3.1803 \(\int (A+B x) \sqrt{d+e x} (a^2+2 a b x+b^2 x^2)^3 \, dx\)
Optimal. Leaf size=308 \[ -\frac{2 b^5 (d+e x)^{15/2} (-6 a B e-A b e+7 b B d)}{15 e^8}+\frac{6 b^4 (d+e x)^{13/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{13 e^8}-\frac{10 b^3 (d+e x)^{11/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{11 e^8}+\frac{10 b^2 (d+e x)^{9/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{9 e^8}-\frac{6 b (d+e x)^{7/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{7 e^8}+\frac{2 (d+e x)^{5/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{5 e^8}-\frac{2 (d+e x)^{3/2} (b d-a e)^6 (B d-A e)}{3 e^8}+\frac{2 b^6 B (d+e x)^{17/2}}{17 e^8} \]
[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)
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Rubi [A] time = 0.144463, 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} \[ -\frac{2 b^5 (d+e x)^{15/2} (-6 a B e-A b e+7 b B d)}{15 e^8}+\frac{6 b^4 (d+e x)^{13/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{13 e^8}-\frac{10 b^3 (d+e x)^{11/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{11 e^8}+\frac{10 b^2 (d+e x)^{9/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{9 e^8}-\frac{6 b (d+e x)^{7/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{7 e^8}+\frac{2 (d+e x)^{5/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{5 e^8}-\frac{2 (d+e x)^{3/2} (b d-a e)^6 (B d-A e)}{3 e^8}+\frac{2 b^6 B (d+e x)^{17/2}}{17 e^8} \]
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*}
Mathematica [A] time = 0.325707, size = 259, normalized size = 0.84 \[ \frac{2 (d+e x)^{3/2} \left (-51051 b^5 (d+e x)^6 (-6 a B e-A b e+7 b B d)+176715 b^4 (d+e x)^5 (b d-a e) (-5 a B e-2 A b e+7 b B d)-348075 b^3 (d+e x)^4 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)+425425 b^2 (d+e x)^3 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)-328185 b (d+e x)^2 (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)+153153 (d+e x) (b d-a e)^5 (-a B e-6 A b e+7 b B d)-255255 (b d-a e)^6 (B d-A e)+45045 b^6 B (d+e x)^7\right )}{765765 e^8} \]
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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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)*(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 = 1.00229, 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)*(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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Fricas [B] time = 1.32202, size = 2136, normalized size = 6.94 \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)*(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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Sympy [B] time = 11.9011, size = 969, normalized size = 3.15 \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)*(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
________________________________________________________________________________________
Giac [B] time = 1.22566, size = 1214, normalized size = 3.94 \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)*(b^2*x^2+2*a*b*x+a^2)^3*(e*x+d)^(1/2),x, algorithm="giac")
[Out]
2/765765*(51051*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^6*e^(-1) + 306306*(3*(x*e + d)^(5/2) - 5*(x*e +
d)^(3/2)*d)*A*a^5*b*e^(-1) + 43758*(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*e^(-2) + 109395*(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*e^(-2) + 3646
5*(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*e
^(-3) + 48620*(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*e^(-3) + 4420*(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*e^(-4) + 3315*(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*e^(-4) + 1275
*(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 + 900
9*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*B*a^2*b^4*e^(-5) + 510*(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*e^(-5) + 102*(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*e^(-6) + 17*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100
100*(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*e^(-6) + 7*(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*b^6*e^(-7) + 255255*(x*e + d)^(3/2)*A*a^6)*e^(-1)