Integrand size = 24, antiderivative size = 262 \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 c^3 \left (b^2-4 a c\right ) x}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {3 \left (5 b^2-4 a c\right ) x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2} \sqrt {a x^2+b x^3+c x^4}} \]
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Time = 0.32 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1937, 1963, 12, 1928, 635, 212} \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {3 x \left (5 b^2-4 a c\right ) \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2} \sqrt {a x^2+b x^3+c x^4}}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 c^3 x \left (b^2-4 a c\right )}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}+\frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )} \]
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Rule 12
Rule 212
Rule 635
Rule 1928
Rule 1937
Rule 1963
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 \int \frac {x^3 (6 a+3 b x)}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{b^2-4 a c} \\ & = \frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {2 \int \frac {x^2 \left (6 a b+\frac {3}{2} \left (5 b^2-12 a c\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{3 c \left (b^2-4 a c\right )} \\ & = \frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}-\frac {\int \frac {x \left (\frac {3}{2} a \left (5 b^2-12 a c\right )+\frac {3}{4} b \left (15 b^2-52 a c\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{3 c^2 \left (b^2-4 a c\right )} \\ & = \frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 c^3 \left (b^2-4 a c\right ) x}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\int \frac {9 \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) x}{8 \sqrt {a x^2+b x^3+c x^4}} \, dx}{3 c^3 \left (b^2-4 a c\right )} \\ & = \frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 c^3 \left (b^2-4 a c\right ) x}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\left (3 \left (5 b^2-4 a c\right )\right ) \int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{8 c^3} \\ & = \frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 c^3 \left (b^2-4 a c\right ) x}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\left (3 \left (5 b^2-4 a c\right ) x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 c^3 \sqrt {a x^2+b x^3+c x^4}} \\ & = \frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 c^3 \left (b^2-4 a c\right ) x}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\left (3 \left (5 b^2-4 a c\right ) x \sqrt {a+b x+c x^2}\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{4 c^3 \sqrt {a x^2+b x^3+c x^4}} \\ & = \frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 c^3 \left (b^2-4 a c\right ) x}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {3 \left (5 b^2-4 a c\right ) x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2} \sqrt {a x^2+b x^3+c x^4}} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.71 \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {x \left (2 \sqrt {c} \left (4 a^2 c (-13 b+6 c x)+b^2 x \left (15 b^2+5 b c x-2 c^2 x^2\right )+a \left (15 b^3-62 b^2 c x-20 b c^2 x^2+8 c^3 x^3\right )\right )+3 \left (5 b^4-24 a b^2 c+16 a^2 c^2\right ) \sqrt {a+x (b+c x)} \log \left (c^3 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )\right )}{8 c^{7/2} \left (-b^2+4 a c\right ) \sqrt {x^2 (a+x (b+c x))}} \]
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Time = 0.22 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.63
method | result | size |
pseudoelliptic | \(-\frac {48 \left (\left (-\frac {5}{24} b^{3} x^{2}+\frac {31}{12} b^{2} a x +\frac {13}{6} a^{2} b \right ) c^{\frac {3}{2}}+\left (\frac {1}{12} b^{2} x^{3}+\frac {5}{6} a b \,x^{2}-a^{2} x \right ) c^{\frac {5}{2}}-\frac {a \,c^{\frac {7}{2}} x^{3}}{3}-\frac {5 b^{3} \sqrt {c}\, \left (b x +a \right )}{8}+\frac {\ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}\, \left (16 a^{2} c^{2}-24 a \,b^{2} c +5 b^{4}\right )}{16}\right )}{\sqrt {c \,x^{2}+b x +a}\, c^{\frac {7}{2}} \left (32 a c -8 b^{2}\right )}\) | \(164\) |
default | \(\frac {x^{3} \left (c \,x^{2}+b x +a \right ) \left (16 c^{\frac {9}{2}} a \,x^{3}-4 c^{\frac {7}{2}} b^{2} x^{3}-40 c^{\frac {7}{2}} a b \,x^{2}+48 c^{\frac {7}{2}} a^{2} x +10 c^{\frac {5}{2}} b^{3} x^{2}-124 c^{\frac {5}{2}} a \,b^{2} x -104 c^{\frac {5}{2}} a^{2} b +30 c^{\frac {3}{2}} b^{4} x +30 c^{\frac {3}{2}} a \,b^{3}-48 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{3}+72 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{2}-15 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) \sqrt {c \,x^{2}+b x +a}\, b^{4} c \right )}{8 c^{\frac {9}{2}} \left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )}\) | \(283\) |
risch | \(-\frac {\left (-2 c x +7 b \right ) \left (c \,x^{2}+b x +a \right ) x}{4 c^{3} \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}-\frac {\left (\frac {8 c \,a^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {14 b^{2} a \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\left (-4 a b c -7 b^{3}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+\left (12 a \,c^{2}-15 b^{2} c \right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )\right ) x \sqrt {c \,x^{2}+b x +a}}{8 c^{3} \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}\) | \(337\) |
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Time = 0.33 (sec) , antiderivative size = 616, normalized size of antiderivative = 2.35 \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{3} + {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2} + {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x\right )} \sqrt {c} \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \, {\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2} - 2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + 5 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + {\left (15 \, b^{4} c - 62 \, a b^{2} c^{2} + 24 \, a^{2} c^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{16 \, {\left ({\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{3} + {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{2} + {\left (a b^{2} c^{4} - 4 \, a^{2} c^{5}\right )} x\right )}}, -\frac {3 \, {\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{3} + {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2} + {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) + 2 \, {\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2} - 2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + 5 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + {\left (15 \, b^{4} c - 62 \, a b^{2} c^{2} + 24 \, a^{2} c^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{8 \, {\left ({\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{3} + {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{2} + {\left (a b^{2} c^{4} - 4 \, a^{2} c^{5}\right )} x\right )}}\right ] \]
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\[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int \frac {x^{7}}{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int { \frac {x^{7}}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 0.34 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.21 \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {{\left (15 \, b^{4} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 72 \, a b^{2} c \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 48 \, a^{2} c^{2} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 30 \, \sqrt {a} b^{3} \sqrt {c} - 104 \, a^{\frac {3}{2}} b c^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{8 \, {\left (b^{2} c^{\frac {7}{2}} - 4 \, a c^{\frac {9}{2}}\right )}} + \frac {{\left ({\left (\frac {2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x}{b^{2} c^{3} \mathrm {sgn}\left (x\right ) - 4 \, a c^{4} \mathrm {sgn}\left (x\right )} - \frac {5 \, {\left (b^{3} c - 4 \, a b c^{2}\right )}}{b^{2} c^{3} \mathrm {sgn}\left (x\right ) - 4 \, a c^{4} \mathrm {sgn}\left (x\right )}\right )} x - \frac {15 \, b^{4} - 62 \, a b^{2} c + 24 \, a^{2} c^{2}}{b^{2} c^{3} \mathrm {sgn}\left (x\right ) - 4 \, a c^{4} \mathrm {sgn}\left (x\right )}\right )} x - \frac {15 \, a b^{3} - 52 \, a^{2} b c}{b^{2} c^{3} \mathrm {sgn}\left (x\right ) - 4 \, a c^{4} \mathrm {sgn}\left (x\right )}}{4 \, \sqrt {c x^{2} + b x + a}} - \frac {3 \, {\left (5 \, b^{2} - 4 \, a c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{8 \, c^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int \frac {x^7}{{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}} \,d x \]
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