3.23.48 \(\int \frac {(A+B x) (d+e x)^4}{(a+b x+c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=210 \[ \frac {128 \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d) (-2 a B e+A b e-2 A c d+b B d)}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^4 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {16 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d) (-2 a B e+A b e-2 A c d+b B d)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \]

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Rubi [A]  time = 0.14, antiderivative size = 210, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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]

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 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]

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*}

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Mathematica [B]  time = 6.30, size = 1196, normalized size = 5.70 \begin {gather*} \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}} \end {gather*}

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))

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IntegrateAlgebraic [B]  time = 21.88, size = 1893, normalized size = 9.01 \begin {gather*} -\frac {2 \left (-256 B e^4 a^5+128 A b e^4 a^4+512 b B d e^3 a^4-256 A c d e^3 a^4-384 B c d^2 e^2 a^4-640 B c e^4 x^2 a^4-640 b B e^4 x a^4-96 B c^2 d^4 a^3-480 B c^2 e^4 x^4 a^3-192 A b^2 d e^3 a^3-960 b B c e^4 x^3 a^3-288 b^2 B d^2 e^2 a^3+576 A b c d^2 e^2 a^3-480 b^2 B e^4 x^2 a^3+320 A b c e^4 x^2 a^3-640 A c^2 d e^3 x^2 a^3+1280 b B c d e^3 x^2 a^3-960 B c^2 d^2 e^2 x^2 a^3-384 A c^2 d^3 e a^3+384 b B c d^3 e a^3+320 A b^2 e^4 x a^3+1280 b^2 B d e^3 x a^3-640 A b c d e^3 x a^3-960 b B c d^2 e^2 x a^3+96 A c^3 e^4 x^5 a^2-240 b B c^2 e^4 x^5 a^2+384 B c^3 d e^3 x^5 a^2+240 A b c^2 d^4 a^2-48 b^2 B c d^4 a^2+240 A b c^2 e^4 x^4 a^2-240 b^2 B c e^4 x^4 a^2+960 b B c^2 d e^3 x^4 a^2-80 b^3 B e^4 x^3 a^2+480 A b^2 c e^4 x^3 a^2-960 A b c^2 d e^3 x^3 a^2+1920 b^2 B c d e^3 x^3 a^2+960 A c^3 d^2 e^2 x^3 a^2-1440 b B c^2 d^2 e^2 x^3 a^2+640 B c^3 d^3 e x^3 a^2+48 A b^3 d^2 e^2 a^2+240 A b^3 e^4 x^2 a^2+960 b^3 B d e^3 x^2 a^2-960 A b^2 c d e^3 x^2 a^2+1440 A b c^2 d^2 e^2 x^2 a^2-1440 b^2 B c d^2 e^2 x^2 a^2+960 b B c^2 d^3 e x^2 a^2+32 b^3 B d^3 e a^2-192 A b^2 c d^3 e a^2+480 A c^3 d^4 x a^2-240 b B c^2 d^4 x a^2-480 A b^3 d e^3 x a^2-720 b^3 B d^2 e^2 x a^2+1440 A b^2 c d^2 e^2 x a^2-960 A b c^2 d^3 e x a^2+960 b^2 B c d^3 e x a^2+48 A b^2 c^2 e^4 x^5 a+40 b^3 B c e^4 x^5 a-384 A b c^3 d e^3 x^5 a+192 b^2 B c^2 d e^3 x^5 a+384 A c^4 d^2 e^2 x^5 a-576 b B c^3 d^2 e^2 x^5 a+256 B c^4 d^3 e x^5 a+2 b^4 B d^4 a-40 A b^3 c d^4 a+10 b^4 B e^4 x^4 a+120 A b^3 c e^4 x^4 a-960 A b^2 c^2 d e^3 x^4 a+480 b^3 B c d e^3 x^4 a+960 A b c^3 d^2 e^2 x^4 a-1440 b^2 B c^2 d^2 e^2 x^4 a+640 b B c^3 d^3 e x^4 a+640 A c^4 d^4 x^3 a-320 b B c^3 d^4 x^3 a+40 A b^4 e^4 x^3 a+160 b^4 B d e^3 x^3 a-800 A b^3 c d e^3 x^3 a+1440 A b^2 c^2 d^2 e^2 x^3 a-1200 b^3 B c d^2 e^2 x^3 a-1280 A b c^3 d^3 e x^3 a+960 b^2 B c^2 d^3 e x^3 a+960 A b c^3 d^4 x^2 a-480 b^2 B c^2 d^4 x^2 a-360 A b^4 d e^3 x^2 a-540 b^4 B d^2 e^2 x^2 a+1200 A b^3 c d^2 e^2 x^2 a-1920 A b^2 c^2 d^3 e x^2 a+800 b^3 B c d^3 e x^2 a+8 A b^4 d^3 e a+240 A b^2 c^2 d^4 x a-120 b^3 B c d^4 x a+120 A b^4 d^2 e^2 x a+80 b^4 B d^3 e x a-480 A b^3 c d^3 e x a+256 A c^5 d^4 x^5-128 b B c^4 d^4 x^5-3 b^5 B e^4 x^5-2 A b^4 c e^4 x^5-32 A b^3 c^2 d e^3 x^5-8 b^4 B c d e^3 x^5+288 A b^2 c^3 d^2 e^2 x^5-48 b^3 B c^2 d^2 e^2 x^5-512 A b c^4 d^3 e x^5+192 b^2 B c^3 d^3 e x^5+3 A b^5 d^4+640 A b c^4 d^4 x^4-320 b^2 B c^3 d^4 x^4-5 A b^5 e^4 x^4-20 b^5 B d e^3 x^4-80 A b^4 c d e^3 x^4+720 A b^3 c^2 d^2 e^2 x^4-120 b^4 B c d^2 e^2 x^4-1280 A b^2 c^3 d^3 e x^4+480 b^3 B c^2 d^3 e x^4+480 A b^2 c^3 d^4 x^3-240 b^3 B c^2 d^4 x^3-60 A b^5 d e^3 x^3-90 b^5 B d^2 e^2 x^3+540 A b^4 c d^2 e^2 x^3-960 A b^3 c^2 d^3 e x^3+360 b^4 B c d^3 e x^3+80 A b^3 c^2 d^4 x^2-40 b^4 B c d^4 x^2+90 A b^5 d^2 e^2 x^2+60 b^5 B d^3 e x^2-160 A b^4 c d^3 e x^2+5 b^5 B d^4 x-10 A b^4 c d^4 x+20 A b^5 d^3 e x\right )}{15 \left (b^2-4 a c\right )^3 \left (c x^2+b x+a\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^4)/(a + b*x + c*x^2)^(7/2),x]

[Out]

(-2*(3*A*b^5*d^4 + 2*a*b^4*B*d^4 - 40*a*A*b^3*c*d^4 - 48*a^2*b^2*B*c*d^4 + 240*a^2*A*b*c^2*d^4 - 96*a^3*B*c^2*
d^4 + 8*a*A*b^4*d^3*e + 32*a^2*b^3*B*d^3*e - 192*a^2*A*b^2*c*d^3*e + 384*a^3*b*B*c*d^3*e - 384*a^3*A*c^2*d^3*e
+ 48*a^2*A*b^3*d^2*e^2 - 288*a^3*b^2*B*d^2*e^2 + 576*a^3*A*b*c*d^2*e^2 - 384*a^4*B*c*d^2*e^2 - 192*a^3*A*b^2*
d*e^3 + 512*a^4*b*B*d*e^3 - 256*a^4*A*c*d*e^3 + 128*a^4*A*b*e^4 - 256*a^5*B*e^4 + 5*b^5*B*d^4*x - 10*A*b^4*c*d
^4*x - 120*a*b^3*B*c*d^4*x + 240*a*A*b^2*c^2*d^4*x - 240*a^2*b*B*c^2*d^4*x + 480*a^2*A*c^3*d^4*x + 20*A*b^5*d^
3*e*x + 80*a*b^4*B*d^3*e*x - 480*a*A*b^3*c*d^3*e*x + 960*a^2*b^2*B*c*d^3*e*x - 960*a^2*A*b*c^2*d^3*e*x + 120*a
*A*b^4*d^2*e^2*x - 720*a^2*b^3*B*d^2*e^2*x + 1440*a^2*A*b^2*c*d^2*e^2*x - 960*a^3*b*B*c*d^2*e^2*x - 480*a^2*A*
b^3*d*e^3*x + 1280*a^3*b^2*B*d*e^3*x - 640*a^3*A*b*c*d*e^3*x + 320*a^3*A*b^2*e^4*x - 640*a^4*b*B*e^4*x - 40*b^
4*B*c*d^4*x^2 + 80*A*b^3*c^2*d^4*x^2 - 480*a*b^2*B*c^2*d^4*x^2 + 960*a*A*b*c^3*d^4*x^2 + 60*b^5*B*d^3*e*x^2 -
160*A*b^4*c*d^3*e*x^2 + 800*a*b^3*B*c*d^3*e*x^2 - 1920*a*A*b^2*c^2*d^3*e*x^2 + 960*a^2*b*B*c^2*d^3*e*x^2 + 90*
A*b^5*d^2*e^2*x^2 - 540*a*b^4*B*d^2*e^2*x^2 + 1200*a*A*b^3*c*d^2*e^2*x^2 - 1440*a^2*b^2*B*c*d^2*e^2*x^2 + 1440
*a^2*A*b*c^2*d^2*e^2*x^2 - 960*a^3*B*c^2*d^2*e^2*x^2 - 360*a*A*b^4*d*e^3*x^2 + 960*a^2*b^3*B*d*e^3*x^2 - 960*a
^2*A*b^2*c*d*e^3*x^2 + 1280*a^3*b*B*c*d*e^3*x^2 - 640*a^3*A*c^2*d*e^3*x^2 + 240*a^2*A*b^3*e^4*x^2 - 480*a^3*b^
2*B*e^4*x^2 + 320*a^3*A*b*c*e^4*x^2 - 640*a^4*B*c*e^4*x^2 - 240*b^3*B*c^2*d^4*x^3 + 480*A*b^2*c^3*d^4*x^3 - 32
0*a*b*B*c^3*d^4*x^3 + 640*a*A*c^4*d^4*x^3 + 360*b^4*B*c*d^3*e*x^3 - 960*A*b^3*c^2*d^3*e*x^3 + 960*a*b^2*B*c^2*
d^3*e*x^3 - 1280*a*A*b*c^3*d^3*e*x^3 + 640*a^2*B*c^3*d^3*e*x^3 - 90*b^5*B*d^2*e^2*x^3 + 540*A*b^4*c*d^2*e^2*x^
3 - 1200*a*b^3*B*c*d^2*e^2*x^3 + 1440*a*A*b^2*c^2*d^2*e^2*x^3 - 1440*a^2*b*B*c^2*d^2*e^2*x^3 + 960*a^2*A*c^3*d
^2*e^2*x^3 - 60*A*b^5*d*e^3*x^3 + 160*a*b^4*B*d*e^3*x^3 - 800*a*A*b^3*c*d*e^3*x^3 + 1920*a^2*b^2*B*c*d*e^3*x^3
- 960*a^2*A*b*c^2*d*e^3*x^3 + 40*a*A*b^4*e^4*x^3 - 80*a^2*b^3*B*e^4*x^3 + 480*a^2*A*b^2*c*e^4*x^3 - 960*a^3*b
*B*c*e^4*x^3 - 320*b^2*B*c^3*d^4*x^4 + 640*A*b*c^4*d^4*x^4 + 480*b^3*B*c^2*d^3*e*x^4 - 1280*A*b^2*c^3*d^3*e*x^
4 + 640*a*b*B*c^3*d^3*e*x^4 - 120*b^4*B*c*d^2*e^2*x^4 + 720*A*b^3*c^2*d^2*e^2*x^4 - 1440*a*b^2*B*c^2*d^2*e^2*x
^4 + 960*a*A*b*c^3*d^2*e^2*x^4 - 20*b^5*B*d*e^3*x^4 - 80*A*b^4*c*d*e^3*x^4 + 480*a*b^3*B*c*d*e^3*x^4 - 960*a*A
*b^2*c^2*d*e^3*x^4 + 960*a^2*b*B*c^2*d*e^3*x^4 - 5*A*b^5*e^4*x^4 + 10*a*b^4*B*e^4*x^4 + 120*a*A*b^3*c*e^4*x^4
- 240*a^2*b^2*B*c*e^4*x^4 + 240*a^2*A*b*c^2*e^4*x^4 - 480*a^3*B*c^2*e^4*x^4 - 128*b*B*c^4*d^4*x^5 + 256*A*c^5*
d^4*x^5 + 192*b^2*B*c^3*d^3*e*x^5 - 512*A*b*c^4*d^3*e*x^5 + 256*a*B*c^4*d^3*e*x^5 - 48*b^3*B*c^2*d^2*e^2*x^5 +
288*A*b^2*c^3*d^2*e^2*x^5 - 576*a*b*B*c^3*d^2*e^2*x^5 + 384*a*A*c^4*d^2*e^2*x^5 - 8*b^4*B*c*d*e^3*x^5 - 32*A*
b^3*c^2*d*e^3*x^5 + 192*a*b^2*B*c^2*d*e^3*x^5 - 384*a*A*b*c^3*d*e^3*x^5 + 384*a^2*B*c^3*d*e^3*x^5 - 3*b^5*B*e^
4*x^5 - 2*A*b^4*c*e^4*x^5 + 40*a*b^3*B*c*e^4*x^5 + 48*a*A*b^2*c^2*e^4*x^5 - 240*a^2*b*B*c^2*e^4*x^5 + 96*a^2*A
*c^3*e^4*x^5))/(15*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)^(5/2))

________________________________________________________________________________________

fricas [B]  time = 66.93, size = 1628, normalized size = 7.75

result too large to display

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]

2/15*((128*(B*b*c^4 - 2*A*c^5)*d^4 - 64*(3*B*b^2*c^3 + 4*(B*a - 2*A*b)*c^4)*d^3*e + 48*(B*b^3*c^2 - 8*A*a*c^4
+ 6*(2*B*a*b - A*b^2)*c^3)*d^2*e^2 + 8*(B*b^4*c - 48*(B*a^2 - A*a*b)*c^3 - 4*(6*B*a*b^2 - A*b^3)*c^2)*d*e^3 +
(3*B*b^5 - 96*A*a^2*c^3 + 48*(5*B*a^2*b - A*a*b^2)*c^2 - 2*(20*B*a*b^3 - A*b^4)*c)*e^4)*x^5 - (2*B*a*b^4 + 3*A
*b^5 - 48*(2*B*a^3 - 5*A*a^2*b)*c^2 - 8*(6*B*a^2*b^2 + 5*A*a*b^3)*c)*d^4 - 8*(4*B*a^2*b^3 + A*a*b^4 - 48*A*a^3
*c^2 + 24*(2*B*a^3*b - A*a^2*b^2)*c)*d^3*e + 48*(6*B*a^3*b^2 - A*a^2*b^3 + 4*(2*B*a^4 - 3*A*a^3*b)*c)*d^2*e^2
- 64*(8*B*a^4*b - 3*A*a^3*b^2 - 4*A*a^4*c)*d*e^3 + 128*(2*B*a^5 - A*a^4*b)*e^4 + 5*(64*(B*b^2*c^3 - 2*A*b*c^4)
*d^4 - 32*(3*B*b^3*c^2 + 4*(B*a*b - 2*A*b^2)*c^3)*d^3*e + 24*(B*b^4*c - 8*A*a*b*c^3 + 6*(2*B*a*b^2 - A*b^3)*c^
2)*d^2*e^2 + 4*(B*b^5 - 48*(B*a^2*b - A*a*b^2)*c^2 - 4*(6*B*a*b^3 - A*b^4)*c)*d*e^3 - (2*B*a*b^4 - A*b^5 - 48*
(2*B*a^3 - A*a^2*b)*c^2 - 24*(2*B*a^2*b^2 - A*a*b^3)*c)*e^4)*x^4 + 10*(8*(3*B*b^3*c^2 - 8*A*a*c^4 + 2*(2*B*a*b
- 3*A*b^2)*c^3)*d^4 - 4*(9*B*b^4*c + 16*(B*a^2 - 2*A*a*b)*c^3 + 24*(B*a*b^2 - A*b^3)*c^2)*d^3*e + 3*(3*B*b^5
- 32*A*a^2*c^3 + 48*(B*a^2*b - A*a*b^2)*c^2 + 2*(20*B*a*b^3 - 9*A*b^4)*c)*d^2*e^2 - 2*(8*B*a*b^4 - 3*A*b^5 - 4
8*A*a^2*b*c^2 + 8*(12*B*a^2*b^2 - 5*A*a*b^3)*c)*d*e^3 + 4*(2*B*a^2*b^3 - A*a*b^4 + 12*(2*B*a^3*b - A*a^2*b^2)*
c)*e^4)*x^3 + 10*(4*(B*b^4*c - 24*A*a*b*c^3 + 2*(6*B*a*b^2 - A*b^3)*c^2)*d^4 - 2*(3*B*b^5 + 48*(B*a^2*b - 2*A*
a*b^2)*c^2 + 8*(5*B*a*b^3 - A*b^4)*c)*d^3*e + 3*(18*B*a*b^4 - 3*A*b^5 + 16*(2*B*a^3 - 3*A*a^2*b)*c^2 + 8*(6*B*
a^2*b^2 - 5*A*a*b^3)*c)*d^2*e^2 - 4*(24*B*a^2*b^3 - 9*A*a*b^4 - 16*A*a^3*c^2 + 8*(4*B*a^3*b - 3*A*a^2*b^2)*c)*
d*e^3 + 8*(6*B*a^3*b^2 - 3*A*a^2*b^3 + 4*(2*B*a^4 - A*a^3*b)*c)*e^4)*x^2 - 5*((B*b^5 + 96*A*a^2*c^3 - 48*(B*a^
2*b - A*a*b^2)*c^2 - 2*(12*B*a*b^3 + A*b^4)*c)*d^4 + 4*(4*B*a*b^4 + A*b^5 - 48*A*a^2*b*c^2 + 24*(2*B*a^2*b^2 -
A*a*b^3)*c)*d^3*e - 24*(6*B*a^2*b^3 - A*a*b^4 + 4*(2*B*a^3*b - 3*A*a^2*b^2)*c)*d^2*e^2 + 32*(8*B*a^3*b^2 - 3*
A*a^2*b^3 - 4*A*a^3*b*c)*d*e^3 - 64*(2*B*a^4*b - A*a^3*b^2)*e^4)*x)*sqrt(c*x^2 + b*x + a)/(a^3*b^6 - 12*a^4*b^
4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*x^6 + 3*(b^7*c^2 -
12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^5 + 3*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4
- 64*a^4*c^5)*x^4 + (b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*x^3 + 3*(a*b^8 - 11*a
^2*b^6*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*x^2 + 3*(a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64
*a^5*b*c^3)*x)

________________________________________________________________________________________

giac [B]  time = 0.37, size = 1763, normalized size = 8.40

result too large to display

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]

2/15*((((((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 - 12*a*b^4*c + 4
8*a^2*b^2*c^2 - 64*a^3*c^3) + 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 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x + 10*(24*B*b^3*c^2*d^4 + 32*B*a*b*c^3*d^4 - 48*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 - 14
4*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 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*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 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x - 5*(B*b^5*d^4 - 24*B*a*b^3*c*d^4 - 2*A*b^4*c*d^4 - 4
8*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 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*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 - 25
6*A*a^4*c*d*e^3 - 256*B*a^5*e^4 + 128*A*a^4*b*e^4)/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))/(c*x^2 +
b*x + a)^(5/2)

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maple [B]  time = 0.02, size = 1914, normalized size = 9.11

result too large to display

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 zero or nonzero?

________________________________________________________________________________________

mupad [B]  time = 5.48, size = 7972, normalized size = 37.96

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^4)/(a + b*x + c*x^2)^(7/2),x)

[Out]

((a*((b*((16*c*e^3*(A*c*e - B*b*e + 4*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c*e^4)/(5*(4*a*c^2
- b^2*c)*(4*a*c - b^2))))/c + (2*(24*A*b*c^2*e^4 + 48*B*a*c^2*e^4 - 30*B*b^2*c*e^4 - 96*A*c^3*d*e^3 - 144*B*c^
3*d^2*e^2 + 96*B*b*c^2*d*e^3))/(15*c*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*b*e^3*(A*c*e - B*b*e + 4*B*c*d))/(5
*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (16*B*a*c*e^4)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - x*((a*((16*c*e^3*
(A*c*e - B*b*e + 4*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c*e^4)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b
^2))))/c - (b*((b*((16*c*e^3*(A*c*e - B*b*e + 4*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c*e^4)/(5
*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (2*(24*A*b*c^2*e^4 + 48*B*a*c^2*e^4 - 30*B*b^2*c*e^4 - 96*A*c^3*d*e^3
- 144*B*c^3*d^2*e^2 + 96*B*b*c^2*d*e^3))/(15*c*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*b*e^3*(A*c*e - B*b*e + 4*
B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (16*B*a*c*e^4)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (2*(4*B
*b^3*e^4 - 16*A*a*c^2*e^4 - 2*A*b^2*c*e^4 + 32*B*c^3*d^3*e + 48*A*c^3*d^2*e^2 + 16*B*a*b*c*e^4 - 64*B*a*c^2*d*
e^3 - 8*B*b^2*c*d*e^3))/(15*c*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (b*(24*A*b*c^2*e^4 + 48*B*a*c^2*e^4 - 30*B*b^
2*c*e^4 - 96*A*c^3*d*e^3 - 144*B*c^3*d^2*e^2 + 96*B*b*c^2*d*e^3))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a*c - b^2))) +
(b*(4*B*b^3*e^4 - 16*A*a*c^2*e^4 - 2*A*b^2*c*e^4 + 32*B*c^3*d^3*e + 48*A*c^3*d^2*e^2 + 16*B*a*b*c*e^4 - 64*B*a
*c^2*d*e^3 - 8*B*b^2*c*d*e^3))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a*c - b^2)))/(a + b*x + c*x^2)^(1/2) - ((a*((2*A*b
*c^3*e^4 + 4*B*a*c^3*e^4 - 16*A*c^4*d*e^3 - 2*B*b^2*c^2*e^4 - 24*B*c^4*d^2*e^2 + 8*B*b*c^3*d*e^3)/(15*c^4*(4*a
*c - b^2)) + (b*((2*e^3*(2*A*c*e - B*b*e + 8*B*c*d))/(15*c*(4*a*c - b^2)) - (4*B*b*e^4)/(15*c*(4*a*c - b^2))))
/c + (4*B*a*e^4)/(15*c*(4*a*c - b^2))))/c - x*((4*A*a*c^3*e^4 + 2*B*b^3*c*e^4 - 16*B*c^4*d^3*e - 2*A*b^2*c^2*e
^4 - 24*A*c^4*d^2*e^2 + 12*B*b*c^3*d^2*e^2 - 8*B*b^2*c^2*d*e^3 - 6*B*a*b*c^2*e^4 + 8*A*b*c^3*d*e^3 + 16*B*a*c^
3*d*e^3)/(15*c^4*(4*a*c - b^2)) - (b*((2*A*b*c^3*e^4 + 4*B*a*c^3*e^4 - 16*A*c^4*d*e^3 - 2*B*b^2*c^2*e^4 - 24*B
*c^4*d^2*e^2 + 8*B*b*c^3*d*e^3)/(15*c^4*(4*a*c - b^2)) + (b*((2*e^3*(2*A*c*e - B*b*e + 8*B*c*d))/(15*c*(4*a*c
- b^2)) - (4*B*b*e^4)/(15*c*(4*a*c - b^2))))/c + (4*B*a*e^4)/(15*c*(4*a*c - b^2))))/c + (a*((2*e^3*(2*A*c*e -
B*b*e + 8*B*c*d))/(15*c*(4*a*c - b^2)) - (4*B*b*e^4)/(15*c*(4*a*c - b^2))))/c) + (2*B*b^4*e^4 + 4*B*c^4*d^4 -
2*A*b^3*c*e^4 + 16*A*c^4*d^3*e + 4*B*a^2*c^2*e^4 - 12*A*b*c^3*d^2*e^2 + 8*A*b^2*c^2*d*e^3 - 24*B*a*c^3*d^2*e^2
+ 12*B*b^2*c^2*d^2*e^2 + 6*A*a*b*c^2*e^4 - 8*B*a*b^2*c*e^4 - 16*A*a*c^3*d*e^3 - 8*B*b*c^3*d^3*e - 8*B*b^3*c*d
*e^3 + 24*B*a*b*c^2*d*e^3)/(15*c^4*(4*a*c - b^2)))/(a + b*x + c*x^2)^(3/2) + (x*((a*((b*((b*((16*c^3*e^3*(A*e
+ 4*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e^4)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (2
*(32*A*c^5*d*e^3 - 48*B*a*c^4*e^4 + 12*B*b^2*c^3*e^4 + 48*B*c^5*d^2*e^2))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a*c - b
^2)) + (16*B*a*c^2*e^4)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*b*c^2*e^3*(A*e + 4*B*d))/(15*(4*a*c^2 - b^2*
c)*(4*a*c - b^2))))/c - (a*((16*c^3*e^3*(A*e + 4*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e^4)/
(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (2*(32*B*c^5*d^3*e - 48*A*a*c^4*e^4 + 12*A*b^2*c^3*e^4 - 12*B*b^3*c
^2*e^4 + 48*A*c^5*d^2*e^2 + 48*B*b^2*c^3*d*e^3 + 48*B*a*b*c^3*e^4 - 192*B*a*c^4*d*e^3))/(15*c^2*(4*a*c^2 - b^2
*c)*(4*a*c - b^2)) + (b*(32*A*c^5*d*e^3 - 48*B*a*c^4*e^4 + 12*B*b^2*c^3*e^4 + 48*B*c^5*d^2*e^2))/(15*c^3*(4*a*
c^2 - b^2*c)*(4*a*c - b^2))))/c + (b*((a*((b*((16*c^3*e^3*(A*e + 4*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))
- (8*B*b*c^2*e^4)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (2*(32*A*c^5*d*e^3 - 48*B*a*c^4*e^4 + 12*B*b^2*c^
3*e^4 + 48*B*c^5*d^2*e^2))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (16*B*a*c^2*e^4)/(15*(4*a*c^2 - b^2*c)*(
4*a*c - b^2)) - (8*b*c^2*e^3*(A*e + 4*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (b*((b*((b*((16*c^3*e^3
*(A*e + 4*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e^4)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/
c - (2*(32*A*c^5*d*e^3 - 48*B*a*c^4*e^4 + 12*B*b^2*c^3*e^4 + 48*B*c^5*d^2*e^2))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a
*c - b^2)) + (16*B*a*c^2*e^4)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*b*c^2*e^3*(A*e + 4*B*d))/(15*(4*a*c^2
- b^2*c)*(4*a*c - b^2))))/c - (a*((16*c^3*e^3*(A*e + 4*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2
*e^4)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (2*(32*B*c^5*d^3*e - 48*A*a*c^4*e^4 + 12*A*b^2*c^3*e^4 - 12*B
*b^3*c^2*e^4 + 48*A*c^5*d^2*e^2 + 48*B*b^2*c^3*d*e^3 + 48*B*a*b*c^3*e^4 - 192*B*a*c^4*d*e^3))/(15*c^2*(4*a*c^2
- b^2*c)*(4*a*c - b^2)) + (b*(32*A*c^5*d*e^3 - 48*B*a*c^4*e^4 + 12*B*b^2*c^3*e^4 + 48*B*c^5*d^2*e^2))/(15*c^3
*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (2*(8*B*c^5*d^4 + 12*B*b^4*c*e^4 + 32*A*c^5*d^3*e - 12*A*b^3*c^2*e^4 +
48*B*a^2*c^3*e^4 - 60*B*a*b^2*c^2*e^4 + 48*A*b^2*c^3*d*e^3 - 288*B*a*c^4*d^2*e^2 - 48*B*b^3*c^2*d*e^3 + 72*B*
b^2*c^3*d^2*e^2 + 48*A*a*b*c^3*e^4 - 192*A*a*c^4*d*e^3 + 192*B*a*b*c^3*d*e^3))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a*
c - b^2)) + (b*(32*B*c^5*d^3*e - 48*A*a*c^4*e^4 + 12*A*b^2*c^3*e^4 - 12*B*b^3*c^2*e^4 + 48*A*c^5*d^2*e^2 + 48*
B*b^2*c^3*d*e^3 + 48*B*a*b*c^3*e^4 - 192*B*a*c^4*d*e^3))/(15*c^3*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (2*(32
*A*c^5*d^4 - 8*B*b^5*e^4 + 8*A*b^4*c*e^4 - 20*B*b*c^4*d^4 - 8*A*a^2*c^3*e^4 - 20*A*a*b^2*c^2*e^4 - 4*B*a^2*b*c
^2*e^4 + 48*A*a*c^4*d^2*e^2 - 32*A*b^3*c^2*d*e^3 - 32*B*a^2*c^3*d*e^3 + 32*B*b^2*c^3*d^3*e + 48*A*b^2*c^3*d^2*
e^2 - 48*B*b^3*c^2*d^2*e^2 + 28*B*a*b^3*c*e^4 - 80*A*b*c^4*d^3*e + 32*B*a*c^4*d^3*e + 32*B*b^4*c*d*e^3 + 48*A*
a*b*c^3*d*e^3 + 72*B*a*b*c^3*d^2*e^2 - 80*B*a*b^2*c^2*d*e^3))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (b*(8
*B*c^5*d^4 + 12*B*b^4*c*e^4 + 32*A*c^5*d^3*e - 12*A*b^3*c^2*e^4 + 48*B*a^2*c^3*e^4 - 60*B*a*b^2*c^2*e^4 + 48*A
*b^2*c^3*d*e^3 - 288*B*a*c^4*d^2*e^2 - 48*B*b^3*c^2*d*e^3 + 72*B*b^2*c^3*d^2*e^2 + 48*A*a*b*c^3*e^4 - 192*A*a*
c^4*d*e^3 + 192*B*a*b*c^3*d*e^3))/(15*c^3*(4*a*c^2 - b^2*c)*(4*a*c - b^2))) + (a*((a*((b*((16*c^3*e^3*(A*e + 4
*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e^4)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (2*(3
2*A*c^5*d*e^3 - 48*B*a*c^4*e^4 + 12*B*b^2*c^3*e^4 + 48*B*c^5*d^2*e^2))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a*c - b^2)
) + (16*B*a*c^2*e^4)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*b*c^2*e^3*(A*e + 4*B*d))/(15*(4*a*c^2 - b^2*c)*
(4*a*c - b^2))))/c - (b*((b*((b*((16*c^3*e^3*(A*e + 4*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*
e^4)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (2*(32*A*c^5*d*e^3 - 48*B*a*c^4*e^4 + 12*B*b^2*c^3*e^4 + 48*B*
c^5*d^2*e^2))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (16*B*a*c^2*e^4)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))
- (8*b*c^2*e^3*(A*e + 4*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (a*((16*c^3*e^3*(A*e + 4*B*d))/(15*(
4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e^4)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (2*(32*B*c^5*d^3*
e - 48*A*a*c^4*e^4 + 12*A*b^2*c^3*e^4 - 12*B*b^3*c^2*e^4 + 48*A*c^5*d^2*e^2 + 48*B*b^2*c^3*d*e^3 + 48*B*a*b*c^
3*e^4 - 192*B*a*c^4*d*e^3))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (b*(32*A*c^5*d*e^3 - 48*B*a*c^4*e^4 + 1
2*B*b^2*c^3*e^4 + 48*B*c^5*d^2*e^2))/(15*c^3*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (2*(8*B*c^5*d^4 + 12*B*b^4
*c*e^4 + 32*A*c^5*d^3*e - 12*A*b^3*c^2*e^4 + 48*B*a^2*c^3*e^4 - 60*B*a*b^2*c^2*e^4 + 48*A*b^2*c^3*d*e^3 - 288*
B*a*c^4*d^2*e^2 - 48*B*b^3*c^2*d*e^3 + 72*B*b^2*c^3*d^2*e^2 + 48*A*a*b*c^3*e^4 - 192*A*a*c^4*d*e^3 + 192*B*a*b
*c^3*d*e^3))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (b*(32*B*c^5*d^3*e - 48*A*a*c^4*e^4 + 12*A*b^2*c^3*e^4
- 12*B*b^3*c^2*e^4 + 48*A*c^5*d^2*e^2 + 48*B*b^2*c^3*d*e^3 + 48*B*a*b*c^3*e^4 - 192*B*a*c^4*d*e^3))/(15*c^3*(
4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (b*(32*A*c^5*d^4 - 8*B*b^5*e^4 + 8*A*b^4*c*e^4 - 20*B*b*c^4*d^4 - 8*A*a^
2*c^3*e^4 - 20*A*a*b^2*c^2*e^4 - 4*B*a^2*b*c^2*e^4 + 48*A*a*c^4*d^2*e^2 - 32*A*b^3*c^2*d*e^3 - 32*B*a^2*c^3*d*
e^3 + 32*B*b^2*c^3*d^3*e + 48*A*b^2*c^3*d^2*e^2 - 48*B*b^3*c^2*d^2*e^2 + 28*B*a*b^3*c*e^4 - 80*A*b*c^4*d^3*e +
32*B*a*c^4*d^3*e + 32*B*b^4*c*d*e^3 + 48*A*a*b*c^3*d*e^3 + 72*B*a*b*c^3*d^2*e^2 - 80*B*a*b^2*c^2*d*e^3))/(15*
c^3*(4*a*c^2 - b^2*c)*(4*a*c - b^2)))/(a + b*x + c*x^2)^(3/2) + (x*((a*((16*c*e^3*(A*c*e - B*b*e + 4*B*c*d))/(
5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c*e^4)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (b*((b*((16*c*e^3
*(A*c*e - B*b*e + 4*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c*e^4)/(5*(4*a*c^2 - b^2*c)*(4*a*c -
b^2))))/c - (8*e^2*(5*B*b^2*e^2 + 12*B*c^2*d^2 - 2*A*b*c*e^2 - 14*B*a*c*e^2 + 8*A*c^2*d*e - 8*B*b*c*d*e))/(5*(
4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*b*e^3*(A*c*e - B*b*e + 4*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (1
6*B*a*c*e^4)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (2*(256*A*c^5*d^4 + 16*B*b^5*e^4 - 4*A*b^4*c*e^4 - 128*
B*b*c^4*d^4 + 160*A*a^2*c^3*e^4 + 40*A*a*b^2*c^2*e^4 - 64*B*a^2*b*c^2*e^4 + 192*A*a*c^4*d^2*e^2 - 32*A*b^3*c^2
*d*e^3 + 640*B*a^2*c^3*d*e^3 + 224*B*b^2*c^3*d^3*e + 336*A*b^2*c^3*d^2*e^2 - 48*B*b^3*c^2*d^2*e^2 - 80*B*a*b^3
*c*e^4 - 512*A*b*c^4*d^3*e + 128*B*a*c^4*d^3*e - 16*B*b^4*c*d*e^3 - 384*A*a*b*c^3*d*e^3 - 576*B*a*b*c^3*d^2*e^
2 + 160*B*a*b^2*c^2*d*e^3))/(15*c*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^2) - (4*b*e^2*(5*B*b^2*e^2 + 12*B*c^2*d^2 -
2*A*b*c*e^2 - 14*B*a*c*e^2 + 8*A*c^2*d*e - 8*B*b*c*d*e))/(5*c*(4*a*c^2 - b^2*c)*(4*a*c - b^2))) - (a*((b*((16*
c*e^3*(A*c*e - B*b*e + 4*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c*e^4)/(5*(4*a*c^2 - b^2*c)*(4*a
*c - b^2))))/c - (8*e^2*(5*B*b^2*e^2 + 12*B*c^2*d^2 - 2*A*b*c*e^2 - 14*B*a*c*e^2 + 8*A*c^2*d*e - 8*B*b*c*d*e))
/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*b*e^3*(A*c*e - B*b*e + 4*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))
+ (16*B*a*c*e^4)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (b*(256*A*c^5*d^4 + 16*B*b^5*e^4 - 4*A*b^4*c*e^4 -
128*B*b*c^4*d^4 + 160*A*a^2*c^3*e^4 + 40*A*a*b^2*c^2*e^4 - 64*B*a^2*b*c^2*e^4 + 192*A*a*c^4*d^2*e^2 - 32*A*b^
3*c^2*d*e^3 + 640*B*a^2*c^3*d*e^3 + 224*B*b^2*c^3*d^3*e + 336*A*b^2*c^3*d^2*e^2 - 48*B*b^3*c^2*d^2*e^2 - 80*B*
a*b^3*c*e^4 - 512*A*b*c^4*d^3*e + 128*B*a*c^4*d^3*e - 16*B*b^4*c*d*e^3 - 384*A*a*b*c^3*d*e^3 - 576*B*a*b*c^3*d
^2*e^2 + 160*B*a*b^2*c^2*d*e^3))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^2))/(a + b*x + c*x^2)^(1/2) - (x*((b*
((a*((b*((2*c^2*((2*A*e^4)/5 + (8*B*d*e^3)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e^4)/(5*(4*a*c^2 - b^2*c))))/c - (
b*c*((2*A*e^4)/5 + (8*B*d*e^3)/5))/(4*a*c^2 - b^2*c) - (8*c^2*d*e^2*(2*A*e + 3*B*d))/(5*(4*a*c^2 - b^2*c)) + (
4*B*a*c*e^4)/(5*(4*a*c^2 - b^2*c))))/c - (b*((b*((b*((2*c^2*((2*A*e^4)/5 + (8*B*d*e^3)/5))/(4*a*c^2 - b^2*c) -
(2*B*b*c*e^4)/(5*(4*a*c^2 - b^2*c))))/c - (b*c*((2*A*e^4)/5 + (8*B*d*e^3)/5))/(4*a*c^2 - b^2*c) - (8*c^2*d*e^
2*(2*A*e + 3*B*d))/(5*(4*a*c^2 - b^2*c)) + (4*B*a*c*e^4)/(5*(4*a*c^2 - b^2*c))))/c - (a*((2*c^2*((2*A*e^4)/5 +
(8*B*d*e^3)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e^4)/(5*(4*a*c^2 - b^2*c))))/c + (8*c^2*d^2*e*(3*A*e + 2*B*d))/(
5*(4*a*c^2 - b^2*c)) + (4*b*c*d*e^2*(2*A*e + 3*B*d))/(5*(4*a*c^2 - b^2*c))))/c + (2*c^2*((2*B*d^4)/5 + (8*A*d^
3*e)/5))/(4*a*c^2 - b^2*c) + (4*b*c*d^2*e*(3*A*e + 2*B*d))/(5*(4*a*c^2 - b^2*c))))/c + (a*((b*((b*((2*c^2*((2*
A*e^4)/5 + (8*B*d*e^3)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e^4)/(5*(4*a*c^2 - b^2*c))))/c - (b*c*((2*A*e^4)/5 + (
8*B*d*e^3)/5))/(4*a*c^2 - b^2*c) - (8*c^2*d*e^2*(2*A*e + 3*B*d))/(5*(4*a*c^2 - b^2*c)) + (4*B*a*c*e^4)/(5*(4*a
*c^2 - b^2*c))))/c - (a*((2*c^2*((2*A*e^4)/5 + (8*B*d*e^3)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e^4)/(5*(4*a*c^2 -
b^2*c))))/c + (8*c^2*d^2*e*(3*A*e + 2*B*d))/(5*(4*a*c^2 - b^2*c)) + (4*b*c*d*e^2*(2*A*e + 3*B*d))/(5*(4*a*c^2
- b^2*c))))/c - (b*c*((2*B*d^4)/5 + (8*A*d^3*e)/5))/(4*a*c^2 - b^2*c) - (4*A*c^2*d^4)/(5*(4*a*c^2 - b^2*c)))
+ (a*((a*((b*((2*c^2*((2*A*e^4)/5 + (8*B*d*e^3)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e^4)/(5*(4*a*c^2 - b^2*c))))/
c - (b*c*((2*A*e^4)/5 + (8*B*d*e^3)/5))/(4*a*c^2 - b^2*c) - (8*c^2*d*e^2*(2*A*e + 3*B*d))/(5*(4*a*c^2 - b^2*c)
) + (4*B*a*c*e^4)/(5*(4*a*c^2 - b^2*c))))/c - (b*((b*((b*((2*c^2*((2*A*e^4)/5 + (8*B*d*e^3)/5))/(4*a*c^2 - b^2
*c) - (2*B*b*c*e^4)/(5*(4*a*c^2 - b^2*c))))/c - (b*c*((2*A*e^4)/5 + (8*B*d*e^3)/5))/(4*a*c^2 - b^2*c) - (8*c^2
*d*e^2*(2*A*e + 3*B*d))/(5*(4*a*c^2 - b^2*c)) + (4*B*a*c*e^4)/(5*(4*a*c^2 - b^2*c))))/c - (a*((2*c^2*((2*A*e^4
)/5 + (8*B*d*e^3)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e^4)/(5*(4*a*c^2 - b^2*c))))/c + (8*c^2*d^2*e*(3*A*e + 2*B*
d))/(5*(4*a*c^2 - b^2*c)) + (4*b*c*d*e^2*(2*A*e + 3*B*d))/(5*(4*a*c^2 - b^2*c))))/c + (2*c^2*((2*B*d^4)/5 + (8
*A*d^3*e)/5))/(4*a*c^2 - b^2*c) + (4*b*c*d^2*e*(3*A*e + 2*B*d))/(5*(4*a*c^2 - b^2*c))))/c - (2*A*b*c*d^4)/(5*(
4*a*c^2 - b^2*c)))/(a + b*x + c*x^2)^(5/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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