3.17.7 \(\int (b+2 c x) (d+e x)^{5/2} (a+b x+c x^2)^3 \, dx\) [1607]
Optimal. Leaf size=427 \[ -\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{7/2}}{7 e^8}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{9/2}}{9 e^8}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{11/2}}{11 e^8}+\frac {2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{13/2}}{13 e^8}-\frac {2 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{15/2}}{3 e^8}+\frac {6 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{17/2}}{17 e^8}-\frac {14 c^3 (2 c d-b e) (d+e x)^{19/2}}{19 e^8}+\frac {4 c^4 (d+e x)^{21/2}}{21 e^8} \]
[Out]
-2/7*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^(7/2)/e^8+2/9*(a*e^2-b*d*e+c*d^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*
e*(-a*e+7*b*d))*(e*x+d)^(9/2)/e^8-6/11*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))
*(e*x+d)^(11/2)/e^8+2/13*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(-3*a*e+5*b*d)-20*c^3*d^2*e*(-3*a*e+7*b*d)+6*c^2*e^2*
(a^2*e^2-10*a*b*d*e+15*b^2*d^2))*(e*x+d)^(13/2)/e^8-2/3*c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*
(e*x+d)^(15/2)/e^8+6/17*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^(17/2)/e^8-14/19*c^3*(-b*e+2*c*d
)*(e*x+d)^(19/2)/e^8+4/21*c^4*(e*x+d)^(21/2)/e^8
________________________________________________________________________________________
Rubi [A]
time = 0.20, antiderivative size = 427, normalized size of antiderivative = 1.00, 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 = {785}
\begin {gather*} \frac {2 (d+e x)^{13/2} \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 )}{13 e^8}+\frac {6 c^2 (d+e x)^{17/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{17 e^8}-\frac {2 c (d+e x)^{15/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^8}-\frac {6 (d+e x)^{11/2} (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 )}{11 e^8}+\frac {2 (d+e x)^{9/2} \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 )}{9 e^8}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{7 e^8}-\frac {14 c^3 (d+e x)^{19/2} (2 c d-b e)}{19 e^8}+\frac {4 c^4 (d+e x)^{21/2}}{21 e^8} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(b + 2*c*x)*(d + e*x)^(5/2)*(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)^(7/2))/(7*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)^(9/2))/(9*e^8) - (6*(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)^(11/2))/(11*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)^(13/2))
/(13*e^8) - (2*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)^(15/2))/(3*e^8) + (6*c^2*
(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(17/2))/(17*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(1
9/2))/(19*e^8) + (4*c^4*(d + e*x)^(21/2))/(21*e^8)
Rule 785
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)^{5/2} \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)^{5/2}}{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)^{7/2}}{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)^{9/2}}{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)^{11/2}}{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)^{13/2}}{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)^{15/2}}{e^7}-\frac {7 c^3 (2 c d-b e) (d+e x)^{17/2}}{e^7}+\frac {2 c^4 (d+e x)^{19/2}}{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)^{7/2}}{7 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)^{9/2}}{9 e^8}-\frac {6 (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)^{11/2}}{11 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)^{13/2}}{13 e^8}-\frac {2 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)^{15/2}}{3 e^8}+\frac {6 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)^{17/2}}{17 e^8}-\frac {14 c^3 (2 c d-b e) (d+e x)^{19/2}}{19 e^8}+\frac {4 c^4 (d+e x)^{21/2}}{21 e^8}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.45, size = 600, normalized size = 1.41 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (-2 c^4 \left (2048 d^7-7168 d^6 e x+16128 d^5 e^2 x^2-29568 d^4 e^3 x^3+48048 d^3 e^4 x^4-72072 d^2 e^5 x^5+102102 d e^6 x^6-138567 e^7 x^7\right )+969 b e^4 \left (429 a^3 e^3+143 a^2 b e^2 (-2 d+7 e x)+13 a b^2 e \left (8 d^2-28 d e x+63 e^2 x^2\right )+b^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )\right )+323 c e^3 \left (286 a^3 e^3 (-2 d+7 e x)+117 a^2 b e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+36 a b^2 e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b^3 \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 )-57 c^2 e^2 \left (102 a^2 e^2 \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )-17 a b e \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 )+3 b^2 \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 )\right )+3 c^3 e \left (38 a e \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 )+7 b \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )\right )\right )}{2909907 e^8} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(b + 2*c*x)*(d + e*x)^(5/2)*(a + b*x + c*x^2)^3,x]
[Out]
(2*(d + e*x)^(7/2)*(-2*c^4*(2048*d^7 - 7168*d^6*e*x + 16128*d^5*e^2*x^2 - 29568*d^4*e^3*x^3 + 48048*d^3*e^4*x^
4 - 72072*d^2*e^5*x^5 + 102102*d*e^6*x^6 - 138567*e^7*x^7) + 969*b*e^4*(429*a^3*e^3 + 143*a^2*b*e^2*(-2*d + 7*
e*x) + 13*a*b^2*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + b^3*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3))
+ 323*c*e^3*(286*a^3*e^3*(-2*d + 7*e*x) + 117*a^2*b*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 36*a*b^2*e*(-16*d^3
+ 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + b^3*(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)) - 57*c^2*e^2*(102*a^2*e^2*(16*d^3 - 56*d^2*e*x + 126*d*e^2*x^2 - 231*e^3*x^3) - 17*a*b*e*(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) + 3*b^2*(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)) + 3*c^3*e*(38*a*e*(-256*d^5 + 896*d^4*e*x - 201
6*d^3*e^2*x^2 + 3696*d^2*e^3*x^3 - 6006*d*e^4*x^4 + 9009*e^5*x^5) + 7*b*(1024*d^6 - 3584*d^5*e*x + 8064*d^4*e^
2*x^2 - 14784*d^3*e^3*x^3 + 24024*d^2*e^4*x^4 - 36036*d*e^5*x^5 + 51051*e^6*x^6))))/(2909907*e^8)
________________________________________________________________________________________
Maple [A]
time = 0.88, size = 713, normalized size = 1.67 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
[Out]
2/e^8*(2/21*c^4*(e*x+d)^(21/2)+7/19*(b*e-2*c*d)*c^3*(e*x+d)^(19/2)+1/17*(3*(b*e-2*c*d)^2*c^2+2*c*((a*e^2-b*d*e
+c*d^2)*c^2+2*(b*e-2*c*d)^2*c+c*((b*e-2*c*d)^2+2*c*(a*e^2-b*d*e+c*d^2))))*(e*x+d)^(17/2)+1/15*((b*e-2*c*d)*((a
*e^2-b*d*e+c*d^2)*c^2+2*(b*e-2*c*d)^2*c+c*((b*e-2*c*d)^2+2*c*(a*e^2-b*d*e+c*d^2)))+2*c*((b*e-2*c*d)*((b*e-2*c*
d)^2+2*c*(a*e^2-b*d*e+c*d^2))+4*c*(a*e^2-b*d*e+c*d^2)*(b*e-2*c*d)))*(e*x+d)^(15/2)+1/13*((b*e-2*c*d)*((b*e-2*c
*d)*((b*e-2*c*d)^2+2*c*(a*e^2-b*d*e+c*d^2))+4*c*(a*e^2-b*d*e+c*d^2)*(b*e-2*c*d))+2*c*((a*e^2-b*d*e+c*d^2)*((b*
e-2*c*d)^2+2*c*(a*e^2-b*d*e+c*d^2))+2*(b*e-2*c*d)^2*(a*e^2-b*d*e+c*d^2)+c*(a*e^2-b*d*e+c*d^2)^2))*(e*x+d)^(13/
2)+1/11*((b*e-2*c*d)*((a*e^2-b*d*e+c*d^2)*((b*e-2*c*d)^2+2*c*(a*e^2-b*d*e+c*d^2))+2*(b*e-2*c*d)^2*(a*e^2-b*d*e
+c*d^2)+c*(a*e^2-b*d*e+c*d^2)^2)+6*c*(a*e^2-b*d*e+c*d^2)^2*(b*e-2*c*d))*(e*x+d)^(11/2)+1/9*(3*(b*e-2*c*d)^2*(a
*e^2-b*d*e+c*d^2)^2+2*c*(a*e^2-b*d*e+c*d^2)^3)*(e*x+d)^(9/2)+1/7*(b*e-2*c*d)*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^(7/
2))
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 680, normalized size = 1.59 \begin {gather*} \frac {2}{2909907} \, {\left (277134 \, {\left (x e + d\right )}^{\frac {21}{2}} c^{4} - 1072071 \, {\left (2 \, c^{4} d - b c^{3} e\right )} {\left (x e + d\right )}^{\frac {19}{2}} + 513513 \, {\left (14 \, c^{4} d^{2} - 14 \, b c^{3} d e + 3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} {\left (x e + d\right )}^{\frac {17}{2}} - 969969 \, {\left (14 \, c^{4} d^{3} - 21 \, b c^{3} d^{2} e - b^{3} c e^{3} - 3 \, a b c^{2} e^{3} + 3 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d\right )} {\left (x e + d\right )}^{\frac {15}{2}} + 223839 \, {\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4} + 30 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{2} - 20 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d\right )} {\left (x e + d\right )}^{\frac {13}{2}} - 793611 \, {\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e - a b^{3} e^{5} - 3 \, a^{2} b c e^{5} + 10 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{3} - 10 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{2} + {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d\right )} {\left (x e + d\right )}^{\frac {11}{2}} + 323323 \, {\left (14 \, c^{4} d^{6} - 42 \, b c^{3} d^{5} e + 15 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{4} + 3 \, a^{2} b^{2} e^{6} + 2 \, a^{3} c e^{6} - 20 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{3} + 3 \, {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d^{2} - 6 \, {\left (a b^{3} e^{5} + 3 \, a^{2} b c e^{5}\right )} d\right )} {\left (x e + d\right )}^{\frac {9}{2}} - 415701 \, {\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e + 3 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{5} - 5 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{4} - a^{3} b e^{7} + {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d^{3} - 3 \, {\left (a b^{3} e^{5} + 3 \, a^{2} b c e^{5}\right )} d^{2} + {\left (3 \, a^{2} b^{2} e^{6} + 2 \, a^{3} c e^{6}\right )} d\right )} {\left (x e + d\right )}^{\frac {7}{2}}\right )} e^{\left (-8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x, algorithm="maxima")
[Out]
2/2909907*(277134*(x*e + d)^(21/2)*c^4 - 1072071*(2*c^4*d - b*c^3*e)*(x*e + d)^(19/2) + 513513*(14*c^4*d^2 - 1
4*b*c^3*d*e + 3*b^2*c^2*e^2 + 2*a*c^3*e^2)*(x*e + d)^(17/2) - 969969*(14*c^4*d^3 - 21*b*c^3*d^2*e - b^3*c*e^3
- 3*a*b*c^2*e^3 + 3*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d)*(x*e + d)^(15/2) + 223839*(70*c^4*d^4 - 140*b*c^3*d^3*e +
b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4 + 30*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d^2 - 20*(b^3*c*e^3 + 3*a*b*c^2*e
^3)*d)*(x*e + d)^(13/2) - 793611*(14*c^4*d^5 - 35*b*c^3*d^4*e - a*b^3*e^5 - 3*a^2*b*c*e^5 + 10*(3*b^2*c^2*e^2
+ 2*a*c^3*e^2)*d^3 - 10*(b^3*c*e^3 + 3*a*b*c^2*e^3)*d^2 + (b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*d)*(x*e +
d)^(11/2) + 323323*(14*c^4*d^6 - 42*b*c^3*d^5*e + 15*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d^4 + 3*a^2*b^2*e^6 + 2*a^
3*c*e^6 - 20*(b^3*c*e^3 + 3*a*b*c^2*e^3)*d^3 + 3*(b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*d^2 - 6*(a*b^3*e^5
+ 3*a^2*b*c*e^5)*d)*(x*e + d)^(9/2) - 415701*(2*c^4*d^7 - 7*b*c^3*d^6*e + 3*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d^5
- 5*(b^3*c*e^3 + 3*a*b*c^2*e^3)*d^4 - a^3*b*e^7 + (b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*d^3 - 3*(a*b^3*e
^5 + 3*a^2*b*c*e^5)*d^2 + (3*a^2*b^2*e^6 + 2*a^3*c*e^6)*d)*(x*e + d)^(7/2))*e^(-8)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1077 vs.
\(2 (403) = 806\).
time = 1.08, size = 1077, normalized size = 2.52 \begin {gather*} -\frac {2}{2909907} \, {\left (4096 \, c^{4} d^{10} - {\left (277134 \, c^{4} x^{10} + 1072071 \, b c^{3} x^{9} + 513513 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{8} + 969969 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} x^{7} + 415701 \, a^{3} b x^{3} + 223839 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{6} + 793611 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} x^{5} + 323323 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} x^{4}\right )} e^{10} - {\left (627198 \, c^{4} d x^{9} + 2459457 \, b c^{3} d x^{8} + 1198197 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d x^{7} + 2313003 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d x^{6} + 1247103 \, a^{3} b d x^{2} + 549423 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d x^{5} + 2028117 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d x^{4} + 877591 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d x^{3}\right )} e^{9} - 3 \, {\left (120978 \, c^{4} d^{2} x^{8} + 483483 \, b c^{3} d^{2} x^{7} + 241395 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{6} + 481593 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} x^{5} + 415701 \, a^{3} b d^{2} x + 119833 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} x^{4} + 474487 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} x^{3} + 230945 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} x^{2}\right )} e^{8} - {\left (858 \, c^{4} d^{3} x^{7} + 4851 \, b c^{3} d^{3} x^{6} + 3591 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} x^{5} + 11305 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} x^{4} + 415701 \, a^{3} b d^{3} + 4845 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} x^{3} + 37791 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} x^{2} + 46189 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3} x\right )} e^{7} + 2 \, {\left (462 \, c^{4} d^{4} x^{6} + 2646 \, b c^{3} d^{4} x^{5} + 1995 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x^{4} + 6460 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} x^{3} + 2907 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} x^{2} + 25194 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{4} x + 46189 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{4}\right )} e^{6} - 24 \, {\left (42 \, c^{4} d^{5} x^{5} + 245 \, b c^{3} d^{5} x^{4} + 190 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} x^{3} + 646 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} x^{2} + 323 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{5} x + 4199 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{5}\right )} e^{5} + 16 \, {\left (70 \, c^{4} d^{6} x^{4} + 420 \, b c^{3} d^{6} x^{3} + 342 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} x^{2} + 1292 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{6} x + 969 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{6}\right )} e^{4} - 128 \, {\left (10 \, c^{4} d^{7} x^{3} + 63 \, b c^{3} d^{7} x^{2} + 57 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{7} x + 323 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{7}\right )} e^{3} + 768 \, {\left (2 \, c^{4} d^{8} x^{2} + 14 \, b c^{3} d^{8} x + 19 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{8}\right )} e^{2} - 1024 \, {\left (2 \, c^{4} d^{9} x + 21 \, b c^{3} d^{9}\right )} e\right )} \sqrt {x e + d} e^{\left (-8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x, algorithm="fricas")
[Out]
-2/2909907*(4096*c^4*d^10 - (277134*c^4*x^10 + 1072071*b*c^3*x^9 + 513513*(3*b^2*c^2 + 2*a*c^3)*x^8 + 969969*(
b^3*c + 3*a*b*c^2)*x^7 + 415701*a^3*b*x^3 + 223839*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*x^6 + 793611*(a*b^3 + 3*a^2*
b*c)*x^5 + 323323*(3*a^2*b^2 + 2*a^3*c)*x^4)*e^10 - (627198*c^4*d*x^9 + 2459457*b*c^3*d*x^8 + 1198197*(3*b^2*c
^2 + 2*a*c^3)*d*x^7 + 2313003*(b^3*c + 3*a*b*c^2)*d*x^6 + 1247103*a^3*b*d*x^2 + 549423*(b^4 + 12*a*b^2*c + 6*a
^2*c^2)*d*x^5 + 2028117*(a*b^3 + 3*a^2*b*c)*d*x^4 + 877591*(3*a^2*b^2 + 2*a^3*c)*d*x^3)*e^9 - 3*(120978*c^4*d^
2*x^8 + 483483*b*c^3*d^2*x^7 + 241395*(3*b^2*c^2 + 2*a*c^3)*d^2*x^6 + 481593*(b^3*c + 3*a*b*c^2)*d^2*x^5 + 415
701*a^3*b*d^2*x + 119833*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*x^4 + 474487*(a*b^3 + 3*a^2*b*c)*d^2*x^3 + 230945*
(3*a^2*b^2 + 2*a^3*c)*d^2*x^2)*e^8 - (858*c^4*d^3*x^7 + 4851*b*c^3*d^3*x^6 + 3591*(3*b^2*c^2 + 2*a*c^3)*d^3*x^
5 + 11305*(b^3*c + 3*a*b*c^2)*d^3*x^4 + 415701*a^3*b*d^3 + 4845*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*x^3 + 37791
*(a*b^3 + 3*a^2*b*c)*d^3*x^2 + 46189*(3*a^2*b^2 + 2*a^3*c)*d^3*x)*e^7 + 2*(462*c^4*d^4*x^6 + 2646*b*c^3*d^4*x^
5 + 1995*(3*b^2*c^2 + 2*a*c^3)*d^4*x^4 + 6460*(b^3*c + 3*a*b*c^2)*d^4*x^3 + 2907*(b^4 + 12*a*b^2*c + 6*a^2*c^2
)*d^4*x^2 + 25194*(a*b^3 + 3*a^2*b*c)*d^4*x + 46189*(3*a^2*b^2 + 2*a^3*c)*d^4)*e^6 - 24*(42*c^4*d^5*x^5 + 245*
b*c^3*d^5*x^4 + 190*(3*b^2*c^2 + 2*a*c^3)*d^5*x^3 + 646*(b^3*c + 3*a*b*c^2)*d^5*x^2 + 323*(b^4 + 12*a*b^2*c +
6*a^2*c^2)*d^5*x + 4199*(a*b^3 + 3*a^2*b*c)*d^5)*e^5 + 16*(70*c^4*d^6*x^4 + 420*b*c^3*d^6*x^3 + 342*(3*b^2*c^2
+ 2*a*c^3)*d^6*x^2 + 1292*(b^3*c + 3*a*b*c^2)*d^6*x + 969*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6)*e^4 - 128*(10*c
^4*d^7*x^3 + 63*b*c^3*d^7*x^2 + 57*(3*b^2*c^2 + 2*a*c^3)*d^7*x + 323*(b^3*c + 3*a*b*c^2)*d^7)*e^3 + 768*(2*c^4
*d^8*x^2 + 14*b*c^3*d^8*x + 19*(3*b^2*c^2 + 2*a*c^3)*d^8)*e^2 - 1024*(2*c^4*d^9*x + 21*b*c^3*d^9)*e)*sqrt(x*e
+ d)*e^(-8)
________________________________________________________________________________________
Sympy [A]
time = 62.14, size = 3529, normalized size = 8.26 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*x+b)*(e*x+d)**(5/2)*(c*x**2+b*x+a)**3,x)
[Out]
a**3*b*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**3*b*d*(-d*(d + e*x)**(3/
2)/3 + (d + e*x)**(5/2)/5)/e + 2*a**3*b*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7
)/e + 4*a**3*c*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 8*a**3*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**2 + 4*a**3*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**2 + 6*a**2*b**2*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)
**(5/2)/5)/e**2 + 12*a**2*b**2*d*(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**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**2 + 18*a**2*b*c*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 36
*a**2*b*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**3 + 18*a**2*b*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**3 + 12*a**2*c**2*d**2*(-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**2*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**4 + 12*
a**2*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**4 + 6*a*b**3*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d
*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 12*a*b**3*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**3 + 6*a*b**3*(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**3 + 24*a*b**2*c*d**
2*(-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 +
48*a*b**2*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**4 + 24*a*b**2*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**4 + 3
0*a*b*c**2*d**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**5 + 60*a*b*c**2*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 1
0*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
+ 30*a*b*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**5 + 12*a*
c**3*d**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**6 + 24*a*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**6 + 12*a*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) + 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**6 + 2*b**4*d**2*(-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 + 4*b**4*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**4 + 2*b**4*(-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**4 + 10*b**3*c*d**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**5 + 20*b**3*c*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**5 + 10*b**3*c*(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)**(1
1/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5 + 18*b**2*c**2*d**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**6 + 36*b**2*c**2*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**6 + 18*b**2*c**2*(-d**7*(d...
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4552 vs.
\(2 (403) = 806\).
time = 1.67, size = 4552, normalized size = 10.66 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x, algorithm="giac")
[Out]
2/14549535*(14549535*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b^2*d^3*e^(-1) + 9699690*((x*e + d)^(3/2) - 3*s
qrt(x*e + d)*d)*a^3*c*d^3*e^(-1) + 2909907*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a
*b^3*d^3*e^(-2) + 8729721*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*b*c*d^3*e^(-2)
+ 415701*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*b^4*d^3*e
^(-3) + 4988412*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a*b
^2*c*d^3*e^(-3) + 2494206*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d
)*d^3)*a^2*c^2*d^3*e^(-3) + 230945*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420
*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*b^3*c*d^3*e^(-4) + 692835*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7
/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*b*c^2*d^3*e^(-4) + 188955
*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e
+ d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b^2*c^2*d^3*e^(-5) + 125970*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2
)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a
*c^3*d^3*e^(-5) + 33915*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e
+ d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b*c^3*d^3*e^(-
6) + 4522*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)
*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*
d^7)*c^4*d^3*e^(-7) + 8729721*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*b^2*d^2*e^
(-1) + 5819814*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*c*d^2*e^(-1) + 3741309*(5
*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a*b^3*d^2*e^(-2) + 11
223927*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^2*b*c*d^2*
e^(-2) + 138567*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^
3 + 315*sqrt(x*e + d)*d^4)*b^4*d^2*e^(-3) + 1662804*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d
)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*b^2*c*d^2*e^(-3) + 831402*(35*(x*e + d)^(9/2)
- 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^2*c^2*
d^2*e^(-3) + 314925*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5
/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b^3*c*d^2*e^(-4) + 944775*(63*(x*e + d)^(11/2) - 3
85*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqr
t(x*e + d)*d^5)*a*b*c^2*d^2*e^(-4) + 130815*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(
9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)
*d^6)*b^2*c^2*d^2*e^(-5) + 87210*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 -
8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*a*c^3
*d^2*e^(-5) + 47481*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e
+ d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt
(x*e + d)*d^7)*b*c^3*d^2*e^(-6) + 798*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13
/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e
+ d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*c^4*d^2*e^(-7) + 14549535*sqrt(x*e + d
)*a^3*b*d^3 + 14549535*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*b*d^2 + 3741309*(5*(x*e + d)^(7/2) - 21*(x*e
+ d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^2*b^2*d*e^(-1) + 2494206*(5*(x*e + d)^(7/2) -
21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^3*c*d*e^(-1) + 415701*(35*(x*e + d)^(9
/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*b^3
*d*e^(-2) + 1247103*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2
)*d^3 + 315*sqrt(x*e + d)*d^4)*a^2*b*c*d*e^(-2) + 62985*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*
e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b^4*d*e^(-3) +
755820*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 11
55*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a*b^2*c*d*e^(-3) + 377910*(63*(x*e + d)^(11/2) - 385*(x*e + d)
^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e ...
________________________________________________________________________________________
Mupad [B]
time = 1.96, size = 444, normalized size = 1.04 \begin {gather*} \frac {{\left (d+e\,x\right )}^{17/2}\,\left (18\,b^2\,c^2\,e^2-84\,b\,c^3\,d\,e+84\,c^4\,d^2+12\,a\,c^3\,e^2\right )}{17\,e^8}+\frac {4\,c^4\,{\left (d+e\,x\right )}^{21/2}}{21\,e^8}-\frac {\left (28\,c^4\,d-14\,b\,c^3\,e\right )\,{\left (d+e\,x\right )}^{19/2}}{19\,e^8}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (12\,a^2\,c^2\,e^4+24\,a\,b^2\,c\,e^4-120\,a\,b\,c^2\,d\,e^3+120\,a\,c^3\,d^2\,e^2+2\,b^4\,e^4-40\,b^3\,c\,d\,e^3+180\,b^2\,c^2\,d^2\,e^2-280\,b\,c^3\,d^3\,e+140\,c^4\,d^4\right )}{13\,e^8}+\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 )}^3}{7\,e^8}+\frac {6\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (3\,a^2\,c\,e^4+a\,b^2\,e^4-10\,a\,b\,c\,d\,e^3+10\,a\,c^2\,d^2\,e^2-b^3\,d\,e^3+8\,b^2\,c\,d^2\,e^2-14\,b\,c^2\,d^3\,e+7\,c^3\,d^4\right )}{11\,e^8}+\frac {2\,{\left (d+e\,x\right )}^{9/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{9\,e^8}+\frac {2\,c\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{15/2}\,\left (b^2\,e^2-7\,b\,c\,d\,e+7\,c^2\,d^2+3\,a\,c\,e^2\right )}{3\,e^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b + 2*c*x)*(d + e*x)^(5/2)*(a + b*x + c*x^2)^3,x)
[Out]
((d + e*x)^(17/2)*(84*c^4*d^2 + 12*a*c^3*e^2 + 18*b^2*c^2*e^2 - 84*b*c^3*d*e))/(17*e^8) + (4*c^4*(d + e*x)^(21
/2))/(21*e^8) - ((28*c^4*d - 14*b*c^3*e)*(d + e*x)^(19/2))/(19*e^8) + ((d + e*x)^(13/2)*(2*b^4*e^4 + 140*c^4*d
^4 + 12*a^2*c^2*e^4 + 120*a*c^3*d^2*e^2 + 180*b^2*c^2*d^2*e^2 + 24*a*b^2*c*e^4 - 280*b*c^3*d^3*e - 40*b^3*c*d*
e^3 - 120*a*b*c^2*d*e^3))/(13*e^8) + (2*(b*e - 2*c*d)*(d + e*x)^(7/2)*(a*e^2 + c*d^2 - b*d*e)^3)/(7*e^8) + (6*
(b*e - 2*c*d)*(d + e*x)^(11/2)*(7*c^3*d^4 + a*b^2*e^4 + 3*a^2*c*e^4 - b^3*d*e^3 + 10*a*c^2*d^2*e^2 + 8*b^2*c*d
^2*e^2 - 14*b*c^2*d^3*e - 10*a*b*c*d*e^3))/(11*e^8) + (2*(d + e*x)^(9/2)*(a*e^2 + c*d^2 - b*d*e)^2*(3*b^2*e^2
+ 14*c^2*d^2 + 2*a*c*e^2 - 14*b*c*d*e))/(9*e^8) + (2*c*(b*e - 2*c*d)*(d + e*x)^(15/2)*(b^2*e^2 + 7*c^2*d^2 + 3
*a*c*e^2 - 7*b*c*d*e))/(3*e^8)
________________________________________________________________________________________