Integrand size = 17, antiderivative size = 184 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx=-\frac {\left (c d^2+a e^2\right )^3}{6 e^7 (d+e x)^6}+\frac {6 c d \left (c d^2+a e^2\right )^2}{5 e^7 (d+e x)^5}-\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{4 e^7 (d+e x)^4}+\frac {4 c^2 d \left (5 c d^2+3 a e^2\right )}{3 e^7 (d+e x)^3}-\frac {3 c^2 \left (5 c d^2+a e^2\right )}{2 e^7 (d+e x)^2}+\frac {6 c^3 d}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7} \]
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Time = 0.09 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711} \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx=-\frac {3 c^2 \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^2}+\frac {4 c^2 d \left (3 a e^2+5 c d^2\right )}{3 e^7 (d+e x)^3}-\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{4 e^7 (d+e x)^4}+\frac {6 c d \left (a e^2+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac {\left (a e^2+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac {6 c^3 d}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7} \]
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Rule 711
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^7}-\frac {6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^6}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{e^6 (d+e x)^5}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^6 (d+e x)^4}+\frac {3 c^2 \left (5 c d^2+a e^2\right )}{e^6 (d+e x)^3}-\frac {6 c^3 d}{e^6 (d+e x)^2}+\frac {c^3}{e^6 (d+e x)}\right ) \, dx \\ & = -\frac {\left (c d^2+a e^2\right )^3}{6 e^7 (d+e x)^6}+\frac {6 c d \left (c d^2+a e^2\right )^2}{5 e^7 (d+e x)^5}-\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{4 e^7 (d+e x)^4}+\frac {4 c^2 d \left (5 c d^2+3 a e^2\right )}{3 e^7 (d+e x)^3}-\frac {3 c^2 \left (5 c d^2+a e^2\right )}{2 e^7 (d+e x)^2}+\frac {6 c^3 d}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {-10 a^3 e^6-3 a^2 c e^4 \left (d^2+6 d e x+15 e^2 x^2\right )-6 a c^2 e^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+c^3 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6} \]
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Time = 2.15 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.09
method | result | size |
risch | \(\frac {\frac {6 c^{3} d \,x^{5}}{e^{2}}-\frac {3 c^{2} \left (e^{2} a -15 c \,d^{2}\right ) x^{4}}{2 e^{3}}-\frac {2 c^{2} d \left (3 e^{2} a -55 c \,d^{2}\right ) x^{3}}{3 e^{4}}-\frac {c \left (3 a^{2} e^{4}+6 a c \,d^{2} e^{2}-125 c^{2} d^{4}\right ) x^{2}}{4 e^{5}}-\frac {c d \left (3 a^{2} e^{4}+6 a c \,d^{2} e^{2}-137 c^{2} d^{4}\right ) x}{10 e^{6}}-\frac {10 e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +6 d^{4} e^{2} c^{2} a -147 c^{3} d^{6}}{60 e^{7}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(201\) |
norman | \(\frac {-\frac {10 e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +6 d^{4} e^{2} c^{2} a -147 c^{3} d^{6}}{60 e^{7}}-\frac {3 \left (e^{2} c^{2} a -15 c^{3} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (3 e^{4} a^{2} c +6 d^{2} e^{2} c^{2} a -125 d^{4} c^{3}\right ) x^{2}}{4 e^{5}}+\frac {6 c^{3} d \,x^{5}}{e^{2}}-\frac {2 d \left (3 e^{2} c^{2} a -55 c^{3} d^{2}\right ) x^{3}}{3 e^{4}}-\frac {d \left (3 e^{4} a^{2} c +6 d^{2} e^{2} c^{2} a -137 d^{4} c^{3}\right ) x}{10 e^{6}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(209\) |
default | \(-\frac {e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{6 e^{7} \left (e x +d \right )^{6}}+\frac {6 c^{3} d}{e^{7} \left (e x +d \right )}+\frac {4 c^{2} d \left (3 e^{2} a +5 c \,d^{2}\right )}{3 e^{7} \left (e x +d \right )^{3}}-\frac {3 c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right )}{4 e^{7} \left (e x +d \right )^{4}}-\frac {3 c^{2} \left (e^{2} a +5 c \,d^{2}\right )}{2 e^{7} \left (e x +d \right )^{2}}+\frac {6 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{5 e^{7} \left (e x +d \right )^{5}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(216\) |
parallelrisch | \(\frac {-6 d^{4} e^{2} c^{2} a -3 d^{2} e^{4} a^{2} c +1350 x^{4} c^{3} d^{2} e^{4}+2200 x^{3} c^{3} d^{3} e^{3}+1875 x^{2} c^{3} d^{4} e^{2}+822 x \,c^{3} d^{5} e +147 c^{3} d^{6}+900 \ln \left (e x +d \right ) x^{4} c^{3} d^{2} e^{4}-18 x \,a^{2} c d \,e^{5}-36 x a \,c^{2} d^{3} e^{3}+360 \ln \left (e x +d \right ) x \,c^{3} d^{5} e -10 e^{6} a^{3}+900 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}-120 x^{3} a \,c^{2} d \,e^{5}-90 x^{2} a \,c^{2} d^{2} e^{4}+1200 \ln \left (e x +d \right ) x^{3} c^{3} d^{3} e^{3}+360 \ln \left (e x +d \right ) x^{5} c^{3} d \,e^{5}+360 x^{5} c^{3} d \,e^{5}-90 x^{4} a \,c^{2} e^{6}-45 x^{2} a^{2} c \,e^{6}+60 \ln \left (e x +d \right ) x^{6} c^{3} e^{6}+60 \ln \left (e x +d \right ) c^{3} d^{6}}{60 e^{7} \left (e x +d \right )^{6}}\) | \(319\) |
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Time = 0.27 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.84 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {360 \, c^{3} d e^{5} x^{5} + 147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - a c^{2} e^{6}\right )} x^{4} + 40 \, {\left (55 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 6 \, a c^{2} d^{2} e^{4} - 3 \, a^{2} c e^{6}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x + 60 \, {\left (c^{3} e^{6} x^{6} + 6 \, c^{3} d e^{5} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} + 6 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \]
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Time = 24.87 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^{3} \log {\left (d + e x \right )}}{e^{7}} + \frac {- 10 a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} - 6 a c^{2} d^{4} e^{2} + 147 c^{3} d^{6} + 360 c^{3} d e^{5} x^{5} + x^{4} \left (- 90 a c^{2} e^{6} + 1350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 120 a c^{2} d e^{5} + 2200 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 45 a^{2} c e^{6} - 90 a c^{2} d^{2} e^{4} + 1875 c^{3} d^{4} e^{2}\right ) + x \left (- 18 a^{2} c d e^{5} - 36 a c^{2} d^{3} e^{3} + 822 c^{3} d^{5} e\right )}{60 d^{6} e^{7} + 360 d^{5} e^{8} x + 900 d^{4} e^{9} x^{2} + 1200 d^{3} e^{10} x^{3} + 900 d^{2} e^{11} x^{4} + 360 d e^{12} x^{5} + 60 e^{13} x^{6}} \]
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Time = 0.20 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {360 \, c^{3} d e^{5} x^{5} + 147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - a c^{2} e^{6}\right )} x^{4} + 40 \, {\left (55 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 6 \, a c^{2} d^{2} e^{4} - 3 \, a^{2} c e^{6}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac {c^{3} \log \left (e x + d\right )}{e^{7}} \]
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Time = 0.26 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^{3} \log \left ({\left | e x + d \right |}\right )}{e^{7}} + \frac {360 \, c^{3} d e^{4} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{3} - a c^{2} e^{5}\right )} x^{4} + 40 \, {\left (55 \, c^{3} d^{3} e^{2} - 3 \, a c^{2} d e^{4}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e - 6 \, a c^{2} d^{2} e^{3} - 3 \, a^{2} c e^{5}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} - 6 \, a c^{2} d^{3} e^{2} - 3 \, a^{2} c d e^{4}\right )} x + \frac {147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6}}{e}}{60 \, {\left (e x + d\right )}^{6} e^{6}} \]
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Time = 9.58 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^3\,\ln \left (d+e\,x\right )}{e^7}-\frac {\frac {10\,a^3\,e^6+3\,a^2\,c\,d^2\,e^4+6\,a\,c^2\,d^4\,e^2-147\,c^3\,d^6}{60\,e^7}+\frac {x^2\,\left (3\,a^2\,c\,e^4+6\,a\,c^2\,d^2\,e^2-125\,c^3\,d^4\right )}{4\,e^5}+\frac {x\,\left (3\,a^2\,c\,d\,e^4+6\,a\,c^2\,d^3\,e^2-137\,c^3\,d^5\right )}{10\,e^6}-\frac {2\,x^3\,\left (55\,c^3\,d^3-3\,a\,c^2\,d\,e^2\right )}{3\,e^4}-\frac {6\,c^3\,d\,x^5}{e^2}+\frac {3\,c^2\,x^4\,\left (a\,e^2-15\,c\,d^2\right )}{2\,e^3}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \]
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