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

Optimal. Leaf size=324 \[ -\frac {2 (d+e x)^2 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {8 (b+2 c x) \left (4 b c \left (3 a B e^2+8 A c d e+4 B c d^2\right )-8 c^2 \left (a A e^2+2 a B d e+4 A c d^2\right )-6 b^2 c e (A e+2 B d)+b^3 B e^2\right )}{15 c \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}-\frac {8 \left (b^2 \left (a B e^2+A c d e+2 B c d^2\right )+x \left (-4 c^2 \left (-a A e^2+a B d e+2 A c d^2\right )-3 b^2 c e (A e+B d)+4 b c^2 d (2 A e+B d)+b^3 B e^2\right )-4 b c \left (a A e^2+2 a B d e+A c d^2\right )+4 a c e (a B e+3 A c d)\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \]

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Rubi [A]  time = 0.39, antiderivative size = 324, 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 = {820, 777, 613} \begin {gather*} \frac {8 (b+2 c x) \left (4 b c \left (3 a B e^2+8 A c d e+4 B c d^2\right )-8 c^2 \left (a A e^2+2 a B d e+4 A c d^2\right )-6 b^2 c e (A e+2 B d)+b^3 B e^2\right )}{15 c \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}-\frac {8 \left (x \left (-4 c^2 \left (-a A e^2+a B d e+2 A c d^2\right )-3 b^2 c e (A e+B d)+4 b c^2 d (2 A e+B d)+b^3 B e^2\right )+b^2 \left (a B e^2+A c d e+2 B c d^2\right )-4 b c \left (a A e^2+2 a B d e+A c d^2\right )+4 a c e (a B e+3 A c d)\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 (d+e x)^2 (-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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^2)/(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)^2)/(5*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) - (8*(4*a*c*e*(3*A*
c*d + a*B*e) - 4*b*c*(A*c*d^2 + 2*a*B*d*e + a*A*e^2) + b^2*(2*B*c*d^2 + A*c*d*e + a*B*e^2) + (b^3*B*e^2 - 3*b^
2*c*e*(B*d + A*e) + 4*b*c^2*d*(B*d + 2*A*e) - 4*c^2*(2*A*c*d^2 + a*B*d*e - a*A*e^2))*x))/(15*c*(b^2 - 4*a*c)^2
*(a + b*x + c*x^2)^(3/2)) + (8*(b^3*B*e^2 - 6*b^2*c*e*(2*B*d + A*e) - 8*c^2*(4*A*c*d^2 + 2*a*B*d*e + a*A*e^2)
+ 4*b*c*(4*B*c*d^2 + 8*A*c*d*e + 3*a*B*e^2))*(b + 2*c*x))/(15*c*(b^2 - 4*a*c)^3*Sqrt[a + b*x + c*x^2])

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 777

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((2
*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x)*(a + b*x + c*x^2)^
(p + 1))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p
+ 3))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && N
eQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 820

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)*(f*b - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/
((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g*(2*a*e*m + b*d*(2*p + 3)) - f*
(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], 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] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p]
|| IntegersQ[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {(A+B x) (d+e x)^2}{\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)^2}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {2 \int \frac {(d+e x) (2 (4 A c d+2 a B e-b (2 B d+A e))-2 (b B-2 A c) e x)}{\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)^2}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {8 \left (4 a c e (3 A c d+a B e)-4 b c \left (A c d^2+2 a B d e+a A e^2\right )+b^2 \left (2 B c d^2+A c d e+a B e^2\right )+\left (b^3 B e^2-3 b^2 c e (B d+A e)+4 b c^2 d (B d+2 A e)-4 c^2 \left (2 A c d^2+a B d e-a A e^2\right )\right ) x\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {\left (4 \left (b^3 B e^2-6 b^2 c e (2 B d+A e)-8 c^2 \left (4 A c d^2+2 a B d e+a A e^2\right )+4 b c \left (4 B c d^2+8 A c d e+3 a B e^2\right )\right )\right ) \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{15 c \left (b^2-4 a c\right )^2}\\ &=-\frac {2 (A b-2 a B-(b B-2 A c) x) (d+e x)^2}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {8 \left (4 a c e (3 A c d+a B e)-4 b c \left (A c d^2+2 a B d e+a A e^2\right )+b^2 \left (2 B c d^2+A c d e+a B e^2\right )+\left (b^3 B e^2-3 b^2 c e (B d+A e)+4 b c^2 d (B d+2 A e)-4 c^2 \left (2 A c d^2+a B d e-a A e^2\right )\right ) x\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {8 \left (b^3 B e^2-6 b^2 c e (2 B d+A e)-8 c^2 \left (4 A c d^2+2 a B d e+a A e^2\right )+4 b c \left (4 B c d^2+8 A c d e+3 a B e^2\right )\right ) (b+2 c x)}{15 c \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [B]  time = 1.41, size = 711, normalized size = 2.19 \begin {gather*} \frac {2 B \left (64 a^4 c e^2+16 a^3 \left (3 b^2 e^2+2 b c e (5 e x-6 d)+2 c^2 \left (3 d^2+5 e^2 x^2\right )\right )-8 a^2 \left (b^3 e (2 d-15 e x)-6 b^2 c \left (d^2-10 d e x+5 e^2 x^2\right )-30 b c^2 x (d-e x)^2+40 c^3 d e x^3\right )-2 a \left (b^4 \left (d^2+20 d e x-45 e^2 x^2\right )-20 b^3 c x \left (3 d^2-10 d e x+5 e^2 x^2\right )-120 b^2 c^2 x^2 \left (2 d^2-2 d e x+e^2 x^2\right )-16 b c^3 x^3 \left (10 d^2-10 d e x+3 e^2 x^2\right )+64 c^4 d e x^5\right )+b x \left (-5 b^4 \left (d^2+6 d e x-3 e^2 x^2\right )+20 b^3 c x \left (2 d^2-9 d e x+e^2 x^2\right )+8 b^2 c^2 x^2 \left (30 d^2-30 d e x+e^2 x^2\right )+32 b c^3 d x^3 (10 d-3 e x)+128 c^4 d^2 x^4\right )\right )-2 A \left (8 b^3 \left (a^2 e^2-5 a c \left (d^2+6 d e x-5 e^2 x^2\right )+5 c^2 x^2 \left (2 d^2-12 d e x+3 e^2 x^2\right )\right )+16 b^2 c \left (3 a^2 e (5 e x-2 d)+15 a c x \left (d^2-4 d e x+e^2 x^2\right )+c^2 x^3 \left (30 d^2-40 d e x+3 e^2 x^2\right )\right )+16 b c \left (6 a^3 e^2+15 a^2 c (d-e x)^2+10 a c^2 x^2 \left (6 d^2-4 d e x+e^2 x^2\right )+8 c^3 d x^4 (5 d-2 e x)\right )+32 c^2 \left (-6 a^3 d e+5 a^2 c x \left (3 d^2+e^2 x^2\right )+2 a c^2 x^3 \left (10 d^2+e^2 x^2\right )+8 c^3 d^2 x^5\right )+2 b^4 \left (2 a e (d+5 e x)-5 c x \left (d^2+8 d e x-9 e^2 x^2\right )\right )+b^5 \left (3 d^2+10 d e x+15 e^2 x^2\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)^2)/(a + b*x + c*x^2)^(7/2),x]

[Out]

(-2*A*(b^5*(3*d^2 + 10*d*e*x + 15*e^2*x^2) + 2*b^4*(2*a*e*(d + 5*e*x) - 5*c*x*(d^2 + 8*d*e*x - 9*e^2*x^2)) + 3
2*c^2*(-6*a^3*d*e + 8*c^3*d^2*x^5 + 5*a^2*c*x*(3*d^2 + e^2*x^2) + 2*a*c^2*x^3*(10*d^2 + e^2*x^2)) + 16*b*c*(6*
a^3*e^2 + 8*c^3*d*x^4*(5*d - 2*e*x) + 15*a^2*c*(d - e*x)^2 + 10*a*c^2*x^2*(6*d^2 - 4*d*e*x + e^2*x^2)) + 16*b^
2*c*(3*a^2*e*(-2*d + 5*e*x) + 15*a*c*x*(d^2 - 4*d*e*x + e^2*x^2) + c^2*x^3*(30*d^2 - 40*d*e*x + 3*e^2*x^2)) +
8*b^3*(a^2*e^2 - 5*a*c*(d^2 + 6*d*e*x - 5*e^2*x^2) + 5*c^2*x^2*(2*d^2 - 12*d*e*x + 3*e^2*x^2))) + 2*B*(64*a^4*
c*e^2 + b*x*(128*c^4*d^2*x^4 + 32*b*c^3*d*x^3*(10*d - 3*e*x) - 5*b^4*(d^2 + 6*d*e*x - 3*e^2*x^2) + 8*b^2*c^2*x
^2*(30*d^2 - 30*d*e*x + e^2*x^2) + 20*b^3*c*x*(2*d^2 - 9*d*e*x + e^2*x^2)) + 16*a^3*(3*b^2*e^2 + 2*b*c*e*(-6*d
+ 5*e*x) + 2*c^2*(3*d^2 + 5*e^2*x^2)) - 8*a^2*(40*c^3*d*e*x^3 + b^3*e*(2*d - 15*e*x) - 30*b*c^2*x*(d - e*x)^2
- 6*b^2*c*(d^2 - 10*d*e*x + 5*e^2*x^2)) - 2*a*(64*c^4*d*e*x^5 + b^4*(d^2 + 20*d*e*x - 45*e^2*x^2) - 120*b^2*c
^2*x^2*(2*d^2 - 2*d*e*x + e^2*x^2) - 16*b*c^3*x^3*(10*d^2 - 10*d*e*x + 3*e^2*x^2) - 20*b^3*c*x*(3*d^2 - 10*d*e
*x + 5*e^2*x^2))))/(15*(b^2 - 4*a*c)^3*(a + x*(b + c*x))^(5/2))

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IntegrateAlgebraic [B]  time = 10.23, size = 1043, normalized size = 3.22 \begin {gather*} -\frac {2 \left (-15 B e^2 x^3 b^5+3 A d^2 b^5+15 A e^2 x^2 b^5+30 B d e x^2 b^5+5 B d^2 x b^5+10 A d e x b^5-20 B c e^2 x^4 b^4+90 A c e^2 x^3 b^4+180 B c d e x^3 b^4+2 a B d^2 b^4-40 B c d^2 x^2 b^4-90 a B e^2 x^2 b^4-80 A c d e x^2 b^4+4 a A d e b^4-10 A c d^2 x b^4+20 a A e^2 x b^4+40 a B d e x b^4-8 B c^2 e^2 x^5 b^3+120 A c^2 e^2 x^4 b^3+240 B c^2 d e x^4 b^3-240 B c^2 d^2 x^3 b^3-200 a B c e^2 x^3 b^3-480 A c^2 d e x^3 b^3-40 a A c d^2 b^3+8 a^2 A e^2 b^3+80 A c^2 d^2 x^2 b^3+200 a A c e^2 x^2 b^3+400 a B c d e x^2 b^3+16 a^2 B d e b^3-120 a B c d^2 x b^3-120 a^2 B e^2 x b^3-240 a A c d e x b^3+48 A c^3 e^2 x^5 b^2+96 B c^3 d e x^5 b^2-320 B c^3 d^2 x^4 b^2-240 a B c^2 e^2 x^4 b^2-640 A c^3 d e x^4 b^2+480 A c^3 d^2 x^3 b^2+240 a A c^2 e^2 x^3 b^2+480 a B c^2 d e x^3 b^2-48 a^2 B c d^2 b^2-48 a^3 B e^2 b^2-480 a B c^2 d^2 x^2 b^2-240 a^2 B c e^2 x^2 b^2-960 a A c^2 d e x^2 b^2-96 a^2 A c d e b^2+240 a A c^2 d^2 x b^2+240 a^2 A c e^2 x b^2+480 a^2 B c d e x b^2-128 B c^4 d^2 x^5 b-96 a B c^3 e^2 x^5 b-256 A c^4 d e x^5 b+640 A c^4 d^2 x^4 b+160 a A c^3 e^2 x^4 b+320 a B c^3 d e x^4 b-320 a B c^3 d^2 x^3 b-240 a^2 B c^2 e^2 x^3 b-640 a A c^3 d e x^3 b+240 a^2 A c^2 d^2 b+96 a^3 A c e^2 b+960 a A c^3 d^2 x^2 b+240 a^2 A c^2 e^2 x^2 b+480 a^2 B c^2 d e x^2 b+192 a^3 B c d e b-240 a^2 B c^2 d^2 x b-160 a^3 B c e^2 x b-480 a^2 A c^2 d e x b+256 A c^5 d^2 x^5+64 a A c^4 e^2 x^5+128 a B c^4 d e x^5+640 a A c^4 d^2 x^3+160 a^2 A c^3 e^2 x^3+320 a^2 B c^3 d e x^3-96 a^3 B c^2 d^2-64 a^4 B c e^2-160 a^3 B c^2 e^2 x^2-192 a^3 A c^2 d e+480 a^2 A c^3 d^2 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)^2)/(a + b*x + c*x^2)^(7/2),x]

[Out]

(-2*(3*A*b^5*d^2 + 2*a*b^4*B*d^2 - 40*a*A*b^3*c*d^2 - 48*a^2*b^2*B*c*d^2 + 240*a^2*A*b*c^2*d^2 - 96*a^3*B*c^2*
d^2 + 4*a*A*b^4*d*e + 16*a^2*b^3*B*d*e - 96*a^2*A*b^2*c*d*e + 192*a^3*b*B*c*d*e - 192*a^3*A*c^2*d*e + 8*a^2*A*
b^3*e^2 - 48*a^3*b^2*B*e^2 + 96*a^3*A*b*c*e^2 - 64*a^4*B*c*e^2 + 5*b^5*B*d^2*x - 10*A*b^4*c*d^2*x - 120*a*b^3*
B*c*d^2*x + 240*a*A*b^2*c^2*d^2*x - 240*a^2*b*B*c^2*d^2*x + 480*a^2*A*c^3*d^2*x + 10*A*b^5*d*e*x + 40*a*b^4*B*
d*e*x - 240*a*A*b^3*c*d*e*x + 480*a^2*b^2*B*c*d*e*x - 480*a^2*A*b*c^2*d*e*x + 20*a*A*b^4*e^2*x - 120*a^2*b^3*B
*e^2*x + 240*a^2*A*b^2*c*e^2*x - 160*a^3*b*B*c*e^2*x - 40*b^4*B*c*d^2*x^2 + 80*A*b^3*c^2*d^2*x^2 - 480*a*b^2*B
*c^2*d^2*x^2 + 960*a*A*b*c^3*d^2*x^2 + 30*b^5*B*d*e*x^2 - 80*A*b^4*c*d*e*x^2 + 400*a*b^3*B*c*d*e*x^2 - 960*a*A
*b^2*c^2*d*e*x^2 + 480*a^2*b*B*c^2*d*e*x^2 + 15*A*b^5*e^2*x^2 - 90*a*b^4*B*e^2*x^2 + 200*a*A*b^3*c*e^2*x^2 - 2
40*a^2*b^2*B*c*e^2*x^2 + 240*a^2*A*b*c^2*e^2*x^2 - 160*a^3*B*c^2*e^2*x^2 - 240*b^3*B*c^2*d^2*x^3 + 480*A*b^2*c
^3*d^2*x^3 - 320*a*b*B*c^3*d^2*x^3 + 640*a*A*c^4*d^2*x^3 + 180*b^4*B*c*d*e*x^3 - 480*A*b^3*c^2*d*e*x^3 + 480*a
*b^2*B*c^2*d*e*x^3 - 640*a*A*b*c^3*d*e*x^3 + 320*a^2*B*c^3*d*e*x^3 - 15*b^5*B*e^2*x^3 + 90*A*b^4*c*e^2*x^3 - 2
00*a*b^3*B*c*e^2*x^3 + 240*a*A*b^2*c^2*e^2*x^3 - 240*a^2*b*B*c^2*e^2*x^3 + 160*a^2*A*c^3*e^2*x^3 - 320*b^2*B*c
^3*d^2*x^4 + 640*A*b*c^4*d^2*x^4 + 240*b^3*B*c^2*d*e*x^4 - 640*A*b^2*c^3*d*e*x^4 + 320*a*b*B*c^3*d*e*x^4 - 20*
b^4*B*c*e^2*x^4 + 120*A*b^3*c^2*e^2*x^4 - 240*a*b^2*B*c^2*e^2*x^4 + 160*a*A*b*c^3*e^2*x^4 - 128*b*B*c^4*d^2*x^
5 + 256*A*c^5*d^2*x^5 + 96*b^2*B*c^3*d*e*x^5 - 256*A*b*c^4*d*e*x^5 + 128*a*B*c^4*d*e*x^5 - 8*b^3*B*c^2*e^2*x^5
+ 48*A*b^2*c^3*e^2*x^5 - 96*a*b*B*c^3*e^2*x^5 + 64*a*A*c^4*e^2*x^5))/(15*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)^(5
/2))

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fricas [B]  time = 130.73, size = 1095, normalized size = 3.38

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^2/(c*x^2+b*x+a)^(7/2),x, algorithm="fricas")

[Out]

2/15*(8*(16*(B*b*c^4 - 2*A*c^5)*d^2 - 4*(3*B*b^2*c^3 + 4*(B*a - 2*A*b)*c^4)*d*e + (B*b^3*c^2 - 8*A*a*c^4 + 6*(
2*B*a*b - A*b^2)*c^3)*e^2)*x^5 + 20*(16*(B*b^2*c^3 - 2*A*b*c^4)*d^2 - 4*(3*B*b^3*c^2 + 4*(B*a*b - 2*A*b^2)*c^3
)*d*e + (B*b^4*c - 8*A*a*b*c^3 + 6*(2*B*a*b^2 - A*b^3)*c^2)*e^2)*x^4 + 5*(16*(3*B*b^3*c^2 - 8*A*a*c^4 + 2*(2*B
*a*b - 3*A*b^2)*c^3)*d^2 - 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*e + (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)*e^2)*x^3 - (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^2 - 4*(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*e + 8*(6*B*a^3*b^2 - A*a^2*b^3 + 4*(2*B*a^4 - 3*A*a^3*b)*c)*e^2 + 5*(8*(B*b^4*c
- 24*A*a*b*c^3 + 2*(6*B*a*b^2 - A*b^3)*c^2)*d^2 - 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*e + (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)*e^2)*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^2 + 2*(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*e - 4*(6*B*a^2*b^3 - A*a*b^4 + 4*(2*B*a^3*b - 3*A*a^2*b^2)
*c)*e^2)*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)

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giac [B]  time = 0.34, size = 1089, normalized size = 3.36 \begin {gather*} \frac {2 \, {\left ({\left ({\left ({\left (4 \, {\left (\frac {2 \, {\left (16 \, B b c^{4} d^{2} - 32 \, A c^{5} d^{2} - 12 \, B b^{2} c^{3} d e - 16 \, B a c^{4} d e + 32 \, A b c^{4} d e + B b^{3} c^{2} e^{2} + 12 \, B a b c^{3} e^{2} - 6 \, A b^{2} c^{3} e^{2} - 8 \, A a c^{4} e^{2}\right )} x}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}} + \frac {5 \, {\left (16 \, B b^{2} c^{3} d^{2} - 32 \, A b c^{4} d^{2} - 12 \, B b^{3} c^{2} d e - 16 \, B a b c^{3} d e + 32 \, A b^{2} c^{3} d e + B b^{4} c e^{2} + 12 \, B a b^{2} c^{2} e^{2} - 6 \, A b^{3} c^{2} e^{2} - 8 \, A a b c^{3} e^{2}\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {5 \, {\left (48 \, B b^{3} c^{2} d^{2} + 64 \, B a b c^{3} d^{2} - 96 \, A b^{2} c^{3} d^{2} - 128 \, A a c^{4} d^{2} - 36 \, B b^{4} c d e - 96 \, B a b^{2} c^{2} d e + 96 \, A b^{3} c^{2} d e - 64 \, B a^{2} c^{3} d e + 128 \, A a b c^{3} d e + 3 \, B b^{5} e^{2} + 40 \, B a b^{3} c e^{2} - 18 \, A b^{4} c e^{2} + 48 \, B a^{2} b c^{2} e^{2} - 48 \, A a b^{2} c^{2} e^{2} - 32 \, A a^{2} c^{3} e^{2}\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {5 \, {\left (8 \, B b^{4} c d^{2} + 96 \, B a b^{2} c^{2} d^{2} - 16 \, A b^{3} c^{2} d^{2} - 192 \, A a b c^{3} d^{2} - 6 \, B b^{5} d e - 80 \, B a b^{3} c d e + 16 \, A b^{4} c d e - 96 \, B a^{2} b c^{2} d e + 192 \, A a b^{2} c^{2} d e + 18 \, B a b^{4} e^{2} - 3 \, A b^{5} e^{2} + 48 \, B a^{2} b^{2} c e^{2} - 40 \, A a b^{3} c e^{2} + 32 \, B a^{3} c^{2} e^{2} - 48 \, A a^{2} b c^{2} e^{2}\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x - \frac {5 \, {\left (B b^{5} d^{2} - 24 \, B a b^{3} c d^{2} - 2 \, A b^{4} c d^{2} - 48 \, B a^{2} b c^{2} d^{2} + 48 \, A a b^{2} c^{2} d^{2} + 96 \, A a^{2} c^{3} d^{2} + 8 \, B a b^{4} d e + 2 \, A b^{5} d e + 96 \, B a^{2} b^{2} c d e - 48 \, A a b^{3} c d e - 96 \, A a^{2} b c^{2} d e - 24 \, B a^{2} b^{3} e^{2} + 4 \, A a b^{4} e^{2} - 32 \, B a^{3} b c e^{2} + 48 \, A a^{2} b^{2} c e^{2}\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x - \frac {2 \, B a b^{4} d^{2} + 3 \, A b^{5} d^{2} - 48 \, B a^{2} b^{2} c d^{2} - 40 \, A a b^{3} c d^{2} - 96 \, B a^{3} c^{2} d^{2} + 240 \, A a^{2} b c^{2} d^{2} + 16 \, B a^{2} b^{3} d e + 4 \, A a b^{4} d e + 192 \, B a^{3} b c d e - 96 \, A a^{2} b^{2} c d e - 192 \, A a^{3} c^{2} d e - 48 \, B a^{3} b^{2} e^{2} + 8 \, A a^{2} b^{3} e^{2} - 64 \, B a^{4} c e^{2} + 96 \, A a^{3} b c e^{2}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )}}{15 \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^2/(c*x^2+b*x+a)^(7/2),x, algorithm="giac")

[Out]

2/15*((((4*(2*(16*B*b*c^4*d^2 - 32*A*c^5*d^2 - 12*B*b^2*c^3*d*e - 16*B*a*c^4*d*e + 32*A*b*c^4*d*e + B*b^3*c^2*
e^2 + 12*B*a*b*c^3*e^2 - 6*A*b^2*c^3*e^2 - 8*A*a*c^4*e^2)*x/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3) +
5*(16*B*b^2*c^3*d^2 - 32*A*b*c^4*d^2 - 12*B*b^3*c^2*d*e - 16*B*a*b*c^3*d*e + 32*A*b^2*c^3*d*e + B*b^4*c*e^2 +
12*B*a*b^2*c^2*e^2 - 6*A*b^3*c^2*e^2 - 8*A*a*b*c^3*e^2)/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x +
5*(48*B*b^3*c^2*d^2 + 64*B*a*b*c^3*d^2 - 96*A*b^2*c^3*d^2 - 128*A*a*c^4*d^2 - 36*B*b^4*c*d*e - 96*B*a*b^2*c^2
*d*e + 96*A*b^3*c^2*d*e - 64*B*a^2*c^3*d*e + 128*A*a*b*c^3*d*e + 3*B*b^5*e^2 + 40*B*a*b^3*c*e^2 - 18*A*b^4*c*e
^2 + 48*B*a^2*b*c^2*e^2 - 48*A*a*b^2*c^2*e^2 - 32*A*a^2*c^3*e^2)/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c
^3))*x + 5*(8*B*b^4*c*d^2 + 96*B*a*b^2*c^2*d^2 - 16*A*b^3*c^2*d^2 - 192*A*a*b*c^3*d^2 - 6*B*b^5*d*e - 80*B*a*b
^3*c*d*e + 16*A*b^4*c*d*e - 96*B*a^2*b*c^2*d*e + 192*A*a*b^2*c^2*d*e + 18*B*a*b^4*e^2 - 3*A*b^5*e^2 + 48*B*a^2
*b^2*c*e^2 - 40*A*a*b^3*c*e^2 + 32*B*a^3*c^2*e^2 - 48*A*a^2*b*c^2*e^2)/(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^2 - 24*B*a*b^3*c*d^2 - 2*A*b^4*c*d^2 - 48*B*a^2*b*c^2*d^2 + 48*A*a*b^2*c^2*d^2 + 96*
A*a^2*c^3*d^2 + 8*B*a*b^4*d*e + 2*A*b^5*d*e + 96*B*a^2*b^2*c*d*e - 48*A*a*b^3*c*d*e - 96*A*a^2*b*c^2*d*e - 24*
B*a^2*b^3*e^2 + 4*A*a*b^4*e^2 - 32*B*a^3*b*c*e^2 + 48*A*a^2*b^2*c*e^2)/(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^2 + 3*A*b^5*d^2 - 48*B*a^2*b^2*c*d^2 - 40*A*a*b^3*c*d^2 - 96*B*a^3*c^2*d^2 + 240*A
*a^2*b*c^2*d^2 + 16*B*a^2*b^3*d*e + 4*A*a*b^4*d*e + 192*B*a^3*b*c*d*e - 96*A*a^2*b^2*c*d*e - 192*A*a^3*c^2*d*e
- 48*B*a^3*b^2*e^2 + 8*A*a^2*b^3*e^2 - 64*B*a^4*c*e^2 + 96*A*a^3*b*c*e^2)/(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 = 1064, normalized size = 3.28 \begin {gather*} \frac {-32 B \,a^{2} b \,c^{2} d^{2} x +\frac {16}{3} B a \,b^{4} d e x -16 B a \,b^{3} c \,d^{2} x +32 A \,a^{2} b^{2} c \,e^{2} x +32 A a \,b^{2} c^{2} d^{2} x -\frac {64}{3} B \,a^{3} b c \,e^{2} x -64 B a \,b^{2} c^{2} d^{2} x^{2}-32 B \,a^{2} b^{2} c \,e^{2} x^{2}+128 A a b \,c^{3} d^{2} x^{2}-\frac {32}{3} A \,b^{4} c d e \,x^{2}+32 A \,a^{2} b \,c^{2} e^{2} x^{2}+\frac {80}{3} A a \,b^{3} c \,e^{2} x^{2}-\frac {128}{3} B a b \,c^{3} d^{2} x^{3}+24 B \,b^{4} c d e \,x^{3}+\frac {128}{3} B \,a^{2} c^{3} d e \,x^{3}-\frac {80}{3} B a \,b^{3} c \,e^{2} x^{3}-32 B \,a^{2} b \,c^{2} e^{2} x^{3}-64 A \,b^{3} c^{2} d e \,x^{3}+32 A a \,b^{2} c^{2} e^{2} x^{3}+\frac {64}{3} A a b \,c^{3} e^{2} x^{4}-\frac {256}{3} A \,b^{2} c^{3} d e \,x^{4}-32 B a \,b^{2} c^{2} e^{2} x^{4}+32 B \,b^{3} c^{2} d e \,x^{4}-\frac {512}{15} A b \,c^{4} d e \,x^{5}-\frac {64}{5} B a b \,c^{3} e^{2} x^{5}+\frac {256}{15} B a \,c^{4} d e \,x^{5}+\frac {64}{5} B \,b^{2} c^{3} d e \,x^{5}+\frac {128}{5} B \,a^{3} b c d e -\frac {128}{5} A \,a^{3} c^{2} d e -\frac {64}{5} A \,a^{2} b^{2} c d e +\frac {128}{3} B a b \,c^{3} d e \,x^{4}+\frac {512}{15} A \,c^{5} d^{2} x^{5}-2 B \,b^{5} e^{2} x^{3}+2 A \,b^{5} e^{2} x^{2}+\frac {2}{3} B \,b^{5} d^{2} x +\frac {16}{15} A \,a^{2} b^{3} e^{2}-\frac {32}{5} B \,a^{3} b^{2} e^{2}+\frac {4}{15} B a \,b^{4} d^{2}-\frac {256}{3} A a b \,c^{3} d e \,x^{3}+64 B a \,b^{2} c^{2} d e \,x^{3}-128 A a \,b^{2} c^{2} d e \,x^{2}+64 B \,a^{2} b \,c^{2} d e \,x^{2}+\frac {160}{3} B a \,b^{3} c d e \,x^{2}-64 A \,a^{2} b \,c^{2} d e x -32 A a \,b^{3} c d e x +64 B \,a^{2} b^{2} c d e x +\frac {2}{5} A \,b^{5} d^{2}+4 B \,b^{5} d e \,x^{2}-\frac {16}{3} B \,b^{4} c \,d^{2} x^{2}+64 A \,a^{2} c^{3} d^{2} x +\frac {8}{3} A a \,b^{4} e^{2} x +\frac {4}{3} A \,b^{5} d e x -\frac {4}{3} A \,b^{4} c \,d^{2} x -16 B \,a^{2} b^{3} e^{2} x +\frac {128}{15} A a \,c^{4} e^{2} x^{5}+\frac {32}{5} A \,b^{2} c^{3} e^{2} x^{5}-\frac {16}{15} B \,b^{3} c^{2} e^{2} x^{5}-\frac {256}{15} B b \,c^{4} d^{2} x^{5}+16 A \,b^{3} c^{2} e^{2} x^{4}+\frac {256}{3} A b \,c^{4} d^{2} x^{4}-\frac {8}{3} B \,b^{4} c \,e^{2} x^{4}-\frac {128}{3} B \,b^{2} c^{3} d^{2} x^{4}+\frac {64}{3} A \,a^{2} c^{3} e^{2} x^{3}+\frac {256}{3} A a \,c^{4} d^{2} x^{3}+12 A \,b^{4} c \,e^{2} x^{3}+64 A \,b^{2} c^{3} d^{2} x^{3}-32 B \,b^{3} c^{2} d^{2} x^{3}+\frac {32}{3} A \,b^{3} c^{2} d^{2} x^{2}-\frac {64}{3} B \,a^{3} c^{2} e^{2} x^{2}-12 B a \,b^{4} e^{2} x^{2}-\frac {16}{3} A a \,b^{3} c \,d^{2}+\frac {64}{5} A \,a^{3} b c \,e^{2}+\frac {8}{15} A a \,b^{4} d e -\frac {128}{15} B \,a^{4} c \,e^{2}-\frac {64}{5} B \,a^{3} c^{2} d^{2}+\frac {32}{15} B \,a^{2} b^{3} d e -\frac {32}{5} B \,a^{2} b^{2} c \,d^{2}+32 A \,a^{2} b \,c^{2} d^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15/(c*x^2+b*x+a)^(5/2)*(64*A*a*c^4*e^2*x^5+48*A*b^2*c^3*e^2*x^5-256*A*b*c^4*d*e*x^5+256*A*c^5*d^2*x^5-96*B*a
*b*c^3*e^2*x^5+128*B*a*c^4*d*e*x^5-8*B*b^3*c^2*e^2*x^5+96*B*b^2*c^3*d*e*x^5-128*B*b*c^4*d^2*x^5+160*A*a*b*c^3*
e^2*x^4+120*A*b^3*c^2*e^2*x^4-640*A*b^2*c^3*d*e*x^4+640*A*b*c^4*d^2*x^4-240*B*a*b^2*c^2*e^2*x^4+320*B*a*b*c^3*
d*e*x^4-20*B*b^4*c*e^2*x^4+240*B*b^3*c^2*d*e*x^4-320*B*b^2*c^3*d^2*x^4+160*A*a^2*c^3*e^2*x^3+240*A*a*b^2*c^2*e
^2*x^3-640*A*a*b*c^3*d*e*x^3+640*A*a*c^4*d^2*x^3+90*A*b^4*c*e^2*x^3-480*A*b^3*c^2*d*e*x^3+480*A*b^2*c^3*d^2*x^
3-240*B*a^2*b*c^2*e^2*x^3+320*B*a^2*c^3*d*e*x^3-200*B*a*b^3*c*e^2*x^3+480*B*a*b^2*c^2*d*e*x^3-320*B*a*b*c^3*d^
2*x^3-15*B*b^5*e^2*x^3+180*B*b^4*c*d*e*x^3-240*B*b^3*c^2*d^2*x^3+240*A*a^2*b*c^2*e^2*x^2+200*A*a*b^3*c*e^2*x^2
-960*A*a*b^2*c^2*d*e*x^2+960*A*a*b*c^3*d^2*x^2+15*A*b^5*e^2*x^2-80*A*b^4*c*d*e*x^2+80*A*b^3*c^2*d^2*x^2-160*B*
a^3*c^2*e^2*x^2-240*B*a^2*b^2*c*e^2*x^2+480*B*a^2*b*c^2*d*e*x^2-90*B*a*b^4*e^2*x^2+400*B*a*b^3*c*d*e*x^2-480*B
*a*b^2*c^2*d^2*x^2+30*B*b^5*d*e*x^2-40*B*b^4*c*d^2*x^2+240*A*a^2*b^2*c*e^2*x-480*A*a^2*b*c^2*d*e*x+480*A*a^2*c
^3*d^2*x+20*A*a*b^4*e^2*x-240*A*a*b^3*c*d*e*x+240*A*a*b^2*c^2*d^2*x+10*A*b^5*d*e*x-10*A*b^4*c*d^2*x-160*B*a^3*
b*c*e^2*x-120*B*a^2*b^3*e^2*x+480*B*a^2*b^2*c*d*e*x-240*B*a^2*b*c^2*d^2*x+40*B*a*b^4*d*e*x-120*B*a*b^3*c*d^2*x
+5*B*b^5*d^2*x+96*A*a^3*b*c*e^2-192*A*a^3*c^2*d*e+8*A*a^2*b^3*e^2-96*A*a^2*b^2*c*d*e+240*A*a^2*b*c^2*d^2+4*A*a
*b^4*d*e-40*A*a*b^3*c*d^2+3*A*b^5*d^2-64*B*a^4*c*e^2-48*B*a^3*b^2*e^2+192*B*a^3*b*c*d*e-96*B*a^3*c^2*d^2+16*B*
a^2*b^3*d*e-48*B*a^2*b^2*c*d^2+2*B*a*b^4*d^2)/(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)^2/(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?

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mupad [B]  time = 3.77, size = 1996, normalized size = 6.16

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*((2*c^2*(8*A*e^2 + 16*B*d*e))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c*e^2)/(5*(4*a*c^2 - b^2*c)*(4*
a*c - b^2))) + (b*c*(8*A*e^2 + 16*B*d*e))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (16*B*a*c*e^2)/(5*(4*a*c^2 -
b^2*c)*(4*a*c - b^2)))/(a + b*x + c*x^2)^(1/2) - (x*((a*((2*c^2*((2*A*e^2)/5 + (4*B*d*e)/5))/(4*a*c^2 - b^2*c)
- (2*B*b*c*e^2)/(5*(4*a*c^2 - b^2*c))))/c + (b*((2*c^2*((2*B*d^2)/5 + (4*A*d*e)/5))/(4*a*c^2 - b^2*c) - (b*((
2*c^2*((2*A*e^2)/5 + (4*B*d*e)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e^2)/(5*(4*a*c^2 - b^2*c))))/c + (b*c*((2*A*e^
2)/5 + (4*B*d*e)/5))/(4*a*c^2 - b^2*c) - (4*B*a*c*e^2)/(5*(4*a*c^2 - b^2*c))))/c - (b*c*((2*B*d^2)/5 + (4*A*d*
e)/5))/(4*a*c^2 - b^2*c) - (4*A*c^2*d^2)/(5*(4*a*c^2 - b^2*c))) + (a*((2*c^2*((2*B*d^2)/5 + (4*A*d*e)/5))/(4*a
*c^2 - b^2*c) - (b*((2*c^2*((2*A*e^2)/5 + (4*B*d*e)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e^2)/(5*(4*a*c^2 - b^2*c)
)))/c + (b*c*((2*A*e^2)/5 + (4*B*d*e)/5))/(4*a*c^2 - b^2*c) - (4*B*a*c*e^2)/(5*(4*a*c^2 - b^2*c))))/c - (2*A*b
*c*d^2)/(5*(4*a*c^2 - b^2*c)))/(a + b*x + c*x^2)^(5/2) - (x*((2*e*(2*A*c*e - B*b*e + 4*B*c*d))/(15*c*(4*a*c -
b^2)) - (4*B*b*e^2)/(15*c*(4*a*c - b^2))) + (2*B*b^2*e^2 + 4*B*c^2*d^2 - 2*A*b*c*e^2 - 4*B*a*c*e^2 + 8*A*c^2*d
*e - 4*B*b*c*d*e)/(15*c^2*(4*a*c - b^2)) - (4*B*a*e^2)/(15*c*(4*a*c - b^2)))/(a + b*x + c*x^2)^(3/2) + (x*((b*
((2*(8*B*c^3*d^2 + 16*A*c^3*d*e - 48*B*a*c^2*e^2 + 12*B*b^2*c*e^2))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (b*
((16*c^3*e*(A*e + 2*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e^2)/(15*(4*a*c^2 - b^2*c)*(4*a*c
- b^2))))/c + (8*b*c^2*e*(A*e + 2*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (16*B*a*c^2*e^2)/(15*(4*a*c^2 -
b^2*c)*(4*a*c - b^2))))/c + (2*(32*A*c^3*d^2 - 8*B*b^3*e^2 + 8*A*a*c^2*e^2 + 8*A*b^2*c*e^2 - 20*B*b*c^2*d^2 +
12*B*a*b*c*e^2 - 40*A*b*c^2*d*e + 16*B*a*c^2*d*e + 16*B*b^2*c*d*e))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (a
*((16*c^3*e*(A*e + 2*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e^2)/(15*(4*a*c^2 - b^2*c)*(4*a*c
- b^2))))/c - (b*(8*B*c^3*d^2 + 16*A*c^3*d*e - 48*B*a*c^2*e^2 + 12*B*b^2*c*e^2))/(15*c*(4*a*c^2 - b^2*c)*(4*a
*c - b^2))) + (a*((2*(8*B*c^3*d^2 + 16*A*c^3*d*e - 48*B*a*c^2*e^2 + 12*B*b^2*c*e^2))/(15*(4*a*c^2 - b^2*c)*(4*
a*c - b^2)) - (b*((16*c^3*e*(A*e + 2*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e^2)/(15*(4*a*c^2
- b^2*c)*(4*a*c - b^2))))/c + (8*b*c^2*e*(A*e + 2*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (16*B*a*c^2*e^
2)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (b*(32*A*c^3*d^2 - 8*B*b^3*e^2 + 8*A*a*c^2*e^2 + 8*A*b^2*c*e^2 -
20*B*b*c^2*d^2 + 12*B*a*b*c*e^2 - 40*A*b*c^2*d*e + 16*B*a*c^2*d*e + 16*B*b^2*c*d*e))/(15*c*(4*a*c^2 - b^2*c)*
(4*a*c - b^2)))/(a + b*x + c*x^2)^(3/2) + (x*((2*c*(256*A*c^3*d^2 - 8*B*b^3*e^2 + 32*A*a*c^2*e^2 + 56*A*b^2*c*
e^2 - 128*B*b*c^2*d^2 - 96*B*a*b*c*e^2 - 256*A*b*c^2*d*e + 64*B*a*c^2*d*e + 112*B*b^2*c*d*e))/(15*(4*a*c^2 - b
^2*c)*(4*a*c - b^2)^2) + (8*B*b*c*e^2)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))) + (b*(256*A*c^3*d^2 - 8*B*b^3*e^2
+ 32*A*a*c^2*e^2 + 56*A*b^2*c*e^2 - 128*B*b*c^2*d^2 - 96*B*a*b*c*e^2 - 256*A*b*c^2*d*e + 64*B*a*c^2*d*e + 112*
B*b^2*c*d*e))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^2) + (16*B*a*c*e^2)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)))/(a
+ b*x + c*x^2)^(1/2)

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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)**2/(c*x**2+b*x+a)**(7/2),x)

[Out]

Timed out

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