Integrand size = 22, antiderivative size = 282 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}{e^7}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}{e^7}+\frac {6 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{5/2}}{5 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{7/2}}{7 e^7}+\frac {2 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{3 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{11/2}}{11 e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {712} \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 c (d+e x)^{9/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^7}+\frac {6 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7}+\frac {2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac {6 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7} \]
[In]
[Out]
Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2-b d e+a e^2\right )^3}{e^6 \sqrt {d+e x}}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}{e^6}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^{3/2}}{e^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^{5/2}}{e^6}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{7/2}}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{9/2}}{e^6}+\frac {c^3 (d+e x)^{11/2}}{e^6}\right ) \, dx \\ & = \frac {2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}{e^7}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}{e^7}+\frac {6 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{5/2}}{5 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{7/2}}{7 e^7}+\frac {2 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{3 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{11/2}}{11 e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (5 c^3 \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )+429 e^3 \left (35 a^3 e^3+35 a^2 b e^2 (-2 d+e x)+7 a b^2 e \left (8 d^2-4 d e x+3 e^2 x^2\right )+b^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )\right )+143 c e^2 \left (21 a^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+18 a b e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+b^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )-13 c^2 e \left (-11 a e \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+5 b \left (256 d^5-128 d^4 e x+96 d^3 e^2 x^2-80 d^2 e^3 x^3+70 d e^4 x^4-63 e^5 x^5\right )\right )\right )}{15015 e^7} \]
[In]
[Out]
Time = 0.29 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.16
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {e x +d}\, \left (\left (\frac {c^{3} x^{6}}{13}+\frac {\left (\frac {9 b x}{11}+a \right ) x^{4} c^{2}}{3}+\left (\frac {1}{3} b^{2} x^{4}+\frac {3}{5} a^{2} x^{2}+\frac {6}{7} a b \,x^{3}\right ) c +a^{3}+\frac {3 a \,b^{2} x^{2}}{5}+a^{2} b x +\frac {b^{3} x^{3}}{7}\right ) e^{6}-2 \left (\frac {6 c^{3} x^{5}}{143}+\left (\frac {5}{33} b \,x^{4}+\frac {4}{21} a \,x^{3}\right ) c^{2}+\left (\frac {4}{21} b^{2} x^{3}+\frac {18}{35} a b \,x^{2}+\frac {2}{5} a^{2} x \right ) c +b \left (\frac {3}{35} b^{2} x^{2}+\frac {2}{5} a b x +a^{2}\right )\right ) d \,e^{5}+\frac {8 \left (\frac {25 c^{3} x^{4}}{429}+\frac {2 x^{2} \left (\frac {25 b x}{33}+a \right ) c^{2}}{7}+\left (\frac {2}{7} b^{2} x^{2}+\frac {6}{7} a b x +a^{2}\right ) c +b^{2} \left (\frac {b x}{7}+a \right )\right ) d^{2} e^{4}}{5}-\frac {96 \left (\frac {50 c^{3} x^{3}}{1287}+\left (\frac {5}{33} b \,x^{2}+\frac {2}{9} a x \right ) c^{2}+b \left (\frac {2 b x}{9}+a \right ) c +\frac {b^{3}}{6}\right ) d^{3} e^{3}}{35}+\frac {128 c \left (\frac {15 c^{2} x^{2}}{143}+\left (\frac {5 b x}{11}+a \right ) c +b^{2}\right ) d^{4} e^{2}}{105}-\frac {256 \left (\frac {2 c x}{13}+b \right ) c^{2} d^{5} e}{231}+\frac {1024 c^{3} d^{6}}{3003}\right )}{e^{7}}\) | \(326\) |
derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c \left (b e -2 c d \right )+\left (b e -2 c d \right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )+2 \left (b e -2 c d \right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )+c \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {3}{2}}+2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {e x +d}}{e^{7}}\) | \(355\) |
default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c \left (b e -2 c d \right )+\left (b e -2 c d \right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )+2 \left (b e -2 c d \right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )+c \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {3}{2}}+2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {e x +d}}{e^{7}}\) | \(355\) |
gosper | \(\frac {2 \sqrt {e x +d}\, \left (1155 c^{3} x^{6} e^{6}+4095 b \,c^{2} e^{6} x^{5}-1260 c^{3} d \,e^{5} x^{5}+5005 a \,c^{2} e^{6} x^{4}+5005 b^{2} c \,e^{6} x^{4}-4550 b \,c^{2} d \,e^{5} x^{4}+1400 c^{3} d^{2} e^{4} x^{4}+12870 a b c \,e^{6} x^{3}-5720 a \,c^{2} d \,e^{5} x^{3}+2145 b^{3} e^{6} x^{3}-5720 b^{2} c d \,e^{5} x^{3}+5200 b \,c^{2} d^{2} e^{4} x^{3}-1600 c^{3} d^{3} e^{3} x^{3}+9009 a^{2} c \,e^{6} x^{2}+9009 a \,b^{2} e^{6} x^{2}-15444 a b c d \,e^{5} x^{2}+6864 a \,c^{2} d^{2} e^{4} x^{2}-2574 b^{3} d \,e^{5} x^{2}+6864 b^{2} c \,d^{2} e^{4} x^{2}-6240 b \,c^{2} d^{3} e^{3} x^{2}+1920 c^{3} d^{4} e^{2} x^{2}+15015 a^{2} b \,e^{6} x -12012 a^{2} c d \,e^{5} x -12012 a \,b^{2} d \,e^{5} x +20592 a b c \,d^{2} e^{4} x -9152 a \,c^{2} d^{3} e^{3} x +3432 b^{3} d^{2} e^{4} x -9152 b^{2} c \,d^{3} e^{3} x +8320 b \,c^{2} d^{4} e^{2} x -2560 c^{3} d^{5} e x +15015 a^{3} e^{6}-30030 a^{2} b d \,e^{5}+24024 a^{2} c \,d^{2} e^{4}+24024 a \,b^{2} d^{2} e^{4}-41184 a b c \,d^{3} e^{3}+18304 a \,c^{2} d^{4} e^{2}-6864 b^{3} d^{3} e^{3}+18304 b^{2} c \,d^{4} e^{2}-16640 b \,c^{2} d^{5} e +5120 c^{3} d^{6}\right )}{15015 e^{7}}\) | \(495\) |
trager | \(\frac {2 \sqrt {e x +d}\, \left (1155 c^{3} x^{6} e^{6}+4095 b \,c^{2} e^{6} x^{5}-1260 c^{3} d \,e^{5} x^{5}+5005 a \,c^{2} e^{6} x^{4}+5005 b^{2} c \,e^{6} x^{4}-4550 b \,c^{2} d \,e^{5} x^{4}+1400 c^{3} d^{2} e^{4} x^{4}+12870 a b c \,e^{6} x^{3}-5720 a \,c^{2} d \,e^{5} x^{3}+2145 b^{3} e^{6} x^{3}-5720 b^{2} c d \,e^{5} x^{3}+5200 b \,c^{2} d^{2} e^{4} x^{3}-1600 c^{3} d^{3} e^{3} x^{3}+9009 a^{2} c \,e^{6} x^{2}+9009 a \,b^{2} e^{6} x^{2}-15444 a b c d \,e^{5} x^{2}+6864 a \,c^{2} d^{2} e^{4} x^{2}-2574 b^{3} d \,e^{5} x^{2}+6864 b^{2} c \,d^{2} e^{4} x^{2}-6240 b \,c^{2} d^{3} e^{3} x^{2}+1920 c^{3} d^{4} e^{2} x^{2}+15015 a^{2} b \,e^{6} x -12012 a^{2} c d \,e^{5} x -12012 a \,b^{2} d \,e^{5} x +20592 a b c \,d^{2} e^{4} x -9152 a \,c^{2} d^{3} e^{3} x +3432 b^{3} d^{2} e^{4} x -9152 b^{2} c \,d^{3} e^{3} x +8320 b \,c^{2} d^{4} e^{2} x -2560 c^{3} d^{5} e x +15015 a^{3} e^{6}-30030 a^{2} b d \,e^{5}+24024 a^{2} c \,d^{2} e^{4}+24024 a \,b^{2} d^{2} e^{4}-41184 a b c \,d^{3} e^{3}+18304 a \,c^{2} d^{4} e^{2}-6864 b^{3} d^{3} e^{3}+18304 b^{2} c \,d^{4} e^{2}-16640 b \,c^{2} d^{5} e +5120 c^{3} d^{6}\right )}{15015 e^{7}}\) | \(495\) |
risch | \(\frac {2 \sqrt {e x +d}\, \left (1155 c^{3} x^{6} e^{6}+4095 b \,c^{2} e^{6} x^{5}-1260 c^{3} d \,e^{5} x^{5}+5005 a \,c^{2} e^{6} x^{4}+5005 b^{2} c \,e^{6} x^{4}-4550 b \,c^{2} d \,e^{5} x^{4}+1400 c^{3} d^{2} e^{4} x^{4}+12870 a b c \,e^{6} x^{3}-5720 a \,c^{2} d \,e^{5} x^{3}+2145 b^{3} e^{6} x^{3}-5720 b^{2} c d \,e^{5} x^{3}+5200 b \,c^{2} d^{2} e^{4} x^{3}-1600 c^{3} d^{3} e^{3} x^{3}+9009 a^{2} c \,e^{6} x^{2}+9009 a \,b^{2} e^{6} x^{2}-15444 a b c d \,e^{5} x^{2}+6864 a \,c^{2} d^{2} e^{4} x^{2}-2574 b^{3} d \,e^{5} x^{2}+6864 b^{2} c \,d^{2} e^{4} x^{2}-6240 b \,c^{2} d^{3} e^{3} x^{2}+1920 c^{3} d^{4} e^{2} x^{2}+15015 a^{2} b \,e^{6} x -12012 a^{2} c d \,e^{5} x -12012 a \,b^{2} d \,e^{5} x +20592 a b c \,d^{2} e^{4} x -9152 a \,c^{2} d^{3} e^{3} x +3432 b^{3} d^{2} e^{4} x -9152 b^{2} c \,d^{3} e^{3} x +8320 b \,c^{2} d^{4} e^{2} x -2560 c^{3} d^{5} e x +15015 a^{3} e^{6}-30030 a^{2} b d \,e^{5}+24024 a^{2} c \,d^{2} e^{4}+24024 a \,b^{2} d^{2} e^{4}-41184 a b c \,d^{3} e^{3}+18304 a \,c^{2} d^{4} e^{2}-6864 b^{3} d^{3} e^{3}+18304 b^{2} c \,d^{4} e^{2}-16640 b \,c^{2} d^{5} e +5120 c^{3} d^{6}\right )}{15015 e^{7}}\) | \(495\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (1155 \, c^{3} e^{6} x^{6} + 5120 \, c^{3} d^{6} - 16640 \, b c^{2} d^{5} e - 30030 \, a^{2} b d e^{5} + 15015 \, a^{3} e^{6} + 18304 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 6864 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 24024 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 315 \, {\left (4 \, c^{3} d e^{5} - 13 \, b c^{2} e^{6}\right )} x^{5} + 35 \, {\left (40 \, c^{3} d^{2} e^{4} - 130 \, b c^{2} d e^{5} + 143 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 5 \, {\left (320 \, c^{3} d^{3} e^{3} - 1040 \, b c^{2} d^{2} e^{4} + 1144 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - 429 \, {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 3 \, {\left (640 \, c^{3} d^{4} e^{2} - 2080 \, b c^{2} d^{3} e^{3} + 2288 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 858 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 3003 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - {\left (2560 \, c^{3} d^{5} e - 8320 \, b c^{2} d^{4} e^{2} - 15015 \, a^{2} b e^{6} + 9152 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 3432 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 12012 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt {e x + d}}{15015 \, e^{7}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (279) = 558\).
Time = 1.15 (sec) , antiderivative size = 631, normalized size of antiderivative = 2.24 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (3 b c^{2} e - 6 c^{3} d\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (3 a^{2} c e^{4} + 3 a b^{2} e^{4} - 18 a b c d e^{3} + 18 a c^{2} d^{2} e^{2} - 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (3 a^{2} b e^{5} - 6 a^{2} c d e^{4} - 6 a b^{2} d e^{4} + 18 a b c d^{2} e^{3} - 12 a c^{2} d^{3} e^{2} + 3 b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 15 b c^{2} d^{4} e - 6 c^{3} d^{5}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (a^{3} e^{6} - 3 a^{2} b d e^{5} + 3 a^{2} c d^{2} e^{4} + 3 a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 3 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}\right )}{e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{3} x + \frac {3 a^{2} b x^{2}}{2} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} + \frac {x^{5} \cdot \left (3 a c^{2} + 3 b^{2} c\right )}{5} + \frac {x^{4} \cdot \left (6 a b c + b^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 a^{2} c + 3 a b^{2}\right )}{3}}{\sqrt {d}} & \text {otherwise} \end {cases} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 525 vs. \(2 (258) = 516\).
Time = 0.20 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.86 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (15015 \, \sqrt {e x + d} a^{3} + 3003 \, a^{2} {\left (\frac {5 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} 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 )} c}{e^{2}}\right )} + 143 \, a {\left (\frac {21 \, {\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}} + \frac {18 \, {\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 )} b c}{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 )} c^{2}}{e^{4}}\right )} + \frac {429 \, {\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 )} 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 )} b^{2} c}{e^{4}} + \frac {65 \, {\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 )} b c^{2}}{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 )} c^{3}}{e^{6}}\right )}}{15015 \, e} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (258) = 516\).
Time = 0.27 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.87 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (15015 \, \sqrt {e x + d} a^{3} + \frac {15015 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{2} 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 b^{2}}{e^{2}} + \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^{2} c}{e^{2}} + \frac {429 \, {\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 )} b^{3}}{e^{3}} + \frac {2574 \, {\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 c}{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 )} b^{2} c}{e^{4}} + \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 c^{2}}{e^{4}} + \frac {65 \, {\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 )} b c^{2}}{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 )} c^{3}}{e^{6}}\right )}}{15015 \, e} \]
[In]
[Out]
Time = 9.78 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {{\left (d+e\,x\right )}^{5/2}\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{5\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{9\,e^7}+\frac {2\,\sqrt {d+e\,x}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{7\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{e^7} \]
[In]
[Out]