Integrand size = 26, antiderivative size = 66 \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 d^2 (b+2 c x)}{\sqrt {a+b x+c x^2}}+4 \sqrt {c} d^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 66, 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 = {700, 635, 212} \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=4 \sqrt {c} d^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )-\frac {2 d^2 (b+2 c x)}{\sqrt {a+b x+c x^2}} \]
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Rule 212
Rule 635
Rule 700
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d^2 (b+2 c x)}{\sqrt {a+b x+c x^2}}+\left (4 c d^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx \\ & = -\frac {2 d^2 (b+2 c x)}{\sqrt {a+b x+c x^2}}+\left (8 c d^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 ) \\ & = -\frac {2 d^2 (b+2 c x)}{\sqrt {a+b x+c x^2}}+4 \sqrt {c} d^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.14 \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 d^2 \left (b+2 c x-4 \sqrt {c} \sqrt {a+x (b+c x)} \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )\right )}{\sqrt {a+x (b+c x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(211\) vs. \(2(56)=112\).
Time = 2.54 (sec) , antiderivative size = 212, normalized size of antiderivative = 3.21
method | result | size |
default | \(d^{2} \left (\frac {2 b^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+4 c^{2} \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+4 b c \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )\right )\) | \(212\) |
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none
Time = 0.33 (sec) , antiderivative size = 225, normalized size of antiderivative = 3.41 \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\left [\frac {2 \, {\left ({\left (c d^{2} x^{2} + b d^{2} x + a d^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - {\left (2 \, c d^{2} x + b d^{2}\right )} \sqrt {c x^{2} + b x + a}\right )}}{c x^{2} + b x + a}, -\frac {2 \, {\left (2 \, {\left (c d^{2} x^{2} + b d^{2} x + a d^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + {\left (2 \, c d^{2} x + b d^{2}\right )} \sqrt {c x^{2} + b x + a}\right )}}{c x^{2} + b x + a}\right ] \]
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\[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=d^{2} \left (\int \frac {b^{2}}{a \sqrt {a + b x + c x^{2}} + b x \sqrt {a + b x + c x^{2}} + c x^{2} \sqrt {a + b x + c x^{2}}}\, dx + \int \frac {4 c^{2} x^{2}}{a \sqrt {a + b x + c x^{2}} + b x \sqrt {a + b x + c x^{2}} + c x^{2} \sqrt {a + b x + c x^{2}}}\, dx + \int \frac {4 b c x}{a \sqrt {a + b x + c x^{2}} + b x \sqrt {a + b x + c x^{2}} + c x^{2} \sqrt {a + b x + c x^{2}}}\, dx\right ) \]
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Exception generated. \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.68 \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-4 \, \sqrt {c} d^{2} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right ) - \frac {2 \, {\left (\frac {2 \, {\left (b^{2} c d^{2} - 4 \, a c^{2} d^{2}\right )} x}{b^{2} - 4 \, a c} + \frac {b^{3} d^{2} - 4 \, a b c d^{2}}{b^{2} - 4 \, a c}\right )}}{\sqrt {c x^{2} + b x + a}} \]
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Time = 9.72 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.36 \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=4\,\sqrt {c}\,d^2\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )+\frac {b^2\,d^2\,\left (\frac {b}{2}+c\,x\right )}{\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^2+b\,x+a}}+\frac {4\,c\,d^2\,\left (\frac {a\,b}{2}-x\,\left (a\,c-\frac {b^2}{2}\right )\right )}{\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^2+b\,x+a}}-\frac {4\,b\,c\,d^2\,\left (4\,a+2\,b\,x\right )}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^2+b\,x+a}} \]
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