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3.23.49 \(\int \frac {(A+B x) (d+e x)^3}{(a+b x+c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=264 \[ -\frac {16 (-2 a e+x (2 c d-b e)+b d) \left (-8 b \left (a B e^2+2 A c d e+B c d^2\right )+4 c \left (a A e^2+3 a B d e+4 A c d^2\right )+b^2 e (3 A e+5 B d)\right )}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^3 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {4 (d+e x)^2 \left (-x \left (4 c (3 a B e+4 A c d)-8 b c (A e+B d)+b^2 B e\right )-8 b (a B e+A c d)+4 a A c e+b^2 (3 A e+4 B d)\right )}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \]

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Rubi [A]  time = 0.36, antiderivative size = 264, 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, 804, 636} \begin {gather*} -\frac {16 (-2 a e+x (2 c d-b e)+b d) \left (-8 b \left (a B e^2+2 A c d e+B c d^2\right )+4 c \left (a A e^2+3 a B d e+4 A c d^2\right )+b^2 e (3 A e+5 B d)\right )}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^3 (-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 {4 (d+e x)^2 \left (-x \left (4 c (3 a B e+4 A c d)-8 b c (A e+B d)+b^2 B e\right )-8 b (a B e+A c d)+4 a A c e+b^2 (3 A e+4 B d)\right )}{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)^3)/(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)^3)/(5*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) - (4*(d + e*x)^2*(4
*a*A*c*e + b^2*(4*B*d + 3*A*e) - 8*b*(A*c*d + a*B*e) - (b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + 3*a*B*e))
*x))/(15*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) - (16*(b^2*e*(5*B*d + 3*A*e) + 4*c*(4*A*c*d^2 + 3*a*B*d*e +
a*A*e^2) - 8*b*(B*c*d^2 + 2*A*c*d*e + a*B*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 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 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)^3}{\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)^3}{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 b B d+8 A c d-3 A b e+6 a B e-(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)^3}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {4 (d+e x)^2 \left (4 a A c e+b^2 (4 B d+3 A e)-8 b (A c d+a B e)-\left (b^2 B e-8 b c (B d+A e)+4 c (4 A c d+3 a B e)\right ) x\right )}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {\left (8 \left (b^2 e (5 B d+3 A e)+4 c \left (4 A c d^2+3 a B d e+a A e^2\right )-8 b \left (B c d^2+2 A c d e+a B e^2\right )\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)^3}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {4 (d+e x)^2 \left (4 a A c e+b^2 (4 B d+3 A e)-8 b (A c d+a B e)-\left (b^2 B e-8 b c (B d+A e)+4 c (4 A c d+3 a B e)\right ) x\right )}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {16 \left (b^2 e (5 B d+3 A e)+4 c \left (4 A c d^2+3 a B d e+a A e^2\right )-8 b \left (B c d^2+2 A c d e+a B e^2\right )\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 = 1.62, size = 963, normalized size = 3.65 \begin {gather*} \frac {2 \left (A \left (-3 \left (d^3+5 e x d^2+15 e^2 x^2 d-5 e^3 x^3\right ) b^5+\left (10 c x \left (d^3+12 e x d^2-27 e^2 x^2 d+2 e^3 x^3\right )-6 a e \left (d^2+10 e x d-15 e^2 x^2\right )\right ) b^4-8 \left (3 a^2 (d-5 e x) e^2+c^2 x^2 \left (10 d^3-90 e x d^2+45 e^2 x^2 d-e^3 x^3\right )-5 a c \left (d^3+9 e x d^2-15 e^2 x^2 d+5 e^3 x^3\right )\right ) b^3+48 \left (a^3 e^3+a^2 c \left (3 d^2-15 e x d+5 e^2 x^2\right ) e+c^3 d x^3 \left (-10 d^2+20 e x d-3 e^2 x^2\right )+5 a c^2 x \left (-d^3+6 e x d^2-3 e^2 x^2 d+e^3 x^3\right )\right ) b^2+16 c \left (8 c^3 d^2 (3 e x-5 d) x^4+6 a c^2 \left (-10 d^3+10 e x d^2-5 e^2 x^2 d+e^3 x^3\right ) x^2-15 a^2 c (d-e x)^3+2 a^3 e^2 (5 e x-9 d)\right ) b-32 c \left (8 c^4 d^3 x^5+2 a c^3 d \left (10 d^2+3 e^2 x^2\right ) x^3+15 a^2 c^2 d \left (d^2+e^2 x^2\right ) x-2 a^4 e^3-a^3 c e \left (9 d^2+5 e^2 x^2\right )\right )\right )+B \left (64 e^2 (3 c d-2 b e) a^4+16 \left (6 \left (d^3+5 e^2 x^2 d\right ) c^2-2 b e \left (9 d^2-15 e x d+10 e^2 x^2\right ) c+b^2 e^2 (9 d-20 e x)\right ) a^3-24 \left (e \left (d^2-15 e x d+10 e^2 x^2\right ) b^3-2 c \left (d^3-15 e x d^2+15 e^2 x^2 d-10 e^3 x^3\right ) b^2+10 c^2 x (e x-d)^3 b+4 c^3 e x^3 \left (5 d^2+e^2 x^2\right )\right ) a^2-2 \left (96 c^4 d^2 e x^5-16 b c^3 d \left (10 d^2-15 e x d+9 e^2 x^2\right ) x^3+24 b^2 c^2 \left (-10 d^3+15 e x d^2-15 e^2 x^2 d+e^3 x^3\right ) x^2+60 b^3 c \left (-d^3+5 e x d^2-5 e^2 x^2 d+e^3 x^3\right ) x+b^4 \left (d^3+30 e x d^2-135 e^2 x^2 d+20 e^3 x^3\right )\right ) a+b x \left (-5 \left (d^3+9 e x d^2-9 e^2 x^2 d-e^3 x^3\right ) b^4+2 c x \left (20 d^3-135 e x d^2+30 e^2 x^2 d+e^3 x^3\right ) b^3+24 c^2 d x^2 \left (10 d^2-15 e x d+e^2 x^2\right ) b^2+16 c^3 d^2 x^3 (20 d-9 e x) b+128 c^4 d^3 x^4\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)^3)/(a + b*x + c*x^2)^(7/2),x]

[Out]

(2*(A*(-3*b^5*(d^3 + 5*d^2*e*x + 15*d*e^2*x^2 - 5*e^3*x^3) - 32*c*(-2*a^4*e^3 + 8*c^4*d^3*x^5 + 15*a^2*c^2*d*x
*(d^2 + e^2*x^2) + 2*a*c^3*d*x^3*(10*d^2 + 3*e^2*x^2) - a^3*c*e*(9*d^2 + 5*e^2*x^2)) + 16*b*c*(-15*a^2*c*(d -
e*x)^3 + 8*c^3*d^2*x^4*(-5*d + 3*e*x) + 2*a^3*e^2*(-9*d + 5*e*x) + 6*a*c^2*x^2*(-10*d^3 + 10*d^2*e*x - 5*d*e^2
*x^2 + e^3*x^3)) + 48*b^2*(a^3*e^3 + c^3*d*x^3*(-10*d^2 + 20*d*e*x - 3*e^2*x^2) + a^2*c*e*(3*d^2 - 15*d*e*x +
5*e^2*x^2) + 5*a*c^2*x*(-d^3 + 6*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3)) + b^4*(-6*a*e*(d^2 + 10*d*e*x - 15*e^2*x^2)
 + 10*c*x*(d^3 + 12*d^2*e*x - 27*d*e^2*x^2 + 2*e^3*x^3)) - 8*b^3*(3*a^2*e^2*(d - 5*e*x) + c^2*x^2*(10*d^3 - 90
*d^2*e*x + 45*d*e^2*x^2 - e^3*x^3) - 5*a*c*(d^3 + 9*d^2*e*x - 15*d*e^2*x^2 + 5*e^3*x^3))) + B*(64*a^4*e^2*(3*c
*d - 2*b*e) + 16*a^3*(b^2*e^2*(9*d - 20*e*x) - 2*b*c*e*(9*d^2 - 15*d*e*x + 10*e^2*x^2) + 6*c^2*(d^3 + 5*d*e^2*
x^2)) - 24*a^2*(10*b*c^2*x*(-d + e*x)^3 + 4*c^3*e*x^3*(5*d^2 + e^2*x^2) + b^3*e*(d^2 - 15*d*e*x + 10*e^2*x^2)
- 2*b^2*c*(d^3 - 15*d^2*e*x + 15*d*e^2*x^2 - 10*e^3*x^3)) + b*x*(128*c^4*d^3*x^4 + 16*b*c^3*d^2*x^3*(20*d - 9*
e*x) + 24*b^2*c^2*d*x^2*(10*d^2 - 15*d*e*x + e^2*x^2) - 5*b^4*(d^3 + 9*d^2*e*x - 9*d*e^2*x^2 - e^3*x^3) + 2*b^
3*c*x*(20*d^3 - 135*d^2*e*x + 30*d*e^2*x^2 + e^3*x^3)) - 2*a*(96*c^4*d^2*e*x^5 - 16*b*c^3*d*x^3*(10*d^2 - 15*d
*e*x + 9*e^2*x^2) + 24*b^2*c^2*x^2*(-10*d^3 + 15*d^2*e*x - 15*d*e^2*x^2 + e^3*x^3) + 60*b^3*c*x*(-d^3 + 5*d^2*
e*x - 5*d*e^2*x^2 + e^3*x^3) + b^4*(d^3 + 30*d^2*e*x - 135*d*e^2*x^2 + 20*e^3*x^3)))))/(15*(b^2 - 4*a*c)^3*(a
+ x*(b + c*x))^(5/2))

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IntegrateAlgebraic [B]  time = 14.92, size = 1481, normalized size = 5.61 \begin {gather*} -\frac {2 \left (-5 B e^3 x^4 b^5+3 A d^3 b^5-15 A e^3 x^3 b^5-45 B d e^2 x^3 b^5+45 A d e^2 x^2 b^5+45 B d^2 e x^2 b^5+5 B d^3 x b^5+15 A d^2 e x b^5-2 B c e^3 x^5 b^4-20 A c e^3 x^4 b^4-60 B c d e^2 x^4 b^4+2 a B d^3 b^4+40 a B e^3 x^3 b^4+270 A c d e^2 x^3 b^4+270 B c d^2 e x^3 b^4-40 B c d^3 x^2 b^4-90 a A e^3 x^2 b^4-270 a B d e^2 x^2 b^4-120 A c d^2 e x^2 b^4+6 a A d^2 e b^4-10 A c d^3 x b^4+60 a A d e^2 x b^4+60 a B d^2 e x b^4-8 A c^2 e^3 x^5 b^3-24 B c^2 d e^2 x^5 b^3+120 a B c e^3 x^4 b^3+360 A c^2 d e^2 x^4 b^3+360 B c^2 d^2 e x^4 b^3-40 a A c d^3 b^3-240 B c^2 d^3 x^3 b^3-200 a A c e^3 x^3 b^3-600 a B c d e^2 x^3 b^3-720 A c^2 d^2 e x^3 b^3+24 a^2 A d e^2 b^3+80 A c^2 d^3 x^2 b^3+240 a^2 B e^3 x^2 b^3+600 a A c d e^2 x^2 b^3+600 a B c d^2 e x^2 b^3+24 a^2 B d^2 e b^3-120 a B c d^3 x b^3-120 a^2 A e^3 x b^3-360 a^2 B d e^2 x b^3-360 a A c d^2 e x b^3+48 a B c^2 e^3 x^5 b^2+144 A c^3 d e^2 x^5 b^2+144 B c^3 d^2 e x^5 b^2-320 B c^3 d^3 x^4 b^2-240 a A c^2 e^3 x^4 b^2-720 a B c^2 d e^2 x^4 b^2-960 A c^3 d^2 e x^4 b^2-48 a^2 B c d^3 b^2-48 a^3 A e^3 b^2+480 A c^3 d^3 x^3 b^2+480 a^2 B c e^3 x^3 b^2+720 a A c^2 d e^2 x^3 b^2+720 a B c^2 d^2 e x^3 b^2-144 a^3 B d e^2 b^2-480 a B c^2 d^3 x^2 b^2-240 a^2 A c e^3 x^2 b^2-720 a^2 B c d e^2 x^2 b^2-1440 a A c^2 d^2 e x^2 b^2-144 a^2 A c d^2 e b^2+240 a A c^2 d^3 x b^2+320 a^3 B e^3 x b^2+720 a^2 A c d e^2 x b^2+720 a^2 B c d^2 e x b^2-128 B c^4 d^3 x^5 b-96 a A c^3 e^3 x^5 b-288 a B c^3 d e^2 x^5 b-384 A c^4 d^2 e x^5 b+640 A c^4 d^3 x^4 b+240 a^2 B c^2 e^3 x^4 b+480 a A c^3 d e^2 x^4 b+480 a B c^3 d^2 e x^4 b+240 a^2 A c^2 d^3 b+128 a^4 B e^3 b-320 a B c^3 d^3 x^3 b-240 a^2 A c^2 e^3 x^3 b-720 a^2 B c^2 d e^2 x^3 b-960 a A c^3 d^2 e x^3 b+288 a^3 A c d e^2 b+960 a A c^3 d^3 x^2 b+320 a^3 B c e^3 x^2 b+720 a^2 A c^2 d e^2 x^2 b+720 a^2 B c^2 d^2 e x^2 b+288 a^3 B c d^2 e b-240 a^2 B c^2 d^3 x b-160 a^3 A c e^3 x b-480 a^3 B c d e^2 x b-720 a^2 A c^2 d^2 e x b+256 A c^5 d^3 x^5+96 a^2 B c^3 e^3 x^5+192 a A c^4 d e^2 x^5+192 a B c^4 d^2 e x^5-96 a^3 B c^2 d^3-64 a^4 A c e^3+640 a A c^4 d^3 x^3+480 a^2 A c^3 d e^2 x^3+480 a^2 B c^3 d^2 e x^3-192 a^4 B c d e^2-160 a^3 A c^2 e^3 x^2-480 a^3 B c^2 d e^2 x^2-288 a^3 A c^2 d^2 e+480 a^2 A c^3 d^3 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)^3)/(a + b*x + c*x^2)^(7/2),x]

[Out]

(-2*(3*A*b^5*d^3 + 2*a*b^4*B*d^3 - 40*a*A*b^3*c*d^3 - 48*a^2*b^2*B*c*d^3 + 240*a^2*A*b*c^2*d^3 - 96*a^3*B*c^2*
d^3 + 6*a*A*b^4*d^2*e + 24*a^2*b^3*B*d^2*e - 144*a^2*A*b^2*c*d^2*e + 288*a^3*b*B*c*d^2*e - 288*a^3*A*c^2*d^2*e
 + 24*a^2*A*b^3*d*e^2 - 144*a^3*b^2*B*d*e^2 + 288*a^3*A*b*c*d*e^2 - 192*a^4*B*c*d*e^2 - 48*a^3*A*b^2*e^3 + 128
*a^4*b*B*e^3 - 64*a^4*A*c*e^3 + 5*b^5*B*d^3*x - 10*A*b^4*c*d^3*x - 120*a*b^3*B*c*d^3*x + 240*a*A*b^2*c^2*d^3*x
 - 240*a^2*b*B*c^2*d^3*x + 480*a^2*A*c^3*d^3*x + 15*A*b^5*d^2*e*x + 60*a*b^4*B*d^2*e*x - 360*a*A*b^3*c*d^2*e*x
 + 720*a^2*b^2*B*c*d^2*e*x - 720*a^2*A*b*c^2*d^2*e*x + 60*a*A*b^4*d*e^2*x - 360*a^2*b^3*B*d*e^2*x + 720*a^2*A*
b^2*c*d*e^2*x - 480*a^3*b*B*c*d*e^2*x - 120*a^2*A*b^3*e^3*x + 320*a^3*b^2*B*e^3*x - 160*a^3*A*b*c*e^3*x - 40*b
^4*B*c*d^3*x^2 + 80*A*b^3*c^2*d^3*x^2 - 480*a*b^2*B*c^2*d^3*x^2 + 960*a*A*b*c^3*d^3*x^2 + 45*b^5*B*d^2*e*x^2 -
 120*A*b^4*c*d^2*e*x^2 + 600*a*b^3*B*c*d^2*e*x^2 - 1440*a*A*b^2*c^2*d^2*e*x^2 + 720*a^2*b*B*c^2*d^2*e*x^2 + 45
*A*b^5*d*e^2*x^2 - 270*a*b^4*B*d*e^2*x^2 + 600*a*A*b^3*c*d*e^2*x^2 - 720*a^2*b^2*B*c*d*e^2*x^2 + 720*a^2*A*b*c
^2*d*e^2*x^2 - 480*a^3*B*c^2*d*e^2*x^2 - 90*a*A*b^4*e^3*x^2 + 240*a^2*b^3*B*e^3*x^2 - 240*a^2*A*b^2*c*e^3*x^2
+ 320*a^3*b*B*c*e^3*x^2 - 160*a^3*A*c^2*e^3*x^2 - 240*b^3*B*c^2*d^3*x^3 + 480*A*b^2*c^3*d^3*x^3 - 320*a*b*B*c^
3*d^3*x^3 + 640*a*A*c^4*d^3*x^3 + 270*b^4*B*c*d^2*e*x^3 - 720*A*b^3*c^2*d^2*e*x^3 + 720*a*b^2*B*c^2*d^2*e*x^3
- 960*a*A*b*c^3*d^2*e*x^3 + 480*a^2*B*c^3*d^2*e*x^3 - 45*b^5*B*d*e^2*x^3 + 270*A*b^4*c*d*e^2*x^3 - 600*a*b^3*B
*c*d*e^2*x^3 + 720*a*A*b^2*c^2*d*e^2*x^3 - 720*a^2*b*B*c^2*d*e^2*x^3 + 480*a^2*A*c^3*d*e^2*x^3 - 15*A*b^5*e^3*
x^3 + 40*a*b^4*B*e^3*x^3 - 200*a*A*b^3*c*e^3*x^3 + 480*a^2*b^2*B*c*e^3*x^3 - 240*a^2*A*b*c^2*e^3*x^3 - 320*b^2
*B*c^3*d^3*x^4 + 640*A*b*c^4*d^3*x^4 + 360*b^3*B*c^2*d^2*e*x^4 - 960*A*b^2*c^3*d^2*e*x^4 + 480*a*b*B*c^3*d^2*e
*x^4 - 60*b^4*B*c*d*e^2*x^4 + 360*A*b^3*c^2*d*e^2*x^4 - 720*a*b^2*B*c^2*d*e^2*x^4 + 480*a*A*b*c^3*d*e^2*x^4 -
5*b^5*B*e^3*x^4 - 20*A*b^4*c*e^3*x^4 + 120*a*b^3*B*c*e^3*x^4 - 240*a*A*b^2*c^2*e^3*x^4 + 240*a^2*b*B*c^2*e^3*x
^4 - 128*b*B*c^4*d^3*x^5 + 256*A*c^5*d^3*x^5 + 144*b^2*B*c^3*d^2*e*x^5 - 384*A*b*c^4*d^2*e*x^5 + 192*a*B*c^4*d
^2*e*x^5 - 24*b^3*B*c^2*d*e^2*x^5 + 144*A*b^2*c^3*d*e^2*x^5 - 288*a*b*B*c^3*d*e^2*x^5 + 192*a*A*c^4*d*e^2*x^5
- 2*b^4*B*c*e^3*x^5 - 8*A*b^3*c^2*e^3*x^5 + 48*a*b^2*B*c^2*e^3*x^5 - 96*a*A*b*c^3*e^3*x^5 + 96*a^2*B*c^3*e^3*x
^5))/(15*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)^(5/2))

________________________________________________________________________________________

fricas [B]  time = 128.73, size = 1370, normalized size = 5.19

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(7/2),x, algorithm="fricas")

[Out]

2/15*(2*(64*(B*b*c^4 - 2*A*c^5)*d^3 - 24*(3*B*b^2*c^3 + 4*(B*a - 2*A*b)*c^4)*d^2*e + 12*(B*b^3*c^2 - 8*A*a*c^4
 + 6*(2*B*a*b - A*b^2)*c^3)*d*e^2 + (B*b^4*c - 48*(B*a^2 - A*a*b)*c^3 - 4*(6*B*a*b^2 - A*b^3)*c^2)*e^3)*x^5 +
5*(64*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - 24*(3*B*b^3*c^2 + 4*(B*a*b - 2*A*b^2)*c^3)*d^2*e + 12*(B*b^4*c - 8*A*a*b*c
^3 + 6*(2*B*a*b^2 - A*b^3)*c^2)*d*e^2 + (B*b^5 - 48*(B*a^2*b - A*a*b^2)*c^2 - 4*(6*B*a*b^3 - A*b^4)*c)*e^3)*x^
4 - (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^3 - 6*(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^2*e + 24*(6*B*a^3*b^2 - A*a^2*b^3 + 4*(2*B*a^4 - 3*
A*a^3*b)*c)*d*e^2 - 16*(8*B*a^4*b - 3*A*a^3*b^2 - 4*A*a^4*c)*e^3 + 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^3 - 6*(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^2*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*e^2 - (8*B*a*b^4 - 3*A*b^5 - 48*A*
a^2*b*c^2 + 8*(12*B*a^2*b^2 - 5*A*a*b^3)*c)*e^3)*x^3 + 5*(8*(B*b^4*c - 24*A*a*b*c^3 + 2*(6*B*a*b^2 - A*b^3)*c^
2)*d^3 - 3*(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^2*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*e^2 - 2*(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)*e^3)*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^3 + 3*(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^2*e - 12*(6
*B*a^2*b^3 - A*a*b^4 + 4*(2*B*a^3*b - 3*A*a^2*b^2)*c)*d*e^2 + 8*(8*B*a^3*b^2 - 3*A*a^2*b^3 - 4*A*a^3*b*c)*e^3)
*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 - 1
1*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 = 0.36, size = 1437, normalized size = 5.44

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(7/2),x, algorithm="giac")

[Out]

2/15*(((((2*(64*B*b*c^4*d^3 - 128*A*c^5*d^3 - 72*B*b^2*c^3*d^2*e - 96*B*a*c^4*d^2*e + 192*A*b*c^4*d^2*e + 12*B
*b^3*c^2*d*e^2 + 144*B*a*b*c^3*d*e^2 - 72*A*b^2*c^3*d*e^2 - 96*A*a*c^4*d*e^2 + B*b^4*c*e^3 - 24*B*a*b^2*c^2*e^
3 + 4*A*b^3*c^2*e^3 - 48*B*a^2*c^3*e^3 + 48*A*a*b*c^3*e^3)*x/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)
+ 5*(64*B*b^2*c^3*d^3 - 128*A*b*c^4*d^3 - 72*B*b^3*c^2*d^2*e - 96*B*a*b*c^3*d^2*e + 192*A*b^2*c^3*d^2*e + 12*B
*b^4*c*d*e^2 + 144*B*a*b^2*c^2*d*e^2 - 72*A*b^3*c^2*d*e^2 - 96*A*a*b*c^3*d*e^2 + B*b^5*e^3 - 24*B*a*b^3*c*e^3
+ 4*A*b^4*c*e^3 - 48*B*a^2*b*c^2*e^3 + 48*A*a*b^2*c^2*e^3)/(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^3 + 64*B*a*b*c^3*d^3 - 96*A*b^2*c^3*d^3 - 128*A*a*c^4*d^3 - 54*B*b^4*c*d^2*e - 144*B*a*b^
2*c^2*d^2*e + 144*A*b^3*c^2*d^2*e - 96*B*a^2*c^3*d^2*e + 192*A*a*b*c^3*d^2*e + 9*B*b^5*d*e^2 + 120*B*a*b^3*c*d
*e^2 - 54*A*b^4*c*d*e^2 + 144*B*a^2*b*c^2*d*e^2 - 144*A*a*b^2*c^2*d*e^2 - 96*A*a^2*c^3*d*e^2 - 8*B*a*b^4*e^3 +
 3*A*b^5*e^3 - 96*B*a^2*b^2*c*e^3 + 40*A*a*b^3*c*e^3 + 48*A*a^2*b*c^2*e^3)/(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^3 + 96*B*a*b^2*c^2*d^3 - 16*A*b^3*c^2*d^3 - 192*A*a*b*c^3*d^3 - 9*B*b^5*d^2*
e - 120*B*a*b^3*c*d^2*e + 24*A*b^4*c*d^2*e - 144*B*a^2*b*c^2*d^2*e + 288*A*a*b^2*c^2*d^2*e + 54*B*a*b^4*d*e^2
- 9*A*b^5*d*e^2 + 144*B*a^2*b^2*c*d*e^2 - 120*A*a*b^3*c*d*e^2 + 96*B*a^3*c^2*d*e^2 - 144*A*a^2*b*c^2*d*e^2 - 4
8*B*a^2*b^3*e^3 + 18*A*a*b^4*e^3 - 64*B*a^3*b*c*e^3 + 48*A*a^2*b^2*c*e^3 + 32*A*a^3*c^2*e^3)/(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^3 - 24*B*a*b^3*c*d^3 - 2*A*b^4*c*d^3 - 48*B*a^2*b*c^2*d^3 + 48
*A*a*b^2*c^2*d^3 + 96*A*a^2*c^3*d^3 + 12*B*a*b^4*d^2*e + 3*A*b^5*d^2*e + 144*B*a^2*b^2*c*d^2*e - 72*A*a*b^3*c*
d^2*e - 144*A*a^2*b*c^2*d^2*e - 72*B*a^2*b^3*d*e^2 + 12*A*a*b^4*d*e^2 - 96*B*a^3*b*c*d*e^2 + 144*A*a^2*b^2*c*d
*e^2 + 64*B*a^3*b^2*e^3 - 24*A*a^2*b^3*e^3 - 32*A*a^3*b*c*e^3)/(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^3 + 3*A*b^5*d^3 - 48*B*a^2*b^2*c*d^3 - 40*A*a*b^3*c*d^3 - 96*B*a^3*c^2*d^3 + 240*A*a^2*b*c
^2*d^3 + 24*B*a^2*b^3*d^2*e + 6*A*a*b^4*d^2*e + 288*B*a^3*b*c*d^2*e - 144*A*a^2*b^2*c*d^2*e - 288*A*a^3*c^2*d^
2*e - 144*B*a^3*b^2*d*e^2 + 24*A*a^2*b^3*d*e^2 - 192*B*a^4*c*d*e^2 + 288*A*a^3*b*c*d*e^2 + 128*B*a^4*b*e^3 - 4
8*A*a^3*b^2*e^3 - 64*A*a^4*c*e^3)/(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)

________________________________________________________________________________________

maple [B]  time = 0.02, size = 1502, normalized size = 5.69

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(7/2),x)

[Out]

-2/15/(c*x^2+b*x+a)^(5/2)*(96*A*a*b*c^3*e^3*x^5-192*A*a*c^4*d*e^2*x^5+8*A*b^3*c^2*e^3*x^5-144*A*b^2*c^3*d*e^2*
x^5+384*A*b*c^4*d^2*e*x^5-256*A*c^5*d^3*x^5-96*B*a^2*c^3*e^3*x^5-48*B*a*b^2*c^2*e^3*x^5+288*B*a*b*c^3*d*e^2*x^
5-192*B*a*c^4*d^2*e*x^5+2*B*b^4*c*e^3*x^5+24*B*b^3*c^2*d*e^2*x^5-144*B*b^2*c^3*d^2*e*x^5+128*B*b*c^4*d^3*x^5+2
40*A*a*b^2*c^2*e^3*x^4-480*A*a*b*c^3*d*e^2*x^4+20*A*b^4*c*e^3*x^4-360*A*b^3*c^2*d*e^2*x^4+960*A*b^2*c^3*d^2*e*
x^4-640*A*b*c^4*d^3*x^4-240*B*a^2*b*c^2*e^3*x^4-120*B*a*b^3*c*e^3*x^4+720*B*a*b^2*c^2*d*e^2*x^4-480*B*a*b*c^3*
d^2*e*x^4+5*B*b^5*e^3*x^4+60*B*b^4*c*d*e^2*x^4-360*B*b^3*c^2*d^2*e*x^4+320*B*b^2*c^3*d^3*x^4+240*A*a^2*b*c^2*e
^3*x^3-480*A*a^2*c^3*d*e^2*x^3+200*A*a*b^3*c*e^3*x^3-720*A*a*b^2*c^2*d*e^2*x^3+960*A*a*b*c^3*d^2*e*x^3-640*A*a
*c^4*d^3*x^3+15*A*b^5*e^3*x^3-270*A*b^4*c*d*e^2*x^3+720*A*b^3*c^2*d^2*e*x^3-480*A*b^2*c^3*d^3*x^3-480*B*a^2*b^
2*c*e^3*x^3+720*B*a^2*b*c^2*d*e^2*x^3-480*B*a^2*c^3*d^2*e*x^3-40*B*a*b^4*e^3*x^3+600*B*a*b^3*c*d*e^2*x^3-720*B
*a*b^2*c^2*d^2*e*x^3+320*B*a*b*c^3*d^3*x^3+45*B*b^5*d*e^2*x^3-270*B*b^4*c*d^2*e*x^3+240*B*b^3*c^2*d^3*x^3+160*
A*a^3*c^2*e^3*x^2+240*A*a^2*b^2*c*e^3*x^2-720*A*a^2*b*c^2*d*e^2*x^2+90*A*a*b^4*e^3*x^2-600*A*a*b^3*c*d*e^2*x^2
+1440*A*a*b^2*c^2*d^2*e*x^2-960*A*a*b*c^3*d^3*x^2-45*A*b^5*d*e^2*x^2+120*A*b^4*c*d^2*e*x^2-80*A*b^3*c^2*d^3*x^
2-320*B*a^3*b*c*e^3*x^2+480*B*a^3*c^2*d*e^2*x^2-240*B*a^2*b^3*e^3*x^2+720*B*a^2*b^2*c*d*e^2*x^2-720*B*a^2*b*c^
2*d^2*e*x^2+270*B*a*b^4*d*e^2*x^2-600*B*a*b^3*c*d^2*e*x^2+480*B*a*b^2*c^2*d^3*x^2-45*B*b^5*d^2*e*x^2+40*B*b^4*
c*d^3*x^2+160*A*a^3*b*c*e^3*x+120*A*a^2*b^3*e^3*x-720*A*a^2*b^2*c*d*e^2*x+720*A*a^2*b*c^2*d^2*e*x-480*A*a^2*c^
3*d^3*x-60*A*a*b^4*d*e^2*x+360*A*a*b^3*c*d^2*e*x-240*A*a*b^2*c^2*d^3*x-15*A*b^5*d^2*e*x+10*A*b^4*c*d^3*x-320*B
*a^3*b^2*e^3*x+480*B*a^3*b*c*d*e^2*x+360*B*a^2*b^3*d*e^2*x-720*B*a^2*b^2*c*d^2*e*x+240*B*a^2*b*c^2*d^3*x-60*B*
a*b^4*d^2*e*x+120*B*a*b^3*c*d^3*x-5*B*b^5*d^3*x+64*A*a^4*c*e^3+48*A*a^3*b^2*e^3-288*A*a^3*b*c*d*e^2+288*A*a^3*
c^2*d^2*e-24*A*a^2*b^3*d*e^2+144*A*a^2*b^2*c*d^2*e-240*A*a^2*b*c^2*d^3-6*A*a*b^4*d^2*e+40*A*a*b^3*c*d^3-3*A*b^
5*d^3-128*B*a^4*b*e^3+192*B*a^4*c*d*e^2+144*B*a^3*b^2*d*e^2-288*B*a^3*b*c*d^2*e+96*B*a^3*c^2*d^3-24*B*a^2*b^3*
d^2*e+48*B*a^2*b^2*c*d^3-2*B*a*b^4*d^3)/(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)^3/(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 = 4.43, size = 4090, normalized size = 15.49

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^3)/(a + b*x + c*x^2)^(7/2),x)

[Out]

(x*((2*A*b*c^2*e^3 + 4*B*a*c^2*e^3 - 2*B*b^2*c*e^3 - 12*A*c^3*d*e^2 - 12*B*c^3*d^2*e + 6*B*b*c^2*d*e^2)/(15*c^
3*(4*a*c - b^2)) + (b*((2*e^2*(2*A*c*e - B*b*e + 6*B*c*d))/(15*c*(4*a*c - b^2)) - (4*B*b*e^3)/(15*c*(4*a*c - b
^2))))/c + (4*B*a*e^3)/(15*c*(4*a*c - b^2))) + (2*B*b^3*e^3 - 4*B*c^3*d^3 + 4*A*a*c^2*e^3 - 2*A*b^2*c*e^3 - 12
*A*c^3*d^2*e - 6*B*a*b*c*e^3 + 6*A*b*c^2*d*e^2 + 12*B*a*c^2*d*e^2 + 6*B*b*c^2*d^2*e - 6*B*b^2*c*d*e^2)/(15*c^3
*(4*a*c - b^2)) + (a*((2*e^2*(2*A*c*e - B*b*e + 6*B*c*d))/(15*c*(4*a*c - b^2)) - (4*B*b*e^3)/(15*c*(4*a*c - b^
2))))/c)/(a + b*x + c*x^2)^(3/2) - (x*((b*((16*c*e^2*(A*c*e - B*b*e + 3*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c -
b^2)) - (8*B*b*c*e^3)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (2*(2*B*b^2*e^3 + 16*B*a*c*e^3 - 24*A*c^2*d*e^
2 - 24*B*c^2*d^2*e))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*b*e^2*(A*c*e - B*b*e + 3*B*c*d))/(5*(4*a*c^2 -
b^2*c)*(4*a*c - b^2)) + (16*B*a*c*e^3)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))) + (a*((16*c*e^2*(A*c*e - B*b*e + 3
*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c*e^3)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (b*(2*B
*b^2*e^3 + 16*B*a*c*e^3 - 24*A*c^2*d*e^2 - 24*B*c^2*d^2*e))/(15*c*(4*a*c^2 - b^2*c)*(4*a*c - b^2)))/(a + b*x +
 c*x^2)^(1/2) + (x*((b*((b*((b*((16*c^3*e^2*(A*e + 3*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e
^3)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (2*(24*A*c^4*d*e^2 - 48*B*a*c^3*e^3 + 24*B*c^4*d^2*e + 12*B*b^2
*c^2*e^3))/(15*c*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (16*B*a*c^2*e^3)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8
*b*c^2*e^2*(A*e + 3*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (a*((16*c^3*e^2*(A*e + 3*B*d))/(15*(4*a*c
^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e^3)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (2*(8*B*c^4*d^3 - 48*A
*a*c^3*e^3 - 12*B*b^3*c*e^3 + 24*A*c^4*d^2*e + 12*A*b^2*c^2*e^3 + 36*B*b^2*c^2*d*e^2 + 48*B*a*b*c^2*e^3 - 144*
B*a*c^3*d*e^2))/(15*c*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (b*(24*A*c^4*d*e^2 - 48*B*a*c^3*e^3 + 24*B*c^4*d^2*e
+ 12*B*b^2*c^2*e^3))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (a*((b*((16*c^3*e^2*(A*e + 3*B*d))/(15*(4*
a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e^3)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (2*(24*A*c^4*d*e^2
- 48*B*a*c^3*e^3 + 24*B*c^4*d^2*e + 12*B*b^2*c^2*e^3))/(15*c*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (16*B*a*c^2*e^
3)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*b*c^2*e^2*(A*e + 3*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c
 + (2*(32*A*c^4*d^3 + 8*B*b^4*e^3 - 8*A*b^3*c*e^3 - 20*B*b*c^3*d^3 - 8*B*a^2*c^2*e^3 + 24*A*b^2*c^2*d*e^2 + 24
*B*b^2*c^2*d^2*e + 12*A*a*b*c^2*e^3 - 20*B*a*b^2*c*e^3 + 24*A*a*c^3*d*e^2 - 60*A*b*c^3*d^2*e + 24*B*a*c^3*d^2*
e - 24*B*b^3*c*d*e^2 + 36*B*a*b*c^2*d*e^2))/(15*c*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (b*(8*B*c^4*d^3 - 48*A*a*
c^3*e^3 - 12*B*b^3*c*e^3 + 24*A*c^4*d^2*e + 12*A*b^2*c^2*e^3 + 36*B*b^2*c^2*d*e^2 + 48*B*a*b*c^2*e^3 - 144*B*a
*c^3*d*e^2))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a*c - b^2))) + (a*((b*((b*((16*c^3*e^2*(A*e + 3*B*d))/(15*(4*a*c^2 -
 b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e^3)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (2*(24*A*c^4*d*e^2 - 48*B*
a*c^3*e^3 + 24*B*c^4*d^2*e + 12*B*b^2*c^2*e^3))/(15*c*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (16*B*a*c^2*e^3)/(15*
(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*b*c^2*e^2*(A*e + 3*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (a*(
(16*c^3*e^2*(A*e + 3*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e^3)/(15*(4*a*c^2 - b^2*c)*(4*a*c
 - b^2))))/c + (2*(8*B*c^4*d^3 - 48*A*a*c^3*e^3 - 12*B*b^3*c*e^3 + 24*A*c^4*d^2*e + 12*A*b^2*c^2*e^3 + 36*B*b^
2*c^2*d*e^2 + 48*B*a*b*c^2*e^3 - 144*B*a*c^3*d*e^2))/(15*c*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (b*(24*A*c^4*d*e
^2 - 48*B*a*c^3*e^3 + 24*B*c^4*d^2*e + 12*B*b^2*c^2*e^3))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (b*(3
2*A*c^4*d^3 + 8*B*b^4*e^3 - 8*A*b^3*c*e^3 - 20*B*b*c^3*d^3 - 8*B*a^2*c^2*e^3 + 24*A*b^2*c^2*d*e^2 + 24*B*b^2*c
^2*d^2*e + 12*A*a*b*c^2*e^3 - 20*B*a*b^2*c*e^3 + 24*A*a*c^3*d*e^2 - 60*A*b*c^3*d^2*e + 24*B*a*c^3*d^2*e - 24*B
*b^3*c*d*e^2 + 36*B*a*b*c^2*d*e^2))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a*c - b^2)))/(a + b*x + c*x^2)^(3/2) + (x*((a
*((b*((2*c^2*((2*A*e^3)/5 + (6*B*d*e^2)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e^3)/(5*(4*a*c^2 - b^2*c))))/c - (b*c
*((2*A*e^3)/5 + (6*B*d*e^2)/5))/(4*a*c^2 - b^2*c) - (12*c^2*d*e*(A*e + B*d))/(5*(4*a*c^2 - b^2*c)) + (4*B*a*c*
e^3)/(5*(4*a*c^2 - b^2*c))))/c - (b*((2*c^2*((2*B*d^3)/5 + (6*A*d^2*e)/5))/(4*a*c^2 - b^2*c) - (a*((2*c^2*((2*
A*e^3)/5 + (6*B*d*e^2)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e^3)/(5*(4*a*c^2 - b^2*c))))/c + (b*((b*((2*c^2*((2*A*
e^3)/5 + (6*B*d*e^2)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e^3)/(5*(4*a*c^2 - b^2*c))))/c - (b*c*((2*A*e^3)/5 + (6*
B*d*e^2)/5))/(4*a*c^2 - b^2*c) - (12*c^2*d*e*(A*e + B*d))/(5*(4*a*c^2 - b^2*c)) + (4*B*a*c*e^3)/(5*(4*a*c^2 -
b^2*c))))/c + (6*b*c*d*e*(A*e + B*d))/(5*(4*a*c^2 - b^2*c))))/c + (b*c*((2*B*d^3)/5 + (6*A*d^2*e)/5))/(4*a*c^2
 - b^2*c) + (4*A*c^2*d^3)/(5*(4*a*c^2 - b^2*c))) - (a*((2*c^2*((2*B*d^3)/5 + (6*A*d^2*e)/5))/(4*a*c^2 - b^2*c)
 - (a*((2*c^2*((2*A*e^3)/5 + (6*B*d*e^2)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e^3)/(5*(4*a*c^2 - b^2*c))))/c + (b*
((b*((2*c^2*((2*A*e^3)/5 + (6*B*d*e^2)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e^3)/(5*(4*a*c^2 - b^2*c))))/c - (b*c*
((2*A*e^3)/5 + (6*B*d*e^2)/5))/(4*a*c^2 - b^2*c) - (12*c^2*d*e*(A*e + B*d))/(5*(4*a*c^2 - b^2*c)) + (4*B*a*c*e
^3)/(5*(4*a*c^2 - b^2*c))))/c + (6*b*c*d*e*(A*e + B*d))/(5*(4*a*c^2 - b^2*c))))/c + (2*A*b*c*d^3)/(5*(4*a*c^2
- b^2*c)))/(a + b*x + c*x^2)^(5/2) + (x*((2*(256*A*c^4*d^3 - 4*B*b^4*e^3 - 8*A*b^3*c*e^3 - 128*B*b*c^3*d^3 + 1
60*B*a^2*c^2*e^3 + 168*A*b^2*c^2*d*e^2 + 168*B*b^2*c^2*d^2*e - 96*A*a*b*c^2*e^3 + 40*B*a*b^2*c*e^3 + 96*A*a*c^
3*d*e^2 - 384*A*b*c^3*d^2*e + 96*B*a*c^3*d^2*e - 24*B*b^3*c*d*e^2 - 288*B*a*b*c^2*d*e^2))/(15*(4*a*c^2 - b^2*c
)*(4*a*c - b^2)^2) + (b*((16*c*e^2*(A*c*e - B*b*e + 3*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c*e
^3)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (8*b*e^2*(A*c*e - B*b*e + 3*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c
- b^2)) + (16*B*a*c*e^3)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))) + (a*((16*c*e^2*(A*c*e - B*b*e + 3*B*c*d))/(5*(4
*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c*e^3)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (b*(256*A*c^4*d^3 - 4
*B*b^4*e^3 - 8*A*b^3*c*e^3 - 128*B*b*c^3*d^3 + 160*B*a^2*c^2*e^3 + 168*A*b^2*c^2*d*e^2 + 168*B*b^2*c^2*d^2*e -
 96*A*a*b*c^2*e^3 + 40*B*a*b^2*c*e^3 + 96*A*a*c^3*d*e^2 - 384*A*b*c^3*d^2*e + 96*B*a*c^3*d^2*e - 24*B*b^3*c*d*
e^2 - 288*B*a*b*c^2*d*e^2))/(15*c*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^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)**3/(c*x**2+b*x+a)**(7/2),x)

[Out]

Timed out

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