Integrand size = 28, antiderivative size = 181 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 (b d-a e)^6 \sqrt {d+e x}}{e^7}-\frac {4 b (b d-a e)^5 (d+e x)^{3/2}}{e^7}+\frac {6 b^2 (b d-a e)^4 (d+e x)^{5/2}}{e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{7/2}}{7 e^7}+\frac {10 b^4 (b d-a e)^2 (d+e x)^{9/2}}{3 e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{11/2}}{11 e^7}+\frac {2 b^6 (d+e x)^{13/2}}{13 e^7} \]
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Time = 0.04 (sec) , antiderivative size = 181, 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} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=-\frac {12 b^5 (d+e x)^{11/2} (b d-a e)}{11 e^7}+\frac {10 b^4 (d+e x)^{9/2} (b d-a e)^2}{3 e^7}-\frac {40 b^3 (d+e x)^{7/2} (b d-a e)^3}{7 e^7}+\frac {6 b^2 (d+e x)^{5/2} (b d-a e)^4}{e^7}-\frac {4 b (d+e x)^{3/2} (b d-a e)^5}{e^7}+\frac {2 \sqrt {d+e x} (b d-a e)^6}{e^7}+\frac {2 b^6 (d+e x)^{13/2}}{13 e^7} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^6}{\sqrt {d+e x}} \, dx \\ & = \int \left (\frac {(-b d+a e)^6}{e^6 \sqrt {d+e x}}-\frac {6 b (b d-a e)^5 \sqrt {d+e x}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{3/2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{5/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{7/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{9/2}}{e^6}+\frac {b^6 (d+e x)^{11/2}}{e^6}\right ) \, dx \\ & = \frac {2 (b d-a e)^6 \sqrt {d+e x}}{e^7}-\frac {4 b (b d-a e)^5 (d+e x)^{3/2}}{e^7}+\frac {6 b^2 (b d-a e)^4 (d+e x)^{5/2}}{e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{7/2}}{7 e^7}+\frac {10 b^4 (b d-a e)^2 (d+e x)^{9/2}}{3 e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{11/2}}{11 e^7}+\frac {2 b^6 (d+e x)^{13/2}}{13 e^7} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (3003 a^6 e^6+6006 a^5 b e^5 (-2 d+e x)+3003 a^4 b^2 e^4 \left (8 d^2-4 d e x+3 e^2 x^2\right )+1716 a^3 b^3 e^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+143 a^2 b^4 e^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+26 a b^5 e \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )+b^6 \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )\right )}{3003 e^7} \]
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Time = 2.36 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.52
method | result | size |
pseudoelliptic | \(\frac {2 \left (\left (\frac {1}{13} b^{6} x^{6}+a^{6}+\frac {6}{11} a \,x^{5} b^{5}+\frac {5}{3} a^{2} x^{4} b^{4}+\frac {20}{7} a^{3} x^{3} b^{3}+3 a^{4} x^{2} b^{2}+2 a^{5} x b \right ) e^{6}-4 \left (\frac {3}{143} b^{5} x^{5}+\frac {5}{33} a \,b^{4} x^{4}+\frac {10}{21} a^{2} b^{3} x^{3}+\frac {6}{7} a^{3} b^{2} x^{2}+a^{4} b x +a^{5}\right ) b d \,e^{5}+8 b^{2} d^{2} \left (\frac {5}{429} b^{4} x^{4}+\frac {20}{231} a \,b^{3} x^{3}+\frac {2}{7} a^{2} b^{2} x^{2}+\frac {4}{7} a^{3} b x +a^{4}\right ) e^{4}-\frac {64 b^{3} \left (\frac {5}{429} b^{3} x^{3}+\frac {1}{11} a \,b^{2} x^{2}+\frac {1}{3} a^{2} b x +a^{3}\right ) d^{3} e^{3}}{7}+\frac {128 b^{4} d^{4} \left (\frac {3}{143} b^{2} x^{2}+\frac {2}{11} a b x +a^{2}\right ) e^{2}}{21}-\frac {512 \left (\frac {b x}{13}+a \right ) b^{5} d^{5} e}{231}+\frac {1024 b^{6} d^{6}}{3003}\right ) \sqrt {e x +d}}{e^{7}}\) | \(275\) |
gosper | \(\frac {2 \left (231 x^{6} b^{6} e^{6}+1638 x^{5} a \,b^{5} e^{6}-252 x^{5} b^{6} d \,e^{5}+5005 x^{4} a^{2} b^{4} e^{6}-1820 x^{4} a \,b^{5} d \,e^{5}+280 x^{4} b^{6} d^{2} e^{4}+8580 x^{3} a^{3} b^{3} e^{6}-5720 x^{3} a^{2} b^{4} d \,e^{5}+2080 x^{3} a \,b^{5} d^{2} e^{4}-320 x^{3} b^{6} d^{3} e^{3}+9009 x^{2} a^{4} b^{2} e^{6}-10296 x^{2} a^{3} b^{3} d \,e^{5}+6864 x^{2} a^{2} b^{4} d^{2} e^{4}-2496 x^{2} a \,b^{5} d^{3} e^{3}+384 x^{2} b^{6} d^{4} e^{2}+6006 x \,a^{5} b \,e^{6}-12012 x \,a^{4} b^{2} d \,e^{5}+13728 x \,a^{3} b^{3} d^{2} e^{4}-9152 x \,a^{2} b^{4} d^{3} e^{3}+3328 x a \,b^{5} d^{4} e^{2}-512 x \,b^{6} d^{5} e +3003 a^{6} e^{6}-12012 a^{5} b d \,e^{5}+24024 a^{4} b^{2} d^{2} e^{4}-27456 a^{3} b^{3} d^{3} e^{3}+18304 a^{2} b^{4} d^{4} e^{2}-6656 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \sqrt {e x +d}}{3003 e^{7}}\) | \(377\) |
trager | \(\frac {2 \left (231 x^{6} b^{6} e^{6}+1638 x^{5} a \,b^{5} e^{6}-252 x^{5} b^{6} d \,e^{5}+5005 x^{4} a^{2} b^{4} e^{6}-1820 x^{4} a \,b^{5} d \,e^{5}+280 x^{4} b^{6} d^{2} e^{4}+8580 x^{3} a^{3} b^{3} e^{6}-5720 x^{3} a^{2} b^{4} d \,e^{5}+2080 x^{3} a \,b^{5} d^{2} e^{4}-320 x^{3} b^{6} d^{3} e^{3}+9009 x^{2} a^{4} b^{2} e^{6}-10296 x^{2} a^{3} b^{3} d \,e^{5}+6864 x^{2} a^{2} b^{4} d^{2} e^{4}-2496 x^{2} a \,b^{5} d^{3} e^{3}+384 x^{2} b^{6} d^{4} e^{2}+6006 x \,a^{5} b \,e^{6}-12012 x \,a^{4} b^{2} d \,e^{5}+13728 x \,a^{3} b^{3} d^{2} e^{4}-9152 x \,a^{2} b^{4} d^{3} e^{3}+3328 x a \,b^{5} d^{4} e^{2}-512 x \,b^{6} d^{5} e +3003 a^{6} e^{6}-12012 a^{5} b d \,e^{5}+24024 a^{4} b^{2} d^{2} e^{4}-27456 a^{3} b^{3} d^{3} e^{3}+18304 a^{2} b^{4} d^{4} e^{2}-6656 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \sqrt {e x +d}}{3003 e^{7}}\) | \(377\) |
risch | \(\frac {2 \left (231 x^{6} b^{6} e^{6}+1638 x^{5} a \,b^{5} e^{6}-252 x^{5} b^{6} d \,e^{5}+5005 x^{4} a^{2} b^{4} e^{6}-1820 x^{4} a \,b^{5} d \,e^{5}+280 x^{4} b^{6} d^{2} e^{4}+8580 x^{3} a^{3} b^{3} e^{6}-5720 x^{3} a^{2} b^{4} d \,e^{5}+2080 x^{3} a \,b^{5} d^{2} e^{4}-320 x^{3} b^{6} d^{3} e^{3}+9009 x^{2} a^{4} b^{2} e^{6}-10296 x^{2} a^{3} b^{3} d \,e^{5}+6864 x^{2} a^{2} b^{4} d^{2} e^{4}-2496 x^{2} a \,b^{5} d^{3} e^{3}+384 x^{2} b^{6} d^{4} e^{2}+6006 x \,a^{5} b \,e^{6}-12012 x \,a^{4} b^{2} d \,e^{5}+13728 x \,a^{3} b^{3} d^{2} e^{4}-9152 x \,a^{2} b^{4} d^{3} e^{3}+3328 x a \,b^{5} d^{4} e^{2}-512 x \,b^{6} d^{5} e +3003 a^{6} e^{6}-12012 a^{5} b d \,e^{5}+24024 a^{4} b^{2} d^{2} e^{4}-27456 a^{3} b^{3} d^{3} e^{3}+18304 a^{2} b^{4} d^{4} e^{2}-6656 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \sqrt {e x +d}}{3003 e^{7}}\) | \(377\) |
derivativedivides | \(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (2 a e b -2 b^{2} d \right ) b^{4} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a e b -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) b^{2}+\left (2 a e b -2 b^{2} d \right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )+2 \left (2 a e b -2 b^{2} d \right )^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {3}{2}}+2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \sqrt {e x +d}}{e^{7}}\) | \(455\) |
default | \(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (2 a e b -2 b^{2} d \right ) b^{4} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a e b -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) b^{2}+\left (2 a e b -2 b^{2} d \right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )+2 \left (2 a e b -2 b^{2} d \right )^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {3}{2}}+2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \sqrt {e x +d}}{e^{7}}\) | \(455\) |
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Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (159) = 318\).
Time = 0.34 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.97 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (231 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 6656 \, a b^{5} d^{5} e + 18304 \, a^{2} b^{4} d^{4} e^{2} - 27456 \, a^{3} b^{3} d^{3} e^{3} + 24024 \, a^{4} b^{2} d^{2} e^{4} - 12012 \, a^{5} b d e^{5} + 3003 \, a^{6} e^{6} - 126 \, {\left (2 \, b^{6} d e^{5} - 13 \, a b^{5} e^{6}\right )} x^{5} + 35 \, {\left (8 \, b^{6} d^{2} e^{4} - 52 \, a b^{5} d e^{5} + 143 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (16 \, b^{6} d^{3} e^{3} - 104 \, a b^{5} d^{2} e^{4} + 286 \, a^{2} b^{4} d e^{5} - 429 \, a^{3} b^{3} e^{6}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{4} e^{2} - 832 \, a b^{5} d^{3} e^{3} + 2288 \, a^{2} b^{4} d^{2} e^{4} - 3432 \, a^{3} b^{3} d e^{5} + 3003 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \, {\left (256 \, b^{6} d^{5} e - 1664 \, a b^{5} d^{4} e^{2} + 4576 \, a^{2} b^{4} d^{3} e^{3} - 6864 \, a^{3} b^{3} d^{2} e^{4} + 6006 \, a^{4} b^{2} d e^{5} - 3003 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{3003 \, e^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (168) = 336\).
Time = 1.23 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.72 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{6} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (6 a b^{5} e - 6 b^{6} d\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (15 a^{2} b^{4} e^{2} - 30 a b^{5} d e + 15 b^{6} d^{2}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (20 a^{3} b^{3} e^{3} - 60 a^{2} b^{4} d e^{2} + 60 a b^{5} d^{2} e - 20 b^{6} d^{3}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (15 a^{4} b^{2} e^{4} - 60 a^{3} b^{3} d e^{3} + 90 a^{2} b^{4} d^{2} e^{2} - 60 a b^{5} d^{3} e + 15 b^{6} d^{4}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (6 a^{5} b e^{5} - 30 a^{4} b^{2} d e^{4} + 60 a^{3} b^{3} d^{2} e^{3} - 60 a^{2} b^{4} d^{3} e^{2} + 30 a b^{5} d^{4} e - 6 b^{6} d^{5}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (a^{6} e^{6} - 6 a^{5} b d e^{5} + 15 a^{4} b^{2} d^{2} e^{4} - 20 a^{3} b^{3} d^{3} e^{3} + 15 a^{2} b^{4} d^{4} e^{2} - 6 a b^{5} d^{5} e + b^{6} d^{6}\right )}{e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{6} x + 3 a^{5} b x^{2} + 5 a^{4} b^{2} x^{3} + 5 a^{3} b^{3} x^{4} + 3 a^{2} b^{4} x^{5} + a b^{5} x^{6} + \frac {b^{6} x^{7}}{7}}{\sqrt {d}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (159) = 318\).
Time = 0.22 (sec) , antiderivative size = 540, normalized size of antiderivative = 2.98 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (15015 \, \sqrt {e x + d} a^{6} + 3003 \, {\left (\frac {10 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a b}{e} + \frac {{\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} b^{2}}{e^{2}}\right )} a^{4} + \frac {3432 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a^{3} b^{3}}{e^{3}} + 143 \, {\left (\frac {84 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{2} b^{2}}{e^{2}} + \frac {36 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a b^{3}}{e^{3}} + \frac {{\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{4}}{e^{4}}\right )} a^{2} + \frac {572 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a^{2} b^{4}}{e^{4}} + \frac {130 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} a b^{5}}{e^{5}} + \frac {5 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} b^{6}}{e^{6}}\right )}}{15015 \, e} \]
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Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (159) = 318\).
Time = 0.28 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.07 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (3003 \, \sqrt {e x + d} a^{6} + \frac {6006 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{5} b}{e} + \frac {3003 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{4} b^{2}}{e^{2}} + \frac {1716 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a^{3} b^{3}}{e^{3}} + \frac {143 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a^{2} b^{4}}{e^{4}} + \frac {26 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} a b^{5}}{e^{5}} + \frac {{\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} b^{6}}{e^{6}}\right )}}{3003 \, e} \]
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Time = 10.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2\,b^6\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {2\,{\left (a\,e-b\,d\right )}^6\,\sqrt {d+e\,x}}{e^7}+\frac {6\,b^2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{5/2}}{e^7}+\frac {40\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {10\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{3\,e^7}+\frac {4\,b\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{3/2}}{e^7} \]
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