Integrand size = 27, antiderivative size = 226 \[ \int \frac {A+B x+C x^2}{(d+e x) \left (a+c x^2\right )^2} \, dx=-\frac {a (B c d-A c e+a C e)-c (A c d-a C d+a B e) x}{2 a c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\left (a (C d-B e) \left (c d^2-a e^2\right )+A c d \left (c d^2+3 a e^2\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c} \left (c d^2+a e^2\right )^2}+\frac {e \left (C d^2-B d e+A e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac {e \left (C d^2-B d e+A e^2\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2} \]
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Time = 0.26 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1661, 815, 649, 211, 266} \[ \int \frac {A+B x+C x^2}{(d+e x) \left (a+c x^2\right )^2} \, dx=\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )+a \left (c d^2-a e^2\right ) (C d-B e)\right )}{2 a^{3/2} \sqrt {c} \left (a e^2+c d^2\right )^2}-\frac {a (a C e-A c e+B c d)-c x (a B e-a C d+A c d)}{2 a c \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac {e \log \left (a+c x^2\right ) \left (A e^2-B d e+C d^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac {e \log (d+e x) \left (A e^2-B d e+C d^2\right )}{\left (a e^2+c d^2\right )^2} \]
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Rule 211
Rule 266
Rule 649
Rule 815
Rule 1661
Rubi steps \begin{align*} \text {integral}& = -\frac {a (B c d-A c e+a C e)-c (A c d-a C d+a B e) x}{2 a c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \frac {-\frac {c \left (a d (C d-B e)+A \left (c d^2+2 a e^2\right )\right )}{c d^2+a e^2}-\frac {c e (A c d-a C d+a B e) x}{c d^2+a e^2}}{(d+e x) \left (a+c x^2\right )} \, dx}{2 a c} \\ & = -\frac {a (B c d-A c e+a C e)-c (A c d-a C d+a B e) x}{2 a c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \left (-\frac {2 a c e^2 \left (C d^2-B d e+A e^2\right )}{\left (c d^2+a e^2\right )^2 (d+e x)}+\frac {c \left (-a (C d-B e) \left (c d^2-a e^2\right )-A c d \left (c d^2+3 a e^2\right )+2 a c e \left (C d^2-B d e+A e^2\right ) x\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx}{2 a c} \\ & = -\frac {a (B c d-A c e+a C e)-c (A c d-a C d+a B e) x}{2 a c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e \left (C d^2-B d e+A e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac {\int \frac {-a (C d-B e) \left (c d^2-a e^2\right )-A c d \left (c d^2+3 a e^2\right )+2 a c e \left (C d^2-B d e+A e^2\right ) x}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )^2} \\ & = -\frac {a (B c d-A c e+a C e)-c (A c d-a C d+a B e) x}{2 a c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e \left (C d^2-B d e+A e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac {\left (c e \left (C d^2-B d e+A e^2\right )\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^2}+\frac {\left (a (C d-B e) \left (c d^2-a e^2\right )+A c d \left (c d^2+3 a e^2\right )\right ) \int \frac {1}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )^2} \\ & = -\frac {a (B c d-A c e+a C e)-c (A c d-a C d+a B e) x}{2 a c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\left (a (C d-B e) \left (c d^2-a e^2\right )+A c d \left (c d^2+3 a e^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c} \left (c d^2+a e^2\right )^2}+\frac {e \left (C d^2-B d e+A e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac {e \left (C d^2-B d e+A e^2\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x+C x^2}{(d+e x) \left (a+c x^2\right )^2} \, dx=\frac {\frac {\left (c d^2+a e^2\right ) \left (-a^2 C e+A c^2 d x+a c (-B d+A e-C d x+B e x)\right )}{a c \left (a+c x^2\right )}+\frac {\left (a (C d-B e) \left (c d^2-a e^2\right )+A c d \left (c d^2+3 a e^2\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {c}}+2 e \left (C d^2+e (-B d+A e)\right ) \log (d+e x)-e \left (C d^2+e (-B d+A e)\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2} \]
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Time = 0.61 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.30
method | result | size |
default | \(\frac {\frac {\frac {\left (A a c d \,e^{2}+A \,d^{3} c^{2}+a^{2} B \,e^{3}+B a c \,d^{2} e -C \,a^{2} d \,e^{2}-C a c \,d^{3}\right ) x}{2 a}+\frac {A a c \,e^{3}+A \,c^{2} d^{2} e -B a c d \,e^{2}-B \,c^{2} d^{3}-C \,a^{2} e^{3}-C a c \,d^{2} e}{2 c}}{c \,x^{2}+a}+\frac {\frac {\left (-2 A a c \,e^{3}+2 B a c d \,e^{2}-2 C a c \,d^{2} e \right ) \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {\left (3 A a c d \,e^{2}+A \,d^{3} c^{2}+a^{2} B \,e^{3}-B a c \,d^{2} e -C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 a}}{\left (e^{2} a +c \,d^{2}\right )^{2}}+\frac {e \left (A \,e^{2}-B d e +C \,d^{2}\right ) \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{2}}\) | \(293\) |
risch | \(\text {Expression too large to display}\) | \(15144\) |
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Leaf count of result is larger than twice the leaf count of optimal. 499 vs. \(2 (212) = 424\).
Time = 21.75 (sec) , antiderivative size = 1024, normalized size of antiderivative = 4.53 \[ \int \frac {A+B x+C x^2}{(d+e x) \left (a+c x^2\right )^2} \, dx=\left [-\frac {2 \, B a^{2} c^{2} d^{3} + 2 \, B a^{3} c d e^{2} + 2 \, {\left (C a^{3} c - A a^{2} c^{2}\right )} d^{2} e + 2 \, {\left (C a^{4} - A a^{3} c\right )} e^{3} - {\left (B a^{2} c d^{2} e - B a^{3} e^{3} - {\left (C a^{2} c + A a c^{2}\right )} d^{3} + {\left (C a^{3} - 3 \, A a^{2} c\right )} d e^{2} + {\left (B a c^{2} d^{2} e - B a^{2} c e^{3} - {\left (C a c^{2} + A c^{3}\right )} d^{3} + {\left (C a^{2} c - 3 \, A a c^{2}\right )} d e^{2}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (B a^{2} c^{2} d^{2} e + B a^{3} c e^{3} - {\left (C a^{2} c^{2} - A a c^{3}\right )} d^{3} - {\left (C a^{3} c - A a^{2} c^{2}\right )} d e^{2}\right )} x + 2 \, {\left (C a^{3} c d^{2} e - B a^{3} c d e^{2} + A a^{3} c e^{3} + {\left (C a^{2} c^{2} d^{2} e - B a^{2} c^{2} d e^{2} + A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right ) - 4 \, {\left (C a^{3} c d^{2} e - B a^{3} c d e^{2} + A a^{3} c e^{3} + {\left (C a^{2} c^{2} d^{2} e - B a^{2} c^{2} d e^{2} + A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (e x + d\right )}{4 \, {\left (a^{3} c^{3} d^{4} + 2 \, a^{4} c^{2} d^{2} e^{2} + a^{5} c e^{4} + {\left (a^{2} c^{4} d^{4} + 2 \, a^{3} c^{3} d^{2} e^{2} + a^{4} c^{2} e^{4}\right )} x^{2}\right )}}, -\frac {B a^{2} c^{2} d^{3} + B a^{3} c d e^{2} + {\left (C a^{3} c - A a^{2} c^{2}\right )} d^{2} e + {\left (C a^{4} - A a^{3} c\right )} e^{3} + {\left (B a^{2} c d^{2} e - B a^{3} e^{3} - {\left (C a^{2} c + A a c^{2}\right )} d^{3} + {\left (C a^{3} - 3 \, A a^{2} c\right )} d e^{2} + {\left (B a c^{2} d^{2} e - B a^{2} c e^{3} - {\left (C a c^{2} + A c^{3}\right )} d^{3} + {\left (C a^{2} c - 3 \, A a c^{2}\right )} d e^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (B a^{2} c^{2} d^{2} e + B a^{3} c e^{3} - {\left (C a^{2} c^{2} - A a c^{3}\right )} d^{3} - {\left (C a^{3} c - A a^{2} c^{2}\right )} d e^{2}\right )} x + {\left (C a^{3} c d^{2} e - B a^{3} c d e^{2} + A a^{3} c e^{3} + {\left (C a^{2} c^{2} d^{2} e - B a^{2} c^{2} d e^{2} + A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right ) - 2 \, {\left (C a^{3} c d^{2} e - B a^{3} c d e^{2} + A a^{3} c e^{3} + {\left (C a^{2} c^{2} d^{2} e - B a^{2} c^{2} d e^{2} + A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (e x + d\right )}{2 \, {\left (a^{3} c^{3} d^{4} + 2 \, a^{4} c^{2} d^{2} e^{2} + a^{5} c e^{4} + {\left (a^{2} c^{4} d^{4} + 2 \, a^{3} c^{3} d^{2} e^{2} + a^{4} c^{2} e^{4}\right )} x^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {A+B x+C x^2}{(d+e x) \left (a+c x^2\right )^2} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.30 \[ \int \frac {A+B x+C x^2}{(d+e x) \left (a+c x^2\right )^2} \, dx=-\frac {{\left (C d^{2} e - B d e^{2} + A e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac {{\left (C d^{2} e - B d e^{2} + A e^{3}\right )} \log \left (e x + d\right )}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac {{\left (B a c d^{2} e - B a^{2} e^{3} - {\left (C a c + A c^{2}\right )} d^{3} + {\left (C a^{2} - 3 \, A a c\right )} d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt {a c}} - \frac {B a c d + {\left (C a^{2} - A a c\right )} e - {\left (B a c e - {\left (C a c - A c^{2}\right )} d\right )} x}{2 \, {\left (a^{2} c^{2} d^{2} + a^{3} c e^{2} + {\left (a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} x^{2}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.61 \[ \int \frac {A+B x+C x^2}{(d+e x) \left (a+c x^2\right )^2} \, dx=-\frac {{\left (C d^{2} e - B d e^{2} + A e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac {{\left (C d^{2} e^{2} - B d e^{3} + A e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}} + \frac {{\left (C a c d^{3} + A c^{2} d^{3} - B a c d^{2} e - C a^{2} d e^{2} + 3 \, A a c d e^{2} + B a^{2} e^{3}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt {a c}} - \frac {B a c^{2} d^{3} + C a^{2} c d^{2} e - A a c^{2} d^{2} e + B a^{2} c d e^{2} + C a^{3} e^{3} - A a^{2} c e^{3} + {\left (C a c^{2} d^{3} - A c^{3} d^{3} - B a c^{2} d^{2} e + C a^{2} c d e^{2} - A a c^{2} d e^{2} - B a^{2} c e^{3}\right )} x}{2 \, {\left (c d^{2} + a e^{2}\right )}^{2} {\left (c x^{2} + a\right )} a c} \]
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Time = 17.72 (sec) , antiderivative size = 1493, normalized size of antiderivative = 6.61 \[ \int \frac {A+B x+C x^2}{(d+e x) \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]
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