Integrand size = 27, antiderivative size = 214 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )} \, dx=-\frac {C d^2-B d e+A e^2}{e \left (c d^2+a e^2\right ) (d+e x)}+\frac {\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c} \left (c d^2+a e^2\right )^2}-\frac {\left (B c d^2-2 A c d e+2 a C d e-a B e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}+\frac {\left (B c d^2-2 A c d e+2 a C d e-a B e^2\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2} \]
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Time = 0.20 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1643, 649, 211, 266} \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{\sqrt {a} \sqrt {c} \left (a e^2+c d^2\right )^2}+\frac {\log \left (a+c x^2\right ) \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )}{2 \left (a e^2+c d^2\right )^2}-\frac {A e^2-B d e+C d^2}{e (d+e x) \left (a e^2+c d^2\right )}-\frac {\log (d+e x) \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )}{\left (a e^2+c d^2\right )^2} \]
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Rule 211
Rule 266
Rule 649
Rule 1643
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {C d^2-B d e+A e^2}{\left (c d^2+a e^2\right ) (d+e x)^2}+\frac {e \left (-B c d^2+2 A c d e-2 a C d e+a B e^2\right )}{\left (c d^2+a e^2\right )^2 (d+e x)}+\frac {A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )+c \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right ) x}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx \\ & = -\frac {C d^2-B d e+A e^2}{e \left (c d^2+a e^2\right ) (d+e x)}-\frac {\left (B c d^2-2 A c d e+2 a C d e-a B e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}+\frac {\int \frac {A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )+c \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right ) x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^2} \\ & = -\frac {C d^2-B d e+A e^2}{e \left (c d^2+a e^2\right ) (d+e x)}-\frac {\left (B c d^2-2 A c d e+2 a C d e-a B e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}+\frac {\left (c \left (B c d^2-2 A c d e+2 a C d e-a B 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 \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) \int \frac {1}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^2} \\ & = -\frac {C d^2-B d e+A e^2}{e \left (c d^2+a e^2\right ) (d+e x)}+\frac {\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c} \left (c d^2+a e^2\right )^2}-\frac {\left (B c d^2-2 A c d e+2 a C d e-a B e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}+\frac {\left (B c d^2-2 A c d e+2 a C d e-a B e^2\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\frac {-\frac {2 \left (c d^2+a e^2\right ) \left (C d^2+e (-B d+A e)\right )}{e (d+e x)}+\frac {2 \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2+c d (-C d+2 B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\left (-2 B c d^2+4 A c d e-4 a C d e+2 a B e^2\right ) \log (d+e x)+\left (B c d^2-2 A c d e+2 a C d e-a B e^2\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2} \]
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Time = 0.62 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {\frac {\left (-2 A \,c^{2} d e -B \,e^{2} a c +B \,c^{2} d^{2}+2 a c d e C \right ) \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {\left (-A a c \,e^{2}+A \,c^{2} d^{2}+2 B a c d e +a^{2} C \,e^{2}-C a c \,d^{2}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{\left (e^{2} a +c \,d^{2}\right )^{2}}-\frac {A \,e^{2}-B d e +C \,d^{2}}{\left (e^{2} a +c \,d^{2}\right ) e \left (e x +d \right )}+\frac {\left (2 A c d e +B a \,e^{2}-B c \,d^{2}-2 a d e C \right ) \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{2}}\) | \(204\) |
risch | \(\text {Expression too large to display}\) | \(15347\) |
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Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (204) = 408\).
Time = 25.70 (sec) , antiderivative size = 904, normalized size of antiderivative = 4.22 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\left [-\frac {2 \, C a c^{2} d^{4} - 2 \, B a c^{2} d^{3} e - 2 \, B a^{2} c d e^{3} + 2 \, A a^{2} c e^{4} + 2 \, {\left (C a^{2} c + A a c^{2}\right )} d^{2} e^{2} - {\left (2 \, B a c d^{2} e^{2} - {\left (C a c - A c^{2}\right )} d^{3} e + {\left (C a^{2} - A a c\right )} d e^{3} + {\left (2 \, B a c d e^{3} - {\left (C a c - A c^{2}\right )} d^{2} e^{2} + {\left (C a^{2} - A a c\right )} e^{4}\right )} x\right )} \sqrt {-a c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - {\left (B a c^{2} d^{3} e - B a^{2} c d e^{3} + 2 \, {\left (C a^{2} c - A a c^{2}\right )} d^{2} e^{2} + {\left (B a c^{2} d^{2} e^{2} - B a^{2} c e^{4} + 2 \, {\left (C a^{2} c - A a c^{2}\right )} d e^{3}\right )} x\right )} \log \left (c x^{2} + a\right ) + 2 \, {\left (B a c^{2} d^{3} e - B a^{2} c d e^{3} + 2 \, {\left (C a^{2} c - A a c^{2}\right )} d^{2} e^{2} + {\left (B a c^{2} d^{2} e^{2} - B a^{2} c e^{4} + 2 \, {\left (C a^{2} c - A a c^{2}\right )} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a c^{3} d^{5} e + 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5} + {\left (a c^{3} d^{4} e^{2} + 2 \, a^{2} c^{2} d^{2} e^{4} + a^{3} c e^{6}\right )} x\right )}}, -\frac {2 \, C a c^{2} d^{4} - 2 \, B a c^{2} d^{3} e - 2 \, B a^{2} c d e^{3} + 2 \, A a^{2} c e^{4} + 2 \, {\left (C a^{2} c + A a c^{2}\right )} d^{2} e^{2} - 2 \, {\left (2 \, B a c d^{2} e^{2} - {\left (C a c - A c^{2}\right )} d^{3} e + {\left (C a^{2} - A a c\right )} d e^{3} + {\left (2 \, B a c d e^{3} - {\left (C a c - A c^{2}\right )} d^{2} e^{2} + {\left (C a^{2} - A a c\right )} e^{4}\right )} x\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (B a c^{2} d^{3} e - B a^{2} c d e^{3} + 2 \, {\left (C a^{2} c - A a c^{2}\right )} d^{2} e^{2} + {\left (B a c^{2} d^{2} e^{2} - B a^{2} c e^{4} + 2 \, {\left (C a^{2} c - A a c^{2}\right )} d e^{3}\right )} x\right )} \log \left (c x^{2} + a\right ) + 2 \, {\left (B a c^{2} d^{3} e - B a^{2} c d e^{3} + 2 \, {\left (C a^{2} c - A a c^{2}\right )} d^{2} e^{2} + {\left (B a c^{2} d^{2} e^{2} - B a^{2} c e^{4} + 2 \, {\left (C a^{2} c - A a c^{2}\right )} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a c^{3} d^{5} e + 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5} + {\left (a c^{3} d^{4} e^{2} + 2 \, a^{2} c^{2} d^{2} e^{4} + a^{3} c e^{6}\right )} x\right )}}\right ] \]
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Timed out. \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.19 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\frac {{\left (B c d^{2} - B a e^{2} + 2 \, {\left (C a - A c\right )} d e\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} - \frac {{\left (B c d^{2} - B a e^{2} + 2 \, {\left (C a - A c\right )} d e\right )} \log \left (e x + d\right )}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac {{\left (2 \, B a c d e - {\left (C a c - A c^{2}\right )} d^{2} + {\left (C a^{2} - A a c\right )} e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {a c}} - \frac {C d^{2} - B d e + A e^{2}}{c d^{3} e + a d e^{3} + {\left (c d^{2} e^{2} + a e^{4}\right )} x} \]
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Time = 0.27 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.29 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\frac {{\left (B c d^{2} + 2 \, C a d e - 2 \, A c d e - B a e^{2}\right )} \log \left (c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {a e^{2}}{{\left (e x + d\right )}^{2}}\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} - \frac {\frac {C d^{2} e}{e x + d} - \frac {B d e^{2}}{e x + d} + \frac {A e^{3}}{e x + d}}{c d^{2} e^{2} + a e^{4}} - \frac {{\left (C a c d^{2} e^{2} - A c^{2} d^{2} e^{2} - 2 \, B a c d e^{3} - C a^{2} e^{4} + A a c e^{4}\right )} \arctan \left (\frac {c d - \frac {c d^{2}}{e x + d} - \frac {a e^{2}}{e x + d}}{\sqrt {a c} e}\right )}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {a c} e^{2}} \]
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Time = 16.31 (sec) , antiderivative size = 1199, normalized size of antiderivative = 5.60 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\frac {\ln \left (C\,c\,d^4\,{\left (-a\,c\right )}^{3/2}-A\,a\,e^4\,{\left (-a\,c\right )}^{3/2}+3\,B\,a\,c^3\,d^4+3\,B\,a^3\,c\,e^4+A\,c^4\,d^4\,x+A\,c^3\,d^4\,\sqrt {-a\,c}-C\,a^3\,e^4\,\sqrt {-a\,c}-C\,a\,c^3\,d^4\,x-C\,a^3\,c\,e^4\,x+14\,A\,c\,d^2\,e^2\,{\left (-a\,c\right )}^{3/2}-14\,C\,a\,d^2\,e^2\,{\left (-a\,c\right )}^{3/2}-3\,B\,c^3\,d^4\,x\,\sqrt {-a\,c}+8\,A\,a^2\,c^2\,d\,e^3+8\,C\,a^2\,c^2\,d^3\,e+A\,a^2\,c^2\,e^4\,x-10\,B\,a^2\,c^2\,d^2\,e^2+8\,B\,a\,d\,e^3\,{\left (-a\,c\right )}^{3/2}-8\,B\,c\,d^3\,e\,{\left (-a\,c\right )}^{3/2}+3\,B\,a\,e^4\,x\,{\left (-a\,c\right )}^{3/2}-8\,A\,a\,c^3\,d^3\,e-8\,C\,a^3\,c\,d\,e^3+14\,C\,a^2\,c^2\,d^2\,e^2\,x+8\,A\,c\,d\,e^3\,x\,{\left (-a\,c\right )}^{3/2}-8\,C\,a\,d\,e^3\,x\,{\left (-a\,c\right )}^{3/2}+8\,C\,c\,d^3\,e\,x\,{\left (-a\,c\right )}^{3/2}+8\,B\,a\,c^3\,d^3\,e\,x+8\,A\,c^3\,d^3\,e\,x\,\sqrt {-a\,c}-10\,B\,c\,d^2\,e^2\,x\,{\left (-a\,c\right )}^{3/2}-14\,A\,a\,c^3\,d^2\,e^2\,x-8\,B\,a^2\,c^2\,d\,e^3\,x\right )\,\left (c^2\,\left (a\,\left (\frac {B\,d^2}{2}-A\,d\,e\right )+\frac {A\,d^2\,\sqrt {-a\,c}}{2}\right )-c\,\left (a^2\,\left (\frac {B\,e^2}{2}-C\,d\,e\right )+a\,\left (\frac {A\,e^2\,\sqrt {-a\,c}}{2}+\frac {C\,d^2\,\sqrt {-a\,c}}{2}-B\,d\,e\,\sqrt {-a\,c}\right )\right )+\frac {C\,a^2\,e^2\,\sqrt {-a\,c}}{2}\right )}{a^3\,c\,e^4+2\,a^2\,c^2\,d^2\,e^2+a\,c^3\,d^4}-\frac {\ln \left (d+e\,x\right )\,\left (c\,\left (B\,d^2-2\,A\,d\,e\right )-a\,\left (B\,e^2-2\,C\,d\,e\right )\right )}{a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4}-\frac {\ln \left (A\,a\,e^4\,{\left (-a\,c\right )}^{3/2}-C\,c\,d^4\,{\left (-a\,c\right )}^{3/2}+3\,B\,a\,c^3\,d^4+3\,B\,a^3\,c\,e^4+A\,c^4\,d^4\,x-A\,c^3\,d^4\,\sqrt {-a\,c}+C\,a^3\,e^4\,\sqrt {-a\,c}-C\,a\,c^3\,d^4\,x-C\,a^3\,c\,e^4\,x-14\,A\,c\,d^2\,e^2\,{\left (-a\,c\right )}^{3/2}+14\,C\,a\,d^2\,e^2\,{\left (-a\,c\right )}^{3/2}+3\,B\,c^3\,d^4\,x\,\sqrt {-a\,c}+8\,A\,a^2\,c^2\,d\,e^3+8\,C\,a^2\,c^2\,d^3\,e+A\,a^2\,c^2\,e^4\,x-10\,B\,a^2\,c^2\,d^2\,e^2-8\,B\,a\,d\,e^3\,{\left (-a\,c\right )}^{3/2}+8\,B\,c\,d^3\,e\,{\left (-a\,c\right )}^{3/2}-3\,B\,a\,e^4\,x\,{\left (-a\,c\right )}^{3/2}-8\,A\,a\,c^3\,d^3\,e-8\,C\,a^3\,c\,d\,e^3+14\,C\,a^2\,c^2\,d^2\,e^2\,x-8\,A\,c\,d\,e^3\,x\,{\left (-a\,c\right )}^{3/2}+8\,C\,a\,d\,e^3\,x\,{\left (-a\,c\right )}^{3/2}-8\,C\,c\,d^3\,e\,x\,{\left (-a\,c\right )}^{3/2}+8\,B\,a\,c^3\,d^3\,e\,x-8\,A\,c^3\,d^3\,e\,x\,\sqrt {-a\,c}+10\,B\,c\,d^2\,e^2\,x\,{\left (-a\,c\right )}^{3/2}-14\,A\,a\,c^3\,d^2\,e^2\,x-8\,B\,a^2\,c^2\,d\,e^3\,x\right )\,\left (c\,\left (a^2\,\left (\frac {B\,e^2}{2}-C\,d\,e\right )-a\,\left (\frac {A\,e^2\,\sqrt {-a\,c}}{2}+\frac {C\,d^2\,\sqrt {-a\,c}}{2}-B\,d\,e\,\sqrt {-a\,c}\right )\right )-c^2\,\left (a\,\left (\frac {B\,d^2}{2}-A\,d\,e\right )-\frac {A\,d^2\,\sqrt {-a\,c}}{2}\right )+\frac {C\,a^2\,e^2\,\sqrt {-a\,c}}{2}\right )}{a^3\,c\,e^4+2\,a^2\,c^2\,d^2\,e^2+a\,c^3\,d^4}-\frac {C\,d^2-B\,d\,e+A\,e^2}{e\,\left (c\,d^2+a\,e^2\right )\,\left (d+e\,x\right )} \]
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