Integrand size = 17, antiderivative size = 149 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx=-\frac {3 e^2 \left (c d^2-a e^2\right ) x}{2 a c^2}-\frac {d e^3 x^2}{2 a c}-\frac {(a e-c d x) (d+e x)^3}{2 a c \left (a+c x^2\right )}+\frac {\left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{5/2}}+\frac {2 d e^3 \log \left (a+c x^2\right )}{c^2} \]
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Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {753, 815, 649, 211, 266} \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )}{2 a^{3/2} c^{5/2}}-\frac {3 e^2 x \left (c d^2-a e^2\right )}{2 a c^2}+\frac {2 d e^3 \log \left (a+c x^2\right )}{c^2}-\frac {d e^3 x^2}{2 a c}-\frac {(d+e x)^3 (a e-c d x)}{2 a c \left (a+c x^2\right )} \]
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Rule 211
Rule 266
Rule 649
Rule 753
Rule 815
Rubi steps \begin{align*} \text {integral}& = -\frac {(a e-c d x) (d+e x)^3}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {(d+e x)^2 \left (c d^2+3 a e^2-2 c d e x\right )}{a+c x^2} \, dx}{2 a c} \\ & = -\frac {(a e-c d x) (d+e x)^3}{2 a c \left (a+c x^2\right )}+\frac {\int \left (-3 e^2 \left (d^2-\frac {a e^2}{c}\right )-2 d e^3 x+\frac {c^2 d^4+6 a c d^2 e^2-3 a^2 e^4+8 a c d e^3 x}{c \left (a+c x^2\right )}\right ) \, dx}{2 a c} \\ & = -\frac {3 e^2 \left (c d^2-a e^2\right ) x}{2 a c^2}-\frac {d e^3 x^2}{2 a c}-\frac {(a e-c d x) (d+e x)^3}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {c^2 d^4+6 a c d^2 e^2-3 a^2 e^4+8 a c d e^3 x}{a+c x^2} \, dx}{2 a c^2} \\ & = -\frac {3 e^2 \left (c d^2-a e^2\right ) x}{2 a c^2}-\frac {d e^3 x^2}{2 a c}-\frac {(a e-c d x) (d+e x)^3}{2 a c \left (a+c x^2\right )}+\frac {\left (4 d e^3\right ) \int \frac {x}{a+c x^2} \, dx}{c}+\frac {\left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right ) \int \frac {1}{a+c x^2} \, dx}{2 a c^2} \\ & = -\frac {3 e^2 \left (c d^2-a e^2\right ) x}{2 a c^2}-\frac {d e^3 x^2}{2 a c}-\frac {(a e-c d x) (d+e x)^3}{2 a c \left (a+c x^2\right )}+\frac {\left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{5/2}}+\frac {2 d e^3 \log \left (a+c x^2\right )}{c^2} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\frac {e^4 x}{c^2}+\frac {c^2 d^4 x+a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)}{2 a c^2 \left (a+c x^2\right )}+\frac {\left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{5/2}}+\frac {2 d e^3 \log \left (a+c x^2\right )}{c^2} \]
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Time = 2.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {e^{4} x}{c^{2}}-\frac {\frac {-\frac {\left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{2 a}-2 d e \left (e^{2} a -c \,d^{2}\right )}{c \,x^{2}+a}+\frac {-4 d \,e^{3} a \ln \left (c \,x^{2}+a \right )+\frac {\left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 a}}{c^{2}}\) | \(137\) |
risch | \(\frac {e^{4} x}{c^{2}}+\frac {\frac {\left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{2 a}+2 d e \left (e^{2} a -c \,d^{2}\right )}{c^{2} \left (c \,x^{2}+a \right )}+\frac {2 \ln \left (-3 e^{4} a^{3}+6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a -\sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}\, x \right ) d \,e^{3}}{c^{2}}+\frac {\ln \left (-3 e^{4} a^{3}+6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a -\sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}\, x \right ) \sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}}{4 c^{3} a^{2}}+\frac {2 \ln \left (-3 e^{4} a^{3}+6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a +\sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}\, x \right ) d \,e^{3}}{c^{2}}-\frac {\ln \left (-3 e^{4} a^{3}+6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a +\sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}\, x \right ) \sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}}{4 c^{3} a^{2}}\) | \(445\) |
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Time = 0.48 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.91 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\left [\frac {4 \, a^{2} c^{2} e^{4} x^{3} - 8 \, a^{2} c^{2} d^{3} e + 8 \, a^{3} c d e^{3} + {\left (a c^{2} d^{4} + 6 \, a^{2} c d^{2} e^{2} - 3 \, a^{3} e^{4} + {\left (c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} - 3 \, a^{2} c e^{4}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + 2 \, {\left (a c^{3} d^{4} - 6 \, a^{2} c^{2} d^{2} e^{2} + 3 \, a^{3} c e^{4}\right )} x + 8 \, {\left (a^{2} c^{2} d e^{3} x^{2} + a^{3} c d e^{3}\right )} \log \left (c x^{2} + a\right )}{4 \, {\left (a^{2} c^{4} x^{2} + a^{3} c^{3}\right )}}, \frac {2 \, a^{2} c^{2} e^{4} x^{3} - 4 \, a^{2} c^{2} d^{3} e + 4 \, a^{3} c d e^{3} + {\left (a c^{2} d^{4} + 6 \, a^{2} c d^{2} e^{2} - 3 \, a^{3} e^{4} + {\left (c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} - 3 \, a^{2} c e^{4}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + {\left (a c^{3} d^{4} - 6 \, a^{2} c^{2} d^{2} e^{2} + 3 \, a^{3} c e^{4}\right )} x + 4 \, {\left (a^{2} c^{2} d e^{3} x^{2} + a^{3} c d e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (a^{2} c^{4} x^{2} + a^{3} c^{3}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (138) = 276\).
Time = 0.77 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.70 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\left (\frac {2 d e^{3}}{c^{2}} - \frac {\sqrt {- a^{3} c^{5}} \cdot \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{5}}\right ) \log {\left (x + \frac {- 4 a^{2} c^{2} \cdot \left (\frac {2 d e^{3}}{c^{2}} - \frac {\sqrt {- a^{3} c^{5}} \cdot \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{5}}\right ) + 8 a^{2} d e^{3}}{3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}} \right )} + \left (\frac {2 d e^{3}}{c^{2}} + \frac {\sqrt {- a^{3} c^{5}} \cdot \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{5}}\right ) \log {\left (x + \frac {- 4 a^{2} c^{2} \cdot \left (\frac {2 d e^{3}}{c^{2}} + \frac {\sqrt {- a^{3} c^{5}} \cdot \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{5}}\right ) + 8 a^{2} d e^{3}}{3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}} \right )} + \frac {4 a^{2} d e^{3} - 4 a c d^{3} e + x \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a^{2} c^{2} + 2 a c^{3} x^{2}} + \frac {e^{4} x}{c^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\frac {e^{4} x}{c^{2}} + \frac {2 \, d e^{3} \log \left (c x^{2} + a\right )}{c^{2}} - \frac {4 \, a c d^{3} e - 4 \, a^{2} d e^{3} - {\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x}{2 \, {\left (a c^{3} x^{2} + a^{2} c^{2}\right )}} + \frac {{\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\frac {e^{4} x}{c^{2}} + \frac {2 \, d e^{3} \log \left (c x^{2} + a\right )}{c^{2}} + \frac {{\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c^{2}} - \frac {4 \, a c d^{3} e - 4 \, a^{2} d e^{3} - {\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x}{2 \, {\left (c x^{2} + a\right )} a c^{2}} \]
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Time = 9.59 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\frac {\frac {x\,\left (a^2\,e^4-6\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{2\,a}+2\,a\,d\,e^3-2\,c\,d^3\,e}{c^3\,x^2+a\,c^2}+\frac {e^4\,x}{c^2}+\frac {2\,d\,e^3\,\ln \left (c\,x^2+a\right )}{c^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (-3\,a^2\,e^4+6\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{2\,a^{3/2}\,c^{5/2}} \]
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