Integrand size = 24, antiderivative size = 198 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^2} \, dx=\frac {3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{128 c^3 x}-\frac {b (b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{16 c^2 x^3}+\frac {\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5}-\frac {3 b \left (b^2-4 a c\right )^2 x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{7/2} \sqrt {a x^2+b x^3+c x^4}} \]
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Time = 0.11 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1931, 1932, 1928, 635, 212} \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^2} \, dx=-\frac {3 b x \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{7/2} \sqrt {a x^2+b x^3+c x^4}}+\frac {3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{128 c^3 x}-\frac {b (b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{16 c^2 x^3}+\frac {\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5} \]
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Rule 212
Rule 635
Rule 1928
Rule 1931
Rule 1932
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5}-\frac {b \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^3} \, dx}{2 c} \\ & = -\frac {b (b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{16 c^2 x^3}+\frac {\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5}+\frac {\left (3 b \left (b^2-4 a c\right )\right ) \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x} \, dx}{32 c^2} \\ & = \frac {3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{128 c^3 x}-\frac {b (b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{16 c^2 x^3}+\frac {\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5}-\frac {\left (3 b \left (b^2-4 a c\right )^2\right ) \int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{256 c^3} \\ & = \frac {3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{128 c^3 x}-\frac {b (b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{16 c^2 x^3}+\frac {\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5}-\frac {\left (3 b \left (b^2-4 a c\right )^2 x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^3 \sqrt {a x^2+b x^3+c x^4}} \\ & = \frac {3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{128 c^3 x}-\frac {b (b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{16 c^2 x^3}+\frac {\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5}-\frac {\left (3 b \left (b^2-4 a c\right )^2 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 )}{128 c^3 \sqrt {a x^2+b x^3+c x^4}} \\ & = \frac {3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{128 c^3 x}-\frac {b (b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{16 c^2 x^3}+\frac {\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5}-\frac {3 b \left (b^2-4 a c\right )^2 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 )}{256 c^{7/2} \sqrt {a x^2+b x^3+c x^4}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^2} \, dx=\frac {x \sqrt {a+x (b+c x)} \left (2 \sqrt {c} \sqrt {a+x (b+c x)} \left (15 b^4-10 b^3 c x+128 c^2 \left (a+c x^2\right )^2+4 b^2 c \left (-25 a+2 c x^2\right )+8 b c^2 x \left (7 a+22 c x^2\right )\right )+15 b \left (b^2-4 a c\right )^2 \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{1280 c^{7/2} \sqrt {x^2 (a+x (b+c x))}} \]
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Time = 0.19 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.64
method | result | size |
pseudoelliptic | \(\frac {-\frac {15 b \left (a c -\frac {b^{2}}{4}\right )^{2} \ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right )}{16}+\left (\left (\frac {1}{16} b^{2} x^{2}+\frac {7}{16} a b x +a^{2}\right ) c^{\frac {5}{2}}-\frac {25 b^{2} \left (\frac {b x}{10}+a \right ) c^{\frac {3}{2}}}{32}+\left (\frac {11}{8} b \,x^{3}+2 a \,x^{2}\right ) c^{\frac {7}{2}}+c^{\frac {9}{2}} x^{4}+\frac {15 \sqrt {c}\, b^{4}}{128}\right ) \sqrt {c \,x^{2}+b x +a}}{5 c^{\frac {7}{2}}}\) | \(126\) |
risch | \(\frac {\left (128 c^{4} x^{4}+176 b \,c^{3} x^{3}+256 a \,c^{3} x^{2}+8 b^{2} c^{2} x^{2}+56 a b \,c^{2} x -10 b^{3} c x +128 a^{2} c^{2}-100 a \,b^{2} c +15 b^{4}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{640 c^{3} x}-\frac {3 b \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{256 c^{\frac {7}{2}} x \sqrt {c \,x^{2}+b x +a}}\) | \(180\) |
default | \(\frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (256 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} c^{\frac {7}{2}}-160 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b x -80 c^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2}-240 c^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}\, a b x +60 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, b^{3} x -120 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, a \,b^{2}+30 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, b^{4}-240 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a^{2} b \,c^{3}+120 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a \,b^{3} c^{2}-15 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) b^{5} c \right )}{1280 x^{3} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {9}{2}}}\) | \(289\) |
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Time = 0.30 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^2} \, dx=\left [\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \, {\left (128 \, c^{5} x^{4} + 176 \, b c^{4} x^{3} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} + 8 \, {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{2} - 2 \, {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{2560 \, c^{4} x}, \frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) + 2 \, {\left (128 \, c^{5} x^{4} + 176 \, b c^{4} x^{3} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} + 8 \, {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{2} - 2 \, {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{1280 \, c^{4} x}\right ] \]
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\[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^2} \, dx=\int \frac {\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
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\[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^2} \, dx=\int { \frac {{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \]
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Time = 0.34 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^2} \, dx=\frac {1}{640} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, c x \mathrm {sgn}\left (x\right ) + 11 \, b \mathrm {sgn}\left (x\right )\right )} x + \frac {b^{2} c^{3} \mathrm {sgn}\left (x\right ) + 32 \, a c^{4} \mathrm {sgn}\left (x\right )}{c^{4}}\right )} x - \frac {5 \, b^{3} c^{2} \mathrm {sgn}\left (x\right ) - 28 \, a b c^{3} \mathrm {sgn}\left (x\right )}{c^{4}}\right )} x + \frac {15 \, b^{4} c \mathrm {sgn}\left (x\right ) - 100 \, a b^{2} c^{2} \mathrm {sgn}\left (x\right ) + 128 \, a^{2} c^{3} \mathrm {sgn}\left (x\right )}{c^{4}}\right )} + \frac {3 \, {\left (b^{5} \mathrm {sgn}\left (x\right ) - 8 \, a b^{3} c \mathrm {sgn}\left (x\right ) + 16 \, a^{2} b c^{2} \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {7}{2}}} - \frac {{\left (15 \, b^{5} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 120 \, a b^{3} c \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 240 \, a^{2} b c^{2} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 30 \, \sqrt {a} b^{4} \sqrt {c} - 200 \, a^{\frac {3}{2}} b^{2} c^{\frac {3}{2}} + 256 \, a^{\frac {5}{2}} c^{\frac {5}{2}}\right )} \mathrm {sgn}\left (x\right )}{1280 \, c^{\frac {7}{2}}} \]
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Timed out. \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^2} \, dx=\int \frac {{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}}{x^2} \,d x \]
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