Integrand size = 28, antiderivative size = 252 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^6}+\frac {4 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{9 e^6}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{11/2}}{11 e^6}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{13/2}}{13 e^6}-\frac {2 c^2 (2 c d-b e) (d+e x)^{15/2}}{3 e^6}+\frac {4 c^3 (d+e x)^{17/2}}{17 e^6} \] Output:
-2/7*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(7/2)/e^6+4/9*(a*e^2-b*d*e +c*d^2)*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(9/2)/e^6-2/11*(-b*e+ 2*c*d)*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))*(e*x+d)^(11/2)/e^6+8/13*c *(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(13/2)/e^6-2/3*c^2*(-b*e+2*c *d)*(e*x+d)^(15/2)/e^6+4/17*c^3*(e*x+d)^(17/2)/e^6
Time = 0.34 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.15 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{7/2} \left (-2 c^3 \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )+221 b e^3 \left (99 a^2 e^2+22 a b e (-2 d+7 e x)+b^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-34 c e^2 \left (143 a^2 e^2 (2 d-7 e x)-39 a b e \left (8 d^2-28 d e x+63 e^2 x^2\right )+6 b^2 \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )\right )+17 c^2 e \left (12 a e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )\right )}{153153 e^6} \] Input:
Integrate[(b + 2*c*x)*(d + e*x)^(5/2)*(a + b*x + c*x^2)^2,x]
Output:
(2*(d + e*x)^(7/2)*(-2*c^3*(256*d^5 - 896*d^4*e*x + 2016*d^3*e^2*x^2 - 369 6*d^2*e^3*x^3 + 6006*d*e^4*x^4 - 9009*e^5*x^5) + 221*b*e^3*(99*a^2*e^2 + 2 2*a*b*e*(-2*d + 7*e*x) + b^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) - 34*c*e^2*( 143*a^2*e^2*(2*d - 7*e*x) - 39*a*b*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 6*b ^2*(16*d^3 - 56*d^2*e*x + 126*d*e^2*x^2 - 231*e^3*x^3)) + 17*c^2*e*(12*a*e *(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + b*(128*d^4 - 448*d ^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4))))/(153153*e^6)
Time = 0.76 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1195, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {4 c (d+e x)^{11/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^5}+\frac {(d+e x)^{9/2} (2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^5}+\frac {2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^5}+\frac {(d+e x)^{5/2} (b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^5}-\frac {5 c^2 (d+e x)^{13/2} (2 c d-b e)}{e^5}+\frac {2 c^3 (d+e x)^{15/2}}{e^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {8 c (d+e x)^{13/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{13 e^6}-\frac {2 (d+e x)^{11/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{11 e^6}+\frac {4 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{9 e^6}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^6}-\frac {2 c^2 (d+e x)^{15/2} (2 c d-b e)}{3 e^6}+\frac {4 c^3 (d+e x)^{17/2}}{17 e^6}\) |
Input:
Int[(b + 2*c*x)*(d + e*x)^(5/2)*(a + b*x + c*x^2)^2,x]
Output:
(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(7/2))/(7*e^6) + (4* (c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x )^(9/2))/(9*e^6) - (2*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^(11/2))/(11*e^6) + (8*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b *d - a*e))*(d + e*x)^(13/2))/(13*e^6) - (2*c^2*(2*c*d - b*e)*(d + e*x)^(15 /2))/(3*e^6) + (4*c^3*(d + e*x)^(17/2))/(17*e^6)
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 2.03 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(\frac {2 \left (\left (\frac {14 c^{3} x^{5}}{17}+\left (\frac {7}{3} b \,x^{4}+\frac {28}{13} a \,x^{3}\right ) c^{2}+\left (\frac {28}{13} b^{2} x^{3}+\frac {14}{9} a^{2} x +\frac {42}{11} a b \,x^{2}\right ) c +b \left (\frac {7}{11} b^{2} x^{2}+a^{2}+\frac {14}{9} a b x \right )\right ) e^{5}-\frac {4 d \left (\frac {21 c^{3} x^{4}}{17}+\frac {378 x^{2} \left (\frac {11 b x}{9}+a \right ) c^{2}}{143}+\left (\frac {378}{143} b^{2} x^{2}+\frac {42}{11} a b x +a^{2}\right ) c +b^{2} \left (\frac {7 b x}{11}+a \right )\right ) e^{4}}{9}+\frac {16 d^{2} \left (\frac {154 c^{3} x^{3}}{221}+\frac {14 x \left (\frac {3 b x}{2}+a \right ) c^{2}}{13}+b \left (\frac {14 b x}{13}+a \right ) c +\frac {b^{3}}{6}\right ) e^{3}}{33}-\frac {64 d^{3} \left (\frac {21 c^{2} x^{2}}{17}+\left (\frac {7 b x}{3}+a \right ) c +b^{2}\right ) c \,e^{2}}{429}+\frac {128 d^{4} \left (\frac {14 c x}{17}+b \right ) c^{2} e}{1287}-\frac {512 d^{5} c^{3}}{21879}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7 e^{6}}\) | \(240\) |
derivativedivides | \(\frac {\frac {4 c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {15}{2}}}{3}+\frac {2 \left (2 \left (b e -2 c d \right )^{2} c +2 c \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (b e -2 c d \right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )+4 c \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (b e -2 c d \right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )+2 c \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (b e -2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{6}}\) | \(265\) |
default | \(\frac {\frac {4 c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {15}{2}}}{3}+\frac {2 \left (2 \left (b e -2 c d \right )^{2} c +2 c \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (b e -2 c d \right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )+4 c \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (b e -2 c d \right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )+2 c \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (b e -2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{6}}\) | \(265\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (18018 e^{5} x^{5} c^{3}+51051 e^{5} x^{4} c^{2} b -12012 x^{4} d \,e^{4} c^{3}+47124 e^{5} a \,c^{2} x^{3}+47124 e^{5} x^{3} b^{2} c -31416 x^{3} d \,e^{4} c^{2} b +7392 d^{2} e^{3} c^{3} x^{3}+83538 e^{5} a b c \,x^{2}-25704 d \,e^{4} a \,c^{2} x^{2}+13923 x^{2} e^{5} b^{3}-25704 x^{2} d \,e^{4} b^{2} c +17136 d^{2} e^{3} b \,c^{2} x^{2}-4032 d^{3} e^{2} c^{3} x^{2}+34034 e^{5} a^{2} c x +34034 e^{5} x a \,b^{2}-37128 d \,e^{4} a b c x +11424 d^{2} e^{3} a \,c^{2} x -6188 x d \,e^{4} b^{3}+11424 d^{2} e^{3} b^{2} c x -7616 d^{3} e^{2} c^{2} b x +1792 c^{3} d^{4} e x +21879 a^{2} b \,e^{5}-9724 d \,e^{4} a^{2} c -9724 a \,b^{2} d \,e^{4}+10608 a b c \,d^{2} e^{3}-3264 d^{3} e^{2} a \,c^{2}+1768 b^{3} d^{2} e^{3}-3264 b^{2} c \,d^{3} e^{2}+2176 b \,c^{2} d^{4} e -512 d^{5} c^{3}\right )}{153153 e^{6}}\) | \(359\) |
orering | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (18018 e^{5} x^{5} c^{3}+51051 e^{5} x^{4} c^{2} b -12012 x^{4} d \,e^{4} c^{3}+47124 e^{5} a \,c^{2} x^{3}+47124 e^{5} x^{3} b^{2} c -31416 x^{3} d \,e^{4} c^{2} b +7392 d^{2} e^{3} c^{3} x^{3}+83538 e^{5} a b c \,x^{2}-25704 d \,e^{4} a \,c^{2} x^{2}+13923 x^{2} e^{5} b^{3}-25704 x^{2} d \,e^{4} b^{2} c +17136 d^{2} e^{3} b \,c^{2} x^{2}-4032 d^{3} e^{2} c^{3} x^{2}+34034 e^{5} a^{2} c x +34034 e^{5} x a \,b^{2}-37128 d \,e^{4} a b c x +11424 d^{2} e^{3} a \,c^{2} x -6188 x d \,e^{4} b^{3}+11424 d^{2} e^{3} b^{2} c x -7616 d^{3} e^{2} c^{2} b x +1792 c^{3} d^{4} e x +21879 a^{2} b \,e^{5}-9724 d \,e^{4} a^{2} c -9724 a \,b^{2} d \,e^{4}+10608 a b c \,d^{2} e^{3}-3264 d^{3} e^{2} a \,c^{2}+1768 b^{3} d^{2} e^{3}-3264 b^{2} c \,d^{3} e^{2}+2176 b \,c^{2} d^{4} e -512 d^{5} c^{3}\right )}{153153 e^{6}}\) | \(359\) |
trager | \(\frac {2 \left (18018 c^{3} e^{8} x^{8}+51051 b \,c^{2} e^{8} x^{7}+42042 c^{3} d \,e^{7} x^{7}+47124 a \,c^{2} e^{8} x^{6}+47124 b^{2} c \,e^{8} x^{6}+121737 b \,c^{2} d \,e^{7} x^{6}+25410 c^{3} d^{2} e^{6} x^{6}+83538 a b c \,e^{8} x^{5}+115668 a \,c^{2} d \,e^{7} x^{5}+13923 b^{3} e^{8} x^{5}+115668 b^{2} c d \,e^{7} x^{5}+76041 b \,c^{2} d^{2} e^{6} x^{5}+126 d^{3} c^{3} e^{5} x^{5}+34034 a^{2} c \,e^{8} x^{4}+34034 a \,b^{2} e^{8} x^{4}+213486 a b c d \,e^{7} x^{4}+75684 a \,c^{2} d^{2} e^{6} x^{4}+35581 b^{3} d \,e^{7} x^{4}+75684 b^{2} c \,d^{2} e^{6} x^{4}+595 b \,c^{2} d^{3} e^{5} x^{4}-140 c^{3} d^{4} e^{4} x^{4}+21879 a^{2} b \,e^{8} x^{3}+92378 a^{2} c d \,e^{7} x^{3}+92378 a \,b^{2} d \,e^{7} x^{3}+149838 a b c \,d^{2} e^{6} x^{3}+1020 a \,c^{2} d^{3} e^{5} x^{3}+24973 b^{3} d^{2} e^{6} x^{3}+1020 b^{2} c \,d^{3} e^{5} x^{3}-680 b \,c^{2} d^{4} e^{4} x^{3}+160 d^{5} c^{3} e^{3} x^{3}+65637 a^{2} b d \,e^{7} x^{2}+72930 a^{2} c \,d^{2} e^{6} x^{2}+72930 a \,b^{2} d^{2} e^{6} x^{2}+3978 a b c \,d^{3} e^{5} x^{2}-1224 a \,c^{2} d^{4} e^{4} x^{2}+663 b^{3} d^{3} e^{5} x^{2}-1224 b^{2} c \,d^{4} e^{4} x^{2}+816 b \,c^{2} d^{5} e^{3} x^{2}-192 c^{3} d^{6} e^{2} x^{2}+65637 a^{2} b \,d^{2} e^{6} x +4862 a^{2} c \,d^{3} e^{5} x +4862 a \,b^{2} d^{3} e^{5} x -5304 a b c \,d^{4} e^{4} x +1632 a \,c^{2} d^{5} e^{3} x -884 b^{3} d^{4} e^{4} x +1632 b^{2} c \,d^{5} e^{3} x -1088 b \,c^{2} d^{6} e^{2} x +256 c^{3} d^{7} e x +21879 a^{2} b \,d^{3} e^{5}-9724 a^{2} c \,d^{4} e^{4}-9724 a \,b^{2} d^{4} e^{4}+10608 a b c \,d^{5} e^{3}-3264 a \,c^{2} d^{6} e^{2}+1768 b^{3} d^{5} e^{3}-3264 b^{2} c \,d^{6} e^{2}+2176 b \,c^{2} d^{7} e -512 c^{3} d^{8}\right ) \sqrt {e x +d}}{153153 e^{6}}\) | \(751\) |
risch | \(\frac {2 \left (18018 c^{3} e^{8} x^{8}+51051 b \,c^{2} e^{8} x^{7}+42042 c^{3} d \,e^{7} x^{7}+47124 a \,c^{2} e^{8} x^{6}+47124 b^{2} c \,e^{8} x^{6}+121737 b \,c^{2} d \,e^{7} x^{6}+25410 c^{3} d^{2} e^{6} x^{6}+83538 a b c \,e^{8} x^{5}+115668 a \,c^{2} d \,e^{7} x^{5}+13923 b^{3} e^{8} x^{5}+115668 b^{2} c d \,e^{7} x^{5}+76041 b \,c^{2} d^{2} e^{6} x^{5}+126 d^{3} c^{3} e^{5} x^{5}+34034 a^{2} c \,e^{8} x^{4}+34034 a \,b^{2} e^{8} x^{4}+213486 a b c d \,e^{7} x^{4}+75684 a \,c^{2} d^{2} e^{6} x^{4}+35581 b^{3} d \,e^{7} x^{4}+75684 b^{2} c \,d^{2} e^{6} x^{4}+595 b \,c^{2} d^{3} e^{5} x^{4}-140 c^{3} d^{4} e^{4} x^{4}+21879 a^{2} b \,e^{8} x^{3}+92378 a^{2} c d \,e^{7} x^{3}+92378 a \,b^{2} d \,e^{7} x^{3}+149838 a b c \,d^{2} e^{6} x^{3}+1020 a \,c^{2} d^{3} e^{5} x^{3}+24973 b^{3} d^{2} e^{6} x^{3}+1020 b^{2} c \,d^{3} e^{5} x^{3}-680 b \,c^{2} d^{4} e^{4} x^{3}+160 d^{5} c^{3} e^{3} x^{3}+65637 a^{2} b d \,e^{7} x^{2}+72930 a^{2} c \,d^{2} e^{6} x^{2}+72930 a \,b^{2} d^{2} e^{6} x^{2}+3978 a b c \,d^{3} e^{5} x^{2}-1224 a \,c^{2} d^{4} e^{4} x^{2}+663 b^{3} d^{3} e^{5} x^{2}-1224 b^{2} c \,d^{4} e^{4} x^{2}+816 b \,c^{2} d^{5} e^{3} x^{2}-192 c^{3} d^{6} e^{2} x^{2}+65637 a^{2} b \,d^{2} e^{6} x +4862 a^{2} c \,d^{3} e^{5} x +4862 a \,b^{2} d^{3} e^{5} x -5304 a b c \,d^{4} e^{4} x +1632 a \,c^{2} d^{5} e^{3} x -884 b^{3} d^{4} e^{4} x +1632 b^{2} c \,d^{5} e^{3} x -1088 b \,c^{2} d^{6} e^{2} x +256 c^{3} d^{7} e x +21879 a^{2} b \,d^{3} e^{5}-9724 a^{2} c \,d^{4} e^{4}-9724 a \,b^{2} d^{4} e^{4}+10608 a b c \,d^{5} e^{3}-3264 a \,c^{2} d^{6} e^{2}+1768 b^{3} d^{5} e^{3}-3264 b^{2} c \,d^{6} e^{2}+2176 b \,c^{2} d^{7} e -512 c^{3} d^{8}\right ) \sqrt {e x +d}}{153153 e^{6}}\) | \(751\) |
Input:
int((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
2/7*((14/17*c^3*x^5+(7/3*b*x^4+28/13*a*x^3)*c^2+(28/13*b^2*x^3+14/9*a^2*x+ 42/11*a*b*x^2)*c+b*(7/11*b^2*x^2+a^2+14/9*a*b*x))*e^5-4/9*d*(21/17*c^3*x^4 +378/143*x^2*(11/9*b*x+a)*c^2+(378/143*b^2*x^2+42/11*a*b*x+a^2)*c+b^2*(7/1 1*b*x+a))*e^4+16/33*d^2*(154/221*c^3*x^3+14/13*x*(3/2*b*x+a)*c^2+b*(14/13* b*x+a)*c+1/6*b^3)*e^3-64/429*d^3*(21/17*c^2*x^2+(7/3*b*x+a)*c+b^2)*c*e^2+1 28/1287*d^4*(14/17*c*x+b)*c^2*e-512/21879*d^5*c^3)*(e*x+d)^(7/2)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (228) = 456\).
Time = 0.08 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.34 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (18018 \, c^{3} e^{8} x^{8} - 512 \, c^{3} d^{8} + 2176 \, b c^{2} d^{7} e + 21879 \, a^{2} b d^{3} e^{5} - 3264 \, {\left (b^{2} c + a c^{2}\right )} d^{6} e^{2} + 1768 \, {\left (b^{3} + 6 \, a b c\right )} d^{5} e^{3} - 9724 \, {\left (a b^{2} + a^{2} c\right )} d^{4} e^{4} + 3003 \, {\left (14 \, c^{3} d e^{7} + 17 \, b c^{2} e^{8}\right )} x^{7} + 231 \, {\left (110 \, c^{3} d^{2} e^{6} + 527 \, b c^{2} d e^{7} + 204 \, {\left (b^{2} c + a c^{2}\right )} e^{8}\right )} x^{6} + 63 \, {\left (2 \, c^{3} d^{3} e^{5} + 1207 \, b c^{2} d^{2} e^{6} + 1836 \, {\left (b^{2} c + a c^{2}\right )} d e^{7} + 221 \, {\left (b^{3} + 6 \, a b c\right )} e^{8}\right )} x^{5} - 7 \, {\left (20 \, c^{3} d^{4} e^{4} - 85 \, b c^{2} d^{3} e^{5} - 10812 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{6} - 5083 \, {\left (b^{3} + 6 \, a b c\right )} d e^{7} - 4862 \, {\left (a b^{2} + a^{2} c\right )} e^{8}\right )} x^{4} + {\left (160 \, c^{3} d^{5} e^{3} - 680 \, b c^{2} d^{4} e^{4} + 21879 \, a^{2} b e^{8} + 1020 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{5} + 24973 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{6} + 92378 \, {\left (a b^{2} + a^{2} c\right )} d e^{7}\right )} x^{3} - 3 \, {\left (64 \, c^{3} d^{6} e^{2} - 272 \, b c^{2} d^{5} e^{3} - 21879 \, a^{2} b d e^{7} + 408 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{4} - 221 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{5} - 24310 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{6}\right )} x^{2} + {\left (256 \, c^{3} d^{7} e - 1088 \, b c^{2} d^{6} e^{2} + 65637 \, a^{2} b d^{2} e^{6} + 1632 \, {\left (b^{2} c + a c^{2}\right )} d^{5} e^{3} - 884 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} e^{4} + 4862 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e^{5}\right )} x\right )} \sqrt {e x + d}}{153153 \, e^{6}} \] Input:
integrate((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
2/153153*(18018*c^3*e^8*x^8 - 512*c^3*d^8 + 2176*b*c^2*d^7*e + 21879*a^2*b *d^3*e^5 - 3264*(b^2*c + a*c^2)*d^6*e^2 + 1768*(b^3 + 6*a*b*c)*d^5*e^3 - 9 724*(a*b^2 + a^2*c)*d^4*e^4 + 3003*(14*c^3*d*e^7 + 17*b*c^2*e^8)*x^7 + 231 *(110*c^3*d^2*e^6 + 527*b*c^2*d*e^7 + 204*(b^2*c + a*c^2)*e^8)*x^6 + 63*(2 *c^3*d^3*e^5 + 1207*b*c^2*d^2*e^6 + 1836*(b^2*c + a*c^2)*d*e^7 + 221*(b^3 + 6*a*b*c)*e^8)*x^5 - 7*(20*c^3*d^4*e^4 - 85*b*c^2*d^3*e^5 - 10812*(b^2*c + a*c^2)*d^2*e^6 - 5083*(b^3 + 6*a*b*c)*d*e^7 - 4862*(a*b^2 + a^2*c)*e^8)* x^4 + (160*c^3*d^5*e^3 - 680*b*c^2*d^4*e^4 + 21879*a^2*b*e^8 + 1020*(b^2*c + a*c^2)*d^3*e^5 + 24973*(b^3 + 6*a*b*c)*d^2*e^6 + 92378*(a*b^2 + a^2*c)* d*e^7)*x^3 - 3*(64*c^3*d^6*e^2 - 272*b*c^2*d^5*e^3 - 21879*a^2*b*d*e^7 + 4 08*(b^2*c + a*c^2)*d^4*e^4 - 221*(b^3 + 6*a*b*c)*d^3*e^5 - 24310*(a*b^2 + a^2*c)*d^2*e^6)*x^2 + (256*c^3*d^7*e - 1088*b*c^2*d^6*e^2 + 65637*a^2*b*d^ 2*e^6 + 1632*(b^2*c + a*c^2)*d^5*e^3 - 884*(b^3 + 6*a*b*c)*d^4*e^4 + 4862* (a*b^2 + a^2*c)*d^3*e^5)*x)*sqrt(e*x + d)/e^6
Time = 1.63 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.68 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\begin {cases} \frac {2 \cdot \left (\frac {2 c^{3} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{5}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (5 b c^{2} e - 10 c^{3} d\right )}{15 e^{5}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (4 a c^{2} e^{2} + 4 b^{2} c e^{2} - 20 b c^{2} d e + 20 c^{3} d^{2}\right )}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (2 a^{2} c e^{4} + 2 a b^{2} e^{4} - 12 a b c d e^{3} + 12 a c^{2} d^{2} e^{2} - 2 b^{3} d e^{3} + 12 b^{2} c d^{2} e^{2} - 20 b c^{2} d^{3} e + 10 c^{3} d^{4}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (a^{2} b e^{5} - 2 a^{2} c d e^{4} - 2 a b^{2} d e^{4} + 6 a b c d^{2} e^{3} - 4 a c^{2} d^{3} e^{2} + b^{3} d^{2} e^{3} - 4 b^{2} c d^{3} e^{2} + 5 b c^{2} d^{4} e - 2 c^{3} d^{5}\right )}{7 e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {d^{\frac {5}{2}} \left (a + b x + c x^{2}\right )^{3}}{3} & \text {otherwise} \end {cases} \] Input:
integrate((2*c*x+b)*(e*x+d)**(5/2)*(c*x**2+b*x+a)**2,x)
Output:
Piecewise((2*(2*c**3*(d + e*x)**(17/2)/(17*e**5) + (d + e*x)**(15/2)*(5*b* c**2*e - 10*c**3*d)/(15*e**5) + (d + e*x)**(13/2)*(4*a*c**2*e**2 + 4*b**2* c*e**2 - 20*b*c**2*d*e + 20*c**3*d**2)/(13*e**5) + (d + e*x)**(11/2)*(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)/(11*e**5) + (d + e*x)**(9/2)*(2*a**2*c*e**4 + 2*a*b**2 *e**4 - 12*a*b*c*d*e**3 + 12*a*c**2*d**2*e**2 - 2*b**3*d*e**3 + 12*b**2*c* d**2*e**2 - 20*b*c**2*d**3*e + 10*c**3*d**4)/(9*e**5) + (d + e*x)**(7/2)*( a**2*b*e**5 - 2*a**2*c*d*e**4 - 2*a*b**2*d*e**4 + 6*a*b*c*d**2*e**3 - 4*a* c**2*d**3*e**2 + b**3*d**2*e**3 - 4*b**2*c*d**3*e**2 + 5*b*c**2*d**4*e - 2 *c**3*d**5)/(7*e**5))/e, Ne(e, 0)), (d**(5/2)*(a + b*x + c*x**2)**3/3, Tru e))
Time = 0.04 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.22 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (18018 \, {\left (e x + d\right )}^{\frac {17}{2}} c^{3} - 51051 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 47124 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 13923 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 34034 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 21879 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{153153 \, e^{6}} \] Input:
integrate((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^2,x, algorithm="maxima")
Output:
2/153153*(18018*(e*x + d)^(17/2)*c^3 - 51051*(2*c^3*d - b*c^2*e)*(e*x + d) ^(15/2) + 47124*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^ (13/2) - 13923*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 - ( b^3 + 6*a*b*c)*e^3)*(e*x + d)^(11/2) + 34034*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*(b^2*c + a*c^2)*d^2*e^2 - (b^3 + 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)* (e*x + d)^(9/2) - 21879*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*(e*x + d)^(7/2))/e^6
Leaf count of result is larger than twice the leaf count of optimal. 2306 vs. \(2 (228) = 456\).
Time = 0.22 (sec) , antiderivative size = 2306, normalized size of antiderivative = 9.15 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
2/765765*(765765*sqrt(e*x + d)*a^2*b*d^3 + 765765*((e*x + d)^(3/2) - 3*sqr t(e*x + d)*d)*a^2*b*d^2 + 510510*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a*b ^2*d^3/e + 510510*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*c*d^3/e + 1531 53*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*b *d + 51051*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^ 2)*b^3*d^3/e^2 + 306306*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqr t(e*x + d)*d^2)*a*b*c*d^3/e^2 + 306306*(3*(e*x + d)^(5/2) - 10*(e*x + d)^( 3/2)*d + 15*sqrt(e*x + d)*d^2)*a*b^2*d^2/e + 306306*(3*(e*x + d)^(5/2) - 1 0*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*c*d^2/e + 21879*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d )*d^3)*a^2*b + 87516*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*b^2*c*d^3/e^3 + 87516*(5*(e*x + d)^( 7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^ 3)*a*c^2*d^3/e^3 + 65637*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e *x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*b^3*d^2/e^2 + 393822*(5*(e*x + d )^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d) *d^3)*a*b*c*d^2/e^2 + 131274*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 3 5*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a*b^2*d/e + 131274*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^2*c*d/e + 12155*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d +...
Time = 0.11 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.06 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {{\left (d+e\,x\right )}^{9/2}\,\left (4\,a^2\,c\,e^4+4\,a\,b^2\,e^4-24\,a\,b\,c\,d\,e^3+24\,a\,c^2\,d^2\,e^2-4\,b^3\,d\,e^3+24\,b^2\,c\,d^2\,e^2-40\,b\,c^2\,d^3\,e+20\,c^3\,d^4\right )}{9\,e^6}+\frac {4\,c^3\,{\left (d+e\,x\right )}^{17/2}}{17\,e^6}-\frac {\left (20\,c^3\,d-10\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^6}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (8\,b^2\,c\,e^2-40\,b\,c^2\,d\,e+40\,c^3\,d^2+8\,a\,c^2\,e^2\right )}{13\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{11\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{7\,e^6} \] Input:
int((b + 2*c*x)*(d + e*x)^(5/2)*(a + b*x + c*x^2)^2,x)
Output:
((d + e*x)^(9/2)*(20*c^3*d^4 + 4*a*b^2*e^4 + 4*a^2*c*e^4 - 4*b^3*d*e^3 + 2 4*a*c^2*d^2*e^2 + 24*b^2*c*d^2*e^2 - 40*b*c^2*d^3*e - 24*a*b*c*d*e^3))/(9* e^6) + (4*c^3*(d + e*x)^(17/2))/(17*e^6) - ((20*c^3*d - 10*b*c^2*e)*(d + e *x)^(15/2))/(15*e^6) + ((d + e*x)^(13/2)*(40*c^3*d^2 + 8*a*c^2*e^2 + 8*b^2 *c*e^2 - 40*b*c^2*d*e))/(13*e^6) + (2*(b*e - 2*c*d)*(d + e*x)^(11/2)*(b^2* e^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c*d*e))/(11*e^6) + (2*(b*e - 2*c*d)*(d + e*x)^(7/2)*(a*e^2 + c*d^2 - b*d*e)^2)/(7*e^6)
Time = 0.22 (sec) , antiderivative size = 749, normalized size of antiderivative = 2.97 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \sqrt {e x +d}\, \left (18018 c^{3} e^{8} x^{8}+51051 b \,c^{2} e^{8} x^{7}+42042 c^{3} d \,e^{7} x^{7}+47124 a \,c^{2} e^{8} x^{6}+47124 b^{2} c \,e^{8} x^{6}+121737 b \,c^{2} d \,e^{7} x^{6}+25410 c^{3} d^{2} e^{6} x^{6}+83538 a b c \,e^{8} x^{5}+115668 a \,c^{2} d \,e^{7} x^{5}+13923 b^{3} e^{8} x^{5}+115668 b^{2} c d \,e^{7} x^{5}+76041 b \,c^{2} d^{2} e^{6} x^{5}+126 c^{3} d^{3} e^{5} x^{5}+34034 a^{2} c \,e^{8} x^{4}+34034 a \,b^{2} e^{8} x^{4}+213486 a b c d \,e^{7} x^{4}+75684 a \,c^{2} d^{2} e^{6} x^{4}+35581 b^{3} d \,e^{7} x^{4}+75684 b^{2} c \,d^{2} e^{6} x^{4}+595 b \,c^{2} d^{3} e^{5} x^{4}-140 c^{3} d^{4} e^{4} x^{4}+21879 a^{2} b \,e^{8} x^{3}+92378 a^{2} c d \,e^{7} x^{3}+92378 a \,b^{2} d \,e^{7} x^{3}+149838 a b c \,d^{2} e^{6} x^{3}+1020 a \,c^{2} d^{3} e^{5} x^{3}+24973 b^{3} d^{2} e^{6} x^{3}+1020 b^{2} c \,d^{3} e^{5} x^{3}-680 b \,c^{2} d^{4} e^{4} x^{3}+160 c^{3} d^{5} e^{3} x^{3}+65637 a^{2} b d \,e^{7} x^{2}+72930 a^{2} c \,d^{2} e^{6} x^{2}+72930 a \,b^{2} d^{2} e^{6} x^{2}+3978 a b c \,d^{3} e^{5} x^{2}-1224 a \,c^{2} d^{4} e^{4} x^{2}+663 b^{3} d^{3} e^{5} x^{2}-1224 b^{2} c \,d^{4} e^{4} x^{2}+816 b \,c^{2} d^{5} e^{3} x^{2}-192 c^{3} d^{6} e^{2} x^{2}+65637 a^{2} b \,d^{2} e^{6} x +4862 a^{2} c \,d^{3} e^{5} x +4862 a \,b^{2} d^{3} e^{5} x -5304 a b c \,d^{4} e^{4} x +1632 a \,c^{2} d^{5} e^{3} x -884 b^{3} d^{4} e^{4} x +1632 b^{2} c \,d^{5} e^{3} x -1088 b \,c^{2} d^{6} e^{2} x +256 c^{3} d^{7} e x +21879 a^{2} b \,d^{3} e^{5}-9724 a^{2} c \,d^{4} e^{4}-9724 a \,b^{2} d^{4} e^{4}+10608 a b c \,d^{5} e^{3}-3264 a \,c^{2} d^{6} e^{2}+1768 b^{3} d^{5} e^{3}-3264 b^{2} c \,d^{6} e^{2}+2176 b \,c^{2} d^{7} e -512 c^{3} d^{8}\right )}{153153 e^{6}} \] Input:
int((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^2,x)
Output:
(2*sqrt(d + e*x)*(21879*a**2*b*d**3*e**5 + 65637*a**2*b*d**2*e**6*x + 6563 7*a**2*b*d*e**7*x**2 + 21879*a**2*b*e**8*x**3 - 9724*a**2*c*d**4*e**4 + 48 62*a**2*c*d**3*e**5*x + 72930*a**2*c*d**2*e**6*x**2 + 92378*a**2*c*d*e**7* x**3 + 34034*a**2*c*e**8*x**4 - 9724*a*b**2*d**4*e**4 + 4862*a*b**2*d**3*e **5*x + 72930*a*b**2*d**2*e**6*x**2 + 92378*a*b**2*d*e**7*x**3 + 34034*a*b **2*e**8*x**4 + 10608*a*b*c*d**5*e**3 - 5304*a*b*c*d**4*e**4*x + 3978*a*b* c*d**3*e**5*x**2 + 149838*a*b*c*d**2*e**6*x**3 + 213486*a*b*c*d*e**7*x**4 + 83538*a*b*c*e**8*x**5 - 3264*a*c**2*d**6*e**2 + 1632*a*c**2*d**5*e**3*x - 1224*a*c**2*d**4*e**4*x**2 + 1020*a*c**2*d**3*e**5*x**3 + 75684*a*c**2*d **2*e**6*x**4 + 115668*a*c**2*d*e**7*x**5 + 47124*a*c**2*e**8*x**6 + 1768* b**3*d**5*e**3 - 884*b**3*d**4*e**4*x + 663*b**3*d**3*e**5*x**2 + 24973*b* *3*d**2*e**6*x**3 + 35581*b**3*d*e**7*x**4 + 13923*b**3*e**8*x**5 - 3264*b **2*c*d**6*e**2 + 1632*b**2*c*d**5*e**3*x - 1224*b**2*c*d**4*e**4*x**2 + 1 020*b**2*c*d**3*e**5*x**3 + 75684*b**2*c*d**2*e**6*x**4 + 115668*b**2*c*d* e**7*x**5 + 47124*b**2*c*e**8*x**6 + 2176*b*c**2*d**7*e - 1088*b*c**2*d**6 *e**2*x + 816*b*c**2*d**5*e**3*x**2 - 680*b*c**2*d**4*e**4*x**3 + 595*b*c* *2*d**3*e**5*x**4 + 76041*b*c**2*d**2*e**6*x**5 + 121737*b*c**2*d*e**7*x** 6 + 51051*b*c**2*e**8*x**7 - 512*c**3*d**8 + 256*c**3*d**7*e*x - 192*c**3* d**6*e**2*x**2 + 160*c**3*d**5*e**3*x**3 - 140*c**3*d**4*e**4*x**4 + 126*c **3*d**3*e**5*x**5 + 25410*c**3*d**2*e**6*x**6 + 42042*c**3*d*e**7*x**7...