Integrand size = 28, antiderivative size = 125 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2 (b d-a e)^4}{5 e^5 (d+e x)^{5/2}}+\frac {8 b (b d-a e)^3}{3 e^5 (d+e x)^{3/2}}-\frac {12 b^2 (b d-a e)^2}{e^5 \sqrt {d+e x}}-\frac {8 b^3 (b d-a e) \sqrt {d+e x}}{e^5}+\frac {2 b^4 (d+e x)^{3/2}}{3 e^5} \]
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Time = 0.03 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {8 b^3 \sqrt {d+e x} (b d-a e)}{e^5}-\frac {12 b^2 (b d-a e)^2}{e^5 \sqrt {d+e x}}+\frac {8 b (b d-a e)^3}{3 e^5 (d+e x)^{3/2}}-\frac {2 (b d-a e)^4}{5 e^5 (d+e x)^{5/2}}+\frac {2 b^4 (d+e x)^{3/2}}{3 e^5} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^4}{(d+e x)^{7/2}} \, dx \\ & = \int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^{7/2}}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^{5/2}}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^{3/2}}-\frac {4 b^3 (b d-a e)}{e^4 \sqrt {d+e x}}+\frac {b^4 \sqrt {d+e x}}{e^4}\right ) \, dx \\ & = -\frac {2 (b d-a e)^4}{5 e^5 (d+e x)^{5/2}}+\frac {8 b (b d-a e)^3}{3 e^5 (d+e x)^{3/2}}-\frac {12 b^2 (b d-a e)^2}{e^5 \sqrt {d+e x}}-\frac {8 b^3 (b d-a e) \sqrt {d+e x}}{e^5}+\frac {2 b^4 (d+e x)^{3/2}}{3 e^5} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (3 a^4 e^4+4 a^3 b e^3 (2 d+5 e x)+6 a^2 b^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-12 a b^3 e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+b^4 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{5/2}} \]
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Time = 2.33 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.05
method | result | size |
risch | \(\frac {2 b^{3} \left (b e x +12 a e -11 b d \right ) \sqrt {e x +d}}{3 e^{5}}-\frac {2 \left (90 x^{2} b^{2} e^{2}+20 x a b \,e^{2}+160 b^{2} d e x +3 a^{2} e^{2}+14 a b d e +73 b^{2} d^{2}\right ) \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{15 e^{5} \sqrt {e x +d}\, \left (x^{2} e^{2}+2 d e x +d^{2}\right )}\) | \(131\) |
pseudoelliptic | \(-\frac {2 \left (\left (-\frac {5}{3} b^{4} x^{4}-20 a \,b^{3} x^{3}+30 a^{2} b^{2} x^{2}+\frac {20}{3} a^{3} b x +a^{4}\right ) e^{4}+\frac {8 b d \left (5 b^{3} x^{3}-45 a \,b^{2} x^{2}+15 a^{2} b x +a^{3}\right ) e^{3}}{3}+16 b^{2} d^{2} \left (5 b^{2} x^{2}-10 a b x +a^{2}\right ) e^{2}-64 \left (-\frac {5 b x}{3}+a \right ) b^{3} d^{3} e +\frac {128 b^{4} d^{4}}{3}\right )}{5 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\) | \(143\) |
derivativedivides | \(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {3}{2}}}{3}+8 e \,b^{3} a \sqrt {e x +d}-8 b^{4} d \sqrt {e x +d}-\frac {2 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {12 b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{\sqrt {e x +d}}-\frac {8 b \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{5}}\) | \(183\) |
default | \(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {3}{2}}}{3}+8 e \,b^{3} a \sqrt {e x +d}-8 b^{4} d \sqrt {e x +d}-\frac {2 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {12 b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{\sqrt {e x +d}}-\frac {8 b \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{5}}\) | \(183\) |
gosper | \(-\frac {2 \left (-5 b^{4} x^{4} e^{4}-60 x^{3} a \,b^{3} e^{4}+40 x^{3} b^{4} d \,e^{3}+90 x^{2} a^{2} b^{2} e^{4}-360 x^{2} a \,b^{3} d \,e^{3}+240 x^{2} b^{4} d^{2} e^{2}+20 x \,a^{3} b \,e^{4}+120 x \,a^{2} b^{2} d \,e^{3}-480 x a \,b^{3} d^{2} e^{2}+320 x \,b^{4} d^{3} e +3 e^{4} a^{4}+8 b \,e^{3} d \,a^{3}+48 b^{2} e^{2} d^{2} a^{2}-192 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\) | \(186\) |
trager | \(-\frac {2 \left (-5 b^{4} x^{4} e^{4}-60 x^{3} a \,b^{3} e^{4}+40 x^{3} b^{4} d \,e^{3}+90 x^{2} a^{2} b^{2} e^{4}-360 x^{2} a \,b^{3} d \,e^{3}+240 x^{2} b^{4} d^{2} e^{2}+20 x \,a^{3} b \,e^{4}+120 x \,a^{2} b^{2} d \,e^{3}-480 x a \,b^{3} d^{2} e^{2}+320 x \,b^{4} d^{3} e +3 e^{4} a^{4}+8 b \,e^{3} d \,a^{3}+48 b^{2} e^{2} d^{2} a^{2}-192 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\) | \(186\) |
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Time = 0.30 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.70 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (5 \, b^{4} e^{4} x^{4} - 128 \, b^{4} d^{4} + 192 \, a b^{3} d^{3} e - 48 \, a^{2} b^{2} d^{2} e^{2} - 8 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} - 20 \, {\left (2 \, b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} - 30 \, {\left (8 \, b^{4} d^{2} e^{2} - 12 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} - 20 \, {\left (16 \, b^{4} d^{3} e - 24 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1008 vs. \(2 (116) = 232\).
Time = 0.55 (sec) , antiderivative size = 1008, normalized size of antiderivative = 8.06 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx=\begin {cases} - \frac {6 a^{4} e^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {16 a^{3} b d e^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {40 a^{3} b e^{4} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {96 a^{2} b^{2} d^{2} e^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {240 a^{2} b^{2} d e^{3} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {180 a^{2} b^{2} e^{4} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {384 a b^{3} d^{3} e}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {960 a b^{3} d^{2} e^{2} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {720 a b^{3} d e^{3} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {120 a b^{3} e^{4} x^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {256 b^{4} d^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {640 b^{4} d^{3} e x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {480 b^{4} d^{2} e^{2} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {80 b^{4} d e^{3} x^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {10 b^{4} e^{4} x^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac {b^{4} x^{5}}{5}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (\frac {5 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} b^{4} - 12 \, {\left (b^{4} d - a b^{3} e\right )} \sqrt {e x + d}\right )}}{e^{4}} - \frac {3 \, b^{4} d^{4} - 12 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 12 \, a^{3} b d e^{3} + 3 \, a^{4} e^{4} + 90 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} {\left (e x + d\right )}^{2} - 20 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{4}}\right )}}{15 \, e} \]
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Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (109) = 218\).
Time = 0.29 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.78 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left (90 \, {\left (e x + d\right )}^{2} b^{4} d^{2} - 20 \, {\left (e x + d\right )} b^{4} d^{3} + 3 \, b^{4} d^{4} - 180 \, {\left (e x + d\right )}^{2} a b^{3} d e + 60 \, {\left (e x + d\right )} a b^{3} d^{2} e - 12 \, a b^{3} d^{3} e + 90 \, {\left (e x + d\right )}^{2} a^{2} b^{2} e^{2} - 60 \, {\left (e x + d\right )} a^{2} b^{2} d e^{2} + 18 \, a^{2} b^{2} d^{2} e^{2} + 20 \, {\left (e x + d\right )} a^{3} b e^{3} - 12 \, a^{3} b d e^{3} + 3 \, a^{4} e^{4}\right )}}{15 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{5}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} b^{4} e^{10} - 12 \, \sqrt {e x + d} b^{4} d e^{10} + 12 \, \sqrt {e x + d} a b^{3} e^{11}\right )}}{3 \, e^{15}} \]
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Time = 9.55 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2\,\left (3\,a^4\,e^4+8\,a^3\,b\,d\,e^3+20\,a^3\,b\,e^4\,x+48\,a^2\,b^2\,d^2\,e^2+120\,a^2\,b^2\,d\,e^3\,x+90\,a^2\,b^2\,e^4\,x^2-192\,a\,b^3\,d^3\,e-480\,a\,b^3\,d^2\,e^2\,x-360\,a\,b^3\,d\,e^3\,x^2-60\,a\,b^3\,e^4\,x^3+128\,b^4\,d^4+320\,b^4\,d^3\,e\,x+240\,b^4\,d^2\,e^2\,x^2+40\,b^4\,d\,e^3\,x^3-5\,b^4\,e^4\,x^4\right )}{15\,e^5\,{\left (d+e\,x\right )}^{5/2}} \]
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