3.1802 \(\int (A+B x) (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\)
Optimal. Leaf size=308 \[ -\frac{2 b^5 (d+e x)^{17/2} (-6 a B e-A b e+7 b B d)}{17 e^8}+\frac{2 b^4 (d+e x)^{15/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{5 e^8}-\frac{10 b^3 (d+e x)^{13/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{13 e^8}+\frac{10 b^2 (d+e x)^{11/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{11 e^8}-\frac{2 b (d+e x)^{9/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{3 e^8}+\frac{2 (d+e x)^{7/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{7 e^8}-\frac{2 (d+e x)^{5/2} (b d-a e)^6 (B d-A e)}{5 e^8}+\frac{2 b^6 B (d+e x)^{19/2}}{19 e^8} \]
[Out]
(-2*(b*d - a*e)^6*(B*d - A*e)*(d + e*x)^(5/2))/(5*e^8) + (2*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x
)^(7/2))/(7*e^8) - (2*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^(9/2))/(3*e^8) + (10*b^2*(b*d -
a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^(11/2))/(11*e^8) - (10*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4
*a*B*e)*(d + e*x)^(13/2))/(13*e^8) + (2*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(15/2))/(5*e^8
) - (2*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^(17/2))/(17*e^8) + (2*b^6*B*(d + e*x)^(19/2))/(19*e^8)
________________________________________________________________________________________
Rubi [A] time = 0.141473, antiderivative size = 308, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used =
{27, 77} \[ -\frac{2 b^5 (d+e x)^{17/2} (-6 a B e-A b e+7 b B d)}{17 e^8}+\frac{2 b^4 (d+e x)^{15/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{5 e^8}-\frac{10 b^3 (d+e x)^{13/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{13 e^8}+\frac{10 b^2 (d+e x)^{11/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{11 e^8}-\frac{2 b (d+e x)^{9/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{3 e^8}+\frac{2 (d+e x)^{7/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{7 e^8}-\frac{2 (d+e x)^{5/2} (b d-a e)^6 (B d-A e)}{5 e^8}+\frac{2 b^6 B (d+e x)^{19/2}}{19 e^8} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(-2*(b*d - a*e)^6*(B*d - A*e)*(d + e*x)^(5/2))/(5*e^8) + (2*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x
)^(7/2))/(7*e^8) - (2*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^(9/2))/(3*e^8) + (10*b^2*(b*d -
a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^(11/2))/(11*e^8) - (10*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4
*a*B*e)*(d + e*x)^(13/2))/(13*e^8) + (2*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(15/2))/(5*e^8
) - (2*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^(17/2))/(17*e^8) + (2*b^6*B*(d + e*x)^(19/2))/(19*e^8)
Rule 27
Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Rule 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (A+B x) (d+e x)^{3/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^6 (-B d+A e) (d+e x)^{3/2}}{e^7}+\frac{(-b d+a e)^5 (-7 b B d+6 A b e+a B e) (d+e x)^{5/2}}{e^7}+\frac{3 b (b d-a e)^4 (-7 b B d+5 A b e+2 a B e) (d+e x)^{7/2}}{e^7}-\frac{5 b^2 (b d-a e)^3 (-7 b B d+4 A b e+3 a B e) (d+e x)^{9/2}}{e^7}+\frac{5 b^3 (b d-a e)^2 (-7 b B d+3 A b e+4 a B e) (d+e x)^{11/2}}{e^7}-\frac{3 b^4 (b d-a e) (-7 b B d+2 A b e+5 a B e) (d+e x)^{13/2}}{e^7}+\frac{b^5 (-7 b B d+A b e+6 a B e) (d+e x)^{15/2}}{e^7}+\frac{b^6 B (d+e x)^{17/2}}{e^7}\right ) \, dx\\ &=-\frac{2 (b d-a e)^6 (B d-A e) (d+e x)^{5/2}}{5 e^8}+\frac{2 (b d-a e)^5 (7 b B d-6 A b e-a B e) (d+e x)^{7/2}}{7 e^8}-\frac{2 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) (d+e x)^{9/2}}{3 e^8}+\frac{10 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) (d+e x)^{11/2}}{11 e^8}-\frac{10 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^{13/2}}{13 e^8}+\frac{2 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^{15/2}}{5 e^8}-\frac{2 b^5 (7 b B d-A b e-6 a B e) (d+e x)^{17/2}}{17 e^8}+\frac{2 b^6 B (d+e x)^{19/2}}{19 e^8}\\ \end{align*}
Mathematica [A] time = 0.366343, size = 259, normalized size = 0.84 \[ \frac{2 (d+e x)^{5/2} \left (-285285 b^5 (d+e x)^6 (-6 a B e-A b e+7 b B d)+969969 b^4 (d+e x)^5 (b d-a e) (-5 a B e-2 A b e+7 b B d)-1865325 b^3 (d+e x)^4 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)+2204475 b^2 (d+e x)^3 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)-1616615 b (d+e x)^2 (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)+692835 (d+e x) (b d-a e)^5 (-a B e-6 A b e+7 b B d)-969969 (b d-a e)^6 (B d-A e)+255255 b^6 B (d+e x)^7\right )}{4849845 e^8} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(2*(d + e*x)^(5/2)*(-969969*(b*d - a*e)^6*(B*d - A*e) + 692835*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d +
e*x) - 1616615*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^2 + 2204475*b^2*(b*d - a*e)^3*(7*b*B*d
- 4*A*b*e - 3*a*B*e)*(d + e*x)^3 - 1865325*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^4 + 96996
9*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^5 - 285285*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)
^6 + 255255*b^6*B*(d + e*x)^7))/(4849845*e^8)
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Maple [B] time = 0.009, size = 913, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
2/4849845*(e*x+d)^(5/2)*(255255*B*b^6*e^7*x^7+285285*A*b^6*e^7*x^6+1711710*B*a*b^5*e^7*x^6-210210*B*b^6*d*e^6*
x^6+1939938*A*a*b^5*e^7*x^5-228228*A*b^6*d*e^6*x^5+4849845*B*a^2*b^4*e^7*x^5-1369368*B*a*b^5*d*e^6*x^5+168168*
B*b^6*d^2*e^5*x^5+5595975*A*a^2*b^4*e^7*x^4-1492260*A*a*b^5*d*e^6*x^4+175560*A*b^6*d^2*e^5*x^4+7461300*B*a^3*b
^3*e^7*x^4-3730650*B*a^2*b^4*d*e^6*x^4+1053360*B*a*b^5*d^2*e^5*x^4-129360*B*b^6*d^3*e^4*x^4+8817900*A*a^3*b^3*
e^7*x^3-4069800*A*a^2*b^4*d*e^6*x^3+1085280*A*a*b^5*d^2*e^5*x^3-127680*A*b^6*d^3*e^4*x^3+6613425*B*a^4*b^2*e^7
*x^3-5426400*B*a^3*b^3*d*e^6*x^3+2713200*B*a^2*b^4*d^2*e^5*x^3-766080*B*a*b^5*d^3*e^4*x^3+94080*B*b^6*d^4*e^3*
x^3+8083075*A*a^4*b^2*e^7*x^2-5878600*A*a^3*b^3*d*e^6*x^2+2713200*A*a^2*b^4*d^2*e^5*x^2-723520*A*a*b^5*d^3*e^4
*x^2+85120*A*b^6*d^4*e^3*x^2+3233230*B*a^5*b*e^7*x^2-4408950*B*a^4*b^2*d*e^6*x^2+3617600*B*a^3*b^3*d^2*e^5*x^2
-1808800*B*a^2*b^4*d^3*e^4*x^2+510720*B*a*b^5*d^4*e^3*x^2-62720*B*b^6*d^5*e^2*x^2+4157010*A*a^5*b*e^7*x-461890
0*A*a^4*b^2*d*e^6*x+3359200*A*a^3*b^3*d^2*e^5*x-1550400*A*a^2*b^4*d^3*e^4*x+413440*A*a*b^5*d^4*e^3*x-48640*A*b
^6*d^5*e^2*x+692835*B*a^6*e^7*x-1847560*B*a^5*b*d*e^6*x+2519400*B*a^4*b^2*d^2*e^5*x-2067200*B*a^3*b^3*d^3*e^4*
x+1033600*B*a^2*b^4*d^4*e^3*x-291840*B*a*b^5*d^5*e^2*x+35840*B*b^6*d^6*e*x+969969*A*a^6*e^7-1662804*A*a^5*b*d*
e^6+1847560*A*a^4*b^2*d^2*e^5-1343680*A*a^3*b^3*d^3*e^4+620160*A*a^2*b^4*d^4*e^3-165376*A*a*b^5*d^5*e^2+19456*
A*b^6*d^6*e-277134*B*a^6*d*e^6+739024*B*a^5*b*d^2*e^5-1007760*B*a^4*b^2*d^3*e^4+826880*B*a^3*b^3*d^4*e^3-41344
0*B*a^2*b^4*d^5*e^2+116736*B*a*b^5*d^6*e-14336*B*b^6*d^7)/e^8
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Maxima [B] time = 1.01682, size = 1035, normalized size = 3.36 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
[Out]
2/4849845*(255255*(e*x + d)^(19/2)*B*b^6 - 285285*(7*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*(e*x + d)^(17/2) + 96996
9*(7*B*b^6*d^2 - 2*(6*B*a*b^5 + A*b^6)*d*e + (5*B*a^2*b^4 + 2*A*a*b^5)*e^2)*(e*x + d)^(15/2) - 1865325*(7*B*b^
6*d^3 - 3*(6*B*a*b^5 + A*b^6)*d^2*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^2 - (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^3)*(e*
x + d)^(13/2) + 2204475*(7*B*b^6*d^4 - 4*(6*B*a*b^5 + A*b^6)*d^3*e + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^2 - 4*(
4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^3 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^4)*(e*x + d)^(11/2) - 1616615*(7*B*b^6*d^5 -
5*(6*B*a*b^5 + A*b^6)*d^4*e + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^2 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^3 +
5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^4 - (2*B*a^5*b + 5*A*a^4*b^2)*e^5)*(e*x + d)^(9/2) + 692835*(7*B*b^6*d^6 - 6
*(6*B*a*b^5 + A*b^6)*d^5*e + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^2 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^3 + 1
5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^4 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)*e^6)*(e*x + d)
^(7/2) - 969969*(B*b^6*d^7 - A*a^6*e^7 - (6*B*a*b^5 + A*b^6)*d^6*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 5*(
4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e
^5 + (B*a^6 + 6*A*a^5*b)*d*e^6)*(e*x + d)^(5/2))/e^8
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Fricas [B] time = 1.41069, size = 2556, normalized size = 8.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
[Out]
2/4849845*(255255*B*b^6*e^9*x^9 - 14336*B*b^6*d^9 + 969969*A*a^6*d^2*e^7 + 19456*(6*B*a*b^5 + A*b^6)*d^8*e - 8
2688*(5*B*a^2*b^4 + 2*A*a*b^5)*d^7*e^2 + 206720*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^6*e^3 - 335920*(3*B*a^4*b^2 + 4*
A*a^3*b^3)*d^5*e^4 + 369512*(2*B*a^5*b + 5*A*a^4*b^2)*d^4*e^5 - 277134*(B*a^6 + 6*A*a^5*b)*d^3*e^6 + 15015*(20
*B*b^6*d*e^8 + 19*(6*B*a*b^5 + A*b^6)*e^9)*x^8 + 3003*(B*b^6*d^2*e^7 + 114*(6*B*a*b^5 + A*b^6)*d*e^8 + 323*(5*
B*a^2*b^4 + 2*A*a*b^5)*e^9)*x^7 - 231*(14*B*b^6*d^3*e^6 - 19*(6*B*a*b^5 + A*b^6)*d^2*e^7 - 5168*(5*B*a^2*b^4 +
2*A*a*b^5)*d*e^8 - 8075*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^9)*x^6 + 21*(168*B*b^6*d^4*e^5 - 228*(6*B*a*b^5 + A*b^6
)*d^3*e^6 + 969*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^7 + 113050*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^8 + 104975*(3*B*a^4
*b^2 + 4*A*a^3*b^3)*e^9)*x^5 - 35*(112*B*b^6*d^5*e^4 - 152*(6*B*a*b^5 + A*b^6)*d^4*e^5 + 646*(5*B*a^2*b^4 + 2*
A*a*b^5)*d^3*e^6 - 1615*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^7 - 83980*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^8 - 46189*
(2*B*a^5*b + 5*A*a^4*b^2)*e^9)*x^4 + 5*(896*B*b^6*d^6*e^3 - 1216*(6*B*a*b^5 + A*b^6)*d^5*e^4 + 5168*(5*B*a^2*b
^4 + 2*A*a*b^5)*d^4*e^5 - 12920*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^6 + 20995*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^
7 + 461890*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^8 + 138567*(B*a^6 + 6*A*a^5*b)*e^9)*x^3 - 3*(1792*B*b^6*d^7*e^2 - 323
323*A*a^6*e^9 - 2432*(6*B*a*b^5 + A*b^6)*d^6*e^3 + 10336*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^4 - 25840*(4*B*a^3*b^
3 + 3*A*a^2*b^4)*d^4*e^5 + 41990*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^6 - 46189*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^7
- 369512*(B*a^6 + 6*A*a^5*b)*d*e^8)*x^2 + (7168*B*b^6*d^8*e + 1939938*A*a^6*d*e^8 - 9728*(6*B*a*b^5 + A*b^6)*
d^7*e^2 + 41344*(5*B*a^2*b^4 + 2*A*a*b^5)*d^6*e^3 - 103360*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^5*e^4 + 167960*(3*B*a
^4*b^2 + 4*A*a^3*b^3)*d^4*e^5 - 184756*(2*B*a^5*b + 5*A*a^4*b^2)*d^3*e^6 + 138567*(B*a^6 + 6*A*a^5*b)*d^2*e^7)
*x)*sqrt(e*x + d)/e^8
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Sympy [A] time = 61.2572, size = 2252, normalized size = 7.31 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
A*a**6*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*A*a**6*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 12*A*a**5*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 12*A*a**5*b*(d**2*
(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 30*A*a**4*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**3 + 30*A*a**4*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**3 + 40*A*a**3*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**4 + 40*A*a**3*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**4 + 30*A*a**2*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**5 + 30*A*a**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**5 + 12*A*a*b**5*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*a*b**5*(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 + 1
5*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*A*b**6*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 + 2*A*b**6*(-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**7 + 2*B*a**6*d*(-d*
(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 2*B*a**6*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (
d + e*x)**(7/2)/7)/e**2 + 12*B*a**5*b*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 + 12*B*a**5*b*(-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 + 30*B*a**4*b**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 + 30*B*a**4*b**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 + 40*B*a**3*b**3*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)/1
1)/e**5 + 40*B*a**3*b**3*(-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 + 30*B*a**2*b**4*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 + 30*B*a**2*b**4*(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)**(1
3/2)/13 + (d + e*x)**(15/2)/15)/e**6 + 12*B*a*b**5*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)/1
3 + (d + e*x)**(15/2)/15)/e**7 + 12*B*a*b**5*(-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**7 + 2*B*b**6*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 + 2*B*b**6*(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)/1
1 - 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 = 1.33754, size = 2687, normalized size = 8.72 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
[Out]
2/14549535*(969969*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^6*d*e^(-1) + 5819814*(3*(x*e + d)^(5/2) - 5*(
x*e + d)^(3/2)*d)*A*a^5*b*d*e^(-1) + 831402*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^
2)*B*a^5*b*d*e^(-2) + 2078505*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a^4*b^2*d
*e^(-2) + 692835*(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*a^4*b^2*d*e^(-3) + 923780*(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*a^3*b^3*d*e^(-3) + 83980*(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*a^3*b^3*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*a^2*b^4*d*e^(-4) + 24225*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 1287
0*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*B*a^2*b^4*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*a*b^5*d*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)*B*a*b^5*d*e^(-6) + 323*(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*b^6*d*e^(-6) + 133*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 17
6715*(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*b^6*d*e^(-7) + 4849845*(x*e + d)^(3/2)*A*a^6*d
+ 138567*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a^6*e^(-1) + 831402*(15*(x*e
+ d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a^5*b*e^(-1) + 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)*B*a^5*b*e^(-2) + 692835*(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*a^4*b^2*e^(-2) + 62985*(31
5*(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*a^4*b^2*e^(-3) + 83980*(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*a^3*b^3*e^(-3) + 32300*(693*(x*e + d)^(13/2) - 4
095*(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 - 30
03*(x*e + d)^(3/2)*d^5)*B*a^3*b^3*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*a^2*b^4*e^
(-4) + 4845*(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)*B*a^2*b^4*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*a*b^5*e^(-5) + 7
98*(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*a*b^5*e^(-6) + 133*(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)*A*b^6*e^(-6) + 7*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 38
79876*(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 + 8314020*(x*e + d)^(7/2)*d^6 - 3325608*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8)*B*b^6*e^(
-7) + 969969*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^6)*e^(-1)