Integrand size = 26, antiderivative size = 112 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {(d+e x)^2}{2 \left (a+b x+c x^2\right )^2}-\frac {e (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {2 e (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {782, 652, 632, 212} \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^3} \, dx=\frac {2 e (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {e (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(d+e x)^2}{2 \left (a+b x+c x^2\right )^2} \]
[In]
[Out]
Rule 212
Rule 632
Rule 652
Rule 782
Rubi steps \begin{align*} \text {integral}& = -\frac {(d+e x)^2}{2 \left (a+b x+c x^2\right )^2}+e \int \frac {d+e x}{\left (a+b x+c x^2\right )^2} \, dx \\ & = -\frac {(d+e x)^2}{2 \left (a+b x+c x^2\right )^2}-\frac {e (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 {(e (2 c d-b e)) \int \frac {1}{a+b x+c x^2} \, dx}{b^2-4 a c} \\ & = -\frac {(d+e x)^2}{2 \left (a+b x+c x^2\right )^2}-\frac {e (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 {(2 e (2 c d-b e)) \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)^2}{2 \left (a+b x+c x^2\right )^2}-\frac {e (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 {2 e (2 c d-b e) \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 = 0.14 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.28 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^3} \, dx=\frac {1}{2} \left (\frac {e \left (b^2 e+4 c (-2 a e+c d x)+2 b c (d-e x)\right )}{c \left (-b^2+4 a c\right ) (a+x (b+c x))}+\frac {e^2 (a+b x)-c d (d+2 e x)}{c (a+x (b+c x))^2}-\frac {4 e (-2 c d+b e) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}\right ) \]
[In]
[Out]
Time = 0.66 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.76
method | result | size |
default | \(\frac {-\frac {c e \left (b e -2 c d \right ) x^{3}}{4 a c -b^{2}}-\frac {e \left (8 a c e +b^{2} e -6 b c d \right ) x^{2}}{2 \left (4 a c -b^{2}\right )}-\frac {e \left (3 a b e +2 a c d -2 b^{2} d \right ) x}{4 a c -b^{2}}-\frac {4 e^{2} a^{2}-2 a b d e +4 c \,d^{2} a -b^{2} d^{2}}{2 \left (4 a c -b^{2}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {2 e \left (b e -2 c d \right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\) | \(197\) |
risch | \(\frac {-\frac {c e \left (b e -2 c d \right ) x^{3}}{4 a c -b^{2}}-\frac {e \left (8 a c e +b^{2} e -6 b c d \right ) x^{2}}{2 \left (4 a c -b^{2}\right )}-\frac {e \left (3 a b e +2 a c d -2 b^{2} d \right ) x}{4 a c -b^{2}}-\frac {4 e^{2} a^{2}-2 a b d e +4 c \,d^{2} a -b^{2} d^{2}}{2 \left (4 a c -b^{2}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {e^{2} \ln \left (\left (-8 a \,c^{2}+2 b^{2} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) b}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {2 e \ln \left (\left (-8 a \,c^{2}+2 b^{2} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) c d}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {e^{2} \ln \left (\left (8 a \,c^{2}-2 b^{2} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) b}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}+\frac {2 e \ln \left (\left (8 a \,c^{2}-2 b^{2} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) c d}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}\) | \(367\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 492 vs. \(2 (106) = 212\).
Time = 0.31 (sec) , antiderivative size = 1004, normalized size of antiderivative = 8.96 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^3} \, dx=\left [-\frac {2 \, {\left (2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d e - {\left (b^{3} c - 4 \, a b c^{2}\right )} e^{2}\right )} x^{3} + {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{2} + 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d e - 4 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e^{2} + {\left (6 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e - {\left (b^{4} + 4 \, a b^{2} c - 32 \, a^{2} c^{2}\right )} e^{2}\right )} x^{2} - 2 \, {\left (2 \, a^{2} c d e - a^{2} b e^{2} + {\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} x^{4} + 2 \, {\left (2 \, b c^{2} d e - b^{2} c e^{2}\right )} x^{3} + {\left (2 \, {\left (b^{2} c + 2 \, a c^{2}\right )} d e - {\left (b^{3} + 2 \, a b c\right )} e^{2}\right )} x^{2} + 2 \, {\left (2 \, a b c d e - a b^{2} 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 ) + 2 \, {\left (2 \, {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d e - 3 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} e^{2}\right )} x}{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 {2 \, {\left (2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d e - {\left (b^{3} c - 4 \, a b c^{2}\right )} e^{2}\right )} x^{3} + {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{2} + 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d e - 4 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e^{2} + {\left (6 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e - {\left (b^{4} + 4 \, a b^{2} c - 32 \, a^{2} c^{2}\right )} e^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{2} c d e - a^{2} b e^{2} + {\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} x^{4} + 2 \, {\left (2 \, b c^{2} d e - b^{2} c e^{2}\right )} x^{3} + {\left (2 \, {\left (b^{2} c + 2 \, a c^{2}\right )} d e - {\left (b^{3} + 2 \, a b c\right )} e^{2}\right )} x^{2} + 2 \, {\left (2 \, a b c d e - a b^{2} 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 ) + 2 \, {\left (2 \, {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d e - 3 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} e^{2}\right )} x}{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 ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (102) = 204\).
Time = 4.98 (sec) , antiderivative size = 530, normalized size of antiderivative = 4.73 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^3} \, dx=e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) \log {\left (x + \frac {- 16 a^{2} c^{2} e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + 8 a b^{2} c e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) - b^{4} e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{2} e^{2} - 2 b c d e}{2 b c e^{2} - 4 c^{2} d e} \right )} - e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) \log {\left (x + \frac {16 a^{2} c^{2} e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) - 8 a b^{2} c e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{4} e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{2} e^{2} - 2 b c d e}{2 b c e^{2} - 4 c^{2} d e} \right )} + \frac {- 4 a^{2} e^{2} + 2 a b d e - 4 a c d^{2} + b^{2} d^{2} + x^{3} \left (- 2 b c e^{2} + 4 c^{2} d e\right ) + x^{2} \left (- 8 a c e^{2} - b^{2} e^{2} + 6 b c d e\right ) + x \left (- 6 a b e^{2} - 4 a c d e + 4 b^{2} d e\right )}{8 a^{3} c - 2 a^{2} b^{2} + x^{4} \cdot \left (8 a c^{3} - 2 b^{2} c^{2}\right ) + x^{3} \cdot \left (16 a b c^{2} - 4 b^{3} c\right ) + x^{2} \cdot \left (16 a^{2} c^{2} + 4 a b^{2} c - 2 b^{4}\right ) + x \left (16 a^{2} b c - 4 a b^{3}\right )} \]
[In]
[Out]
Exception generated. \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.63 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {2 \, {\left (2 \, c d e - b e^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {4 \, c^{2} d e x^{3} - 2 \, b c e^{2} x^{3} + 6 \, b c d e x^{2} - b^{2} e^{2} x^{2} - 8 \, a c e^{2} x^{2} + 4 \, b^{2} d e x - 4 \, a c d e x - 6 \, a b e^{2} x + b^{2} d^{2} - 4 \, a c d^{2} + 2 \, a b d e - 4 \, a^{2} e^{2}}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}} \]
[In]
[Out]
Time = 10.84 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.54 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^3} \, dx=\frac {2\,e\,\mathrm {atan}\left (\frac {\left (4\,a\,c-b^2\right )\,\left (\frac {e\,\left (b^3-4\,a\,b\,c\right )\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {2\,c\,e\,x\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}}\right )}{b\,e^2-2\,c\,d\,e}\right )\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}}-\frac {\frac {4\,a^2\,e^2-2\,a\,b\,d\,e+4\,c\,a\,d^2-b^2\,d^2}{2\,\left (4\,a\,c-b^2\right )}-\frac {e\,x^3\,\left (2\,c^2\,d-b\,c\,e\right )}{4\,a\,c-b^2}+\frac {e\,x\,\left (-2\,d\,b^2+3\,a\,e\,b+2\,a\,c\,d\right )}{4\,a\,c-b^2}+\frac {e\,x^2\,\left (e\,b^2-6\,c\,d\,b+8\,a\,c\,e\right )}{2\,\left (4\,a\,c-b^2\right )}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3} \]
[In]
[Out]