3.2223 \(\int \frac{(d+e x)^3}{(a+b x+c x^2)^5} \, dx\)

Optimal. Leaf size=378 \[ \frac{5 (b+2 c x) (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{2 x (2 c d-b e) \left (-c e (13 a e+35 b d)+12 b^2 e^2+35 c^2 d^2\right )-b^2 \left (27 a e^3+49 c d^2 e\right )+2 b c d \left (99 a e^2+35 c d^2\right )-32 a c e \left (a e^2+7 c d^2\right )+3 b^3 d e^2}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac{10 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}-\frac{(b+2 c x) (d+e x)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^2 \left (-16 a c e-3 b^2 e+14 c x (2 c d-b e)+14 b c d\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3} \]

[Out]

-((b + 2*c*x)*(d + e*x)^3)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + ((d + e*x)^2*(14*b*c*d - 3*b^2*e - 16*a*c*e
+ 14*c*(2*c*d - b*e)*x))/(12*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^3) - (3*b^3*d*e^2 - 32*a*c*e*(7*c*d^2 + a*e^2)
+ 2*b*c*d*(35*c*d^2 + 99*a*e^2) - b^2*(49*c*d^2*e + 27*a*e^3) + 2*(2*c*d - b*e)*(35*c^2*d^2 + 12*b^2*e^2 - c*
e*(35*b*d + 13*a*e))*x)/(12*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)^2) + (5*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e
*(7*b*d - 3*a*e))*(b + 2*c*x))/(2*(b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (10*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^
2 - c*e*(7*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(9/2)

________________________________________________________________________________________

Rubi [A]  time = 0.501609, antiderivative size = 378, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {736, 820, 777, 614, 618, 206} \[ \frac{5 (b+2 c x) (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{2 x (2 c d-b e) \left (-c e (13 a e+35 b d)+12 b^2 e^2+35 c^2 d^2\right )-b^2 \left (27 a e^3+49 c d^2 e\right )+2 b c d \left (99 a e^2+35 c d^2\right )-32 a c e \left (a e^2+7 c d^2\right )+3 b^3 d e^2}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac{10 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}-\frac{(b+2 c x) (d+e x)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^2 \left (-16 a c e-3 b^2 e+14 c x (2 c d-b e)+14 b c d\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3/(a + b*x + c*x^2)^5,x]

[Out]

-((b + 2*c*x)*(d + e*x)^3)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + ((d + e*x)^2*(14*b*c*d - 3*b^2*e - 16*a*c*e
+ 14*c*(2*c*d - b*e)*x))/(12*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^3) - (3*b^3*d*e^2 - 32*a*c*e*(7*c*d^2 + a*e^2)
+ 2*b*c*d*(35*c*d^2 + 99*a*e^2) - b^2*(49*c*d^2*e + 27*a*e^3) + 2*(2*c*d - b*e)*(35*c^2*d^2 + 12*b^2*e^2 - c*
e*(35*b*d + 13*a*e))*x)/(12*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)^2) + (5*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e
*(7*b*d - 3*a*e))*(b + 2*c*x))/(2*(b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (10*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^
2 - c*e*(7*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(9/2)

Rule 736

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*(b + 2*
c*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m
- 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(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] && LtQ[p, -1] && GtQ[m
, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, e, m, p, x]

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])

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 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(d+e x)^3}{\left (a+b x+c x^2\right )^5} \, dx &=-\frac{(b+2 c x) (d+e x)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{\int \frac{(d+e x)^2 (-14 c d+3 b e-8 c e x)}{\left (a+b x+c x^2\right )^4} \, dx}{4 \left (b^2-4 a c\right )}\\ &=-\frac{(b+2 c x) (d+e x)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^2 \left (14 b c d-3 b^2 e-16 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{\int \frac{(d+e x) \left (-2 \left (70 c^2 d^2+3 b^2 e^2-c e (49 b d-16 a e)\right )-42 c e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^3} \, dx}{12 \left (b^2-4 a c\right )^2}\\ &=-\frac{(b+2 c x) (d+e x)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^2 \left (14 b c d-3 b^2 e-16 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{3 b^3 d e^2-32 a c e \left (7 c d^2+a e^2\right )+2 b c d \left (35 c d^2+99 a e^2\right )-b^2 \left (49 c d^2 e+27 a e^3\right )+2 (2 c d-b e) \left (35 c^2 d^2+12 b^2 e^2-c e (35 b d+13 a e)\right ) x}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac{\left (5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )\right ) \int \frac{1}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )^3}\\ &=-\frac{(b+2 c x) (d+e x)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^2 \left (14 b c d-3 b^2 e-16 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{3 b^3 d e^2-32 a c e \left (7 c d^2+a e^2\right )+2 b c d \left (35 c d^2+99 a e^2\right )-b^2 \left (49 c d^2 e+27 a e^3\right )+2 (2 c d-b e) \left (35 c^2 d^2+12 b^2 e^2-c e (35 b d+13 a e)\right ) x}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac{5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (b+2 c x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}+\frac{\left (5 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^4}\\ &=-\frac{(b+2 c x) (d+e x)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^2 \left (14 b c d-3 b^2 e-16 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{3 b^3 d e^2-32 a c e \left (7 c d^2+a e^2\right )+2 b c d \left (35 c d^2+99 a e^2\right )-b^2 \left (49 c d^2 e+27 a e^3\right )+2 (2 c d-b e) \left (35 c^2 d^2+12 b^2 e^2-c e (35 b d+13 a e)\right ) x}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac{5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (b+2 c x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{\left (10 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^4}\\ &=-\frac{(b+2 c x) (d+e x)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^2 \left (14 b c d-3 b^2 e-16 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{3 b^3 d e^2-32 a c e \left (7 c d^2+a e^2\right )+2 b c d \left (35 c d^2+99 a e^2\right )-b^2 \left (49 c d^2 e+27 a e^3\right )+2 (2 c d-b e) \left (35 c^2 d^2+12 b^2 e^2-c e (35 b d+13 a e)\right ) x}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac{5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (b+2 c x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{10 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}\\ \end{align*}

Mathematica [A]  time = 1.30144, size = 467, normalized size = 1.24 \[ \frac{1}{12} \left (\frac{4 c^2 \left (-8 a^2 e^3+3 a c d e^2 x+7 c^2 d^3 x\right )+b^2 c e \left (13 a e^2-3 c d (7 d-6 e x)\right )+2 b c^2 \left (3 a e^2 (d-e x)+7 c d^2 (d-3 e x)\right )+b^3 c e^2 (9 d-2 e x)-3 b^4 e^3}{c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^3}+\frac{3 \left (2 c \left (a^2 e^3-3 a c d e (d+e x)+c^2 d^3 x\right )+b^2 e^2 (3 c d x-a e)+b c \left (3 a e^2 (d+e x)+c d^2 (d-3 e x)\right )-b^3 e^3 x\right )}{c^2 \left (4 a c-b^2\right ) (a+x (b+c x))^4}+\frac{30 (b+2 c x) (2 c d-b e) \left (c e (3 a e-7 b d)+b^2 e^2+7 c^2 d^2\right )}{\left (b^2-4 a c\right )^4 (a+x (b+c x))}+\frac{5 (b+2 c x) (2 c d-b e) \left (c e (3 a e-7 b d)+b^2 e^2+7 c^2 d^2\right )}{c \left (4 a c-b^2\right )^3 (a+x (b+c x))^2}+\frac{120 c (2 c d-b e) \left (c e (3 a e-7 b d)+b^2 e^2+7 c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{9/2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3/(a + b*x + c*x^2)^5,x]

[Out]

((5*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 + c*e*(-7*b*d + 3*a*e))*(b + 2*c*x))/(c*(-b^2 + 4*a*c)^3*(a + x*(b + c*
x))^2) + (30*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 + c*e*(-7*b*d + 3*a*e))*(b + 2*c*x))/((b^2 - 4*a*c)^4*(a + x*(
b + c*x))) + (-3*b^4*e^3 + b^3*c*e^2*(9*d - 2*e*x) + 4*c^2*(-8*a^2*e^3 + 7*c^2*d^3*x + 3*a*c*d*e^2*x) + b^2*c*
e*(13*a*e^2 - 3*c*d*(7*d - 6*e*x)) + 2*b*c^2*(7*c*d^2*(d - 3*e*x) + 3*a*e^2*(d - e*x)))/(c^2*(b^2 - 4*a*c)^2*(
a + x*(b + c*x))^3) + (3*(-(b^3*e^3*x) + b^2*e^2*(-(a*e) + 3*c*d*x) + 2*c*(a^2*e^3 + c^2*d^3*x - 3*a*c*d*e*(d
+ e*x)) + b*c*(c*d^2*(d - 3*e*x) + 3*a*e^2*(d + e*x))))/(c^2*(-b^2 + 4*a*c)*(a + x*(b + c*x))^4) + (120*c*(2*c
*d - b*e)*(7*c^2*d^2 + b^2*e^2 + c*e*(-7*b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^
(9/2))/12

________________________________________________________________________________________

Maple [B]  time = 0.173, size = 1742, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3/(c*x^2+b*x+a)^5,x)

[Out]

(-5*c^4*(3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e-14*c^3*d^3)/(256*a^4*c^4-256*a^3*b^2*c
^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^7-35/2*c^3*(3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e
-14*c^3*d^3)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*b*x^6-5/3*c^2*(11*a*c+13*b^2)*(3*a*b*
c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e-14*c^3*d^3)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c
^2-16*a*b^6*c+b^8)*x^5-25/12*b*(22*a*c+5*b^2)*c*(3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*
e-14*c^3*d^3)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^4-1/3*(73*a^2*c^2+101*a*b^2*c+3*b^
4)*(3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e-14*c^3*d^3)/(256*a^4*c^4-256*a^3*b^2*c^3+96
*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^3-1/6*(256*a^4*c^3*e^3+401*a^3*b^2*c^2*e^3-1314*a^3*b*c^3*d*e^2+399*a^2*b^4*c*e
^3-2139*a^2*b^3*c^2*d*e^2+4599*a^2*b^2*c^3*d^2*e-3066*a^2*b*c^4*d^3+9*a*b^6*e^3-246*a*b^5*c*d*e^2+588*a*b^4*c^
2*d^2*e-392*a*b^3*c^3*d^3+9*b^7*d*e^2-21*b^6*c*d^2*e+14*b^5*c^2*d^3)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c
^2-16*a*b^6*c+b^8)*x^2-1/3*(83*a^4*b*c^2*e^3+90*a^4*c^3*d*e^2+151*a^3*b^3*c*e^3-837*a^3*b^2*c^2*d*e^2+837*a^3*
b*c^3*d^2*e-558*a^3*c^4*d^3+3*a^2*b^5*e^3-84*a^2*b^4*c*d*e^2+522*a^2*b^3*c^2*d^2*e-348*a^2*b^2*c^3*d^3+3*a*b^6
*d*e^2-57*a*b^5*c*d^2*e+38*a*b^4*c^2*d^3+3*b^7*d^2*e-2*b^6*c*d^3)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-
16*a*b^6*c+b^8)*x-1/12*(128*a^5*c^2*e^3+166*a^4*b^2*c*e^3-972*a^4*b*c^2*d*e^2+1152*a^4*c^3*d^2*e+3*a^3*b^4*e^3
-84*a^3*b^3*c*d*e^2+522*a^3*b^2*c^2*d^2*e-1116*a^3*b*c^3*d^3+3*a^2*b^5*d*e^2-57*a^2*b^4*c*d^2*e+326*a^2*b^3*c^
2*d^3+3*a*b^6*d^2*e-50*a*b^5*c*d^3+3*b^7*d^3)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8))/(c*
x^2+b*x+a)^4-30*c^2/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*
x+b)/(4*a*c-b^2)^(1/2))*a*b*e^3+60*c^3/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)
^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*d*e^2-10*c/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c
+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*e^3+90*c^2/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2
*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*d*e^2-210*c^3/(256*a^4*c^4-
256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*d^2*e+1
40*c^4/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b
^2)^(1/2))*d^3

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*x^2+b*x+a)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.44002, size = 12459, normalized size = 32.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*x^2+b*x+a)^5,x, algorithm="fricas")

[Out]

[1/12*(60*(14*(b^2*c^7 - 4*a*c^8)*d^3 - 21*(b^3*c^6 - 4*a*b*c^7)*d^2*e + 3*(3*b^4*c^5 - 10*a*b^2*c^6 - 8*a^2*c
^7)*d*e^2 - (b^5*c^4 - a*b^3*c^5 - 12*a^2*b*c^6)*e^3)*x^7 + 210*(14*(b^3*c^6 - 4*a*b*c^7)*d^3 - 21*(b^4*c^5 -
4*a*b^2*c^6)*d^2*e + 3*(3*b^5*c^4 - 10*a*b^3*c^5 - 8*a^2*b*c^6)*d*e^2 - (b^6*c^3 - a*b^4*c^4 - 12*a^2*b^2*c^5)
*e^3)*x^6 + 20*(14*(13*b^4*c^5 - 41*a*b^2*c^6 - 44*a^2*c^7)*d^3 - 21*(13*b^5*c^4 - 41*a*b^3*c^5 - 44*a^2*b*c^6
)*d^2*e + 3*(39*b^6*c^3 - 97*a*b^4*c^4 - 214*a^2*b^2*c^5 - 88*a^3*c^6)*d*e^2 - (13*b^7*c^2 - 2*a*b^5*c^3 - 167
*a^2*b^3*c^4 - 132*a^3*b*c^5)*e^3)*x^5 + 25*(14*(5*b^5*c^4 + 2*a*b^3*c^5 - 88*a^2*b*c^6)*d^3 - 21*(5*b^6*c^3 +
2*a*b^4*c^4 - 88*a^2*b^2*c^5)*d^2*e + 3*(15*b^7*c^2 + 16*a*b^5*c^3 - 260*a^2*b^3*c^4 - 176*a^3*b*c^5)*d*e^2 -
(5*b^8*c + 17*a*b^6*c^2 - 82*a^2*b^4*c^3 - 264*a^3*b^2*c^4)*e^3)*x^4 - (3*b^9 - 62*a*b^7*c + 526*a^2*b^5*c^2
- 2420*a^3*b^3*c^3 + 4464*a^4*b*c^4)*d^3 - 3*(a*b^8 - 23*a^2*b^6*c + 250*a^3*b^4*c^2 - 312*a^4*b^2*c^3 - 1536*
a^5*c^4)*d^2*e - 3*(a^2*b^7 - 32*a^3*b^5*c - 212*a^4*b^3*c^2 + 1296*a^5*b*c^3)*d*e^2 - (3*a^3*b^6 + 154*a^4*b^
4*c - 536*a^5*b^2*c^2 - 512*a^6*c^3)*e^3 + 4*(14*(3*b^6*c^3 + 89*a*b^4*c^4 - 331*a^2*b^2*c^5 - 292*a^3*c^6)*d^
3 - 21*(3*b^7*c^2 + 89*a*b^5*c^3 - 331*a^2*b^3*c^4 - 292*a^3*b*c^5)*d^2*e + 3*(9*b^8*c + 273*a*b^6*c^2 - 815*a
^2*b^4*c^3 - 1538*a^3*b^2*c^4 - 584*a^4*c^5)*d*e^2 - (3*b^9 + 98*a*b^7*c - 64*a^2*b^5*c^2 - 1285*a^3*b^3*c^3 -
876*a^4*b*c^4)*e^3)*x^3 - 2*(14*(b^7*c^2 - 32*a*b^5*c^3 - 107*a^2*b^3*c^4 + 876*a^3*b*c^5)*d^3 - 21*(b^8*c -
32*a*b^6*c^2 - 107*a^2*b^4*c^3 + 876*a^3*b^2*c^4)*d^2*e + 3*(3*b^9 - 94*a*b^7*c - 385*a^2*b^5*c^2 + 2414*a^3*b
^3*c^3 + 1752*a^4*b*c^4)*d*e^2 + (9*a*b^8 + 363*a^2*b^6*c - 1195*a^3*b^4*c^2 - 1348*a^4*b^2*c^3 - 1024*a^5*c^4
)*e^3)*x^2 - 60*(14*a^4*c^4*d^3 - 21*a^4*b*c^3*d^2*e + (14*c^8*d^3 - 21*b*c^7*d^2*e + 3*(3*b^2*c^6 + 2*a*c^7)*
d*e^2 - (b^3*c^5 + 3*a*b*c^6)*e^3)*x^8 + 4*(14*b*c^7*d^3 - 21*b^2*c^6*d^2*e + 3*(3*b^3*c^5 + 2*a*b*c^6)*d*e^2
- (b^4*c^4 + 3*a*b^2*c^5)*e^3)*x^7 + 2*(14*(3*b^2*c^6 + 2*a*c^7)*d^3 - 21*(3*b^3*c^5 + 2*a*b*c^6)*d^2*e + 3*(9
*b^4*c^4 + 12*a*b^2*c^5 + 4*a^2*c^6)*d*e^2 - (3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)*e^3)*x^6 + 4*(14*(b^3*c^
5 + 3*a*b*c^6)*d^3 - 21*(b^4*c^4 + 3*a*b^2*c^5)*d^2*e + 3*(3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)*d*e^2 - (b^
6*c^2 + 6*a*b^4*c^3 + 9*a^2*b^2*c^4)*e^3)*x^5 + (14*(b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*d^3 - 21*(b^5*c^3 + 1
2*a*b^3*c^4 + 6*a^2*b*c^5)*d^2*e + 3*(3*b^6*c^2 + 38*a*b^4*c^3 + 42*a^2*b^2*c^4 + 12*a^3*c^5)*d*e^2 - (b^7*c +
15*a*b^5*c^2 + 42*a^2*b^3*c^3 + 18*a^3*b*c^4)*e^3)*x^4 + 3*(3*a^4*b^2*c^2 + 2*a^5*c^3)*d*e^2 - (a^4*b^3*c + 3
*a^5*b*c^2)*e^3 + 4*(14*(a*b^3*c^4 + 3*a^2*b*c^5)*d^3 - 21*(a*b^4*c^3 + 3*a^2*b^2*c^4)*d^2*e + 3*(3*a*b^5*c^2
+ 11*a^2*b^3*c^3 + 6*a^3*b*c^4)*d*e^2 - (a*b^6*c + 6*a^2*b^4*c^2 + 9*a^3*b^2*c^3)*e^3)*x^3 + 2*(14*(3*a^2*b^2*
c^4 + 2*a^3*c^5)*d^3 - 21*(3*a^2*b^3*c^3 + 2*a^3*b*c^4)*d^2*e + 3*(9*a^2*b^4*c^2 + 12*a^3*b^2*c^3 + 4*a^4*c^4)
*d*e^2 - (3*a^2*b^5*c + 11*a^3*b^3*c^2 + 6*a^4*b*c^3)*e^3)*x^2 + 4*(14*a^3*b*c^4*d^3 - 21*a^3*b^2*c^3*d^2*e +
3*(3*a^3*b^3*c^2 + 2*a^4*b*c^3)*d*e^2 - (a^3*b^4*c + 3*a^4*b^2*c^2)*e^3)*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 +
2*b*c*x + b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + 4*(2*(b^8*c - 23*a*b^6*c^2 + 250*
a^2*b^4*c^3 - 417*a^3*b^2*c^4 - 1116*a^4*c^5)*d^3 - 3*(b^9 - 23*a*b^7*c + 250*a^2*b^5*c^2 - 417*a^3*b^3*c^3 -
1116*a^4*b*c^4)*d^2*e - 3*(a*b^8 - 32*a^2*b^6*c - 167*a^3*b^4*c^2 + 1146*a^4*b^2*c^3 - 120*a^5*c^4)*d*e^2 - (3
*a^2*b^7 + 139*a^3*b^5*c - 521*a^4*b^3*c^2 - 332*a^5*b*c^3)*e^3)*x)/(a^4*b^10 - 20*a^5*b^8*c + 160*a^6*b^6*c^2
- 640*a^7*b^4*c^3 + 1280*a^8*b^2*c^4 - 1024*a^9*c^5 + (b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^
4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*x^8 + 4*(b^11*c^3 - 20*a*b^9*c^4 + 160*a^2*b^7*c^5 - 640*a^3*b^5*c^6
+ 1280*a^4*b^3*c^7 - 1024*a^5*b*c^8)*x^7 + 2*(3*b^12*c^2 - 58*a*b^10*c^3 + 440*a^2*b^8*c^4 - 1600*a^3*b^6*c^5
+ 2560*a^4*b^4*c^6 - 512*a^5*b^2*c^7 - 2048*a^6*c^8)*x^6 + 4*(b^13*c - 17*a*b^11*c^2 + 100*a^2*b^9*c^3 - 160*a
^3*b^7*c^4 - 640*a^4*b^5*c^5 + 2816*a^5*b^3*c^6 - 3072*a^6*b*c^7)*x^5 + (b^14 - 8*a*b^12*c - 74*a^2*b^10*c^2 +
1160*a^3*b^8*c^3 - 5440*a^4*b^6*c^4 + 10496*a^5*b^4*c^5 - 4608*a^6*b^2*c^6 - 6144*a^7*c^7)*x^4 + 4*(a*b^13 -
17*a^2*b^11*c + 100*a^3*b^9*c^2 - 160*a^4*b^7*c^3 - 640*a^5*b^5*c^4 + 2816*a^6*b^3*c^5 - 3072*a^7*b*c^6)*x^3 +
2*(3*a^2*b^12 - 58*a^3*b^10*c + 440*a^4*b^8*c^2 - 1600*a^5*b^6*c^3 + 2560*a^6*b^4*c^4 - 512*a^7*b^2*c^5 - 204
8*a^8*c^6)*x^2 + 4*(a^3*b^11 - 20*a^4*b^9*c + 160*a^5*b^7*c^2 - 640*a^6*b^5*c^3 + 1280*a^7*b^3*c^4 - 1024*a^8*
b*c^5)*x), 1/12*(60*(14*(b^2*c^7 - 4*a*c^8)*d^3 - 21*(b^3*c^6 - 4*a*b*c^7)*d^2*e + 3*(3*b^4*c^5 - 10*a*b^2*c^6
- 8*a^2*c^7)*d*e^2 - (b^5*c^4 - a*b^3*c^5 - 12*a^2*b*c^6)*e^3)*x^7 + 210*(14*(b^3*c^6 - 4*a*b*c^7)*d^3 - 21*(
b^4*c^5 - 4*a*b^2*c^6)*d^2*e + 3*(3*b^5*c^4 - 10*a*b^3*c^5 - 8*a^2*b*c^6)*d*e^2 - (b^6*c^3 - a*b^4*c^4 - 12*a^
2*b^2*c^5)*e^3)*x^6 + 20*(14*(13*b^4*c^5 - 41*a*b^2*c^6 - 44*a^2*c^7)*d^3 - 21*(13*b^5*c^4 - 41*a*b^3*c^5 - 44
*a^2*b*c^6)*d^2*e + 3*(39*b^6*c^3 - 97*a*b^4*c^4 - 214*a^2*b^2*c^5 - 88*a^3*c^6)*d*e^2 - (13*b^7*c^2 - 2*a*b^5
*c^3 - 167*a^2*b^3*c^4 - 132*a^3*b*c^5)*e^3)*x^5 + 25*(14*(5*b^5*c^4 + 2*a*b^3*c^5 - 88*a^2*b*c^6)*d^3 - 21*(5
*b^6*c^3 + 2*a*b^4*c^4 - 88*a^2*b^2*c^5)*d^2*e + 3*(15*b^7*c^2 + 16*a*b^5*c^3 - 260*a^2*b^3*c^4 - 176*a^3*b*c^
5)*d*e^2 - (5*b^8*c + 17*a*b^6*c^2 - 82*a^2*b^4*c^3 - 264*a^3*b^2*c^4)*e^3)*x^4 - (3*b^9 - 62*a*b^7*c + 526*a^
2*b^5*c^2 - 2420*a^3*b^3*c^3 + 4464*a^4*b*c^4)*d^3 - 3*(a*b^8 - 23*a^2*b^6*c + 250*a^3*b^4*c^2 - 312*a^4*b^2*c
^3 - 1536*a^5*c^4)*d^2*e - 3*(a^2*b^7 - 32*a^3*b^5*c - 212*a^4*b^3*c^2 + 1296*a^5*b*c^3)*d*e^2 - (3*a^3*b^6 +
154*a^4*b^4*c - 536*a^5*b^2*c^2 - 512*a^6*c^3)*e^3 + 4*(14*(3*b^6*c^3 + 89*a*b^4*c^4 - 331*a^2*b^2*c^5 - 292*a
^3*c^6)*d^3 - 21*(3*b^7*c^2 + 89*a*b^5*c^3 - 331*a^2*b^3*c^4 - 292*a^3*b*c^5)*d^2*e + 3*(9*b^8*c + 273*a*b^6*c
^2 - 815*a^2*b^4*c^3 - 1538*a^3*b^2*c^4 - 584*a^4*c^5)*d*e^2 - (3*b^9 + 98*a*b^7*c - 64*a^2*b^5*c^2 - 1285*a^3
*b^3*c^3 - 876*a^4*b*c^4)*e^3)*x^3 - 2*(14*(b^7*c^2 - 32*a*b^5*c^3 - 107*a^2*b^3*c^4 + 876*a^3*b*c^5)*d^3 - 21
*(b^8*c - 32*a*b^6*c^2 - 107*a^2*b^4*c^3 + 876*a^3*b^2*c^4)*d^2*e + 3*(3*b^9 - 94*a*b^7*c - 385*a^2*b^5*c^2 +
2414*a^3*b^3*c^3 + 1752*a^4*b*c^4)*d*e^2 + (9*a*b^8 + 363*a^2*b^6*c - 1195*a^3*b^4*c^2 - 1348*a^4*b^2*c^3 - 10
24*a^5*c^4)*e^3)*x^2 - 120*(14*a^4*c^4*d^3 - 21*a^4*b*c^3*d^2*e + (14*c^8*d^3 - 21*b*c^7*d^2*e + 3*(3*b^2*c^6
+ 2*a*c^7)*d*e^2 - (b^3*c^5 + 3*a*b*c^6)*e^3)*x^8 + 4*(14*b*c^7*d^3 - 21*b^2*c^6*d^2*e + 3*(3*b^3*c^5 + 2*a*b*
c^6)*d*e^2 - (b^4*c^4 + 3*a*b^2*c^5)*e^3)*x^7 + 2*(14*(3*b^2*c^6 + 2*a*c^7)*d^3 - 21*(3*b^3*c^5 + 2*a*b*c^6)*d
^2*e + 3*(9*b^4*c^4 + 12*a*b^2*c^5 + 4*a^2*c^6)*d*e^2 - (3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)*e^3)*x^6 + 4*
(14*(b^3*c^5 + 3*a*b*c^6)*d^3 - 21*(b^4*c^4 + 3*a*b^2*c^5)*d^2*e + 3*(3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)*
d*e^2 - (b^6*c^2 + 6*a*b^4*c^3 + 9*a^2*b^2*c^4)*e^3)*x^5 + (14*(b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*d^3 - 21*(
b^5*c^3 + 12*a*b^3*c^4 + 6*a^2*b*c^5)*d^2*e + 3*(3*b^6*c^2 + 38*a*b^4*c^3 + 42*a^2*b^2*c^4 + 12*a^3*c^5)*d*e^2
- (b^7*c + 15*a*b^5*c^2 + 42*a^2*b^3*c^3 + 18*a^3*b*c^4)*e^3)*x^4 + 3*(3*a^4*b^2*c^2 + 2*a^5*c^3)*d*e^2 - (a^
4*b^3*c + 3*a^5*b*c^2)*e^3 + 4*(14*(a*b^3*c^4 + 3*a^2*b*c^5)*d^3 - 21*(a*b^4*c^3 + 3*a^2*b^2*c^4)*d^2*e + 3*(3
*a*b^5*c^2 + 11*a^2*b^3*c^3 + 6*a^3*b*c^4)*d*e^2 - (a*b^6*c + 6*a^2*b^4*c^2 + 9*a^3*b^2*c^3)*e^3)*x^3 + 2*(14*
(3*a^2*b^2*c^4 + 2*a^3*c^5)*d^3 - 21*(3*a^2*b^3*c^3 + 2*a^3*b*c^4)*d^2*e + 3*(9*a^2*b^4*c^2 + 12*a^3*b^2*c^3 +
4*a^4*c^4)*d*e^2 - (3*a^2*b^5*c + 11*a^3*b^3*c^2 + 6*a^4*b*c^3)*e^3)*x^2 + 4*(14*a^3*b*c^4*d^3 - 21*a^3*b^2*c
^3*d^2*e + 3*(3*a^3*b^3*c^2 + 2*a^4*b*c^3)*d*e^2 - (a^3*b^4*c + 3*a^4*b^2*c^2)*e^3)*x)*sqrt(-b^2 + 4*a*c)*arct
an(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 4*(2*(b^8*c - 23*a*b^6*c^2 + 250*a^2*b^4*c^3 - 417*a^3*b^2
*c^4 - 1116*a^4*c^5)*d^3 - 3*(b^9 - 23*a*b^7*c + 250*a^2*b^5*c^2 - 417*a^3*b^3*c^3 - 1116*a^4*b*c^4)*d^2*e - 3
*(a*b^8 - 32*a^2*b^6*c - 167*a^3*b^4*c^2 + 1146*a^4*b^2*c^3 - 120*a^5*c^4)*d*e^2 - (3*a^2*b^7 + 139*a^3*b^5*c
- 521*a^4*b^3*c^2 - 332*a^5*b*c^3)*e^3)*x)/(a^4*b^10 - 20*a^5*b^8*c + 160*a^6*b^6*c^2 - 640*a^7*b^4*c^3 + 1280
*a^8*b^2*c^4 - 1024*a^9*c^5 + (b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8
- 1024*a^5*c^9)*x^8 + 4*(b^11*c^3 - 20*a*b^9*c^4 + 160*a^2*b^7*c^5 - 640*a^3*b^5*c^6 + 1280*a^4*b^3*c^7 - 1024
*a^5*b*c^8)*x^7 + 2*(3*b^12*c^2 - 58*a*b^10*c^3 + 440*a^2*b^8*c^4 - 1600*a^3*b^6*c^5 + 2560*a^4*b^4*c^6 - 512*
a^5*b^2*c^7 - 2048*a^6*c^8)*x^6 + 4*(b^13*c - 17*a*b^11*c^2 + 100*a^2*b^9*c^3 - 160*a^3*b^7*c^4 - 640*a^4*b^5*
c^5 + 2816*a^5*b^3*c^6 - 3072*a^6*b*c^7)*x^5 + (b^14 - 8*a*b^12*c - 74*a^2*b^10*c^2 + 1160*a^3*b^8*c^3 - 5440*
a^4*b^6*c^4 + 10496*a^5*b^4*c^5 - 4608*a^6*b^2*c^6 - 6144*a^7*c^7)*x^4 + 4*(a*b^13 - 17*a^2*b^11*c + 100*a^3*b
^9*c^2 - 160*a^4*b^7*c^3 - 640*a^5*b^5*c^4 + 2816*a^6*b^3*c^5 - 3072*a^7*b*c^6)*x^3 + 2*(3*a^2*b^12 - 58*a^3*b
^10*c + 440*a^4*b^8*c^2 - 1600*a^5*b^6*c^3 + 2560*a^6*b^4*c^4 - 512*a^7*b^2*c^5 - 2048*a^8*c^6)*x^2 + 4*(a^3*b
^11 - 20*a^4*b^9*c + 160*a^5*b^7*c^2 - 640*a^6*b^5*c^3 + 1280*a^7*b^3*c^4 - 1024*a^8*b*c^5)*x)]

________________________________________________________________________________________

Sympy [B]  time = 58.2152, size = 2994, normalized size = 7.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3/(c*x**2+b*x+a)**5,x)

[Out]

5*c*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d**2)*log(x + (-5120
*a**5*c**6*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d**2) + 6400*
a**4*b**2*c**5*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d**2) - 3
200*a**3*b**4*c**4*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d**2)
+ 800*a**2*b**6*c**3*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d*
*2) - 100*a*b**8*c**2*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d*
*2) + 15*a*b**2*c**2*e**3 - 30*a*b*c**3*d*e**2 + 5*b**10*c*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**
2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d**2) + 5*b**4*c*e**3 - 45*b**3*c**2*d*e**2 + 105*b**2*c**3*d**2*e - 70*b*c
**4*d**3)/(30*a*b*c**3*e**3 - 60*a*c**4*d*e**2 + 10*b**3*c**2*e**3 - 90*b**2*c**3*d*e**2 + 210*b*c**4*d**2*e -
140*c**5*d**3)) - 5*c*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d
**2)*log(x + (5120*a**5*c**6*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*
c**2*d**2) - 6400*a**4*b**2*c**5*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e
+ 7*c**2*d**2) + 3200*a**3*b**4*c**4*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*
d*e + 7*c**2*d**2) - 800*a**2*b**6*c**3*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b
*c*d*e + 7*c**2*d**2) + 100*a*b**8*c**2*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b
*c*d*e + 7*c**2*d**2) + 15*a*b**2*c**2*e**3 - 30*a*b*c**3*d*e**2 - 5*b**10*c*sqrt(-1/(4*a*c - b**2)**9)*(b*e -
2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d**2) + 5*b**4*c*e**3 - 45*b**3*c**2*d*e**2 + 105*b**2*c*
*3*d**2*e - 70*b*c**4*d**3)/(30*a*b*c**3*e**3 - 60*a*c**4*d*e**2 + 10*b**3*c**2*e**3 - 90*b**2*c**3*d*e**2 + 2
10*b*c**4*d**2*e - 140*c**5*d**3)) - (128*a**5*c**2*e**3 + 166*a**4*b**2*c*e**3 - 972*a**4*b*c**2*d*e**2 + 115
2*a**4*c**3*d**2*e + 3*a**3*b**4*e**3 - 84*a**3*b**3*c*d*e**2 + 522*a**3*b**2*c**2*d**2*e - 1116*a**3*b*c**3*d
**3 + 3*a**2*b**5*d*e**2 - 57*a**2*b**4*c*d**2*e + 326*a**2*b**3*c**2*d**3 + 3*a*b**6*d**2*e - 50*a*b**5*c*d**
3 + 3*b**7*d**3 + x**7*(180*a*b*c**5*e**3 - 360*a*c**6*d*e**2 + 60*b**3*c**4*e**3 - 540*b**2*c**5*d*e**2 + 126
0*b*c**6*d**2*e - 840*c**7*d**3) + x**6*(630*a*b**2*c**4*e**3 - 1260*a*b*c**5*d*e**2 + 210*b**4*c**3*e**3 - 18
90*b**3*c**4*d*e**2 + 4410*b**2*c**5*d**2*e - 2940*b*c**6*d**3) + x**5*(660*a**2*b*c**4*e**3 - 1320*a**2*c**5*
d*e**2 + 1000*a*b**3*c**3*e**3 - 3540*a*b**2*c**4*d*e**2 + 4620*a*b*c**5*d**2*e - 3080*a*c**6*d**3 + 260*b**5*
c**2*e**3 - 2340*b**4*c**3*d*e**2 + 5460*b**3*c**4*d**2*e - 3640*b**2*c**5*d**3) + x**4*(1650*a**2*b**2*c**3*e
**3 - 3300*a**2*b*c**4*d*e**2 + 925*a*b**4*c**2*e**3 - 5700*a*b**3*c**3*d*e**2 + 11550*a*b**2*c**4*d**2*e - 77
00*a*b*c**5*d**3 + 125*b**6*c*e**3 - 1125*b**5*c**2*d*e**2 + 2625*b**4*c**3*d**2*e - 1750*b**3*c**4*d**3) + x*
*3*(876*a**3*b*c**3*e**3 - 1752*a**3*c**4*d*e**2 + 1504*a**2*b**3*c**2*e**3 - 5052*a**2*b**2*c**3*d*e**2 + 613
2*a**2*b*c**4*d**2*e - 4088*a**2*c**5*d**3 + 440*a*b**5*c*e**3 - 3708*a*b**4*c**2*d*e**2 + 8484*a*b**3*c**3*d*
*2*e - 5656*a*b**2*c**4*d**3 + 12*b**7*e**3 - 108*b**6*c*d*e**2 + 252*b**5*c**2*d**2*e - 168*b**4*c**3*d**3) +
x**2*(512*a**4*c**3*e**3 + 802*a**3*b**2*c**2*e**3 - 2628*a**3*b*c**3*d*e**2 + 798*a**2*b**4*c*e**3 - 4278*a*
*2*b**3*c**2*d*e**2 + 9198*a**2*b**2*c**3*d**2*e - 6132*a**2*b*c**4*d**3 + 18*a*b**6*e**3 - 492*a*b**5*c*d*e**
2 + 1176*a*b**4*c**2*d**2*e - 784*a*b**3*c**3*d**3 + 18*b**7*d*e**2 - 42*b**6*c*d**2*e + 28*b**5*c**2*d**3) +
x*(332*a**4*b*c**2*e**3 + 360*a**4*c**3*d*e**2 + 604*a**3*b**3*c*e**3 - 3348*a**3*b**2*c**2*d*e**2 + 3348*a**3
*b*c**3*d**2*e - 2232*a**3*c**4*d**3 + 12*a**2*b**5*e**3 - 336*a**2*b**4*c*d*e**2 + 2088*a**2*b**3*c**2*d**2*e
- 1392*a**2*b**2*c**3*d**3 + 12*a*b**6*d*e**2 - 228*a*b**5*c*d**2*e + 152*a*b**4*c**2*d**3 + 12*b**7*d**2*e -
8*b**6*c*d**3))/(3072*a**8*c**4 - 3072*a**7*b**2*c**3 + 1152*a**6*b**4*c**2 - 192*a**5*b**6*c + 12*a**4*b**8
+ x**8*(3072*a**4*c**8 - 3072*a**3*b**2*c**7 + 1152*a**2*b**4*c**6 - 192*a*b**6*c**5 + 12*b**8*c**4) + x**7*(1
2288*a**4*b*c**7 - 12288*a**3*b**3*c**6 + 4608*a**2*b**5*c**5 - 768*a*b**7*c**4 + 48*b**9*c**3) + x**6*(12288*
a**5*c**7 + 6144*a**4*b**2*c**6 - 13824*a**3*b**4*c**5 + 6144*a**2*b**6*c**4 - 1104*a*b**8*c**3 + 72*b**10*c**
2) + x**5*(36864*a**5*b*c**6 - 24576*a**4*b**3*c**5 + 1536*a**3*b**5*c**4 + 2304*a**2*b**7*c**3 - 624*a*b**9*c
**2 + 48*b**11*c) + x**4*(18432*a**6*c**6 + 18432*a**5*b**2*c**5 - 26880*a**4*b**4*c**4 + 9600*a**3*b**6*c**3
- 1080*a**2*b**8*c**2 - 48*a*b**10*c + 12*b**12) + x**3*(36864*a**6*b*c**5 - 24576*a**5*b**3*c**4 + 1536*a**4*
b**5*c**3 + 2304*a**3*b**7*c**2 - 624*a**2*b**9*c + 48*a*b**11) + x**2*(12288*a**7*c**5 + 6144*a**6*b**2*c**4
- 13824*a**5*b**4*c**3 + 6144*a**4*b**6*c**2 - 1104*a**3*b**8*c + 72*a**2*b**10) + x*(12288*a**7*b*c**4 - 1228
8*a**6*b**3*c**3 + 4608*a**5*b**5*c**2 - 768*a**4*b**7*c + 48*a**3*b**9))

________________________________________________________________________________________

Giac [B]  time = 1.12628, size = 1889, normalized size = 5. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*x^2+b*x+a)^5,x, algorithm="giac")

[Out]

10*(14*c^4*d^3 - 21*b*c^3*d^2*e + 9*b^2*c^2*d*e^2 + 6*a*c^3*d*e^2 - b^3*c*e^3 - 3*a*b*c^2*e^3)*arctan((2*c*x +
b)/sqrt(-b^2 + 4*a*c))/((b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*sqrt(-b^2 + 4*a*c
)) + 1/12*(840*c^7*d^3*x^7 - 1260*b*c^6*d^2*x^7*e + 2940*b*c^6*d^3*x^6 + 540*b^2*c^5*d*x^7*e^2 + 360*a*c^6*d*x
^7*e^2 - 4410*b^2*c^5*d^2*x^6*e + 3640*b^2*c^5*d^3*x^5 + 3080*a*c^6*d^3*x^5 - 60*b^3*c^4*x^7*e^3 - 180*a*b*c^5
*x^7*e^3 + 1890*b^3*c^4*d*x^6*e^2 + 1260*a*b*c^5*d*x^6*e^2 - 5460*b^3*c^4*d^2*x^5*e - 4620*a*b*c^5*d^2*x^5*e +
1750*b^3*c^4*d^3*x^4 + 7700*a*b*c^5*d^3*x^4 - 210*b^4*c^3*x^6*e^3 - 630*a*b^2*c^4*x^6*e^3 + 2340*b^4*c^3*d*x^
5*e^2 + 3540*a*b^2*c^4*d*x^5*e^2 + 1320*a^2*c^5*d*x^5*e^2 - 2625*b^4*c^3*d^2*x^4*e - 11550*a*b^2*c^4*d^2*x^4*e
+ 168*b^4*c^3*d^3*x^3 + 5656*a*b^2*c^4*d^3*x^3 + 4088*a^2*c^5*d^3*x^3 - 260*b^5*c^2*x^5*e^3 - 1000*a*b^3*c^3*
x^5*e^3 - 660*a^2*b*c^4*x^5*e^3 + 1125*b^5*c^2*d*x^4*e^2 + 5700*a*b^3*c^3*d*x^4*e^2 + 3300*a^2*b*c^4*d*x^4*e^2
- 252*b^5*c^2*d^2*x^3*e - 8484*a*b^3*c^3*d^2*x^3*e - 6132*a^2*b*c^4*d^2*x^3*e - 28*b^5*c^2*d^3*x^2 + 784*a*b^
3*c^3*d^3*x^2 + 6132*a^2*b*c^4*d^3*x^2 - 125*b^6*c*x^4*e^3 - 925*a*b^4*c^2*x^4*e^3 - 1650*a^2*b^2*c^3*x^4*e^3
+ 108*b^6*c*d*x^3*e^2 + 3708*a*b^4*c^2*d*x^3*e^2 + 5052*a^2*b^2*c^3*d*x^3*e^2 + 1752*a^3*c^4*d*x^3*e^2 + 42*b^
6*c*d^2*x^2*e - 1176*a*b^4*c^2*d^2*x^2*e - 9198*a^2*b^2*c^3*d^2*x^2*e + 8*b^6*c*d^3*x - 152*a*b^4*c^2*d^3*x +
1392*a^2*b^2*c^3*d^3*x + 2232*a^3*c^4*d^3*x - 12*b^7*x^3*e^3 - 440*a*b^5*c*x^3*e^3 - 1504*a^2*b^3*c^2*x^3*e^3
- 876*a^3*b*c^3*x^3*e^3 - 18*b^7*d*x^2*e^2 + 492*a*b^5*c*d*x^2*e^2 + 4278*a^2*b^3*c^2*d*x^2*e^2 + 2628*a^3*b*c
^3*d*x^2*e^2 - 12*b^7*d^2*x*e + 228*a*b^5*c*d^2*x*e - 2088*a^2*b^3*c^2*d^2*x*e - 3348*a^3*b*c^3*d^2*x*e - 3*b^
7*d^3 + 50*a*b^5*c*d^3 - 326*a^2*b^3*c^2*d^3 + 1116*a^3*b*c^3*d^3 - 18*a*b^6*x^2*e^3 - 798*a^2*b^4*c*x^2*e^3 -
802*a^3*b^2*c^2*x^2*e^3 - 512*a^4*c^3*x^2*e^3 - 12*a*b^6*d*x*e^2 + 336*a^2*b^4*c*d*x*e^2 + 3348*a^3*b^2*c^2*d
*x*e^2 - 360*a^4*c^3*d*x*e^2 - 3*a*b^6*d^2*e + 57*a^2*b^4*c*d^2*e - 522*a^3*b^2*c^2*d^2*e - 1152*a^4*c^3*d^2*e
- 12*a^2*b^5*x*e^3 - 604*a^3*b^3*c*x*e^3 - 332*a^4*b*c^2*x*e^3 - 3*a^2*b^5*d*e^2 + 84*a^3*b^3*c*d*e^2 + 972*a
^4*b*c^2*d*e^2 - 3*a^3*b^4*e^3 - 166*a^4*b^2*c*e^3 - 128*a^5*c^2*e^3)/((b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 25
6*a^3*b^2*c^3 + 256*a^4*c^4)*(c*x^2 + b*x + a)^4)