3.2489 \(\int \frac{(A+B x) (d+e x)^4}{(a+b x+c x^2)^{7/2}} \, dx\)
Optimal. Leaf size=210 \[ \frac{128 \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d) (-2 a B e+A b e-2 A c d+b B d)}{15 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^4 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{16 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d) (-2 a B e+A b e-2 A c d+b B d)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \]
[Out]
(-2*(A*b - 2*a*B - (b*B - 2*A*c)*x)*(d + e*x)^4)/(5*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) - (16*(b*B*d - 2*A*
c*d + A*b*e - 2*a*B*e)*(d + e*x)^2*(b*d - 2*a*e + (2*c*d - b*e)*x))/(15*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2
)) + (128*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*(c*d^2 - b*d*e + a*e^2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(15*(b^
2 - 4*a*c)^3*Sqrt[a + b*x + c*x^2])
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Rubi [A] time = 0.135149, antiderivative size = 210, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used =
{804, 722, 636} \[ \frac{128 \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d) (-2 a B e+A b e-2 A c d+b B d)}{15 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^4 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{16 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d) (-2 a B e+A b e-2 A c d+b B d)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(d + e*x)^4)/(a + b*x + c*x^2)^(7/2),x]
[Out]
(-2*(A*b - 2*a*B - (b*B - 2*A*c)*x)*(d + e*x)^4)/(5*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) - (16*(b*B*d - 2*A*
c*d + A*b*e - 2*a*B*e)*(d + e*x)^2*(b*d - 2*a*e + (2*c*d - b*e)*x))/(15*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2
)) + (128*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*(c*d^2 - b*d*e + a*e^2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(15*(b^
2 - 4*a*c)^3*Sqrt[a + b*x + c*x^2])
Rule 804
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(b*f - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[(m
*(b*(e*f + d*g) - 2*(c*d*f + a*e*g)))/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)
, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0] && LtQ[p, -1]
Rule 722
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[(2*(2*p + 3)*(c*d
^2 - b*d*e + a*e^2))/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ
[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && LtQ[p, -1]
Rule 636
Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-2*(b*d - 2*a*e + (2*c*
d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{7/2}} \, dx &=-\frac{2 (A b-2 a B-(b B-2 A c) x) (d+e x)^4}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{(8 (b B d-2 A c d+A b e-2 a B e)) \int \frac{(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx}{5 \left (b^2-4 a c\right )}\\ &=-\frac{2 (A b-2 a B-(b B-2 A c) x) (d+e x)^4}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{16 (b B d-2 A c d+A b e-2 a B e) (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac{\left (64 (b B d-2 A c d+A b e-2 a B e) \left (c d^2-b d e+a e^2\right )\right ) \int \frac{d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{15 \left (b^2-4 a c\right )^2}\\ &=-\frac{2 (A b-2 a B-(b B-2 A c) x) (d+e x)^4}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{16 (b B d-2 A c d+A b e-2 a B e) (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{128 (b B d-2 A c d+A b e-2 a B e) \left (c d^2-b d e+a e^2\right ) (b d-2 a e+(2 c d-b e) x)}{15 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [B] time = 6.7815, size = 1196, normalized size = 5.7 \[ \frac{2 B \left (256 e^4 a^5+128 e^2 \left (b e (5 e x-4 d)+c \left (3 d^2+5 e^2 x^2\right )\right ) a^4+32 \left (3 \left (d^4+10 e^2 x^2 d^2+5 e^4 x^4\right ) c^2+2 b e \left (-6 d^3+15 e x d^2-20 e^2 x^2 d+15 e^3 x^3\right ) c+b^2 e^2 \left (9 d^2-40 e x d+15 e^2 x^2\right )\right ) a^3-16 \left (-15 b c^2 x (d-e x)^4+8 c^3 d e x^3 \left (5 d^2+3 e^2 x^2\right )+b^3 e \left (2 d^3-45 e x d^2+60 e^2 x^2 d-5 e^3 x^3\right )-3 b^2 c \left (d^4-20 e x d^3+30 e^2 x^2 d^2-40 e^3 x^3 d+5 e^4 x^4\right )\right ) a^2-2 \left (128 c^4 d^3 e x^5-32 b c^3 d^2 \left (5 d^2-10 e x d+9 e^2 x^2\right ) x^3+48 b^2 c^2 d \left (-5 d^3+10 e x d^2-15 e^2 x^2 d+2 e^3 x^3\right ) x^2+20 b^3 c (d-e x)^2 \left (-3 d^2+14 e x d+e^2 x^2\right ) x+b^4 \left (d^4+40 e x d^3-270 e^2 x^2 d^2+80 e^3 x^3 d+5 e^4 x^4\right )\right ) a+b x \left (\left (-5 d^4-60 e x d^3+90 e^2 x^2 d^2+20 e^3 x^3 d+3 e^4 x^4\right ) b^4+8 c d x \left (5 d^3-45 e x d^2+15 e^2 x^2 d+e^3 x^3\right ) b^3+48 c^2 d^2 x^2 \left (5 d^2-10 e x d+e^2 x^2\right ) b^2+64 c^3 d^3 x^3 (5 d-3 e x) b+128 c^4 d^4 x^4\right )\right )-2 A \left (\left (3 d^4+20 e x d^3+90 e^2 x^2 d^2-60 e^3 x^3 d-5 e^4 x^4\right ) b^5+2 \left (4 a e \left (d^3+15 e x d^2-45 e^2 x^2 d+5 e^3 x^3\right )-c x \left (5 d^4+80 e x d^3-270 e^2 x^2 d^2+40 e^3 x^3 d+e^4 x^4\right )\right ) b^4+8 \left (6 a^2 \left (d^2-10 e x d+5 e^2 x^2\right ) e^2-5 a c (d-e x)^2 \left (d^2+14 e x d-3 e^2 x^2\right )+2 c^2 d x^2 \left (5 d^3-60 e x d^2+45 e^2 x^2 d-2 e^3 x^3\right )\right ) b^3+16 \left (4 a^3 (5 e x-3 d) e^3+6 a^2 c \left (-2 d^3+15 e x d^2-10 e^2 x^2 d+5 e^3 x^3\right ) e+2 c^3 d^2 x^3 \left (15 d^2-40 e x d+9 e^2 x^2\right )+3 a c^2 x \left (5 d^4-40 e x d^3+30 e^2 x^2 d^2-20 e^3 x^3 d+e^4 x^4\right )\right ) b^2+16 \left (8 a^4 e^4+4 a^3 c \left (9 d^2-10 e x d+5 e^2 x^2\right ) e^2+15 a^2 c^2 (d-e x)^4+8 c^4 d^3 x^4 (5 d-4 e x)+4 a c^3 d x^2 \left (15 d^3-20 e x d^2+15 e^2 x^2 d-6 e^3 x^3\right )\right ) b+32 c \left (8 c^4 d^4 x^5+4 a c^3 d^2 \left (5 d^2+3 e^2 x^2\right ) x^3+3 a^2 c^2 \left (5 d^4+10 e^2 x^2 d^2+e^4 x^4\right ) x-8 a^4 d e^3-4 a^3 c d e \left (3 d^2+5 e^2 x^2\right )\right )\right )}{15 \left (b^2-4 a c\right )^3 (a+x (b+c x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(d + e*x)^4)/(a + b*x + c*x^2)^(7/2),x]
[Out]
(-2*A*(b^5*(3*d^4 + 20*d^3*e*x + 90*d^2*e^2*x^2 - 60*d*e^3*x^3 - 5*e^4*x^4) + 16*b*(8*a^4*e^4 + 8*c^4*d^3*x^4*
(5*d - 4*e*x) + 15*a^2*c^2*(d - e*x)^4 + 4*a^3*c*e^2*(9*d^2 - 10*d*e*x + 5*e^2*x^2) + 4*a*c^3*d*x^2*(15*d^3 -
20*d^2*e*x + 15*d*e^2*x^2 - 6*e^3*x^3)) + 8*b^3*(-5*a*c*(d - e*x)^2*(d^2 + 14*d*e*x - 3*e^2*x^2) + 6*a^2*e^2*(
d^2 - 10*d*e*x + 5*e^2*x^2) + 2*c^2*d*x^2*(5*d^3 - 60*d^2*e*x + 45*d*e^2*x^2 - 2*e^3*x^3)) + 32*c*(-8*a^4*d*e^
3 + 8*c^4*d^4*x^5 + 4*a*c^3*d^2*x^3*(5*d^2 + 3*e^2*x^2) - 4*a^3*c*d*e*(3*d^2 + 5*e^2*x^2) + 3*a^2*c^2*x*(5*d^4
+ 10*d^2*e^2*x^2 + e^4*x^4)) + 16*b^2*(4*a^3*e^3*(-3*d + 5*e*x) + 2*c^3*d^2*x^3*(15*d^2 - 40*d*e*x + 9*e^2*x^
2) + 6*a^2*c*e*(-2*d^3 + 15*d^2*e*x - 10*d*e^2*x^2 + 5*e^3*x^3) + 3*a*c^2*x*(5*d^4 - 40*d^3*e*x + 30*d^2*e^2*x
^2 - 20*d*e^3*x^3 + e^4*x^4)) + 2*b^4*(4*a*e*(d^3 + 15*d^2*e*x - 45*d*e^2*x^2 + 5*e^3*x^3) - c*x*(5*d^4 + 80*d
^3*e*x - 270*d^2*e^2*x^2 + 40*d*e^3*x^3 + e^4*x^4))) + 2*B*(256*a^5*e^4 + 128*a^4*e^2*(b*e*(-4*d + 5*e*x) + c*
(3*d^2 + 5*e^2*x^2)) + b*x*(128*c^4*d^4*x^4 + 64*b*c^3*d^3*x^3*(5*d - 3*e*x) + 48*b^2*c^2*d^2*x^2*(5*d^2 - 10*
d*e*x + e^2*x^2) + 8*b^3*c*d*x*(5*d^3 - 45*d^2*e*x + 15*d*e^2*x^2 + e^3*x^3) + b^4*(-5*d^4 - 60*d^3*e*x + 90*d
^2*e^2*x^2 + 20*d*e^3*x^3 + 3*e^4*x^4)) + 32*a^3*(b^2*e^2*(9*d^2 - 40*d*e*x + 15*e^2*x^2) + 2*b*c*e*(-6*d^3 +
15*d^2*e*x - 20*d*e^2*x^2 + 15*e^3*x^3) + 3*c^2*(d^4 + 10*d^2*e^2*x^2 + 5*e^4*x^4)) - 16*a^2*(-15*b*c^2*x*(d -
e*x)^4 + 8*c^3*d*e*x^3*(5*d^2 + 3*e^2*x^2) + b^3*e*(2*d^3 - 45*d^2*e*x + 60*d*e^2*x^2 - 5*e^3*x^3) - 3*b^2*c*
(d^4 - 20*d^3*e*x + 30*d^2*e^2*x^2 - 40*d*e^3*x^3 + 5*e^4*x^4)) - 2*a*(128*c^4*d^3*e*x^5 + 20*b^3*c*x*(d - e*x
)^2*(-3*d^2 + 14*d*e*x + e^2*x^2) - 32*b*c^3*d^2*x^3*(5*d^2 - 10*d*e*x + 9*e^2*x^2) + 48*b^2*c^2*d*x^2*(-5*d^3
+ 10*d^2*e*x - 15*d*e^2*x^2 + 2*e^3*x^3) + b^4*(d^4 + 40*d^3*e*x - 270*d^2*e^2*x^2 + 80*d*e^3*x^3 + 5*e^4*x^4
))))/(15*(b^2 - 4*a*c)^3*(a + x*(b + c*x))^(5/2))
________________________________________________________________________________________
Maple [B] time = 0.013, size = 1914, normalized size = 9.1 \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)^4/(c*x^2+b*x+a)^(7/2),x)
[Out]
2/15/(c*x^2+b*x+a)^(5/2)*(96*A*a^2*c^3*e^4*x^5+48*A*a*b^2*c^2*e^4*x^5-384*A*a*b*c^3*d*e^3*x^5+384*A*a*c^4*d^2*
e^2*x^5-2*A*b^4*c*e^4*x^5-32*A*b^3*c^2*d*e^3*x^5+288*A*b^2*c^3*d^2*e^2*x^5-512*A*b*c^4*d^3*e*x^5+256*A*c^5*d^4
*x^5-240*B*a^2*b*c^2*e^4*x^5+384*B*a^2*c^3*d*e^3*x^5+40*B*a*b^3*c*e^4*x^5+192*B*a*b^2*c^2*d*e^3*x^5-576*B*a*b*
c^3*d^2*e^2*x^5+256*B*a*c^4*d^3*e*x^5-3*B*b^5*e^4*x^5-8*B*b^4*c*d*e^3*x^5-48*B*b^3*c^2*d^2*e^2*x^5+192*B*b^2*c
^3*d^3*e*x^5-128*B*b*c^4*d^4*x^5+240*A*a^2*b*c^2*e^4*x^4+120*A*a*b^3*c*e^4*x^4-960*A*a*b^2*c^2*d*e^3*x^4+960*A
*a*b*c^3*d^2*e^2*x^4-5*A*b^5*e^4*x^4-80*A*b^4*c*d*e^3*x^4+720*A*b^3*c^2*d^2*e^2*x^4-1280*A*b^2*c^3*d^3*e*x^4+6
40*A*b*c^4*d^4*x^4-480*B*a^3*c^2*e^4*x^4-240*B*a^2*b^2*c*e^4*x^4+960*B*a^2*b*c^2*d*e^3*x^4+10*B*a*b^4*e^4*x^4+
480*B*a*b^3*c*d*e^3*x^4-1440*B*a*b^2*c^2*d^2*e^2*x^4+640*B*a*b*c^3*d^3*e*x^4-20*B*b^5*d*e^3*x^4-120*B*b^4*c*d^
2*e^2*x^4+480*B*b^3*c^2*d^3*e*x^4-320*B*b^2*c^3*d^4*x^4+480*A*a^2*b^2*c*e^4*x^3-960*A*a^2*b*c^2*d*e^3*x^3+960*
A*a^2*c^3*d^2*e^2*x^3+40*A*a*b^4*e^4*x^3-800*A*a*b^3*c*d*e^3*x^3+1440*A*a*b^2*c^2*d^2*e^2*x^3-1280*A*a*b*c^3*d
^3*e*x^3+640*A*a*c^4*d^4*x^3-60*A*b^5*d*e^3*x^3+540*A*b^4*c*d^2*e^2*x^3-960*A*b^3*c^2*d^3*e*x^3+480*A*b^2*c^3*
d^4*x^3-960*B*a^3*b*c*e^4*x^3-80*B*a^2*b^3*e^4*x^3+1920*B*a^2*b^2*c*d*e^3*x^3-1440*B*a^2*b*c^2*d^2*e^2*x^3+640
*B*a^2*c^3*d^3*e*x^3+160*B*a*b^4*d*e^3*x^3-1200*B*a*b^3*c*d^2*e^2*x^3+960*B*a*b^2*c^2*d^3*e*x^3-320*B*a*b*c^3*
d^4*x^3-90*B*b^5*d^2*e^2*x^3+360*B*b^4*c*d^3*e*x^3-240*B*b^3*c^2*d^4*x^3+320*A*a^3*b*c*e^4*x^2-640*A*a^3*c^2*d
*e^3*x^2+240*A*a^2*b^3*e^4*x^2-960*A*a^2*b^2*c*d*e^3*x^2+1440*A*a^2*b*c^2*d^2*e^2*x^2-360*A*a*b^4*d*e^3*x^2+12
00*A*a*b^3*c*d^2*e^2*x^2-1920*A*a*b^2*c^2*d^3*e*x^2+960*A*a*b*c^3*d^4*x^2+90*A*b^5*d^2*e^2*x^2-160*A*b^4*c*d^3
*e*x^2+80*A*b^3*c^2*d^4*x^2-640*B*a^4*c*e^4*x^2-480*B*a^3*b^2*e^4*x^2+1280*B*a^3*b*c*d*e^3*x^2-960*B*a^3*c^2*d
^2*e^2*x^2+960*B*a^2*b^3*d*e^3*x^2-1440*B*a^2*b^2*c*d^2*e^2*x^2+960*B*a^2*b*c^2*d^3*e*x^2-540*B*a*b^4*d^2*e^2*
x^2+800*B*a*b^3*c*d^3*e*x^2-480*B*a*b^2*c^2*d^4*x^2+60*B*b^5*d^3*e*x^2-40*B*b^4*c*d^4*x^2+320*A*a^3*b^2*e^4*x-
640*A*a^3*b*c*d*e^3*x-480*A*a^2*b^3*d*e^3*x+1440*A*a^2*b^2*c*d^2*e^2*x-960*A*a^2*b*c^2*d^3*e*x+480*A*a^2*c^3*d
^4*x+120*A*a*b^4*d^2*e^2*x-480*A*a*b^3*c*d^3*e*x+240*A*a*b^2*c^2*d^4*x+20*A*b^5*d^3*e*x-10*A*b^4*c*d^4*x-640*B
*a^4*b*e^4*x+1280*B*a^3*b^2*d*e^3*x-960*B*a^3*b*c*d^2*e^2*x-720*B*a^2*b^3*d^2*e^2*x+960*B*a^2*b^2*c*d^3*e*x-24
0*B*a^2*b*c^2*d^4*x+80*B*a*b^4*d^3*e*x-120*B*a*b^3*c*d^4*x+5*B*b^5*d^4*x+128*A*a^4*b*e^4-256*A*a^4*c*d*e^3-192
*A*a^3*b^2*d*e^3+576*A*a^3*b*c*d^2*e^2-384*A*a^3*c^2*d^3*e+48*A*a^2*b^3*d^2*e^2-192*A*a^2*b^2*c*d^3*e+240*A*a^
2*b*c^2*d^4+8*A*a*b^4*d^3*e-40*A*a*b^3*c*d^4+3*A*b^5*d^4-256*B*a^5*e^4+512*B*a^4*b*d*e^3-384*B*a^4*c*d^2*e^2-2
88*B*a^3*b^2*d^2*e^2+384*B*a^3*b*c*d^3*e-96*B*a^3*c^2*d^4+32*B*a^2*b^3*d^3*e-48*B*a^2*b^2*c*d^4+2*B*a*b^4*d^4)
/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^4/(c*x^2+b*x+a)^(7/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^4/(c*x^2+b*x+a)^(7/2),x, algorithm="fricas")
[Out]
Timed out
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)**4/(c*x**2+b*x+a)**(7/2),x)
[Out]
Timed out
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Giac [B] time = 1.23468, size = 2429, normalized size = 11.57 \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)^4/(c*x^2+b*x+a)^(7/2),x, algorithm="giac")
[Out]
1/15*((((((128*B*b*c^4*d^4 - 256*A*c^5*d^4 - 192*B*b^2*c^3*d^3*e - 256*B*a*c^4*d^3*e + 512*A*b*c^4*d^3*e + 48*
B*b^3*c^2*d^2*e^2 + 576*B*a*b*c^3*d^2*e^2 - 288*A*b^2*c^3*d^2*e^2 - 384*A*a*c^4*d^2*e^2 + 8*B*b^4*c*d*e^3 - 19
2*B*a*b^2*c^2*d*e^3 + 32*A*b^3*c^2*d*e^3 - 384*B*a^2*c^3*d*e^3 + 384*A*a*b*c^3*d*e^3 + 3*B*b^5*e^4 - 40*B*a*b^
3*c*e^4 + 2*A*b^4*c*e^4 + 240*B*a^2*b*c^2*e^4 - 48*A*a*b^2*c^2*e^4 - 96*A*a^2*c^3*e^4)*x/(b^6*c^3 - 12*a*b^4*c
^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6) + 5*(64*B*b^2*c^3*d^4 - 128*A*b*c^4*d^4 - 96*B*b^3*c^2*d^3*e - 128*B*a*b*c^3
*d^3*e + 256*A*b^2*c^3*d^3*e + 24*B*b^4*c*d^2*e^2 + 288*B*a*b^2*c^2*d^2*e^2 - 144*A*b^3*c^2*d^2*e^2 - 192*A*a*
b*c^3*d^2*e^2 + 4*B*b^5*d*e^3 - 96*B*a*b^3*c*d*e^3 + 16*A*b^4*c*d*e^3 - 192*B*a^2*b*c^2*d*e^3 + 192*A*a*b^2*c^
2*d*e^3 - 2*B*a*b^4*e^4 + A*b^5*e^4 + 48*B*a^2*b^2*c*e^4 - 24*A*a*b^3*c*e^4 + 96*B*a^3*c^2*e^4 - 48*A*a^2*b*c^
2*e^4)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x + 10*(24*B*b^3*c^2*d^4 + 32*B*a*b*c^3*d^4 - 4
8*A*b^2*c^3*d^4 - 64*A*a*c^4*d^4 - 36*B*b^4*c*d^3*e - 96*B*a*b^2*c^2*d^3*e + 96*A*b^3*c^2*d^3*e - 64*B*a^2*c^3
*d^3*e + 128*A*a*b*c^3*d^3*e + 9*B*b^5*d^2*e^2 + 120*B*a*b^3*c*d^2*e^2 - 54*A*b^4*c*d^2*e^2 + 144*B*a^2*b*c^2*
d^2*e^2 - 144*A*a*b^2*c^2*d^2*e^2 - 96*A*a^2*c^3*d^2*e^2 - 16*B*a*b^4*d*e^3 + 6*A*b^5*d*e^3 - 192*B*a^2*b^2*c*
d*e^3 + 80*A*a*b^3*c*d*e^3 + 96*A*a^2*b*c^2*d*e^3 + 8*B*a^2*b^3*e^4 - 4*A*a*b^4*e^4 + 96*B*a^3*b*c*e^4 - 48*A*
a^2*b^2*c*e^4)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x + 10*(4*B*b^4*c*d^4 + 48*B*a*b^2*c^2*
d^4 - 8*A*b^3*c^2*d^4 - 96*A*a*b*c^3*d^4 - 6*B*b^5*d^3*e - 80*B*a*b^3*c*d^3*e + 16*A*b^4*c*d^3*e - 96*B*a^2*b*
c^2*d^3*e + 192*A*a*b^2*c^2*d^3*e + 54*B*a*b^4*d^2*e^2 - 9*A*b^5*d^2*e^2 + 144*B*a^2*b^2*c*d^2*e^2 - 120*A*a*b
^3*c*d^2*e^2 + 96*B*a^3*c^2*d^2*e^2 - 144*A*a^2*b*c^2*d^2*e^2 - 96*B*a^2*b^3*d*e^3 + 36*A*a*b^4*d*e^3 - 128*B*
a^3*b*c*d*e^3 + 96*A*a^2*b^2*c*d*e^3 + 64*A*a^3*c^2*d*e^3 + 48*B*a^3*b^2*e^4 - 24*A*a^2*b^3*e^4 + 64*B*a^4*c*e
^4 - 32*A*a^3*b*c*e^4)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x - 5*(B*b^5*d^4 - 24*B*a*b^3*c
*d^4 - 2*A*b^4*c*d^4 - 48*B*a^2*b*c^2*d^4 + 48*A*a*b^2*c^2*d^4 + 96*A*a^2*c^3*d^4 + 16*B*a*b^4*d^3*e + 4*A*b^5
*d^3*e + 192*B*a^2*b^2*c*d^3*e - 96*A*a*b^3*c*d^3*e - 192*A*a^2*b*c^2*d^3*e - 144*B*a^2*b^3*d^2*e^2 + 24*A*a*b
^4*d^2*e^2 - 192*B*a^3*b*c*d^2*e^2 + 288*A*a^2*b^2*c*d^2*e^2 + 256*B*a^3*b^2*d*e^3 - 96*A*a^2*b^3*d*e^3 - 128*
A*a^3*b*c*d*e^3 - 128*B*a^4*b*e^4 + 64*A*a^3*b^2*e^4)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*
x - (2*B*a*b^4*d^4 + 3*A*b^5*d^4 - 48*B*a^2*b^2*c*d^4 - 40*A*a*b^3*c*d^4 - 96*B*a^3*c^2*d^4 + 240*A*a^2*b*c^2*
d^4 + 32*B*a^2*b^3*d^3*e + 8*A*a*b^4*d^3*e + 384*B*a^3*b*c*d^3*e - 192*A*a^2*b^2*c*d^3*e - 384*A*a^3*c^2*d^3*e
- 288*B*a^3*b^2*d^2*e^2 + 48*A*a^2*b^3*d^2*e^2 - 384*B*a^4*c*d^2*e^2 + 576*A*a^3*b*c*d^2*e^2 + 512*B*a^4*b*d*
e^3 - 192*A*a^3*b^2*d*e^3 - 256*A*a^4*c*d*e^3 - 256*B*a^5*e^4 + 128*A*a^4*b*e^4)/(b^6*c^3 - 12*a*b^4*c^4 + 48*
a^2*b^2*c^5 - 64*a^3*c^6))/(c*x^2 + b*x + a)^(5/2)