Integrand size = 26, antiderivative size = 115 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{b d+2 c d x} \, dx=-\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{8 c^2 d}+\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d}+\frac {\left (b^2-4 a c\right )^{3/2} \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{16 c^{5/2} d} \]
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Time = 0.05 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {699, 702, 211} \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{b d+2 c d x} \, dx=\frac {\left (b^2-4 a c\right )^{3/2} \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{16 c^{5/2} d}-\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{8 c^2 d}+\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d} \]
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Rule 211
Rule 699
Rule 702
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d}-\frac {\left (b^2-4 a c\right ) \int \frac {\sqrt {a+b x+c x^2}}{b d+2 c d x} \, dx}{4 c} \\ & = -\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{8 c^2 d}+\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d}+\frac {\left (b^2-4 a c\right )^2 \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{16 c^2} \\ & = -\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{8 c^2 d}+\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d}+\frac {\left (b^2-4 a c\right )^2 \text {Subst}\left (\int \frac {1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )}{4 c} \\ & = -\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{8 c^2 d}+\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d}+\frac {\left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{16 c^{5/2} d} \\ \end{align*}
Time = 1.19 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{b d+2 c d x} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (-3 b^2+4 b c x+4 c \left (4 a+c x^2\right )\right )-3 \left (-b^2+4 a c\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {-b^2+4 a c} x}{\sqrt {a} (b+2 c x)-b \sqrt {a+x (b+c x)}}\right )}{24 c^{5/2} d} \]
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Time = 2.33 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03
method | result | size |
pseudoelliptic | \(-\frac {\left (-\frac {b^{2}}{4}+a c \right )^{2} \operatorname {arctanh}\left (\frac {2 c \sqrt {c \,x^{2}+b x +a}}{\sqrt {4 c^{2} a -b^{2} c}}\right )-\frac {2 \left (\frac {c^{2} x^{2}}{4}+\left (\frac {b x}{4}+a \right ) c -\frac {3 b^{2}}{16}\right ) \sqrt {4 c^{2} a -b^{2} c}\, \sqrt {c \,x^{2}+b x +a}}{3}}{\sqrt {4 c^{2} a -b^{2} c}\, c^{2} d}\) | \(119\) |
risch | \(\frac {\left (4 c^{2} x^{2}+4 b c x +16 a c -3 b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{24 c^{2} d}-\frac {\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{16 c^{3} \sqrt {\frac {4 a c -b^{2}}{c}}\, d}\) | \(164\) |
default | \(\frac {\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{3}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {\sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 c \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 c}}{2 d c}\) | \(197\) |
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Time = 0.31 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.14 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{b d+2 c d x} \, dx=\left [-\frac {3 \, {\left (b^{2} - 4 \, a c\right )} \sqrt {-\frac {b^{2} - 4 \, a c}{c}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt {c x^{2} + b x + a} c \sqrt {-\frac {b^{2} - 4 \, a c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) - 4 \, {\left (4 \, c^{2} x^{2} + 4 \, b c x - 3 \, b^{2} + 16 \, a c\right )} \sqrt {c x^{2} + b x + a}}{96 \, c^{2} d}, -\frac {3 \, {\left (b^{2} - 4 \, a c\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c}} \arctan \left (\frac {\sqrt {\frac {b^{2} - 4 \, a c}{c}}}{2 \, \sqrt {c x^{2} + b x + a}}\right ) - 2 \, {\left (4 \, c^{2} x^{2} + 4 \, b c x - 3 \, b^{2} + 16 \, a c\right )} \sqrt {c x^{2} + b x + a}}{48 \, c^{2} d}\right ] \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{b d+2 c d x} \, dx=\frac {\int \frac {a \sqrt {a + b x + c x^{2}}}{b + 2 c x}\, dx + \int \frac {b x \sqrt {a + b x + c x^{2}}}{b + 2 c x}\, dx + \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{b + 2 c x}\, dx}{d} \]
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Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{b d+2 c d x} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{b d+2 c d x} \, dx=\frac {1}{24} \, \sqrt {c x^{2} + b x + a} {\left (4 \, x {\left (\frac {x}{d} + \frac {b}{c d}\right )} - \frac {3 \, b^{2} c^{3} d^{3} - 16 \, a c^{4} d^{3}}{c^{5} d^{4}}\right )} + \frac {{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \arctan \left (-\frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c + b \sqrt {c}}{\sqrt {b^{2} c - 4 \, a c^{2}}}\right )}{8 \, \sqrt {b^{2} c - 4 \, a c^{2}} c^{2} d} \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{b d+2 c d x} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{b\,d+2\,c\,d\,x} \,d x \]
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