Integrand size = 26, antiderivative size = 86 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2}{\left (b^2-4 a c\right ) d \sqrt {a+b x+c x^2}}-\frac {4 \sqrt {c} \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} d} \]
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Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {701, 702, 211} \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {4 \sqrt {c} \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{3/2}}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
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Rule 211
Rule 701
Rule 702
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{\left (b^2-4 a c\right ) d \sqrt {a+b x+c x^2}}-\frac {(4 c) \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c} \\ & = -\frac {2}{\left (b^2-4 a c\right ) d \sqrt {a+b x+c x^2}}-\frac {\left (16 c^2\right ) \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 )}{b^2-4 a c} \\ & = -\frac {2}{\left (b^2-4 a c\right ) d \sqrt {a+b x+c x^2}}-\frac {4 \sqrt {c} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} d} \\ \end{align*}
Time = 0.63 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {-\frac {2}{\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}}-\frac {8 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {-b^2+4 a c} x}{\sqrt {a} (b+2 c x)-b \sqrt {a+x (b+c x)}}\right )}{\left (-b^2+4 a c\right )^{3/2}}}{d} \]
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Time = 2.86 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.26
method | result | size |
pseudoelliptic | \(\frac {-4 c \,\operatorname {arctanh}\left (\frac {2 c \sqrt {c \,x^{2}+b x +a}}{\sqrt {4 c^{2} a -b^{2} c}}\right ) \sqrt {c \,x^{2}+b x +a}+2 \sqrt {4 c^{2} a -b^{2} c}}{\sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right ) \sqrt {4 c^{2} a -b^{2} c}\, d}\) | \(108\) |
default | \(\frac {\frac {4 c}{\left (4 a c -b^{2}\right ) \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}-\frac {8 c \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{\left (4 a c -b^{2}\right ) \sqrt {\frac {4 a c -b^{2}}{c}}}}{2 d c}\) | \(162\) |
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none
Time = 0.35 (sec) , antiderivative size = 311, normalized size of antiderivative = 3.62 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\left [-\frac {2 \, {\left ({\left (c x^{2} + b x + a\right )} \sqrt {-\frac {c}{b^{2} - 4 \, a c}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c + 4 \, \sqrt {c x^{2} + b x + a} {\left (b^{2} - 4 \, a c\right )} \sqrt {-\frac {c}{b^{2} - 4 \, a c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + \sqrt {c x^{2} + b x + a}\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} d x^{2} + {\left (b^{3} - 4 \, a b c\right )} d x + {\left (a b^{2} - 4 \, a^{2} c\right )} d}, -\frac {2 \, {\left (2 \, {\left (c x^{2} + b x + a\right )} \sqrt {\frac {c}{b^{2} - 4 \, a c}} \arctan \left (-\frac {\sqrt {c x^{2} + b x + a} {\left (b^{2} - 4 \, a c\right )} \sqrt {\frac {c}{b^{2} - 4 \, a c}}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + \sqrt {c x^{2} + b x + a}\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} d x^{2} + {\left (b^{3} - 4 \, a b c\right )} d x + {\left (a b^{2} - 4 \, a^{2} c\right )} d}\right ] \]
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\[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\int \frac {1}{a b \sqrt {a + b x + c x^{2}} + 2 a c x \sqrt {a + b x + c x^{2}} + b^{2} x \sqrt {a + b x + c x^{2}} + 3 b c x^{2} \sqrt {a + b x + c x^{2}} + 2 c^{2} x^{3} \sqrt {a + b x + c x^{2}}}\, dx}{d} \]
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Exception generated. \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (74) = 148\).
Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.84 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {8 \, c \arctan \left (\frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c + b \sqrt {c}}{\sqrt {b^{2} c - 4 \, a c^{2}}}\right )}{\sqrt {b^{2} c - 4 \, a c^{2}} {\left (b^{2} d - 4 \, a c d\right )}} - \frac {2 \, {\left (b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d\right )}}{{\left (b^{6} d^{2} - 12 \, a b^{4} c d^{2} + 48 \, a^{2} b^{2} c^{2} d^{2} - 64 \, a^{3} c^{3} d^{2}\right )} \sqrt {c x^{2} + b x + a}} \]
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Timed out. \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (b\,d+2\,c\,d\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]
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