Integrand size = 17, antiderivative size = 198 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^3} \, dx=-\frac {d e^2 \left (3 c d^2+7 a e^2\right ) x}{8 a^2 c^2}-\frac {(a e-c d x) (d+e x)^4}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x)^2 \left (2 a e \left (c d^2+2 a e^2\right )-c d \left (3 c d^2+5 a e^2\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{5/2}}+\frac {e^5 \log \left (a+c x^2\right )}{2 c^3} \]
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Time = 0.13 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {753, 833, 788, 649, 211, 266} \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^3} \, dx=-\frac {(d+e x)^2 \left (2 a e \left (2 a e^2+c d^2\right )-c d x \left (5 a e^2+3 c d^2\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac {d e^2 x \left (7 a e^2+3 c d^2\right )}{8 a^2 c^2}+\frac {d \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )}{8 a^{5/2} c^{5/2}}+\frac {e^5 \log \left (a+c x^2\right )}{2 c^3}-\frac {(d+e x)^4 (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \]
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Rule 211
Rule 266
Rule 649
Rule 753
Rule 788
Rule 833
Rubi steps \begin{align*} \text {integral}& = -\frac {(a e-c d x) (d+e x)^4}{4 a c \left (a+c x^2\right )^2}+\frac {\int \frac {(d+e x)^3 \left (3 c d^2+4 a e^2-c d e x\right )}{\left (a+c x^2\right )^2} \, dx}{4 a c} \\ & = -\frac {(a e-c d x) (d+e x)^4}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x)^2 \left (2 a e \left (c d^2+2 a e^2\right )-c d \left (3 c d^2+5 a e^2\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\int \frac {(d+e x) \left (3 c^2 d^4+7 a c d^2 e^2+8 a^2 e^4-c d e \left (3 c d^2+7 a e^2\right ) x\right )}{a+c x^2} \, dx}{8 a^2 c^2} \\ & = -\frac {d e^2 \left (3 c d^2+7 a e^2\right ) x}{8 a^2 c^2}-\frac {(a e-c d x) (d+e x)^4}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x)^2 \left (2 a e \left (c d^2+2 a e^2\right )-c d \left (3 c d^2+5 a e^2\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\int \frac {a c d e^2 \left (3 c d^2+7 a e^2\right )+c d \left (3 c^2 d^4+7 a c d^2 e^2+8 a^2 e^4\right )+c \left (-c d^2 e \left (3 c d^2+7 a e^2\right )+e \left (3 c^2 d^4+7 a c d^2 e^2+8 a^2 e^4\right )\right ) x}{a+c x^2} \, dx}{8 a^2 c^3} \\ & = -\frac {d e^2 \left (3 c d^2+7 a e^2\right ) x}{8 a^2 c^2}-\frac {(a e-c d x) (d+e x)^4}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x)^2 \left (2 a e \left (c d^2+2 a e^2\right )-c d \left (3 c d^2+5 a e^2\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {e^5 \int \frac {x}{a+c x^2} \, dx}{c^2}+\frac {\left (d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right )\right ) \int \frac {1}{a+c x^2} \, dx}{8 a^2 c^2} \\ & = -\frac {d e^2 \left (3 c d^2+7 a e^2\right ) x}{8 a^2 c^2}-\frac {(a e-c d x) (d+e x)^4}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x)^2 \left (2 a e \left (c d^2+2 a e^2\right )-c d \left (3 c d^2+5 a e^2\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{5/2}}+\frac {e^5 \log \left (a+c x^2\right )}{2 c^3} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^3} \, dx=\frac {-\frac {2 \left (a^3 e^5-c^3 d^5 x-5 a^2 c d e^3 (2 d+e x)+5 a c^2 d^3 e (d+2 e x)\right )}{a \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+3 c^3 d^5 x+10 a c^2 d^3 e^2 x-5 a^2 c d e^3 (8 d+5 e x)}{a^2 \left (a+c x^2\right )}+\frac {\sqrt {c} d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{5/2}}+4 e^5 \log \left (a+c x^2\right )}{8 c^3} \]
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Time = 2.20 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.09
method | result | size |
default | \(\frac {-\frac {d \left (25 a^{2} e^{4}-10 a c \,d^{2} e^{2}-3 c^{2} d^{4}\right ) x^{3}}{8 a^{2} c}+\frac {e^{3} \left (e^{2} a -5 c \,d^{2}\right ) x^{2}}{c^{2}}-\frac {5 d \left (3 a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) x}{8 c^{2} a}+\frac {e \left (3 a^{2} e^{4}-10 a c \,d^{2} e^{2}-5 c^{2} d^{4}\right )}{4 c^{3}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {4 a^{2} e^{5} \ln \left (c \,x^{2}+a \right )}{c}+\frac {\left (15 a^{2} d \,e^{4}+10 a c \,d^{3} e^{2}+3 c^{2} d^{5}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{8 a^{2} c^{2}}\) | \(215\) |
risch | \(\frac {-\frac {d \left (25 a^{2} e^{4}-10 a c \,d^{2} e^{2}-3 c^{2} d^{4}\right ) x^{3}}{8 a^{2} c}+\frac {e^{3} \left (e^{2} a -5 c \,d^{2}\right ) x^{2}}{c^{2}}-\frac {5 d \left (3 a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) x}{8 c^{2} a}+\frac {e \left (3 a^{2} e^{4}-10 a c \,d^{2} e^{2}-5 c^{2} d^{4}\right )}{4 c^{3}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\ln \left (15 d \,e^{4} a^{3}+10 d^{3} e^{2} a^{2} c +3 d^{5} c^{2} a -\sqrt {-a c \,d^{2} \left (15 a^{2} e^{4}+10 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right )^{2}}\, x \right ) e^{5}}{2 c^{3}}+\frac {\ln \left (15 d \,e^{4} a^{3}+10 d^{3} e^{2} a^{2} c +3 d^{5} c^{2} a -\sqrt {-a c \,d^{2} \left (15 a^{2} e^{4}+10 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right )^{2}}\, x \right ) \sqrt {-a c \,d^{2} \left (15 a^{2} e^{4}+10 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right )^{2}}}{16 a^{3} c^{3}}+\frac {\ln \left (15 d \,e^{4} a^{3}+10 d^{3} e^{2} a^{2} c +3 d^{5} c^{2} a +\sqrt {-a c \,d^{2} \left (15 a^{2} e^{4}+10 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right )^{2}}\, x \right ) e^{5}}{2 c^{3}}-\frac {\ln \left (15 d \,e^{4} a^{3}+10 d^{3} e^{2} a^{2} c +3 d^{5} c^{2} a +\sqrt {-a c \,d^{2} \left (15 a^{2} e^{4}+10 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right )^{2}}\, x \right ) \sqrt {-a c \,d^{2} \left (15 a^{2} e^{4}+10 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right )^{2}}}{16 c^{3} a^{3}}\) | \(542\) |
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Time = 0.28 (sec) , antiderivative size = 707, normalized size of antiderivative = 3.57 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^3} \, dx=\left [-\frac {20 \, a^{3} c^{2} d^{4} e + 40 \, a^{4} c d^{2} e^{3} - 12 \, a^{5} e^{5} - 2 \, {\left (3 \, a c^{4} d^{5} + 10 \, a^{2} c^{3} d^{3} e^{2} - 25 \, a^{3} c^{2} d e^{4}\right )} x^{3} + 16 \, {\left (5 \, a^{3} c^{2} d^{2} e^{3} - a^{4} c e^{5}\right )} x^{2} + {\left (3 \, a^{2} c^{2} d^{5} + 10 \, a^{3} c d^{3} e^{2} + 15 \, a^{4} d e^{4} + {\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} + 15 \, a^{2} c^{2} d e^{4}\right )} x^{4} + 2 \, {\left (3 \, a c^{3} d^{5} + 10 \, a^{2} c^{2} d^{3} e^{2} + 15 \, a^{3} c d e^{4}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 10 \, {\left (a^{2} c^{3} d^{5} - 2 \, a^{3} c^{2} d^{3} e^{2} - 3 \, a^{4} c d e^{4}\right )} x - 8 \, {\left (a^{3} c^{2} e^{5} x^{4} + 2 \, a^{4} c e^{5} x^{2} + a^{5} e^{5}\right )} \log \left (c x^{2} + a\right )}{16 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}, -\frac {10 \, a^{3} c^{2} d^{4} e + 20 \, a^{4} c d^{2} e^{3} - 6 \, a^{5} e^{5} - {\left (3 \, a c^{4} d^{5} + 10 \, a^{2} c^{3} d^{3} e^{2} - 25 \, a^{3} c^{2} d e^{4}\right )} x^{3} + 8 \, {\left (5 \, a^{3} c^{2} d^{2} e^{3} - a^{4} c e^{5}\right )} x^{2} - {\left (3 \, a^{2} c^{2} d^{5} + 10 \, a^{3} c d^{3} e^{2} + 15 \, a^{4} d e^{4} + {\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} + 15 \, a^{2} c^{2} d e^{4}\right )} x^{4} + 2 \, {\left (3 \, a c^{3} d^{5} + 10 \, a^{2} c^{2} d^{3} e^{2} + 15 \, a^{3} c d e^{4}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - 5 \, {\left (a^{2} c^{3} d^{5} - 2 \, a^{3} c^{2} d^{3} e^{2} - 3 \, a^{4} c d e^{4}\right )} x - 4 \, {\left (a^{3} c^{2} e^{5} x^{4} + 2 \, a^{4} c e^{5} x^{2} + a^{5} e^{5}\right )} \log \left (c x^{2} + a\right )}{8 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (189) = 378\).
Time = 2.08 (sec) , antiderivative size = 520, normalized size of antiderivative = 2.63 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^3} \, dx=\left (\frac {e^{5}}{2 c^{3}} - \frac {d \sqrt {- a^{5} c^{7}} \cdot \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{16 a^{5} c^{6}}\right ) \log {\left (x + \frac {16 a^{3} c^{3} \left (\frac {e^{5}}{2 c^{3}} - \frac {d \sqrt {- a^{5} c^{7}} \cdot \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{16 a^{5} c^{6}}\right ) - 8 a^{3} e^{5}}{15 a^{2} c d e^{4} + 10 a c^{2} d^{3} e^{2} + 3 c^{3} d^{5}} \right )} + \left (\frac {e^{5}}{2 c^{3}} + \frac {d \sqrt {- a^{5} c^{7}} \cdot \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{16 a^{5} c^{6}}\right ) \log {\left (x + \frac {16 a^{3} c^{3} \left (\frac {e^{5}}{2 c^{3}} + \frac {d \sqrt {- a^{5} c^{7}} \cdot \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{16 a^{5} c^{6}}\right ) - 8 a^{3} e^{5}}{15 a^{2} c d e^{4} + 10 a c^{2} d^{3} e^{2} + 3 c^{3} d^{5}} \right )} + \frac {6 a^{4} e^{5} - 20 a^{3} c d^{2} e^{3} - 10 a^{2} c^{2} d^{4} e + x^{3} \left (- 25 a^{2} c^{2} d e^{4} + 10 a c^{3} d^{3} e^{2} + 3 c^{4} d^{5}\right ) + x^{2} \cdot \left (8 a^{3} c e^{5} - 40 a^{2} c^{2} d^{2} e^{3}\right ) + x \left (- 15 a^{3} c d e^{4} - 10 a^{2} c^{2} d^{3} e^{2} + 5 a c^{3} d^{5}\right )}{8 a^{4} c^{3} + 16 a^{3} c^{4} x^{2} + 8 a^{2} c^{5} x^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^3} \, dx=\frac {e^{5} \log \left (c x^{2} + a\right )}{2 \, c^{3}} - \frac {10 \, a^{2} c^{2} d^{4} e + 20 \, a^{3} c d^{2} e^{3} - 6 \, a^{4} e^{5} - {\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} - 25 \, a^{2} c^{2} d e^{4}\right )} x^{3} + 8 \, {\left (5 \, a^{2} c^{2} d^{2} e^{3} - a^{3} c e^{5}\right )} x^{2} - 5 \, {\left (a c^{3} d^{5} - 2 \, a^{2} c^{2} d^{3} e^{2} - 3 \, a^{3} c d e^{4}\right )} x}{8 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}} + \frac {{\left (3 \, c^{2} d^{5} + 10 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} \]
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Time = 0.38 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^3} \, dx=\frac {e^{5} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac {{\left (3 \, c^{2} d^{5} + 10 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} + \frac {{\left (3 \, c^{3} d^{5} + 10 \, a c^{2} d^{3} e^{2} - 25 \, a^{2} c d e^{4}\right )} x^{3} - 8 \, {\left (5 \, a^{2} c d^{2} e^{3} - a^{3} e^{5}\right )} x^{2} + 5 \, {\left (a c^{2} d^{5} - 2 \, a^{2} c d^{3} e^{2} - 3 \, a^{3} d e^{4}\right )} x - \frac {2 \, {\left (5 \, a^{2} c^{2} d^{4} e + 10 \, a^{3} c d^{2} e^{3} - 3 \, a^{4} e^{5}\right )}}{c}}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \]
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Time = 9.77 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.36 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^3} \, dx=\frac {e^5\,\ln \left (c\,x^2+a\right )}{2\,c^3}-\frac {5\,d^4\,e}{4\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {5\,d^5\,x}{8\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}+\frac {3\,a^2\,e^5}{4\,\left (a^2\,c^3+2\,a\,c^4\,x^2+c^5\,x^4\right )}-\frac {5\,a\,d^2\,e^3}{2\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )}+\frac {a\,e^5\,x^2}{a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4}-\frac {5\,d^2\,e^3\,x^2}{a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4}+\frac {3\,d^5\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,a^{5/2}\,\sqrt {c}}+\frac {3\,c\,d^5\,x^3}{8\,\left (a^4+2\,a^3\,c\,x^2+a^2\,c^2\,x^4\right )}+\frac {5\,d^3\,e^2\,x^3}{4\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}-\frac {5\,d^3\,e^2\,x}{4\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}-\frac {25\,d\,e^4\,x^3}{8\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {5\,d^3\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{4\,a^{3/2}\,c^{3/2}}-\frac {15\,a\,d\,e^4\,x}{8\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )}+\frac {15\,d\,e^4\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,\sqrt {a}\,c^{5/2}} \]
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