3.1608 \(\int (b+2 c x) (d+e x)^{3/2} (a+b x+c x^2)^3 \, dx\)
Optimal. Leaf size=427 \[ \frac{2 (d+e x)^{11/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{11 e^8}+\frac{2 c^2 (d+e x)^{15/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^8}-\frac{10 c (d+e x)^{13/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{13 e^8}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^8}+\frac{2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^8}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8}-\frac{14 c^3 (d+e x)^{17/2} (2 c d-b e)}{17 e^8}+\frac{4 c^4 (d+e x)^{19/2}}{19 e^8} \]
[Out]
(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(5/2))/(5*e^8) + (2*(c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^
2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(7/2))/(7*e^8) - (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^
2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(9/2))/(3*e^8) + (2*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*
d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^(11/2))/(
11*e^8) - (10*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(13/2))/(13*e^8) + (2*c^2*
(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(15/2))/(5*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(17
/2))/(17*e^8) + (4*c^4*(d + e*x)^(19/2))/(19*e^8)
________________________________________________________________________________________
Rubi [A] time = 0.242838, antiderivative size = 427, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used =
{771} \[ \frac{2 (d+e x)^{11/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{11 e^8}+\frac{2 c^2 (d+e x)^{15/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^8}-\frac{10 c (d+e x)^{13/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{13 e^8}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^8}+\frac{2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^8}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8}-\frac{14 c^3 (d+e x)^{17/2} (2 c d-b e)}{17 e^8}+\frac{4 c^4 (d+e x)^{19/2}}{19 e^8} \]
Antiderivative was successfully verified.
[In]
Int[(b + 2*c*x)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^3,x]
[Out]
(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(5/2))/(5*e^8) + (2*(c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^
2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(7/2))/(7*e^8) - (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^
2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(9/2))/(3*e^8) + (2*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*
d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^(11/2))/(
11*e^8) - (10*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(13/2))/(13*e^8) + (2*c^2*
(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(15/2))/(5*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(17
/2))/(17*e^8) + (4*c^4*(d + e*x)^(19/2))/(19*e^8)
Rule 771
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))
Rubi steps
\begin{align*} \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}{e^7}+\frac{\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{5/2}}{e^7}+\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right ) (d+e x)^{7/2}}{e^7}+\frac{\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{9/2}}{e^7}+\frac{5 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right ) (d+e x)^{11/2}}{e^7}+\frac{3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{13/2}}{e^7}-\frac{7 c^3 (2 c d-b e) (d+e x)^{15/2}}{e^7}+\frac{2 c^4 (d+e x)^{17/2}}{e^7}\right ) \, dx\\ &=-\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{5/2}}{5 e^8}+\frac{2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{7/2}}{7 e^8}-\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{9/2}}{3 e^8}+\frac{2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{11/2}}{11 e^8}-\frac{10 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{13/2}}{13 e^8}+\frac{2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{15/2}}{5 e^8}-\frac{14 c^3 (2 c d-b e) (d+e x)^{17/2}}{17 e^8}+\frac{4 c^4 (d+e x)^{19/2}}{19 e^8}\\ \end{align*}
Mathematica [A] time = 0.677173, size = 600, normalized size = 1.41 \[ \frac{2 (d+e x)^{5/2} \left (-969 c^2 e^2 \left (26 a^2 e^2 \left (-40 d^2 e x+16 d^3+70 d e^2 x^2-105 e^3 x^3\right )-5 a b e \left (560 d^2 e^2 x^2-320 d^3 e x+128 d^4-840 d e^3 x^3+1155 e^4 x^4\right )+b^2 \left (1120 d^3 e^2 x^2-1680 d^2 e^3 x^3-640 d^4 e x+256 d^5+2310 d e^4 x^4-3003 e^5 x^5\right )\right )+323 c e^3 \left (429 a^2 b e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+858 a^3 e^3 (5 e x-2 d)+156 a b^2 e \left (40 d^2 e x-16 d^3-70 d e^2 x^2+105 e^3 x^3\right )+5 b^3 \left (560 d^2 e^2 x^2-320 d^3 e x+128 d^4-840 d e^3 x^3+1155 e^4 x^4\right )\right )+4199 b e^4 \left (99 a^2 b e^2 (5 e x-2 d)+231 a^3 e^3+11 a b^2 e \left (8 d^2-20 d e x+35 e^2 x^2\right )+b^3 \left (40 d^2 e x-16 d^3-70 d e^2 x^2+105 e^3 x^3\right )\right )+19 c^3 e \left (34 a e \left (-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3+640 d^4 e x-256 d^5-2310 d e^4 x^4+3003 e^5 x^5\right )+7 b \left (4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-2560 d^5 e x+1024 d^6-12012 d e^5 x^5+15015 e^6 x^6\right )\right )-14 c^4 \left (8960 d^5 e^2 x^2-13440 d^4 e^3 x^3+18480 d^3 e^4 x^4-24024 d^2 e^5 x^5-5120 d^6 e x+2048 d^7+30030 d e^6 x^6-36465 e^7 x^7\right )\right )}{4849845 e^8} \]
Antiderivative was successfully verified.
[In]
Integrate[(b + 2*c*x)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^3,x]
[Out]
(2*(d + e*x)^(5/2)*(-14*c^4*(2048*d^7 - 5120*d^6*e*x + 8960*d^5*e^2*x^2 - 13440*d^4*e^3*x^3 + 18480*d^3*e^4*x^
4 - 24024*d^2*e^5*x^5 + 30030*d*e^6*x^6 - 36465*e^7*x^7) + 4199*b*e^4*(231*a^3*e^3 + 99*a^2*b*e^2*(-2*d + 5*e*
x) + 11*a*b^2*e*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + b^3*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3)) + 3
23*c*e^3*(858*a^3*e^3*(-2*d + 5*e*x) + 429*a^2*b*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 156*a*b^2*e*(-16*d^3 +
40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 5*b^3*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 11
55*e^4*x^4)) - 969*c^2*e^2*(26*a^2*e^2*(16*d^3 - 40*d^2*e*x + 70*d*e^2*x^2 - 105*e^3*x^3) - 5*a*b*e*(128*d^4 -
320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4) + b^2*(256*d^5 - 640*d^4*e*x + 1120*d^3*e^2*x^2
- 1680*d^2*e^3*x^3 + 2310*d*e^4*x^4 - 3003*e^5*x^5)) + 19*c^3*e*(34*a*e*(-256*d^5 + 640*d^4*e*x - 1120*d^3*e^
2*x^2 + 1680*d^2*e^3*x^3 - 2310*d*e^4*x^4 + 3003*e^5*x^5) + 7*b*(1024*d^6 - 2560*d^5*e*x + 4480*d^4*e^2*x^2 -
6720*d^3*e^3*x^3 + 9240*d^2*e^4*x^4 - 12012*d*e^5*x^5 + 15015*e^6*x^6))))/(4849845*e^8)
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Maple [B] time = 0.009, size = 795, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*c*x+b)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^3,x)
[Out]
2/4849845*(e*x+d)^(5/2)*(510510*c^4*e^7*x^7+1996995*b*c^3*e^7*x^6-420420*c^4*d*e^6*x^6+1939938*a*c^3*e^7*x^5+2
909907*b^2*c^2*e^7*x^5-1597596*b*c^3*d*e^6*x^5+336336*c^4*d^2*e^5*x^5+5595975*a*b*c^2*e^7*x^4-1492260*a*c^3*d*
e^6*x^4+1865325*b^3*c*e^7*x^4-2238390*b^2*c^2*d*e^6*x^4+1228920*b*c^3*d^2*e^5*x^4-258720*c^4*d^3*e^4*x^4+26453
70*a^2*c^2*e^7*x^3+5290740*a*b^2*c*e^7*x^3-4069800*a*b*c^2*d*e^6*x^3+1085280*a*c^3*d^2*e^5*x^3+440895*b^4*e^7*
x^3-1356600*b^3*c*d*e^6*x^3+1627920*b^2*c^2*d^2*e^5*x^3-893760*b*c^3*d^3*e^4*x^3+188160*c^4*d^4*e^3*x^3+484984
5*a^2*b*c*e^7*x^2-1763580*a^2*c^2*d*e^6*x^2+1616615*a*b^3*e^7*x^2-3527160*a*b^2*c*d*e^6*x^2+2713200*a*b*c^2*d^
2*e^5*x^2-723520*a*c^3*d^3*e^4*x^2-293930*b^4*d*e^6*x^2+904400*b^3*c*d^2*e^5*x^2-1085280*b^2*c^2*d^3*e^4*x^2+5
95840*b*c^3*d^4*e^3*x^2-125440*c^4*d^5*e^2*x^2+1385670*a^3*c*e^7*x+2078505*a^2*b^2*e^7*x-2771340*a^2*b*c*d*e^6
*x+1007760*a^2*c^2*d^2*e^5*x-923780*a*b^3*d*e^6*x+2015520*a*b^2*c*d^2*e^5*x-1550400*a*b*c^2*d^3*e^4*x+413440*a
*c^3*d^4*e^3*x+167960*b^4*d^2*e^5*x-516800*b^3*c*d^3*e^4*x+620160*b^2*c^2*d^4*e^3*x-340480*b*c^3*d^5*e^2*x+716
80*c^4*d^6*e*x+969969*a^3*b*e^7-554268*a^3*c*d*e^6-831402*a^2*b^2*d*e^6+1108536*a^2*b*c*d^2*e^5-403104*a^2*c^2
*d^3*e^4+369512*a*b^3*d^2*e^5-806208*a*b^2*c*d^3*e^4+620160*a*b*c^2*d^4*e^3-165376*a*c^3*d^5*e^2-67184*b^4*d^3
*e^4+206720*b^3*c*d^4*e^3-248064*b^2*c^2*d^5*e^2+136192*b*c^3*d^6*e-28672*c^4*d^7)/e^8
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Maxima [A] time = 1.02785, size = 871, normalized size = 2.04 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*x+b)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^3,x, algorithm="maxima")
[Out]
2/4849845*(510510*(e*x + d)^(19/2)*c^4 - 1996995*(2*c^4*d - b*c^3*e)*(e*x + d)^(17/2) + 969969*(14*c^4*d^2 - 1
4*b*c^3*d*e + (3*b^2*c^2 + 2*a*c^3)*e^2)*(e*x + d)^(15/2) - 1865325*(14*c^4*d^3 - 21*b*c^3*d^2*e + 3*(3*b^2*c^
2 + 2*a*c^3)*d*e^2 - (b^3*c + 3*a*b*c^2)*e^3)*(e*x + d)^(13/2) + 440895*(70*c^4*d^4 - 140*b*c^3*d^3*e + 30*(3*
b^2*c^2 + 2*a*c^3)*d^2*e^2 - 20*(b^3*c + 3*a*b*c^2)*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^4)*(e*x + d)^(11/
2) - 1616615*(14*c^4*d^5 - 35*b*c^3*d^4*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 10*(b^3*c + 3*a*b*c^2)*d^2*e^3
+ (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^4 - (a*b^3 + 3*a^2*b*c)*e^5)*(e*x + d)^(9/2) + 692835*(14*c^4*d^6 - 42*b*
c^3*d^5*e + 15*(3*b^2*c^2 + 2*a*c^3)*d^4*e^2 - 20*(b^3*c + 3*a*b*c^2)*d^3*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^
2)*d^2*e^4 - 6*(a*b^3 + 3*a^2*b*c)*d*e^5 + (3*a^2*b^2 + 2*a^3*c)*e^6)*(e*x + d)^(7/2) - 969969*(2*c^4*d^7 - 7*
b*c^3*d^6*e - a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b^2*c
+ 6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6)*(e*x + d)^(5/2))/e^8
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Fricas [B] time = 1.50862, size = 2242, normalized size = 5.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*x+b)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^3,x, algorithm="fricas")
[Out]
2/4849845*(510510*c^4*e^9*x^9 - 28672*c^4*d^9 + 136192*b*c^3*d^8*e + 969969*a^3*b*d^2*e^7 - 82688*(3*b^2*c^2 +
2*a*c^3)*d^7*e^2 + 206720*(b^3*c + 3*a*b*c^2)*d^6*e^3 - 67184*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^4 + 369512
*(a*b^3 + 3*a^2*b*c)*d^4*e^5 - 277134*(3*a^2*b^2 + 2*a^3*c)*d^3*e^6 + 15015*(40*c^4*d*e^8 + 133*b*c^3*e^9)*x^8
+ 3003*(2*c^4*d^2*e^7 + 798*b*c^3*d*e^8 + 323*(3*b^2*c^2 + 2*a*c^3)*e^9)*x^7 - 231*(28*c^4*d^3*e^6 - 133*b*c^
3*d^2*e^7 - 5168*(3*b^2*c^2 + 2*a*c^3)*d*e^8 - 8075*(b^3*c + 3*a*b*c^2)*e^9)*x^6 + 21*(336*c^4*d^4*e^5 - 1596*
b*c^3*d^3*e^6 + 969*(3*b^2*c^2 + 2*a*c^3)*d^2*e^7 + 113050*(b^3*c + 3*a*b*c^2)*d*e^8 + 20995*(b^4 + 12*a*b^2*c
+ 6*a^2*c^2)*e^9)*x^5 - 35*(224*c^4*d^5*e^4 - 1064*b*c^3*d^4*e^5 + 646*(3*b^2*c^2 + 2*a*c^3)*d^3*e^6 - 1615*(
b^3*c + 3*a*b*c^2)*d^2*e^7 - 16796*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^8 - 46189*(a*b^3 + 3*a^2*b*c)*e^9)*x^4 +
5*(1792*c^4*d^6*e^3 - 8512*b*c^3*d^5*e^4 + 5168*(3*b^2*c^2 + 2*a*c^3)*d^4*e^5 - 12920*(b^3*c + 3*a*b*c^2)*d^3
*e^6 + 4199*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^7 + 461890*(a*b^3 + 3*a^2*b*c)*d*e^8 + 138567*(3*a^2*b^2 + 2*
a^3*c)*e^9)*x^3 - 3*(3584*c^4*d^7*e^2 - 17024*b*c^3*d^6*e^3 - 323323*a^3*b*e^9 + 10336*(3*b^2*c^2 + 2*a*c^3)*d
^5*e^4 - 25840*(b^3*c + 3*a*b*c^2)*d^4*e^5 + 8398*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^6 - 46189*(a*b^3 + 3*a^
2*b*c)*d^2*e^7 - 369512*(3*a^2*b^2 + 2*a^3*c)*d*e^8)*x^2 + (14336*c^4*d^8*e - 68096*b*c^3*d^7*e^2 + 1939938*a^
3*b*d*e^8 + 41344*(3*b^2*c^2 + 2*a*c^3)*d^6*e^3 - 103360*(b^3*c + 3*a*b*c^2)*d^5*e^4 + 33592*(b^4 + 12*a*b^2*c
+ 6*a^2*c^2)*d^4*e^5 - 184756*(a*b^3 + 3*a^2*b*c)*d^3*e^6 + 138567*(3*a^2*b^2 + 2*a^3*c)*d^2*e^7)*x)*sqrt(e*x
+ d)/e^8
________________________________________________________________________________________
Sympy [A] time = 59.3353, size = 2122, normalized size = 4.97 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*x+b)*(e*x+d)**(3/2)*(c*x**2+b*x+a)**3,x)
[Out]
a**3*b*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**3*b*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 4*a**3*c*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 4*a**3*c*(d**2*(d + e
*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 6*a**2*b**2*d*(-d*(d + e*x)**(3/2)/3 + (d +
e*x)**(5/2)/5)/e**2 + 6*a**2*b**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**
2 + 18*a**2*b*c*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 18*a**2*b*c*(
-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 12*
a**2*c**2*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/
9)/e**4 + 12*a**2*c**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*
(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 6*a*b**3*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5
+ (d + e*x)**(7/2)/7)/e**3 + 6*a*b**3*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**
(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 24*a*b**2*c*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d
*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 24*a*b**2*c*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2
)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 30*a*b*c**2*d*(d**4*(d
+ e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)*
*(11/2)/11)/e**5 + 30*a*b*c**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7
+ 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 12*a*c**3*d*(-d**5*(d +
e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*
x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**6 + 12*a*c**3*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 +
15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)
/13 + (d + e*x)**(15/2)/15)/e**6 + 2*b**4*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e
*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 2*b**4*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2
*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 10*b**3*c*d*(d**4*(d + e*x)**(3/2)
/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e*
*5 + 10*b**3*c*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e
*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 18*b**2*c**2*d*(-d**5*(d + e*x)**(3/2)/
3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11
+ (d + e*x)**(13/2)/13)/e**6 + 18*b**2*c**2*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d
+ e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d +
e*x)**(15/2)/15)/e**6 + 14*b*c**3*d*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)*
*(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(
15/2)/15)/e**7 + 14*b*c**3*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 3
5*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2
)/15 + (d + e*x)**(17/2)/17)/e**7 + 4*c**4*d*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d
+ e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*
d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**8 + 4*c**4*(d**8*(d + e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2
)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)*
*(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2)/19)/e**8
________________________________________________________________________________________
Giac [B] time = 1.34616, size = 2541, normalized size = 5.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*x+b)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^3,x, algorithm="giac")
[Out]
2/14549535*(2909907*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2*b^2*d*e^(-1) + 1939938*(3*(x*e + d)^(5/2) -
5*(x*e + d)^(3/2)*d)*a^3*c*d*e^(-1) + 415701*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d
^2)*a*b^3*d*e^(-2) + 1247103*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^2*b*c*d*e^
(-2) + 46189*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*
b^4*d*e^(-3) + 554268*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3
/2)*d^3)*a*b^2*c*d*e^(-3) + 277134*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105
*(x*e + d)^(3/2)*d^3)*a^2*c^2*d*e^(-3) + 20995*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)
^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*b^3*c*d*e^(-4) + 62985*(315*(x*e + d)^(11/2)
- 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*a*
b*c^2*d*e^(-4) + 14535*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*
e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*b^2*c^2*d*e^(-5) + 9690*(693*(x*e + d)
^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/
2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*a*c^3*d*e^(-5) + 2261*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 6
1425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5
+ 15015*(x*e + d)^(3/2)*d^6)*b*c^3*d*e^(-6) + 266*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*
(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 +
153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*c^4*d*e^(-7) + 4849845*(x*e + d)^(3/2)*a^3*b*d + 4157
01*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^2*b^2*e^(-1) + 277134*(15*(x*e + d)^
(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^3*c*e^(-1) + 138567*(35*(x*e + d)^(9/2) - 135*(x*e +
d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a*b^3*e^(-2) + 415701*(35*(x*e + d)^(9/2) - 13
5*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a^2*b*c*e^(-2) + 4199*(315*(x*e + d)^
(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d
^4)*b^4*e^(-3) + 50388*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e +
d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*a*b^2*c*e^(-3) + 25194*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*
d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*a^2*c^2*e^(-3) + 8075*(693
*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*
e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*b^3*c*e^(-4) + 24225*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2
)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*
d^5)*a*b*c^2*e^(-4) + 2907*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 10
0100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*
b^2*c^2*e^(-5) + 1938*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*
(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*a*c^3
*e^(-5) + 931*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(x*e +
d)^(11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(
x*e + d)^(3/2)*d^7)*b*c^3*e^(-6) + 14*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 3879876*(x*e + d)
^(15/2)*d^2 - 8953560*(x*e + d)^(13/2)*d^3 + 13226850*(x*e + d)^(11/2)*d^4 - 12932920*(x*e + d)^(9/2)*d^5 + 83
14020*(x*e + d)^(7/2)*d^6 - 3325608*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8)*c^4*e^(-7) + 969969*(3*(
x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^3*b)*e^(-1)