3.16.82 \(\int (A+B x) (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\)
Optimal. Leaf size=308 \[ -\frac {2 b^5 (d+e x)^{17/2} (-6 a B e-A b e+7 b B d)}{17 e^8}+\frac {2 b^4 (d+e x)^{15/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{5 e^8}-\frac {10 b^3 (d+e x)^{13/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{13 e^8}+\frac {10 b^2 (d+e x)^{11/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{11 e^8}-\frac {2 b (d+e x)^{9/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{3 e^8}+\frac {2 (d+e x)^{7/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{7 e^8}-\frac {2 (d+e x)^{5/2} (b d-a e)^6 (B d-A e)}{5 e^8}+\frac {2 b^6 B (d+e x)^{19/2}}{19 e^8} \]
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Rubi [A] time = 0.14, antiderivative size = 308, normalized size of antiderivative = 1.00,
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} \begin {gather*} -\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} \end {gather*}
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*}
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Mathematica [A] time = 0.24, size = 259, normalized size = 0.84 \begin {gather*} \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} \end {gather*}
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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IntegrateAlgebraic [B] time = 0.46, size = 1069, normalized size = 3.47 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (-969969 b^6 B d^7+969969 A b^6 e d^6+5819814 a b^5 B e d^6+4849845 b^6 B (d+e x) d^6-5819814 a A b^5 e^2 d^5-14549535 a^2 b^4 B e^2 d^5-11316305 b^6 B (d+e x)^2 d^5-4157010 A b^6 e (d+e x) d^5-24942060 a b^5 B e (d+e x) d^5+14549535 a^2 A b^4 e^3 d^4+19399380 a^3 b^3 B e^3 d^4+15431325 b^6 B (d+e x)^3 d^4+8083075 A b^6 e (d+e x)^2 d^4+48498450 a b^5 B e (d+e x)^2 d^4+20785050 a A b^5 e^2 (d+e x) d^4+51962625 a^2 b^4 B e^2 (d+e x) d^4-19399380 a^3 A b^3 e^4 d^3-14549535 a^4 b^2 B e^4 d^3-13057275 b^6 B (d+e x)^4 d^3-8817900 A b^6 e (d+e x)^3 d^3-52907400 a b^5 B e (d+e x)^3 d^3-32332300 a A b^5 e^2 (d+e x)^2 d^3-80830750 a^2 b^4 B e^2 (d+e x)^2 d^3-41570100 a^2 A b^4 e^3 (d+e x) d^3-55426800 a^3 b^3 B e^3 (d+e x) d^3+14549535 a^4 A b^2 e^5 d^2+5819814 a^5 b B e^5 d^2+6789783 b^6 B (d+e x)^5 d^2+5595975 A b^6 e (d+e x)^4 d^2+33575850 a b^5 B e (d+e x)^4 d^2+26453700 a A b^5 e^2 (d+e x)^3 d^2+66134250 a^2 b^4 B e^2 (d+e x)^3 d^2+48498450 a^2 A b^4 e^3 (d+e x)^2 d^2+64664600 a^3 b^3 B e^3 (d+e x)^2 d^2+41570100 a^3 A b^3 e^4 (d+e x) d^2+31177575 a^4 b^2 B e^4 (d+e x) d^2-5819814 a^5 A b e^6 d-969969 a^6 B e^6 d-1996995 b^6 B (d+e x)^6 d-1939938 A b^6 e (d+e x)^5 d-11639628 a b^5 B e (d+e x)^5 d-11191950 a A b^5 e^2 (d+e x)^4 d-27979875 a^2 b^4 B e^2 (d+e x)^4 d-26453700 a^2 A b^4 e^3 (d+e x)^3 d-35271600 a^3 b^3 B e^3 (d+e x)^3 d-32332300 a^3 A b^3 e^4 (d+e x)^2 d-24249225 a^4 b^2 B e^4 (d+e x)^2 d-20785050 a^4 A b^2 e^5 (d+e x) d-8314020 a^5 b B e^5 (d+e x) d+969969 a^6 A e^7+255255 b^6 B (d+e x)^7+285285 A b^6 e (d+e x)^6+1711710 a b^5 B e (d+e x)^6+1939938 a A b^5 e^2 (d+e x)^5+4849845 a^2 b^4 B e^2 (d+e x)^5+5595975 a^2 A b^4 e^3 (d+e x)^4+7461300 a^3 b^3 B e^3 (d+e x)^4+8817900 a^3 A b^3 e^4 (d+e x)^3+6613425 a^4 b^2 B e^4 (d+e x)^3+8083075 a^4 A b^2 e^5 (d+e x)^2+3233230 a^5 b B e^5 (d+e x)^2+4157010 a^5 A b e^6 (d+e x)+692835 a^6 B e^6 (d+e x)\right )}{4849845 e^8} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(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^6*B*d^7 + 969969*A*b^6*d^6*e + 5819814*a*b^5*B*d^6*e - 5819814*a*A*b^5*d^5*e^2 -
14549535*a^2*b^4*B*d^5*e^2 + 14549535*a^2*A*b^4*d^4*e^3 + 19399380*a^3*b^3*B*d^4*e^3 - 19399380*a^3*A*b^3*d^3
*e^4 - 14549535*a^4*b^2*B*d^3*e^4 + 14549535*a^4*A*b^2*d^2*e^5 + 5819814*a^5*b*B*d^2*e^5 - 5819814*a^5*A*b*d*e
^6 - 969969*a^6*B*d*e^6 + 969969*a^6*A*e^7 + 4849845*b^6*B*d^6*(d + e*x) - 4157010*A*b^6*d^5*e*(d + e*x) - 249
42060*a*b^5*B*d^5*e*(d + e*x) + 20785050*a*A*b^5*d^4*e^2*(d + e*x) + 51962625*a^2*b^4*B*d^4*e^2*(d + e*x) - 41
570100*a^2*A*b^4*d^3*e^3*(d + e*x) - 55426800*a^3*b^3*B*d^3*e^3*(d + e*x) + 41570100*a^3*A*b^3*d^2*e^4*(d + e*
x) + 31177575*a^4*b^2*B*d^2*e^4*(d + e*x) - 20785050*a^4*A*b^2*d*e^5*(d + e*x) - 8314020*a^5*b*B*d*e^5*(d + e*
x) + 4157010*a^5*A*b*e^6*(d + e*x) + 692835*a^6*B*e^6*(d + e*x) - 11316305*b^6*B*d^5*(d + e*x)^2 + 8083075*A*b
^6*d^4*e*(d + e*x)^2 + 48498450*a*b^5*B*d^4*e*(d + e*x)^2 - 32332300*a*A*b^5*d^3*e^2*(d + e*x)^2 - 80830750*a^
2*b^4*B*d^3*e^2*(d + e*x)^2 + 48498450*a^2*A*b^4*d^2*e^3*(d + e*x)^2 + 64664600*a^3*b^3*B*d^2*e^3*(d + e*x)^2
- 32332300*a^3*A*b^3*d*e^4*(d + e*x)^2 - 24249225*a^4*b^2*B*d*e^4*(d + e*x)^2 + 8083075*a^4*A*b^2*e^5*(d + e*x
)^2 + 3233230*a^5*b*B*e^5*(d + e*x)^2 + 15431325*b^6*B*d^4*(d + e*x)^3 - 8817900*A*b^6*d^3*e*(d + e*x)^3 - 529
07400*a*b^5*B*d^3*e*(d + e*x)^3 + 26453700*a*A*b^5*d^2*e^2*(d + e*x)^3 + 66134250*a^2*b^4*B*d^2*e^2*(d + e*x)^
3 - 26453700*a^2*A*b^4*d*e^3*(d + e*x)^3 - 35271600*a^3*b^3*B*d*e^3*(d + e*x)^3 + 8817900*a^3*A*b^3*e^4*(d + e
*x)^3 + 6613425*a^4*b^2*B*e^4*(d + e*x)^3 - 13057275*b^6*B*d^3*(d + e*x)^4 + 5595975*A*b^6*d^2*e*(d + e*x)^4 +
33575850*a*b^5*B*d^2*e*(d + e*x)^4 - 11191950*a*A*b^5*d*e^2*(d + e*x)^4 - 27979875*a^2*b^4*B*d*e^2*(d + e*x)^
4 + 5595975*a^2*A*b^4*e^3*(d + e*x)^4 + 7461300*a^3*b^3*B*e^3*(d + e*x)^4 + 6789783*b^6*B*d^2*(d + e*x)^5 - 19
39938*A*b^6*d*e*(d + e*x)^5 - 11639628*a*b^5*B*d*e*(d + e*x)^5 + 1939938*a*A*b^5*e^2*(d + e*x)^5 + 4849845*a^2
*b^4*B*e^2*(d + e*x)^5 - 1996995*b^6*B*d*(d + e*x)^6 + 285285*A*b^6*e*(d + e*x)^6 + 1711710*a*b^5*B*e*(d + e*x
)^6 + 255255*b^6*B*(d + e*x)^7))/(4849845*e^8)
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fricas [B] time = 0.45, size = 1118, normalized size = 3.63
result too large to display
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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giac [B] time = 0.41, size = 3285, normalized size = 10.67
result too large to display
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*(4849845*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^6*d^2*e^(-1) + 29099070*((x*e + d)^(3/2) - 3*sqr
t(x*e + d)*d)*A*a^5*b*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)*B
*a^5*b*d^2*e^(-2) + 14549535*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^4*b^2*d^2*e
^(-2) + 6235515*(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*a
^4*b^2*d^2*e^(-3) + 8314020*(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*a^3*b^3*d^2*e^(-3) + 923780*(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*a^3*b^3*d^2*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*a^2*b^4*d^2*e^(-4
) + 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*a^2*b^4*d^2*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*a*b^5*d^2*e^(-5) + 29070*(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*a*b^5*d^2*e^(-6) + 4845*(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*b^6*d^2*e^
(-6) + 2261*(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*b^6*d^2*e^(-7) + 1939938*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^6*d*e^
(-1) + 11639628*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^5*b*d*e^(-1) + 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)*B*a^5*b*d*e^(-2) + 1
2471030*(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*a^4*b^2*d
*e^(-2) + 1385670*(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*a^4*b^2*d*e^(-3) + 1847560*(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*a^3*b^3*d*e^(-3) + 839800*(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*a^3*b^3*d*e^(-4) + 629850*(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*a^2*b^4*d*e^(
-4) + 145350*(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*a^2*b^4*d*e^(-5) + 5814
0*(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*a*b^5*d*e^(-5) + 27132*(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*a*b^5*d*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)*A*b^6*d*e^(-6) + 266*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 5569
20*(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)*B*b^6*d*e^(-7) + 14549535*sqrt(x*e + d)*A*a^6*d^2 +
9699690*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a^6*d + 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*a^6*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*a^5*b*e^(-1) + 277134*(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*a^5*b*e^(-2) + 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*a^4*b^2*e^(-2) + 314925*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 138
6*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a^4*b^2*e^(-3) + 419900*(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*a^3*b^3*e^(-3) + 96900*(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*sq
rt(x*e + d)*d^6)*B*a^3*b^3*e^(-4) + 72675*(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*a^2*b^4*e^(-4) + 33915*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25
025*(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*a^2*b^4*e^(-5) + 13566*(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)*A*a*b^5*e^(-5) + 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)*B*a*b^5*e^(
-6) + 133*(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)*A*b^6*e^(-6) + 63*(12155*(x*e + d)^(19/2) - 122265*(x*e + d)^(17/2
)*d + 554268*(x*e + d)^(15/2)*d^2 - 1492260*(x*e + d)^(13/2)*d^3 + 2645370*(x*e + d)^(11/2)*d^4 - 3233230*(x*e
+ d)^(9/2)*d^5 + 2771340*(x*e + d)^(7/2)*d^6 - 1662804*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8 - 230
945*sqrt(x*e + d)*d^9)*B*b^6*e^(-7) + 969969*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)
*A*a^6)*e^(-1)
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maple [B] time = 0.06, size = 913, normalized size = 2.96 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (255255 B \,b^{6} x^{7} e^{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 -4618900 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}-413440 B \,a^{2} b^{4} d^{5} e^{2}+116736 B a \,b^{5} d^{6} e -14336 B \,b^{6} d^{7}\right )}{4849845 e^{8}} \end {gather*}
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 = 0.53, size = 767, normalized size = 2.49 \begin {gather*} \frac {2 \, {\left (255255 \, {\left (e x + d\right )}^{\frac {19}{2}} B b^{6} - 285285 \, {\left (7 \, B b^{6} d - {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 969969 \, {\left (7 \, B b^{6} d^{2} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 1865325 \, {\left (7 \, B b^{6} d^{3} - 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{2} - {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 2204475 \, {\left (7 \, B b^{6} d^{4} - 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{2} - 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{3} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 1616615 \, {\left (7 \, B b^{6} d^{5} - 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{2} - 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{4} - {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{5}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 692835 \, {\left (7 \, B b^{6} d^{6} - 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{2} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{3} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{4} - 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{6}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 969969 \, {\left (B b^{6} d^{7} - A a^{6} e^{7} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{4849845 \, e^{8}} \end {gather*}
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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mupad [B] time = 1.93, size = 279, normalized size = 0.91 \begin {gather*} \frac {{\left (d+e\,x\right )}^{17/2}\,\left (2\,A\,b^6\,e-14\,B\,b^6\,d+12\,B\,a\,b^5\,e\right )}{17\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{7/2}\,\left (6\,A\,b\,e+B\,a\,e-7\,B\,b\,d\right )}{7\,e^8}+\frac {2\,B\,b^6\,{\left (d+e\,x\right )}^{19/2}}{19\,e^8}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{5/2}}{5\,e^8}+\frac {2\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{9/2}\,\left (5\,A\,b\,e+2\,B\,a\,e-7\,B\,b\,d\right )}{3\,e^8}+\frac {2\,b^4\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{15/2}\,\left (2\,A\,b\,e+5\,B\,a\,e-7\,B\,b\,d\right )}{5\,e^8}+\frac {10\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{11/2}\,\left (4\,A\,b\,e+3\,B\,a\,e-7\,B\,b\,d\right )}{11\,e^8}+\frac {10\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{13/2}\,\left (3\,A\,b\,e+4\,B\,a\,e-7\,B\,b\,d\right )}{13\,e^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x)*(d + e*x)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)
[Out]
((d + e*x)^(17/2)*(2*A*b^6*e - 14*B*b^6*d + 12*B*a*b^5*e))/(17*e^8) + (2*(a*e - b*d)^5*(d + e*x)^(7/2)*(6*A*b*
e + B*a*e - 7*B*b*d))/(7*e^8) + (2*B*b^6*(d + e*x)^(19/2))/(19*e^8) + (2*(A*e - B*d)*(a*e - b*d)^6*(d + e*x)^(
5/2))/(5*e^8) + (2*b*(a*e - b*d)^4*(d + e*x)^(9/2)*(5*A*b*e + 2*B*a*e - 7*B*b*d))/(3*e^8) + (2*b^4*(a*e - b*d)
*(d + e*x)^(15/2)*(2*A*b*e + 5*B*a*e - 7*B*b*d))/(5*e^8) + (10*b^2*(a*e - b*d)^3*(d + e*x)^(11/2)*(4*A*b*e + 3
*B*a*e - 7*B*b*d))/(11*e^8) + (10*b^3*(a*e - b*d)^2*(d + e*x)^(13/2)*(3*A*b*e + 4*B*a*e - 7*B*b*d))/(13*e^8)
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sympy [A] time = 69.78, size = 2252, normalized size = 7.31
result too large to display
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
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