3.16.83 \(\int (A+B x) \sqrt {d+e x} (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=308 \[ -\frac {2 b^5 (d+e x)^{15/2} (-6 a B e-A b e+7 b B d)}{15 e^8}+\frac {6 b^4 (d+e x)^{13/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{13 e^8}-\frac {10 b^3 (d+e x)^{11/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{11 e^8}+\frac {10 b^2 (d+e x)^{9/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{9 e^8}-\frac {6 b (d+e x)^{7/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{7 e^8}+\frac {2 (d+e x)^{5/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{5 e^8}-\frac {2 (d+e x)^{3/2} (b d-a e)^6 (B d-A e)}{3 e^8}+\frac {2 b^6 B (d+e x)^{17/2}}{17 e^8} \]

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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)^{15/2} (-6 a B e-A b e+7 b B d)}{15 e^8}+\frac {6 b^4 (d+e x)^{13/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{13 e^8}-\frac {10 b^3 (d+e x)^{11/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{11 e^8}+\frac {10 b^2 (d+e x)^{9/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{9 e^8}-\frac {6 b (d+e x)^{7/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{7 e^8}+\frac {2 (d+e x)^{5/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{5 e^8}-\frac {2 (d+e x)^{3/2} (b d-a e)^6 (B d-A e)}{3 e^8}+\frac {2 b^6 B (d+e x)^{17/2}}{17 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*Sqrt[d + e*x]*(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)^(3/2))/(3*e^8) + (2*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x
)^(5/2))/(5*e^8) - (6*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^(7/2))/(7*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)^(9/2))/(9*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)^(11/2))/(11*e^8) + (6*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(13/2))/(13*e^8)
- (2*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^(15/2))/(15*e^8) + (2*b^6*B*(d + e*x)^(17/2))/(17*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) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (A+B x) \sqrt {d+e x} \, dx\\ &=\int \left (\frac {(-b d+a e)^6 (-B d+A e) \sqrt {d+e x}}{e^7}+\frac {(-b d+a e)^5 (-7 b B d+6 A b e+a B e) (d+e x)^{3/2}}{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)^{5/2}}{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)^{7/2}}{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)^{9/2}}{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)^{11/2}}{e^7}+\frac {b^5 (-7 b B d+A b e+6 a B e) (d+e x)^{13/2}}{e^7}+\frac {b^6 B (d+e x)^{15/2}}{e^7}\right ) \, dx\\ &=-\frac {2 (b d-a e)^6 (B d-A e) (d+e x)^{3/2}}{3 e^8}+\frac {2 (b d-a e)^5 (7 b B d-6 A b e-a B e) (d+e x)^{5/2}}{5 e^8}-\frac {6 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) (d+e x)^{7/2}}{7 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)^{9/2}}{9 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)^{11/2}}{11 e^8}+\frac {6 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^{13/2}}{13 e^8}-\frac {2 b^5 (7 b B d-A b e-6 a B e) (d+e x)^{15/2}}{15 e^8}+\frac {2 b^6 B (d+e x)^{17/2}}{17 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)^{3/2} \left (-51051 b^5 (d+e x)^6 (-6 a B e-A b e+7 b B d)+176715 b^4 (d+e x)^5 (b d-a e) (-5 a B e-2 A b e+7 b B d)-348075 b^3 (d+e x)^4 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)+425425 b^2 (d+e x)^3 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)-328185 b (d+e x)^2 (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)+153153 (d+e x) (b d-a e)^5 (-a B e-6 A b e+7 b B d)-255255 (b d-a e)^6 (B d-A e)+45045 b^6 B (d+e x)^7\right )}{765765 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(2*(d + e*x)^(3/2)*(-255255*(b*d - a*e)^6*(B*d - A*e) + 153153*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d +
e*x) - 328185*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^2 + 425425*b^2*(b*d - a*e)^3*(7*b*B*d -
4*A*b*e - 3*a*B*e)*(d + e*x)^3 - 348075*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^4 + 176715*b
^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^5 - 51051*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^6 +
45045*b^6*B*(d + e*x)^7))/(765765*e^8)

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IntegrateAlgebraic [B]  time = 0.40, size = 1069, normalized size = 3.47 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (-255255 b^6 B d^7+255255 A b^6 e d^6+1531530 a b^5 B e d^6+1072071 b^6 B (d+e x) d^6-1531530 a A b^5 e^2 d^5-3828825 a^2 b^4 B e^2 d^5-2297295 b^6 B (d+e x)^2 d^5-918918 A b^6 e (d+e x) d^5-5513508 a b^5 B e (d+e x) d^5+3828825 a^2 A b^4 e^3 d^4+5105100 a^3 b^3 B e^3 d^4+2977975 b^6 B (d+e x)^3 d^4+1640925 A b^6 e (d+e x)^2 d^4+9845550 a b^5 B e (d+e x)^2 d^4+4594590 a A b^5 e^2 (d+e x) d^4+11486475 a^2 b^4 B e^2 (d+e x) d^4-5105100 a^3 A b^3 e^4 d^3-3828825 a^4 b^2 B e^4 d^3-2436525 b^6 B (d+e x)^4 d^3-1701700 A b^6 e (d+e x)^3 d^3-10210200 a b^5 B e (d+e x)^3 d^3-6563700 a A b^5 e^2 (d+e x)^2 d^3-16409250 a^2 b^4 B e^2 (d+e x)^2 d^3-9189180 a^2 A b^4 e^3 (d+e x) d^3-12252240 a^3 b^3 B e^3 (d+e x) d^3+3828825 a^4 A b^2 e^5 d^2+1531530 a^5 b B e^5 d^2+1237005 b^6 B (d+e x)^5 d^2+1044225 A b^6 e (d+e x)^4 d^2+6265350 a b^5 B e (d+e x)^4 d^2+5105100 a A b^5 e^2 (d+e x)^3 d^2+12762750 a^2 b^4 B e^2 (d+e x)^3 d^2+9845550 a^2 A b^4 e^3 (d+e x)^2 d^2+13127400 a^3 b^3 B e^3 (d+e x)^2 d^2+9189180 a^3 A b^3 e^4 (d+e x) d^2+6891885 a^4 b^2 B e^4 (d+e x) d^2-1531530 a^5 A b e^6 d-255255 a^6 B e^6 d-357357 b^6 B (d+e x)^6 d-353430 A b^6 e (d+e x)^5 d-2120580 a b^5 B e (d+e x)^5 d-2088450 a A b^5 e^2 (d+e x)^4 d-5221125 a^2 b^4 B e^2 (d+e x)^4 d-5105100 a^2 A b^4 e^3 (d+e x)^3 d-6806800 a^3 b^3 B e^3 (d+e x)^3 d-6563700 a^3 A b^3 e^4 (d+e x)^2 d-4922775 a^4 b^2 B e^4 (d+e x)^2 d-4594590 a^4 A b^2 e^5 (d+e x) d-1837836 a^5 b B e^5 (d+e x) d+255255 a^6 A e^7+45045 b^6 B (d+e x)^7+51051 A b^6 e (d+e x)^6+306306 a b^5 B e (d+e x)^6+353430 a A b^5 e^2 (d+e x)^5+883575 a^2 b^4 B e^2 (d+e x)^5+1044225 a^2 A b^4 e^3 (d+e x)^4+1392300 a^3 b^3 B e^3 (d+e x)^4+1701700 a^3 A b^3 e^4 (d+e x)^3+1276275 a^4 b^2 B e^4 (d+e x)^3+1640925 a^4 A b^2 e^5 (d+e x)^2+656370 a^5 b B e^5 (d+e x)^2+918918 a^5 A b e^6 (d+e x)+153153 a^6 B e^6 (d+e x)\right )}{765765 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(2*(d + e*x)^(3/2)*(-255255*b^6*B*d^7 + 255255*A*b^6*d^6*e + 1531530*a*b^5*B*d^6*e - 1531530*a*A*b^5*d^5*e^2 -
3828825*a^2*b^4*B*d^5*e^2 + 3828825*a^2*A*b^4*d^4*e^3 + 5105100*a^3*b^3*B*d^4*e^3 - 5105100*a^3*A*b^3*d^3*e^4
- 3828825*a^4*b^2*B*d^3*e^4 + 3828825*a^4*A*b^2*d^2*e^5 + 1531530*a^5*b*B*d^2*e^5 - 1531530*a^5*A*b*d*e^6 - 2
55255*a^6*B*d*e^6 + 255255*a^6*A*e^7 + 1072071*b^6*B*d^6*(d + e*x) - 918918*A*b^6*d^5*e*(d + e*x) - 5513508*a*
b^5*B*d^5*e*(d + e*x) + 4594590*a*A*b^5*d^4*e^2*(d + e*x) + 11486475*a^2*b^4*B*d^4*e^2*(d + e*x) - 9189180*a^2
*A*b^4*d^3*e^3*(d + e*x) - 12252240*a^3*b^3*B*d^3*e^3*(d + e*x) + 9189180*a^3*A*b^3*d^2*e^4*(d + e*x) + 689188
5*a^4*b^2*B*d^2*e^4*(d + e*x) - 4594590*a^4*A*b^2*d*e^5*(d + e*x) - 1837836*a^5*b*B*d*e^5*(d + e*x) + 918918*a
^5*A*b*e^6*(d + e*x) + 153153*a^6*B*e^6*(d + e*x) - 2297295*b^6*B*d^5*(d + e*x)^2 + 1640925*A*b^6*d^4*e*(d + e
*x)^2 + 9845550*a*b^5*B*d^4*e*(d + e*x)^2 - 6563700*a*A*b^5*d^3*e^2*(d + e*x)^2 - 16409250*a^2*b^4*B*d^3*e^2*(
d + e*x)^2 + 9845550*a^2*A*b^4*d^2*e^3*(d + e*x)^2 + 13127400*a^3*b^3*B*d^2*e^3*(d + e*x)^2 - 6563700*a^3*A*b^
3*d*e^4*(d + e*x)^2 - 4922775*a^4*b^2*B*d*e^4*(d + e*x)^2 + 1640925*a^4*A*b^2*e^5*(d + e*x)^2 + 656370*a^5*b*B
*e^5*(d + e*x)^2 + 2977975*b^6*B*d^4*(d + e*x)^3 - 1701700*A*b^6*d^3*e*(d + e*x)^3 - 10210200*a*b^5*B*d^3*e*(d
+ e*x)^3 + 5105100*a*A*b^5*d^2*e^2*(d + e*x)^3 + 12762750*a^2*b^4*B*d^2*e^2*(d + e*x)^3 - 5105100*a^2*A*b^4*d
*e^3*(d + e*x)^3 - 6806800*a^3*b^3*B*d*e^3*(d + e*x)^3 + 1701700*a^3*A*b^3*e^4*(d + e*x)^3 + 1276275*a^4*b^2*B
*e^4*(d + e*x)^3 - 2436525*b^6*B*d^3*(d + e*x)^4 + 1044225*A*b^6*d^2*e*(d + e*x)^4 + 6265350*a*b^5*B*d^2*e*(d
+ e*x)^4 - 2088450*a*A*b^5*d*e^2*(d + e*x)^4 - 5221125*a^2*b^4*B*d*e^2*(d + e*x)^4 + 1044225*a^2*A*b^4*e^3*(d
+ e*x)^4 + 1392300*a^3*b^3*B*e^3*(d + e*x)^4 + 1237005*b^6*B*d^2*(d + e*x)^5 - 353430*A*b^6*d*e*(d + e*x)^5 -
2120580*a*b^5*B*d*e*(d + e*x)^5 + 353430*a*A*b^5*e^2*(d + e*x)^5 + 883575*a^2*b^4*B*e^2*(d + e*x)^5 - 357357*b
^6*B*d*(d + e*x)^6 + 51051*A*b^6*e*(d + e*x)^6 + 306306*a*b^5*B*e*(d + e*x)^6 + 45045*b^6*B*(d + e*x)^7))/(765
765*e^8)

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fricas [B]  time = 0.43, size = 942, normalized size = 3.06 \begin {gather*} \frac {2 \, {\left (45045 \, B b^{6} e^{8} x^{8} - 14336 \, B b^{6} d^{8} + 255255 \, A a^{6} d e^{7} + 17408 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{7} e - 65280 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{6} e^{2} + 141440 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{5} e^{3} - 194480 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{4} e^{4} + 175032 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{3} e^{5} - 102102 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2} e^{6} + 3003 \, {\left (B b^{6} d e^{7} + 17 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{8}\right )} x^{7} - 231 \, {\left (14 \, B b^{6} d^{2} e^{6} - 17 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{7} - 765 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{8}\right )} x^{6} + 63 \, {\left (56 \, B b^{6} d^{3} e^{5} - 68 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{6} + 255 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{7} + 5525 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{8}\right )} x^{5} - 35 \, {\left (112 \, B b^{6} d^{4} e^{4} - 136 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{5} + 510 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{6} - 1105 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{7} - 12155 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{8}\right )} x^{4} + 5 \, {\left (896 \, B b^{6} d^{5} e^{3} - 1088 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{4} + 4080 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{5} - 8840 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{6} + 12155 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{7} + 65637 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{8}\right )} x^{3} - 3 \, {\left (1792 \, B b^{6} d^{6} e^{2} - 2176 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{3} + 8160 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{4} - 17680 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{5} + 24310 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{6} - 21879 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{7} - 51051 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{8}\right )} x^{2} + {\left (7168 \, B b^{6} d^{7} e + 255255 \, A a^{6} e^{8} - 8704 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e^{2} + 32640 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{3} - 70720 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{4} + 97240 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{5} - 87516 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{6} + 51051 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{7}\right )} x\right )} \sqrt {e x + d}}{765765 \, e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/765765*(45045*B*b^6*e^8*x^8 - 14336*B*b^6*d^8 + 255255*A*a^6*d*e^7 + 17408*(6*B*a*b^5 + A*b^6)*d^7*e - 65280
*(5*B*a^2*b^4 + 2*A*a*b^5)*d^6*e^2 + 141440*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^5*e^3 - 194480*(3*B*a^4*b^2 + 4*A*a^
3*b^3)*d^4*e^4 + 175032*(2*B*a^5*b + 5*A*a^4*b^2)*d^3*e^5 - 102102*(B*a^6 + 6*A*a^5*b)*d^2*e^6 + 3003*(B*b^6*d
*e^7 + 17*(6*B*a*b^5 + A*b^6)*e^8)*x^7 - 231*(14*B*b^6*d^2*e^6 - 17*(6*B*a*b^5 + A*b^6)*d*e^7 - 765*(5*B*a^2*b
^4 + 2*A*a*b^5)*e^8)*x^6 + 63*(56*B*b^6*d^3*e^5 - 68*(6*B*a*b^5 + A*b^6)*d^2*e^6 + 255*(5*B*a^2*b^4 + 2*A*a*b^
5)*d*e^7 + 5525*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^8)*x^5 - 35*(112*B*b^6*d^4*e^4 - 136*(6*B*a*b^5 + A*b^6)*d^3*e^5
+ 510*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^6 - 1105*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^7 - 12155*(3*B*a^4*b^2 + 4*A*a
^3*b^3)*e^8)*x^4 + 5*(896*B*b^6*d^5*e^3 - 1088*(6*B*a*b^5 + A*b^6)*d^4*e^4 + 4080*(5*B*a^2*b^4 + 2*A*a*b^5)*d^
3*e^5 - 8840*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^6 + 12155*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^7 + 65637*(2*B*a^5*b
+ 5*A*a^4*b^2)*e^8)*x^3 - 3*(1792*B*b^6*d^6*e^2 - 2176*(6*B*a*b^5 + A*b^6)*d^5*e^3 + 8160*(5*B*a^2*b^4 + 2*A*a
*b^5)*d^4*e^4 - 17680*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^5 + 24310*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^6 - 21879*
(2*B*a^5*b + 5*A*a^4*b^2)*d*e^7 - 51051*(B*a^6 + 6*A*a^5*b)*e^8)*x^2 + (7168*B*b^6*d^7*e + 255255*A*a^6*e^8 -
8704*(6*B*a*b^5 + A*b^6)*d^6*e^2 + 32640*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^3 - 70720*(4*B*a^3*b^3 + 3*A*a^2*b^4)
*d^4*e^4 + 97240*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^5 - 87516*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^6 + 51051*(B*a^6
+ 6*A*a^5*b)*d*e^7)*x)*sqrt(e*x + d)/e^8

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giac [B]  time = 0.29, size = 1984, normalized size = 6.44

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/765765*(255255*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^6*d*e^(-1) + 1531530*((x*e + d)^(3/2) - 3*sqrt(x*e
+ d)*d)*A*a^5*b*d*e^(-1) + 306306*(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*
e^(-2) + 765765*(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*e^(-2) + 328185*
(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*e^(-3)
+ 437580*(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*e^(-3) + 48620*(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*e^(-4) + 36465*(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*e^(-4) + 16575*(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*e^(-5) + 6630*(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*e^(-5) + 15
30*(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 + 900
9*(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*e^(-6) + 255*(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*e^(-6) + 119*(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 - 2702
7*(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*e^(-7) + 51051*(3*(x*e + d
)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^6*e^(-1) + 306306*(3*(x*e + d)^(5/2) - 10*(x*e + d)
^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^5*b*e^(-1) + 131274*(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*e^(-2) + 328185*(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*e^(-2) + 36465*(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*e^(-3) + 48620*(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*e^(-3) + 22100*(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*e^(-4) + 16575*(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*e^(-4) + 3825*(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*e^(-5) + 1530*(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*e^(-5) + 714*(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*e^(-6) + 119*(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*e^(-6) + 7*(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 + 6126
12*(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*e^(-7) + 765765*sqrt(x*e
+ d)*A*a^6*d + 255255*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*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 {3}{2}} \left (45045 B \,b^{6} x^{7} e^{7}+51051 A \,b^{6} e^{7} x^{6}+306306 B a \,b^{5} e^{7} x^{6}-42042 B \,b^{6} d \,e^{6} x^{6}+353430 A a \,b^{5} e^{7} x^{5}-47124 A \,b^{6} d \,e^{6} x^{5}+883575 B \,a^{2} b^{4} e^{7} x^{5}-282744 B a \,b^{5} d \,e^{6} x^{5}+38808 B \,b^{6} d^{2} e^{5} x^{5}+1044225 A \,a^{2} b^{4} e^{7} x^{4}-321300 A a \,b^{5} d \,e^{6} x^{4}+42840 A \,b^{6} d^{2} e^{5} x^{4}+1392300 B \,a^{3} b^{3} e^{7} x^{4}-803250 B \,a^{2} b^{4} d \,e^{6} x^{4}+257040 B a \,b^{5} d^{2} e^{5} x^{4}-35280 B \,b^{6} d^{3} e^{4} x^{4}+1701700 A \,a^{3} b^{3} e^{7} x^{3}-928200 A \,a^{2} b^{4} d \,e^{6} x^{3}+285600 A a \,b^{5} d^{2} e^{5} x^{3}-38080 A \,b^{6} d^{3} e^{4} x^{3}+1276275 B \,a^{4} b^{2} e^{7} x^{3}-1237600 B \,a^{3} b^{3} d \,e^{6} x^{3}+714000 B \,a^{2} b^{4} d^{2} e^{5} x^{3}-228480 B a \,b^{5} d^{3} e^{4} x^{3}+31360 B \,b^{6} d^{4} e^{3} x^{3}+1640925 A \,a^{4} b^{2} e^{7} x^{2}-1458600 A \,a^{3} b^{3} d \,e^{6} x^{2}+795600 A \,a^{2} b^{4} d^{2} e^{5} x^{2}-244800 A a \,b^{5} d^{3} e^{4} x^{2}+32640 A \,b^{6} d^{4} e^{3} x^{2}+656370 B \,a^{5} b \,e^{7} x^{2}-1093950 B \,a^{4} b^{2} d \,e^{6} x^{2}+1060800 B \,a^{3} b^{3} d^{2} e^{5} x^{2}-612000 B \,a^{2} b^{4} d^{3} e^{4} x^{2}+195840 B a \,b^{5} d^{4} e^{3} x^{2}-26880 B \,b^{6} d^{5} e^{2} x^{2}+918918 A \,a^{5} b \,e^{7} x -1312740 A \,a^{4} b^{2} d \,e^{6} x +1166880 A \,a^{3} b^{3} d^{2} e^{5} x -636480 A \,a^{2} b^{4} d^{3} e^{4} x +195840 A a \,b^{5} d^{4} e^{3} x -26112 A \,b^{6} d^{5} e^{2} x +153153 B \,a^{6} e^{7} x -525096 B \,a^{5} b d \,e^{6} x +875160 B \,a^{4} b^{2} d^{2} e^{5} x -848640 B \,a^{3} b^{3} d^{3} e^{4} x +489600 B \,a^{2} b^{4} d^{4} e^{3} x -156672 B a \,b^{5} d^{5} e^{2} x +21504 B \,b^{6} d^{6} e x +255255 A \,a^{6} e^{7}-612612 A \,a^{5} b d \,e^{6}+875160 A \,a^{4} b^{2} d^{2} e^{5}-777920 A \,a^{3} b^{3} d^{3} e^{4}+424320 A \,a^{2} b^{4} d^{4} e^{3}-130560 A a \,b^{5} d^{5} e^{2}+17408 A \,b^{6} d^{6} e -102102 B \,a^{6} d \,e^{6}+350064 B \,a^{5} b \,d^{2} e^{5}-583440 B \,a^{4} b^{2} d^{3} e^{4}+565760 B \,a^{3} b^{3} d^{4} e^{3}-326400 B \,a^{2} b^{4} d^{5} e^{2}+104448 B a \,b^{5} d^{6} e -14336 B \,b^{6} d^{7}\right )}{765765 e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3*(e*x+d)^(1/2),x)

[Out]

2/765765*(e*x+d)^(3/2)*(45045*B*b^6*e^7*x^7+51051*A*b^6*e^7*x^6+306306*B*a*b^5*e^7*x^6-42042*B*b^6*d*e^6*x^6+3
53430*A*a*b^5*e^7*x^5-47124*A*b^6*d*e^6*x^5+883575*B*a^2*b^4*e^7*x^5-282744*B*a*b^5*d*e^6*x^5+38808*B*b^6*d^2*
e^5*x^5+1044225*A*a^2*b^4*e^7*x^4-321300*A*a*b^5*d*e^6*x^4+42840*A*b^6*d^2*e^5*x^4+1392300*B*a^3*b^3*e^7*x^4-8
03250*B*a^2*b^4*d*e^6*x^4+257040*B*a*b^5*d^2*e^5*x^4-35280*B*b^6*d^3*e^4*x^4+1701700*A*a^3*b^3*e^7*x^3-928200*
A*a^2*b^4*d*e^6*x^3+285600*A*a*b^5*d^2*e^5*x^3-38080*A*b^6*d^3*e^4*x^3+1276275*B*a^4*b^2*e^7*x^3-1237600*B*a^3
*b^3*d*e^6*x^3+714000*B*a^2*b^4*d^2*e^5*x^3-228480*B*a*b^5*d^3*e^4*x^3+31360*B*b^6*d^4*e^3*x^3+1640925*A*a^4*b
^2*e^7*x^2-1458600*A*a^3*b^3*d*e^6*x^2+795600*A*a^2*b^4*d^2*e^5*x^2-244800*A*a*b^5*d^3*e^4*x^2+32640*A*b^6*d^4
*e^3*x^2+656370*B*a^5*b*e^7*x^2-1093950*B*a^4*b^2*d*e^6*x^2+1060800*B*a^3*b^3*d^2*e^5*x^2-612000*B*a^2*b^4*d^3
*e^4*x^2+195840*B*a*b^5*d^4*e^3*x^2-26880*B*b^6*d^5*e^2*x^2+918918*A*a^5*b*e^7*x-1312740*A*a^4*b^2*d*e^6*x+116
6880*A*a^3*b^3*d^2*e^5*x-636480*A*a^2*b^4*d^3*e^4*x+195840*A*a*b^5*d^4*e^3*x-26112*A*b^6*d^5*e^2*x+153153*B*a^
6*e^7*x-525096*B*a^5*b*d*e^6*x+875160*B*a^4*b^2*d^2*e^5*x-848640*B*a^3*b^3*d^3*e^4*x+489600*B*a^2*b^4*d^4*e^3*
x-156672*B*a*b^5*d^5*e^2*x+21504*B*b^6*d^6*e*x+255255*A*a^6*e^7-612612*A*a^5*b*d*e^6+875160*A*a^4*b^2*d^2*e^5-
777920*A*a^3*b^3*d^3*e^4+424320*A*a^2*b^4*d^4*e^3-130560*A*a*b^5*d^5*e^2+17408*A*b^6*d^6*e-102102*B*a^6*d*e^6+
350064*B*a^5*b*d^2*e^5-583440*B*a^4*b^2*d^3*e^4+565760*B*a^3*b^3*d^4*e^3-326400*B*a^2*b^4*d^5*e^2+104448*B*a*b
^5*d^6*e-14336*B*b^6*d^7)/e^8

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maxima [B]  time = 0.65, size = 767, normalized size = 2.49 \begin {gather*} \frac {2 \, {\left (45045 \, {\left (e x + d\right )}^{\frac {17}{2}} B b^{6} - 51051 \, {\left (7 \, B b^{6} d - {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 176715 \, {\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 {13}{2}} - 348075 \, {\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 {11}{2}} + 425425 \, {\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 {9}{2}} - 328185 \, {\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 {7}{2}} + 153153 \, {\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 {5}{2}} - 255255 \, {\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 {3}{2}}\right )}}{765765 \, e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/765765*(45045*(e*x + d)^(17/2)*B*b^6 - 51051*(7*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*(e*x + d)^(15/2) + 176715*(
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)^(13/2) - 348075*(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)^(11/2) + 425425*(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)^(9/2) - 328185*(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)^(7/2) + 153153*(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 + 15*(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)^(5/2)
- 255255*(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)^(3/2))/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 )}^{15/2}\,\left (2\,A\,b^6\,e-14\,B\,b^6\,d+12\,B\,a\,b^5\,e\right )}{15\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{5/2}\,\left (6\,A\,b\,e+B\,a\,e-7\,B\,b\,d\right )}{5\,e^8}+\frac {2\,B\,b^6\,{\left (d+e\,x\right )}^{17/2}}{17\,e^8}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{3/2}}{3\,e^8}+\frac {6\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{7/2}\,\left (5\,A\,b\,e+2\,B\,a\,e-7\,B\,b\,d\right )}{7\,e^8}+\frac {6\,b^4\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{13/2}\,\left (2\,A\,b\,e+5\,B\,a\,e-7\,B\,b\,d\right )}{13\,e^8}+\frac {10\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}\,\left (4\,A\,b\,e+3\,B\,a\,e-7\,B\,b\,d\right )}{9\,e^8}+\frac {10\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{11/2}\,\left (3\,A\,b\,e+4\,B\,a\,e-7\,B\,b\,d\right )}{11\,e^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

((d + e*x)^(15/2)*(2*A*b^6*e - 14*B*b^6*d + 12*B*a*b^5*e))/(15*e^8) + (2*(a*e - b*d)^5*(d + e*x)^(5/2)*(6*A*b*
e + B*a*e - 7*B*b*d))/(5*e^8) + (2*B*b^6*(d + e*x)^(17/2))/(17*e^8) + (2*(A*e - B*d)*(a*e - b*d)^6*(d + e*x)^(
3/2))/(3*e^8) + (6*b*(a*e - b*d)^4*(d + e*x)^(7/2)*(5*A*b*e + 2*B*a*e - 7*B*b*d))/(7*e^8) + (6*b^4*(a*e - b*d)
*(d + e*x)^(13/2)*(2*A*b*e + 5*B*a*e - 7*B*b*d))/(13*e^8) + (10*b^2*(a*e - b*d)^3*(d + e*x)^(9/2)*(4*A*b*e + 3
*B*a*e - 7*B*b*d))/(9*e^8) + (10*b^3*(a*e - b*d)^2*(d + e*x)^(11/2)*(3*A*b*e + 4*B*a*e - 7*B*b*d))/(11*e^8)

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sympy [B]  time = 13.35, size = 969, normalized size = 3.15 \begin {gather*} \frac {2 \left (\frac {B b^{6} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{7}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \left (A b^{6} e + 6 B a b^{5} e - 7 B b^{6} d\right )}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \left (6 A a b^{5} e^{2} - 6 A b^{6} d e + 15 B a^{2} b^{4} e^{2} - 36 B a b^{5} d e + 21 B b^{6} d^{2}\right )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (15 A a^{2} b^{4} e^{3} - 30 A a b^{5} d e^{2} + 15 A b^{6} d^{2} e + 20 B a^{3} b^{3} e^{3} - 75 B a^{2} b^{4} d e^{2} + 90 B a b^{5} d^{2} e - 35 B b^{6} d^{3}\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (20 A a^{3} b^{3} e^{4} - 60 A a^{2} b^{4} d e^{3} + 60 A a b^{5} d^{2} e^{2} - 20 A b^{6} d^{3} e + 15 B a^{4} b^{2} e^{4} - 80 B a^{3} b^{3} d e^{3} + 150 B a^{2} b^{4} d^{2} e^{2} - 120 B a b^{5} d^{3} e + 35 B b^{6} d^{4}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (15 A a^{4} b^{2} e^{5} - 60 A a^{3} b^{3} d e^{4} + 90 A a^{2} b^{4} d^{2} e^{3} - 60 A a b^{5} d^{3} e^{2} + 15 A b^{6} d^{4} e + 6 B a^{5} b e^{5} - 45 B a^{4} b^{2} d e^{4} + 120 B a^{3} b^{3} d^{2} e^{3} - 150 B a^{2} b^{4} d^{3} e^{2} + 90 B a b^{5} d^{4} e - 21 B b^{6} d^{5}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (6 A a^{5} b e^{6} - 30 A a^{4} b^{2} d e^{5} + 60 A a^{3} b^{3} d^{2} e^{4} - 60 A a^{2} b^{4} d^{3} e^{3} + 30 A a b^{5} d^{4} e^{2} - 6 A b^{6} d^{5} e + B a^{6} e^{6} - 12 B a^{5} b d e^{5} + 45 B a^{4} b^{2} d^{2} e^{4} - 80 B a^{3} b^{3} d^{3} e^{3} + 75 B a^{2} b^{4} d^{4} e^{2} - 36 B a b^{5} d^{5} e + 7 B b^{6} d^{6}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A a^{6} e^{7} - 6 A a^{5} b d e^{6} + 15 A a^{4} b^{2} d^{2} e^{5} - 20 A a^{3} b^{3} d^{3} e^{4} + 15 A a^{2} b^{4} d^{4} e^{3} - 6 A a b^{5} d^{5} e^{2} + A b^{6} d^{6} e - B a^{6} d e^{6} + 6 B a^{5} b d^{2} e^{5} - 15 B a^{4} b^{2} d^{3} e^{4} + 20 B a^{3} b^{3} d^{4} e^{3} - 15 B a^{2} b^{4} d^{5} e^{2} + 6 B a b^{5} d^{6} e - B b^{6} d^{7}\right )}{3 e^{7}}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3*(e*x+d)**(1/2),x)

[Out]

2*(B*b**6*(d + e*x)**(17/2)/(17*e**7) + (d + e*x)**(15/2)*(A*b**6*e + 6*B*a*b**5*e - 7*B*b**6*d)/(15*e**7) + (
d + e*x)**(13/2)*(6*A*a*b**5*e**2 - 6*A*b**6*d*e + 15*B*a**2*b**4*e**2 - 36*B*a*b**5*d*e + 21*B*b**6*d**2)/(13
*e**7) + (d + e*x)**(11/2)*(15*A*a**2*b**4*e**3 - 30*A*a*b**5*d*e**2 + 15*A*b**6*d**2*e + 20*B*a**3*b**3*e**3
- 75*B*a**2*b**4*d*e**2 + 90*B*a*b**5*d**2*e - 35*B*b**6*d**3)/(11*e**7) + (d + e*x)**(9/2)*(20*A*a**3*b**3*e*
*4 - 60*A*a**2*b**4*d*e**3 + 60*A*a*b**5*d**2*e**2 - 20*A*b**6*d**3*e + 15*B*a**4*b**2*e**4 - 80*B*a**3*b**3*d
*e**3 + 150*B*a**2*b**4*d**2*e**2 - 120*B*a*b**5*d**3*e + 35*B*b**6*d**4)/(9*e**7) + (d + e*x)**(7/2)*(15*A*a*
*4*b**2*e**5 - 60*A*a**3*b**3*d*e**4 + 90*A*a**2*b**4*d**2*e**3 - 60*A*a*b**5*d**3*e**2 + 15*A*b**6*d**4*e + 6
*B*a**5*b*e**5 - 45*B*a**4*b**2*d*e**4 + 120*B*a**3*b**3*d**2*e**3 - 150*B*a**2*b**4*d**3*e**2 + 90*B*a*b**5*d
**4*e - 21*B*b**6*d**5)/(7*e**7) + (d + e*x)**(5/2)*(6*A*a**5*b*e**6 - 30*A*a**4*b**2*d*e**5 + 60*A*a**3*b**3*
d**2*e**4 - 60*A*a**2*b**4*d**3*e**3 + 30*A*a*b**5*d**4*e**2 - 6*A*b**6*d**5*e + B*a**6*e**6 - 12*B*a**5*b*d*e
**5 + 45*B*a**4*b**2*d**2*e**4 - 80*B*a**3*b**3*d**3*e**3 + 75*B*a**2*b**4*d**4*e**2 - 36*B*a*b**5*d**5*e + 7*
B*b**6*d**6)/(5*e**7) + (d + e*x)**(3/2)*(A*a**6*e**7 - 6*A*a**5*b*d*e**6 + 15*A*a**4*b**2*d**2*e**5 - 20*A*a*
*3*b**3*d**3*e**4 + 15*A*a**2*b**4*d**4*e**3 - 6*A*a*b**5*d**5*e**2 + A*b**6*d**6*e - B*a**6*d*e**6 + 6*B*a**5
*b*d**2*e**5 - 15*B*a**4*b**2*d**3*e**4 + 20*B*a**3*b**3*d**4*e**3 - 15*B*a**2*b**4*d**5*e**2 + 6*B*a*b**5*d**
6*e - B*b**6*d**7)/(3*e**7))/e

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