Integrand size = 33, antiderivative size = 103 \[ \int \frac {(d f+e f x)^3}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\frac {f^3 \left (2 a+b (d+e x)^2\right )}{2 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {b f^3 \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} e} \]
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Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1156, 1128, 652, 632, 212} \[ \int \frac {(d f+e f x)^3}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\frac {f^3 \left (2 a+b (d+e x)^2\right )}{2 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {b f^3 \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{e \left (b^2-4 a c\right )^{3/2}} \]
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Rule 212
Rule 632
Rule 652
Rule 1128
Rule 1156
Rubi steps \begin{align*} \text {integral}& = \frac {f^3 \text {Subst}\left (\int \frac {x^3}{\left (a+b x^2+c x^4\right )^2} \, dx,x,d+e x\right )}{e} \\ & = \frac {f^3 \text {Subst}\left (\int \frac {x}{\left (a+b x+c x^2\right )^2} \, dx,x,(d+e x)^2\right )}{2 e} \\ & = \frac {f^3 \left (2 a+b (d+e x)^2\right )}{2 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\left (b f^3\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 \left (b^2-4 a c\right ) e} \\ & = \frac {f^3 \left (2 a+b (d+e x)^2\right )}{2 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\left (b f^3\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right ) e} \\ & = \frac {f^3 \left (2 a+b (d+e x)^2\right )}{2 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {b f^3 \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} e} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00 \[ \int \frac {(d f+e f x)^3}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\frac {f^3 \left (\frac {2 a+b (d+e x)^2}{\left (b^2-4 a c\right ) \left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )}-\frac {2 b \arctan \left (\frac {b+2 c (d+e x)^2}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}\right )}{2 e} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.68 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.72
method | result | size |
default | \(f^{3} \left (\frac {-\frac {x^{2} e b}{2 \left (4 a c -b^{2}\right )}-\frac {x b d}{4 a c -b^{2}}-\frac {b \,d^{2}+2 a}{2 e \left (4 a c -b^{2}\right )}}{c \,x^{4} e^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 b d e x +b \,d^{2}+a}+\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,e^{4} \textit {\_Z}^{4}+4 c d \,e^{3} \textit {\_Z}^{3}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 b d e \right ) \textit {\_Z} +d^{4} c +b \,d^{2}+a \right )}{\sum }\frac {\left (-\textit {\_R} e -d \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 c d \,e^{2} \textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 d^{3} c +b e \textit {\_R} +b d}\right )}{2 \left (4 a c -b^{2}\right ) e}\right )\) | \(280\) |
risch | \(\frac {-\frac {b e \,f^{3} x^{2}}{2 \left (4 a c -b^{2}\right )}-\frac {b d \,f^{3} x}{4 a c -b^{2}}-\frac {f^{3} \left (b \,d^{2}+2 a \right )}{2 e \left (4 a c -b^{2}\right )}}{c \,x^{4} e^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 b d e x +b \,d^{2}+a}+\frac {b \,f^{3} \ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {3}{2}} e^{2}+4 a b \,e^{2} c -b^{3} e^{2}\right ) x^{2}+\left (-2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} d e +8 a b c d e -2 b^{3} d e \right ) x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}} d^{2}+4 b \,d^{2} c a -b^{3} d^{2}+8 c \,a^{2}-2 b^{2} a \right )}{2 e \left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {b \,f^{3} \ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {3}{2}} e^{2}-4 a b \,e^{2} c +b^{3} e^{2}\right ) x^{2}+\left (-2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} d e -8 a b c d e +2 b^{3} d e \right ) x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}} d^{2}-4 b \,d^{2} c a +b^{3} d^{2}-8 c \,a^{2}+2 b^{2} a \right )}{2 e \left (-4 a c +b^{2}\right )^{\frac {3}{2}}}\) | \(401\) |
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Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (97) = 194\).
Time = 0.32 (sec) , antiderivative size = 1077, normalized size of antiderivative = 10.46 \[ \int \frac {(d f+e f x)^3}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\left [\frac {{\left (b^{3} - 4 \, a b c\right )} e^{2} f^{3} x^{2} + 2 \, {\left (b^{3} - 4 \, a b c\right )} d e f^{3} x + {\left (2 \, a b^{2} - 8 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} f^{3} - {\left (b c e^{4} f^{3} x^{4} + 4 \, b c d e^{3} f^{3} x^{3} + {\left (6 \, b c d^{2} + b^{2}\right )} e^{2} f^{3} x^{2} + 2 \, {\left (2 \, b c d^{3} + b^{2} d\right )} e f^{3} x + {\left (b c d^{4} + b^{2} d^{2} + a b\right )} f^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} e^{4} x^{4} + 8 \, c^{2} d e^{3} x^{3} + 2 \, c^{2} d^{4} + 2 \, {\left (6 \, c^{2} d^{2} + b c\right )} e^{2} x^{2} + 2 \, b c d^{2} + 4 \, {\left (2 \, c^{2} d^{3} + b c d\right )} e x + b^{2} - 2 \, a c + {\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a}\right )}{2 \, {\left ({\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} e^{5} x^{4} + 4 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d e^{4} x^{3} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2} + 6 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{2}\right )} e^{3} x^{2} + 2 \, {\left (2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{3} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d\right )} e^{2} x + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{4} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{2}\right )} e\right )}}, \frac {{\left (b^{3} - 4 \, a b c\right )} e^{2} f^{3} x^{2} + 2 \, {\left (b^{3} - 4 \, a b c\right )} d e f^{3} x + {\left (2 \, a b^{2} - 8 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} f^{3} - 2 \, {\left (b c e^{4} f^{3} x^{4} + 4 \, b c d e^{3} f^{3} x^{3} + {\left (6 \, b c d^{2} + b^{2}\right )} e^{2} f^{3} x^{2} + 2 \, {\left (2 \, b c d^{3} + b^{2} d\right )} e f^{3} x + {\left (b c d^{4} + b^{2} d^{2} + a b\right )} f^{3}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{2 \, {\left ({\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} e^{5} x^{4} + 4 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d e^{4} x^{3} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2} + 6 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{2}\right )} e^{3} x^{2} + 2 \, {\left (2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{3} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d\right )} e^{2} x + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{4} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{2}\right )} e\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (88) = 176\).
Time = 2.77 (sec) , antiderivative size = 556, normalized size of antiderivative = 5.40 \[ \int \frac {(d f+e f x)^3}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\frac {b f^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 16 a^{2} b c^{2} f^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 8 a b^{3} c f^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - b^{5} f^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + b^{2} f^{3} + 2 b c d^{2} f^{3}}{2 b c e^{2} f^{3}} \right )}}{2 e} - \frac {b f^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (\frac {2 d x}{e} + x^{2} + \frac {16 a^{2} b c^{2} f^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - 8 a b^{3} c f^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + b^{5} f^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + b^{2} f^{3} + 2 b c d^{2} f^{3}}{2 b c e^{2} f^{3}} \right )}}{2 e} + \frac {- 2 a f^{3} - b d^{2} f^{3} - 2 b d e f^{3} x - b e^{2} f^{3} x^{2}}{8 a^{2} c e - 2 a b^{2} e + 8 a b c d^{2} e + 8 a c^{2} d^{4} e - 2 b^{3} d^{2} e - 2 b^{2} c d^{4} e + x^{4} \cdot \left (8 a c^{2} e^{5} - 2 b^{2} c e^{5}\right ) + x^{3} \cdot \left (32 a c^{2} d e^{4} - 8 b^{2} c d e^{4}\right ) + x^{2} \cdot \left (8 a b c e^{3} + 48 a c^{2} d^{2} e^{3} - 2 b^{3} e^{3} - 12 b^{2} c d^{2} e^{3}\right ) + x \left (16 a b c d e^{2} + 32 a c^{2} d^{3} e^{2} - 4 b^{3} d e^{2} - 8 b^{2} c d^{3} e^{2}\right )} \]
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\[ \int \frac {(d f+e f x)^3}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\int { \frac {{\left (e f x + d f\right )}^{3}}{{\left ({\left (e x + d\right )}^{4} c + {\left (e x + d\right )}^{2} b + a\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (97) = 194\).
Time = 0.29 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.96 \[ \int \frac {(d f+e f x)^3}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\frac {b f^{3} \arctan \left (\frac {2 \, c d^{2} f + 2 \, {\left (e f x^{2} + 2 \, d f x\right )} c e + b f}{\sqrt {-b^{2} + 4 \, a c} f}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c} e} + \frac {b d^{2} f^{5} + {\left (e f x^{2} + 2 \, d f x\right )} b e f^{4} + 2 \, a f^{5}}{2 \, {\left (c d^{4} f^{2} + 2 \, {\left (e f x^{2} + 2 \, d f x\right )} c d^{2} e f + {\left (e f x^{2} + 2 \, d f x\right )}^{2} c e^{2} + b d^{2} f^{2} + {\left (e f x^{2} + 2 \, d f x\right )} b e f + a f^{2}\right )} {\left (b^{2} e - 4 \, a c e\right )}} \]
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Time = 8.60 (sec) , antiderivative size = 460, normalized size of antiderivative = 4.47 \[ \int \frac {(d f+e f x)^3}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\frac {b\,f^3\,\mathrm {atan}\left (\frac {{\left (4\,a\,c-b^2\right )}^4\,\left (x\,\left (\frac {b^3\,f^6\,\left (2\,b^3\,c^2\,d\,e^9-8\,a\,b\,c^3\,d\,e^9\right )}{a\,e^2\,{\left (4\,a\,c-b^2\right )}^{11/2}}-\frac {2\,b^2\,c^2\,d\,e^7\,f^6}{a\,{\left (4\,a\,c-b^2\right )}^{7/2}}\right )+x^2\,\left (\frac {b^3\,f^6\,\left (2\,b^3\,c^2\,e^{10}-8\,a\,b\,c^3\,e^{10}\right )}{2\,a\,e^2\,{\left (4\,a\,c-b^2\right )}^{11/2}}-\frac {b^2\,c^2\,e^8\,f^6}{a\,{\left (4\,a\,c-b^2\right )}^{7/2}}\right )-\frac {b^3\,f^6\,\left (16\,a^2\,c^3\,e^8-4\,a\,b^2\,c^2\,e^8+8\,a\,b\,c^3\,d^2\,e^8-2\,b^3\,c^2\,d^2\,e^8\right )}{2\,a\,e^2\,{\left (4\,a\,c-b^2\right )}^{11/2}}-\frac {b^2\,c^2\,d^2\,e^6\,f^6}{a\,{\left (4\,a\,c-b^2\right )}^{7/2}}\right )}{2\,b^2\,c^2\,e^6\,f^6}\right )}{e\,{\left (4\,a\,c-b^2\right )}^{3/2}}-\frac {\frac {f^3\,\left (b\,d^2+2\,a\right )}{2\,e\,\left (4\,a\,c-b^2\right )}+\frac {b\,d\,f^3\,x}{4\,a\,c-b^2}+\frac {b\,e\,f^3\,x^2}{2\,\left (4\,a\,c-b^2\right )}}{a+x^2\,\left (6\,c\,d^2\,e^2+b\,e^2\right )+b\,d^2+c\,d^4+x\,\left (4\,c\,e\,d^3+2\,b\,e\,d\right )+c\,e^4\,x^4+4\,c\,d\,e^3\,x^3} \]
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