Integrand size = 20, antiderivative size = 99 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {4 \left (c d^2-b d e+a e^2\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {736, 632, 212} \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx=\frac {4 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (a e^2-b d e+c d^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
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Rule 212
Rule 632
Rule 736
Rubi steps \begin{align*} \text {integral}& = -\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (2 \left (c d^2-b d e+a e^2\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{b^2-4 a c} \\ & = -\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (4 \left (c d^2-b d e+a e^2\right )\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+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {4 \left (c d^2-b d e+a e^2\right ) \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 = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.29 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx=\frac {a b e^2+2 c^2 d^2 x+b^2 e^2 x+b c d (d-2 e x)-2 a c e (2 d+e x)}{c \left (-b^2+4 a c\right ) (a+x (b+c x))}+\frac {4 \left (c d^2+e (-b d+a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}} \]
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Time = 20.70 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.51
method | result | size |
default | \(\frac {-\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}\right ) x}{c \left (4 a c -b^{2}\right )}+\frac {a b \,e^{2}-4 a c d e +b c \,d^{2}}{c \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {4 \left (e^{2} a -b d e +c \,d^{2}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\) | \(149\) |
risch | \(\frac {-\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}\right ) x}{c \left (4 a c -b^{2}\right )}+\frac {a b \,e^{2}-4 a c d e +b c \,d^{2}}{c \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {2 \ln \left (\left (-8 c^{2} a +2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) e^{2} a}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {2 \ln \left (\left (-8 c^{2} a +2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) b d e}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}+\frac {2 \ln \left (\left (-8 c^{2} a +2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) c \,d^{2}}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {2 \ln \left (\left (8 c^{2} a -2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) e^{2} a}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}+\frac {2 \ln \left (\left (8 c^{2} a -2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) b d e}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {2 \ln \left (\left (8 c^{2} a -2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) c \,d^{2}}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}\) | \(409\) |
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Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (95) = 190\).
Time = 0.43 (sec) , antiderivative size = 669, normalized size of antiderivative = 6.76 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx=\left [-\frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{2} - 4 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d e + {\left (a b^{3} - 4 \, a^{2} b c\right )} e^{2} + 2 \, {\left (a c^{2} d^{2} - a b c d e + a^{2} c e^{2} + {\left (c^{3} d^{2} - b c^{2} d e + a c^{2} e^{2}\right )} x^{2} + {\left (b c^{2} d^{2} - b^{2} c d e + a b c e^{2}\right )} x\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + {\left (2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e + {\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} e^{2}\right )} x}{a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}, -\frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{2} - 4 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d e + {\left (a b^{3} - 4 \, a^{2} b c\right )} e^{2} - 4 \, {\left (a c^{2} d^{2} - a b c d e + a^{2} c e^{2} + {\left (c^{3} d^{2} - b c^{2} d e + a c^{2} e^{2}\right )} x^{2} + {\left (b c^{2} d^{2} - b^{2} c d e + a b c e^{2}\right )} x\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left (2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e + {\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} e^{2}\right )} x}{a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (95) = 190\).
Time = 0.87 (sec) , antiderivative size = 517, normalized size of antiderivative = 5.22 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx=- 2 \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) \log {\left (x + \frac {- 32 a^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 16 a b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 2 a b e^{2} - 2 b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 2 b^{2} d e + 2 b c d^{2}}{4 a c e^{2} - 4 b c d e + 4 c^{2} d^{2}} \right )} + 2 \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) \log {\left (x + \frac {32 a^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 16 a b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 2 a b e^{2} + 2 b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 2 b^{2} d e + 2 b c d^{2}}{4 a c e^{2} - 4 b c d e + 4 c^{2} d^{2}} \right )} + \frac {a b e^{2} - 4 a c d e + b c d^{2} + x \left (- 2 a c e^{2} + b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{4 a^{2} c^{2} - a b^{2} c + x^{2} \cdot \left (4 a c^{3} - b^{2} c^{2}\right ) + x \left (4 a b c^{2} - b^{3} c\right )} \]
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Exception generated. \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.41 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {4 \, {\left (c d^{2} - b d e + a e^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, c^{2} d^{2} x - 2 \, b c d e x + b^{2} e^{2} x - 2 \, a c e^{2} x + b c d^{2} - 4 \, a c d e + a b e^{2}}{{\left (b^{2} c - 4 \, a c^{2}\right )} {\left (c x^{2} + b x + a\right )}} \]
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Time = 0.16 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.32 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {b\,c\,d^2-4\,a\,c\,d\,e+a\,b\,e^2}{c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (b^2\,e^2-2\,b\,c\,d\,e+2\,c^2\,d^2-2\,a\,c\,e^2\right )}{c\,\left (4\,a\,c-b^2\right )}}{c\,x^2+b\,x+a}-\frac {4\,\mathrm {atan}\left (\frac {\left (\frac {2\,\left (b^3-4\,a\,b\,c\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {4\,c\,x\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (4\,a\,c-b^2\right )}{2\,c\,d^2-2\,b\,d\,e+2\,a\,e^2}\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}} \]
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