 1121
Rule 1928
Rule 1933
Rule 1959
Rule 1963
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}+\frac {3 \int \sqrt {x} \left (-2 a b-\left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5} \, dx}{80 c} \\ & = -\frac {x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}+\frac {\int \frac {x^{3/2} \left (2 a b \left (5 b^2-28 a c\right )+\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) x^2\right )}{\sqrt {a x+b x^3+c x^5}} \, dx}{640 c^2} \\ & = \frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x+b x^3+c x^5}}{1280 c^3 \sqrt {x}}-\frac {x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac {\int \frac {15 b \left (b^2-4 a c\right )^2 x^{3/2}}{\sqrt {a x+b x^3+c x^5}} \, dx}{1280 c^3} \\ & = \frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x+b x^3+c x^5}}{1280 c^3 \sqrt {x}}-\frac {x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac {\left (3 b \left (b^2-4 a c\right )^2\right ) \int \frac {x^{3/2}}{\sqrt {a x+b x^3+c x^5}} \, dx}{256 c^3} \\ & = \frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x+b x^3+c x^5}}{1280 c^3 \sqrt {x}}-\frac {x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac {\left (3 b \left (b^2-4 a c\right )^2 \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {x}{\sqrt {a+b x^2+c x^4}} \, dx}{256 c^3 \sqrt {a x+b x^3+c x^5}} \\ & = \frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x+b x^3+c x^5}}{1280 c^3 \sqrt {x}}-\frac {x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac {\left (3 b \left (b^2-4 a c\right )^2 \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{512 c^3 \sqrt {a x+b x^3+c x^5}} \\ & = \frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x+b x^3+c x^5}}{1280 c^3 \sqrt {x}}-\frac {x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac {\left (3 b \left (b^2-4 a c\right )^2 \sqrt {x} \sqrt {a+b x^2+c x^4}\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 \sqrt {a x+b x^3+c x^5}} \\ & = \frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x+b x^3+c x^5}}{1280 c^3 \sqrt {x}}-\frac {x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac {3 b \left (b^2-4 a c\right )^2 \sqrt {x} \sqrt {a+b x^2+c x^4} \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} \sqrt {a x+b x^3+c x^5}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.74 \[ \int x^{3/2} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\frac {\sqrt {x} \sqrt {a+b x^2+c x^4} \left (2 \sqrt {c} \sqrt {a+b x^2+c x^4} \left (15 b^4-10 b^3 c x^2+128 c^2 \left (a+c x^4\right )^2+4 b^2 c \left (-25 a+2 c x^4\right )+8 b c^2 x^2 \left (7 a+22 c x^4\right )\right )+15 b \left (b^2-4 a c\right )^2 \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )\right )}{2560 c^{7/2} \sqrt {x \left (a+b x^2+c x^4\right )}} \]
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Time = 0.09 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {\left (128 c^{4} x^{8}+176 c^{3} x^{6} b +256 a \,c^{3} x^{4}+8 b^{2} c^{2} x^{4}+56 a b \,c^{2} x^{2}-10 x^{2} c \,b^{3}+128 a^{2} c^{2}-100 a \,b^{2} c +15 b^{4}\right ) \left (c \,x^{4}+b \,x^{2}+a \right ) \sqrt {x}}{1280 c^{3} \sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}}-\frac {3 b \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {x}}{512 c^{\frac {7}{2}} \sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}}\) | \(202\) |
default | \(-\frac {\sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}\, \left (-256 c^{\frac {9}{2}} x^{8} \sqrt {c \,x^{4}+b \,x^{2}+a}-352 b \,c^{\frac {7}{2}} x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}-512 a \,c^{\frac {7}{2}} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}-16 b^{2} c^{\frac {5}{2}} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}-112 a b \,c^{\frac {5}{2}} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}+20 b^{3} c^{\frac {3}{2}} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}+240 \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{2 \sqrt {c}}\right ) a^{2} b \,c^{2}-120 \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{2 \sqrt {c}}\right ) a \,b^{3} c +15 \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{2 \sqrt {c}}\right ) b^{5}-256 a^{2} c^{\frac {5}{2}} \sqrt {c \,x^{4}+b \,x^{2}+a}+200 a \,b^{2} c^{\frac {3}{2}} \sqrt {c \,x^{4}+b \,x^{2}+a}-30 b^{4} \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2560 c^{\frac {7}{2}} \sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\) | \(369\) |
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Time = 0.27 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.62 \[ \int x^{3/2} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\left [\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{5} + 8 \, b c x^{3} - 4 \, \sqrt {c x^{5} + b x^{3} + a x} {\left (2 \, c x^{2} + b\right )} \sqrt {c} \sqrt {x} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \, {\left (128 \, c^{5} x^{8} + 176 \, b c^{4} x^{6} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} + 8 \, {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{4} - 2 \, {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x^{2}\right )} \sqrt {c x^{5} + b x^{3} + a x} \sqrt {x}}{5120 \, c^{4} x}, \frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{5} + b x^{3} + a x} {\left (2 \, c x^{2} + b\right )} \sqrt {-c} \sqrt {x}}{2 \, {\left (c^{2} x^{5} + b c x^{3} + a c x\right )}}\right ) + 2 \, {\left (128 \, c^{5} x^{8} + 176 \, b c^{4} x^{6} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} + 8 \, {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{4} - 2 \, {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x^{2}\right )} \sqrt {c x^{5} + b x^{3} + a x} \sqrt {x}}{2560 \, c^{4} x}\right ] \]
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Timed out. \[ \int x^{3/2} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\text {Timed out} \]
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\[ \int x^{3/2} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\int { {\left (c x^{5} + b x^{3} + a x\right )}^{\frac {3}{2}} x^{\frac {3}{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (210) = 420\).
Time = 0.45 (sec) , antiderivative size = 640, normalized size of antiderivative = 2.62 \[ \int x^{3/2} \left (a x+b x^3+c x^5\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}}} + \frac {3 \, b^{3} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 12 \, a b c \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 6 \, \sqrt {a} b^{2} \sqrt {c} - 16 \, a^{\frac {3}{2}} c^{\frac {3}{2}}}{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}}} - \frac {15 \, b^{4} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 72 \, a b^{2} c \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 48 \, a^{2} c^{2} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 30 \, \sqrt {a} b^{3} \sqrt {c} - 104 \, a^{\frac {3}{2}} b c^{\frac {3}{2}}}{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}}} + \frac {105 \, b^{5} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 600 \, a b^{3} c \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 720 \, a^{2} b c^{2} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 210 \, \sqrt {a} b^{4} \sqrt {c} - 920 \, a^{\frac {3}{2}} b^{2} c^{\frac {3}{2}} + 512 \, a^{\frac {5}{2}} c^{\frac {5}{2}}}{c^{\frac {9}{2}}}\right )} c \]
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Timed out. \[ \int x^{3/2} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\int x^{3/2}\,{\left (c\,x^5+b\,x^3+a\,x\right )}^{3/2} \,d x \]
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