Integrand size = 24, antiderivative size = 100 \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {d^2 (b+2 c x)}{2 \left (a+b x+c x^2\right )^2}-\frac {c d^2 (b+2 c x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {4 c^2 d^2 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {700, 628, 632, 212} \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^3} \, dx=\frac {4 c^2 d^2 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {c d^2 (b+2 c x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {d^2 (b+2 c x)}{2 \left (a+b x+c x^2\right )^2} \]
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Rule 212
Rule 628
Rule 632
Rule 700
Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 (b+2 c x)}{2 \left (a+b x+c x^2\right )^2}+\left (c d^2\right ) \int \frac {1}{\left (a+b x+c x^2\right )^2} \, dx \\ & = -\frac {d^2 (b+2 c x)}{2 \left (a+b x+c x^2\right )^2}-\frac {c d^2 (b+2 c x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (2 c^2 d^2\right ) \int \frac {1}{a+b x+c x^2} \, dx}{b^2-4 a c} \\ & = -\frac {d^2 (b+2 c x)}{2 \left (a+b x+c x^2\right )^2}-\frac {c d^2 (b+2 c x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (4 c^2 d^2\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{b^2-4 a c} \\ & = -\frac {d^2 (b+2 c x)}{2 \left (a+b x+c x^2\right )^2}-\frac {c d^2 (b+2 c x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {4 c^2 d^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.98 \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^3} \, dx=d^2 \left (-\frac {(b+2 c x) \left (b^2+2 b c x+2 c \left (-a+c x^2\right )\right )}{2 \left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac {4 c^2 \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}\right ) \]
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Time = 2.34 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.48
method | result | size |
default | \(d^{2} \left (\frac {\frac {2 c^{3} x^{3}}{4 a c -b^{2}}+\frac {3 b \,c^{2} x^{2}}{4 a c -b^{2}}-\frac {2 c \left (a c -b^{2}\right ) x}{4 a c -b^{2}}-\frac {\left (2 a c -b^{2}\right ) b}{2 \left (4 a c -b^{2}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {4 c^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\right )\) | \(148\) |
risch | \(\frac {\frac {2 c^{3} d^{2} x^{3}}{4 a c -b^{2}}+\frac {3 b \,c^{2} d^{2} x^{2}}{4 a c -b^{2}}-\frac {2 c \,d^{2} \left (a c -b^{2}\right ) x}{4 a c -b^{2}}-\frac {b \,d^{2} \left (2 a c -b^{2}\right )}{2 \left (4 a c -b^{2}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {2 c^{2} d^{2} \ln \left (\left (-8 c^{2} a +2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right )}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {2 c^{2} d^{2} \ln \left (\left (8 c^{2} a -2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right )}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}\) | \(227\) |
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Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (94) = 188\).
Time = 0.30 (sec) , antiderivative size = 713, normalized size of antiderivative = 7.13 \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^3} \, dx=\left [-\frac {4 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{2} x^{3} + 6 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{2} x^{2} + 4 \, {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d^{2} x + {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d^{2} + 4 \, {\left (c^{4} d^{2} x^{4} + 2 \, b c^{3} d^{2} x^{3} + 2 \, a b c^{2} d^{2} x + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{2}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right )}{2 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}}, -\frac {4 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{2} x^{3} + 6 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{2} x^{2} + 4 \, {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d^{2} x + {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d^{2} - 8 \, {\left (c^{4} d^{2} x^{4} + 2 \, b c^{3} d^{2} x^{3} + 2 \, a b c^{2} d^{2} x + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{2}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{2 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (94) = 188\).
Time = 0.91 (sec) , antiderivative size = 430, normalized size of antiderivative = 4.30 \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^3} \, dx=- 2 c^{2} d^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (x + \frac {- 32 a^{2} c^{4} d^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 16 a b^{2} c^{3} d^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - 2 b^{4} c^{2} d^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b c^{2} d^{2}}{4 c^{3} d^{2}} \right )} + 2 c^{2} d^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (x + \frac {32 a^{2} c^{4} d^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - 16 a b^{2} c^{3} d^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b^{4} c^{2} d^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b c^{2} d^{2}}{4 c^{3} d^{2}} \right )} + \frac {- 2 a b c d^{2} + b^{3} d^{2} + 6 b c^{2} d^{2} x^{2} + 4 c^{3} d^{2} x^{3} + x \left (- 4 a c^{2} d^{2} + 4 b^{2} c d^{2}\right )}{8 a^{3} c - 2 a^{2} b^{2} + x^{4} \cdot \left (8 a c^{3} - 2 b^{2} c^{2}\right ) + x^{3} \cdot \left (16 a b c^{2} - 4 b^{3} c\right ) + x^{2} \cdot \left (16 a^{2} c^{2} + 4 a b^{2} c - 2 b^{4}\right ) + x \left (16 a^{2} b c - 4 a b^{3}\right )} \]
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Exception generated. \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.34 \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {4 \, c^{2} d^{2} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {4 \, c^{3} d^{2} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b^{2} c d^{2} x - 4 \, a c^{2} d^{2} x + b^{3} d^{2} - 2 \, a b c d^{2}}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}} \]
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Time = 9.66 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.33 \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^3} \, dx=\frac {4\,c^2\,d^2\,\mathrm {atan}\left (\frac {\left (\frac {4\,c^3\,d^2\,x}{{\left (4\,a\,c-b^2\right )}^{3/2}}-\frac {2\,c^2\,d^2\,\left (b^3-4\,a\,b\,c\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}\right )\,\left (4\,a\,c-b^2\right )}{2\,c^2\,d^2}\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}}-\frac {\frac {b\,d^2\,\left (2\,a\,c-b^2\right )}{2\,\left (4\,a\,c-b^2\right )}-\frac {2\,c^3\,d^2\,x^3}{4\,a\,c-b^2}-\frac {3\,b\,c^2\,d^2\,x^2}{4\,a\,c-b^2}+\frac {2\,c\,d^2\,x\,\left (a\,c-b^2\right )}{4\,a\,c-b^2}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3} \]
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