3.2215 \(\int \frac{(d+e x)^3}{(a+b x+c x^2)^4} \, dx\)

Optimal. Leaf size=306 \[ -\frac{2 x (2 c d-b e) \left (-c e (a e+15 b d)+4 b^2 e^2+15 c^2 d^2\right )-b^2 \left (11 a e^3+25 c d^2 e\right )+6 b c d \left (13 a e^2+5 c d^2\right )-16 a c e \left (a e^2+5 c d^2\right )+3 b^3 d e^2}{3 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{2 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac{(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^2 \left (-8 a c e-3 b^2 e+10 c x (2 c d-b e)+10 b c d\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2} \]

[Out]

-((b + 2*c*x)*(d + e*x)^3)/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + ((d + e*x)^2*(10*b*c*d - 3*b^2*e - 8*a*c*e
+ 10*c*(2*c*d - b*e)*x))/(6*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) - (3*b^3*d*e^2 - 16*a*c*e*(5*c*d^2 + a*e^2) +
6*b*c*d*(5*c*d^2 + 13*a*e^2) - b^2*(25*c*d^2*e + 11*a*e^3) + 2*(2*c*d - b*e)*(15*c^2*d^2 + 4*b^2*e^2 - c*e*(1
5*b*d + a*e))*x)/(3*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)) + (2*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d
- 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(7/2)

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Rubi [A]  time = 0.429487, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {736, 820, 777, 618, 206} \[ -\frac{2 x (2 c d-b e) \left (-c e (a e+15 b d)+4 b^2 e^2+15 c^2 d^2\right )-b^2 \left (11 a e^3+25 c d^2 e\right )+6 b c d \left (13 a e^2+5 c d^2\right )-16 a c e \left (a e^2+5 c d^2\right )+3 b^3 d e^2}{3 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{2 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac{(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^2 \left (-8 a c e-3 b^2 e+10 c x (2 c d-b e)+10 b c d\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3/(a + b*x + c*x^2)^4,x]

[Out]

-((b + 2*c*x)*(d + e*x)^3)/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + ((d + e*x)^2*(10*b*c*d - 3*b^2*e - 8*a*c*e
+ 10*c*(2*c*d - b*e)*x))/(6*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) - (3*b^3*d*e^2 - 16*a*c*e*(5*c*d^2 + a*e^2) +
6*b*c*d*(5*c*d^2 + 13*a*e^2) - b^2*(25*c*d^2*e + 11*a*e^3) + 2*(2*c*d - b*e)*(15*c^2*d^2 + 4*b^2*e^2 - c*e*(1
5*b*d + a*e))*x)/(3*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)) + (2*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d
- 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(7/2)

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 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 )^4} \, dx &=-\frac{(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{\int \frac{(d+e x)^2 (-10 c d+3 b e-4 c e x)}{\left (a+b x+c x^2\right )^3} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac{(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^2 \left (10 b c d-3 b^2 e-8 a c e+10 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{\int \frac{(d+e x) \left (-2 \left (30 c^2 d^2+3 b^2 e^2-c e (25 b d-8 a e)\right )-10 c e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^2} \, dx}{6 \left (b^2-4 a c\right )^2}\\ &=-\frac{(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^2 \left (10 b c d-3 b^2 e-8 a c e+10 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{3 b^3 d e^2-16 a c e \left (5 c d^2+a e^2\right )+6 b c d \left (5 c d^2+13 a e^2\right )-b^2 \left (25 c d^2 e+11 a e^3\right )+2 (2 c d-b e) \left (15 c^2 d^2+4 b^2 e^2-c e (15 b d+a e)\right ) x}{3 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{\left (20 a c^2 e^2 (2 c d-b e)-c \left (-4 c d \left (30 c^2 d^2+3 b^2 e^2-c e (25 b d-8 a e)\right )-b \left (-10 c d e (2 c d-b e)-2 e \left (30 c^2 d^2+3 b^2 e^2-c e (25 b d-8 a e)\right )\right )\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{6 c \left (b^2-4 a c\right )^3}\\ &=-\frac{(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^2 \left (10 b c d-3 b^2 e-8 a c e+10 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{3 b^3 d e^2-16 a c e \left (5 c d^2+a e^2\right )+6 b c d \left (5 c d^2+13 a e^2\right )-b^2 \left (25 c d^2 e+11 a e^3\right )+2 (2 c d-b e) \left (15 c^2 d^2+4 b^2 e^2-c e (15 b d+a e)\right ) x}{3 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{\left (20 a c^2 e^2 (2 c d-b e)-c \left (-4 c d \left (30 c^2 d^2+3 b^2 e^2-c e (25 b d-8 a e)\right )-b \left (-10 c d e (2 c d-b e)-2 e \left (30 c^2 d^2+3 b^2 e^2-c e (25 b d-8 a e)\right )\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{3 c \left (b^2-4 a c\right )^3}\\ &=-\frac{(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^2 \left (10 b c d-3 b^2 e-8 a c e+10 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{3 b^3 d e^2-16 a c e \left (5 c d^2+a e^2\right )+6 b c d \left (5 c d^2+13 a e^2\right )-b^2 \left (25 c d^2 e+11 a e^3\right )+2 (2 c d-b e) \left (15 c^2 d^2+4 b^2 e^2-c e (15 b d+a e)\right ) x}{3 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2+6 a c e^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.822817, size = 401, normalized size = 1.31 \[ \frac{1}{6} \left (\frac{2 \left (2 c \left (a^2 e^3-3 a c d e (d+e x)+c^2 d^3 x\right )+b^2 e^2 (3 c d x-a e)+b c \left (3 a e^2 (d+e x)+c d^2 (d-3 e x)\right )-b^3 e^3 x\right )}{c^2 \left (4 a c-b^2\right ) (a+x (b+c x))^3}+\frac{4 c^2 \left (-6 a^2 e^3+3 a c d e^2 x+5 c^2 d^3 x\right )+3 b^2 c e \left (3 a e^2+c d (4 e x-5 d)\right )+2 b c^2 \left (3 a e^2 (d-e x)+5 c d^2 (d-3 e x)\right )+b^3 c e^2 (6 d-e x)-2 b^4 e^3}{c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+\frac{3 (b+2 c x) (2 c d-b e) \left (2 c e (3 a e-5 b d)+b^2 e^2+10 c^2 d^2\right )}{c \left (4 a c-b^2\right )^3 (a+x (b+c x))}-\frac{12 (b e-2 c d) \left (2 c e (3 a e-5 b d)+b^2 e^2+10 c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{7/2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3/(a + b*x + c*x^2)^4,x]

[Out]

((3*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(-5*b*d + 3*a*e))*(b + 2*c*x))/(c*(-b^2 + 4*a*c)^3*(a + x*(b +
c*x))) + (2*(-(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))^3) + (-2*b^4*e^3 + b^3*c*e^2
*(6*d - e*x) + 4*c^2*(-6*a^2*e^3 + 5*c^2*d^3*x + 3*a*c*d*e^2*x) + 2*b*c^2*(5*c*d^2*(d - 3*e*x) + 3*a*e^2*(d -
e*x)) + 3*b^2*c*e*(3*a*e^2 + c*d*(-5*d + 4*e*x)))/(c^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (12*(-2*c*d + b*
e)*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(-5*b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2
))/6

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Maple [B]  time = 0.165, size = 1213, normalized size = 4. \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)^4,x)

[Out]

(-(6*a*b*c*e^3-12*a*c^2*d*e^2+b^3*e^3-12*b^2*c*d*e^2+30*b*c^2*d^2*e-20*c^3*d^3)/(64*a^3*c^3-48*a^2*b^2*c^2+12*
a*b^4*c-b^6)*c^2*x^5-5/2*(6*a*b*c*e^3-12*a*c^2*d*e^2+b^3*e^3-12*b^2*c*d*e^2+30*b*c^2*d^2*e-20*c^3*d^3)/(64*a^3
*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*b*c*x^4-1/6*(16*a*c+11*b^2)*(6*a*b*c*e^3-12*a*c^2*d*e^2+b^3*e^3-12*b^2*c*d
*e^2+30*b*c^2*d^2*e-20*c^3*d^3)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3-1/2*(32*a^3*c^2*e^3+24*a^2*b^2*
c*e^3-96*a^2*b*c^2*d*e^2+17*a*b^4*e^3-102*a*b^3*c*d*e^2+240*a*b^2*c^2*d^2*e-160*a*b*c^3*d^3-6*b^5*d*e^2+15*b^4
*c*d^2*e-10*b^3*c^2*d^3)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2-1/2*(20*a^3*b*c*e^3+24*a^3*c^2*d*e^2+2
0*a^2*b^3*e^3-132*a^2*b^2*c*d*e^2+132*a^2*b*c^2*d^2*e-88*a^2*c^3*d^3-6*a*b^4*d*e^2+54*a*b^3*c*d^2*e-36*a*b^2*c
^2*d^3-3*b^5*d^2*e+2*b^4*c*d^3)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x-1/6*(32*a^4*c*e^3+22*a^3*b^2*e^3-
156*a^3*b*c*d*e^2+192*a^3*c^2*d^2*e-6*a^2*b^3*d*e^2+54*a^2*b^2*c*d^2*e-132*a^2*b*c^2*d^3-3*a*b^4*d^2*e+26*a*b^
3*c*d^3-2*b^5*d^3)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6))/(c*x^2+b*x+a)^3-12/(64*a^3*c^3-48*a^2*b^2*c^2+1
2*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*c*e^3+24/(64*a^3*c^3-48*a^2*b^2*c^2+1
2*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^2*a*d*e^2-2/(64*a^3*c^3-48*a^2*b^2*c^2+
12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*e^3+24/(64*a^3*c^3-48*a^2*b^2*c^2+12
*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*c*d*e^2-60/(64*a^3*c^3-48*a^2*b^2*c^2+
12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c^2*d^2*e+40/(64*a^3*c^3-48*a^2*b^2*c^
2+12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^3*d^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*x^2+b*x+a)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.6895, size = 7605, normalized size = 24.85 \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)^4,x, algorithm="fricas")

[Out]

[-1/6*(6*(20*(b^2*c^5 - 4*a*c^6)*d^3 - 30*(b^3*c^4 - 4*a*b*c^5)*d^2*e + 12*(b^4*c^3 - 3*a*b^2*c^4 - 4*a^2*c^5)
*d*e^2 - (b^5*c^2 + 2*a*b^3*c^3 - 24*a^2*b*c^4)*e^3)*x^5 + 15*(20*(b^3*c^4 - 4*a*b*c^5)*d^3 - 30*(b^4*c^3 - 4*
a*b^2*c^4)*d^2*e + 12*(b^5*c^2 - 3*a*b^3*c^3 - 4*a^2*b*c^4)*d*e^2 - (b^6*c + 2*a*b^4*c^2 - 24*a^2*b^2*c^3)*e^3
)*x^4 + 2*(b^7 - 17*a*b^5*c + 118*a^2*b^3*c^2 - 264*a^3*b*c^3)*d^3 + 3*(a*b^6 - 22*a^2*b^4*c + 8*a^3*b^2*c^2 +
256*a^4*c^3)*d^2*e + 6*(a^2*b^5 + 22*a^3*b^3*c - 104*a^4*b*c^2)*d*e^2 - 2*(11*a^3*b^4 - 28*a^4*b^2*c - 64*a^5
*c^2)*e^3 + (20*(11*b^4*c^3 - 28*a*b^2*c^4 - 64*a^2*c^5)*d^3 - 30*(11*b^5*c^2 - 28*a*b^3*c^3 - 64*a^2*b*c^4)*d
^2*e + 12*(11*b^6*c - 17*a*b^4*c^2 - 92*a^2*b^2*c^3 - 64*a^3*c^4)*d*e^2 - (11*b^7 + 38*a*b^5*c - 232*a^2*b^3*c
^2 - 384*a^3*b*c^3)*e^3)*x^3 + 3*(10*(b^5*c^2 + 12*a*b^3*c^3 - 64*a^2*b*c^4)*d^3 - 15*(b^6*c + 12*a*b^4*c^2 -
64*a^2*b^2*c^3)*d^2*e + 6*(b^7 + 13*a*b^5*c - 52*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e^2 - (17*a*b^6 - 44*a^2*b^4*c
- 64*a^3*b^2*c^2 - 128*a^4*c^3)*e^3)*x^2 - 6*(20*a^3*c^3*d^3 - 30*a^3*b*c^2*d^2*e + (20*c^6*d^3 - 30*b*c^5*d^2
*e + 12*(b^2*c^4 + a*c^5)*d*e^2 - (b^3*c^3 + 6*a*b*c^4)*e^3)*x^6 + 3*(20*b*c^5*d^3 - 30*b^2*c^4*d^2*e + 12*(b^
3*c^3 + a*b*c^4)*d*e^2 - (b^4*c^2 + 6*a*b^2*c^3)*e^3)*x^5 + 3*(20*(b^2*c^4 + a*c^5)*d^3 - 30*(b^3*c^3 + a*b*c^
4)*d^2*e + 12*(b^4*c^2 + 2*a*b^2*c^3 + a^2*c^4)*d*e^2 - (b^5*c + 7*a*b^3*c^2 + 6*a^2*b*c^3)*e^3)*x^4 + 12*(a^3
*b^2*c + a^4*c^2)*d*e^2 - (a^3*b^3 + 6*a^4*b*c)*e^3 + (20*(b^3*c^3 + 6*a*b*c^4)*d^3 - 30*(b^4*c^2 + 6*a*b^2*c^
3)*d^2*e + 12*(b^5*c + 7*a*b^3*c^2 + 6*a^2*b*c^3)*d*e^2 - (b^6 + 12*a*b^4*c + 36*a^2*b^2*c^2)*e^3)*x^3 + 3*(20
*(a*b^2*c^3 + a^2*c^4)*d^3 - 30*(a*b^3*c^2 + a^2*b*c^3)*d^2*e + 12*(a*b^4*c + 2*a^2*b^2*c^2 + a^3*c^3)*d*e^2 -
(a*b^5 + 7*a^2*b^3*c + 6*a^3*b*c^2)*e^3)*x^2 + 3*(20*a^2*b*c^3*d^3 - 30*a^2*b^2*c^2*d^2*e + 12*(a^2*b^3*c + a
^3*b*c^2)*d*e^2 - (a^2*b^4 + 6*a^3*b^2*c)*e^3)*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + s
qrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) - 3*(2*(b^6*c - 22*a*b^4*c^2 + 28*a^2*b^2*c^3 + 176*a^3*c^4)*
d^3 - 3*(b^7 - 22*a*b^5*c + 28*a^2*b^3*c^2 + 176*a^3*b*c^3)*d^2*e - 6*(a*b^6 + 18*a^2*b^4*c - 92*a^3*b^2*c^2 +
16*a^4*c^3)*d*e^2 + 20*(a^2*b^5 - 3*a^3*b^3*c - 4*a^4*b*c^2)*e^3)*x)/(a^3*b^8 - 16*a^4*b^6*c + 96*a^5*b^4*c^2
- 256*a^6*b^2*c^3 + 256*a^7*c^4 + (b^8*c^3 - 16*a*b^6*c^4 + 96*a^2*b^4*c^5 - 256*a^3*b^2*c^6 + 256*a^4*c^7)*x
^6 + 3*(b^9*c^2 - 16*a*b^7*c^3 + 96*a^2*b^5*c^4 - 256*a^3*b^3*c^5 + 256*a^4*b*c^6)*x^5 + 3*(b^10*c - 15*a*b^8*
c^2 + 80*a^2*b^6*c^3 - 160*a^3*b^4*c^4 + 256*a^5*c^6)*x^4 + (b^11 - 10*a*b^9*c + 320*a^3*b^5*c^3 - 1280*a^4*b^
3*c^4 + 1536*a^5*b*c^5)*x^3 + 3*(a*b^10 - 15*a^2*b^8*c + 80*a^3*b^6*c^2 - 160*a^4*b^4*c^3 + 256*a^6*c^5)*x^2 +
3*(a^2*b^9 - 16*a^3*b^7*c + 96*a^4*b^5*c^2 - 256*a^5*b^3*c^3 + 256*a^6*b*c^4)*x), -1/6*(6*(20*(b^2*c^5 - 4*a*
c^6)*d^3 - 30*(b^3*c^4 - 4*a*b*c^5)*d^2*e + 12*(b^4*c^3 - 3*a*b^2*c^4 - 4*a^2*c^5)*d*e^2 - (b^5*c^2 + 2*a*b^3*
c^3 - 24*a^2*b*c^4)*e^3)*x^5 + 15*(20*(b^3*c^4 - 4*a*b*c^5)*d^3 - 30*(b^4*c^3 - 4*a*b^2*c^4)*d^2*e + 12*(b^5*c
^2 - 3*a*b^3*c^3 - 4*a^2*b*c^4)*d*e^2 - (b^6*c + 2*a*b^4*c^2 - 24*a^2*b^2*c^3)*e^3)*x^4 + 2*(b^7 - 17*a*b^5*c
+ 118*a^2*b^3*c^2 - 264*a^3*b*c^3)*d^3 + 3*(a*b^6 - 22*a^2*b^4*c + 8*a^3*b^2*c^2 + 256*a^4*c^3)*d^2*e + 6*(a^2
*b^5 + 22*a^3*b^3*c - 104*a^4*b*c^2)*d*e^2 - 2*(11*a^3*b^4 - 28*a^4*b^2*c - 64*a^5*c^2)*e^3 + (20*(11*b^4*c^3
- 28*a*b^2*c^4 - 64*a^2*c^5)*d^3 - 30*(11*b^5*c^2 - 28*a*b^3*c^3 - 64*a^2*b*c^4)*d^2*e + 12*(11*b^6*c - 17*a*b
^4*c^2 - 92*a^2*b^2*c^3 - 64*a^3*c^4)*d*e^2 - (11*b^7 + 38*a*b^5*c - 232*a^2*b^3*c^2 - 384*a^3*b*c^3)*e^3)*x^3
+ 3*(10*(b^5*c^2 + 12*a*b^3*c^3 - 64*a^2*b*c^4)*d^3 - 15*(b^6*c + 12*a*b^4*c^2 - 64*a^2*b^2*c^3)*d^2*e + 6*(b
^7 + 13*a*b^5*c - 52*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e^2 - (17*a*b^6 - 44*a^2*b^4*c - 64*a^3*b^2*c^2 - 128*a^4*c
^3)*e^3)*x^2 - 12*(20*a^3*c^3*d^3 - 30*a^3*b*c^2*d^2*e + (20*c^6*d^3 - 30*b*c^5*d^2*e + 12*(b^2*c^4 + a*c^5)*d
*e^2 - (b^3*c^3 + 6*a*b*c^4)*e^3)*x^6 + 3*(20*b*c^5*d^3 - 30*b^2*c^4*d^2*e + 12*(b^3*c^3 + a*b*c^4)*d*e^2 - (b
^4*c^2 + 6*a*b^2*c^3)*e^3)*x^5 + 3*(20*(b^2*c^4 + a*c^5)*d^3 - 30*(b^3*c^3 + a*b*c^4)*d^2*e + 12*(b^4*c^2 + 2*
a*b^2*c^3 + a^2*c^4)*d*e^2 - (b^5*c + 7*a*b^3*c^2 + 6*a^2*b*c^3)*e^3)*x^4 + 12*(a^3*b^2*c + a^4*c^2)*d*e^2 - (
a^3*b^3 + 6*a^4*b*c)*e^3 + (20*(b^3*c^3 + 6*a*b*c^4)*d^3 - 30*(b^4*c^2 + 6*a*b^2*c^3)*d^2*e + 12*(b^5*c + 7*a*
b^3*c^2 + 6*a^2*b*c^3)*d*e^2 - (b^6 + 12*a*b^4*c + 36*a^2*b^2*c^2)*e^3)*x^3 + 3*(20*(a*b^2*c^3 + a^2*c^4)*d^3
- 30*(a*b^3*c^2 + a^2*b*c^3)*d^2*e + 12*(a*b^4*c + 2*a^2*b^2*c^2 + a^3*c^3)*d*e^2 - (a*b^5 + 7*a^2*b^3*c + 6*a
^3*b*c^2)*e^3)*x^2 + 3*(20*a^2*b*c^3*d^3 - 30*a^2*b^2*c^2*d^2*e + 12*(a^2*b^3*c + a^3*b*c^2)*d*e^2 - (a^2*b^4
+ 6*a^3*b^2*c)*e^3)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - 3*(2*(b^6*c
- 22*a*b^4*c^2 + 28*a^2*b^2*c^3 + 176*a^3*c^4)*d^3 - 3*(b^7 - 22*a*b^5*c + 28*a^2*b^3*c^2 + 176*a^3*b*c^3)*d^2
*e - 6*(a*b^6 + 18*a^2*b^4*c - 92*a^3*b^2*c^2 + 16*a^4*c^3)*d*e^2 + 20*(a^2*b^5 - 3*a^3*b^3*c - 4*a^4*b*c^2)*e
^3)*x)/(a^3*b^8 - 16*a^4*b^6*c + 96*a^5*b^4*c^2 - 256*a^6*b^2*c^3 + 256*a^7*c^4 + (b^8*c^3 - 16*a*b^6*c^4 + 96
*a^2*b^4*c^5 - 256*a^3*b^2*c^6 + 256*a^4*c^7)*x^6 + 3*(b^9*c^2 - 16*a*b^7*c^3 + 96*a^2*b^5*c^4 - 256*a^3*b^3*c
^5 + 256*a^4*b*c^6)*x^5 + 3*(b^10*c - 15*a*b^8*c^2 + 80*a^2*b^6*c^3 - 160*a^3*b^4*c^4 + 256*a^5*c^6)*x^4 + (b^
11 - 10*a*b^9*c + 320*a^3*b^5*c^3 - 1280*a^4*b^3*c^4 + 1536*a^5*b*c^5)*x^3 + 3*(a*b^10 - 15*a^2*b^8*c + 80*a^3
*b^6*c^2 - 160*a^4*b^4*c^3 + 256*a^6*c^5)*x^2 + 3*(a^2*b^9 - 16*a^3*b^7*c + 96*a^4*b^5*c^2 - 256*a^5*b^3*c^3 +
256*a^6*b*c^4)*x)]

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Sympy [B]  time = 27.1883, size = 2057, normalized size = 6.72 \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)**4,x)

[Out]

sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2)*log(x + (-256*a*
*4*c**4*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) + 256*a*
*3*b**2*c**3*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) - 9
6*a**2*b**4*c**2*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2)
+ 16*a*b**6*c*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) +
6*a*b**2*c*e**3 - 12*a*b*c**2*d*e**2 - b**8*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2
- 10*b*c*d*e + 10*c**2*d**2) + b**4*e**3 - 12*b**3*c*d*e**2 + 30*b**2*c**2*d**2*e - 20*b*c**3*d**3)/(12*a*b*c*
*2*e**3 - 24*a*c**3*d*e**2 + 2*b**3*c*e**3 - 24*b**2*c**2*d*e**2 + 60*b*c**3*d**2*e - 40*c**4*d**3)) - sqrt(-1
/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2)*log(x + (256*a**4*c**4*
sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) - 256*a**3*b**2*
c**3*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) + 96*a**2*b
**4*c**2*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) - 16*a*
b**6*c*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) + 6*a*b**
2*c*e**3 - 12*a*b*c**2*d*e**2 + b**8*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c
*d*e + 10*c**2*d**2) + b**4*e**3 - 12*b**3*c*d*e**2 + 30*b**2*c**2*d**2*e - 20*b*c**3*d**3)/(12*a*b*c**2*e**3
- 24*a*c**3*d*e**2 + 2*b**3*c*e**3 - 24*b**2*c**2*d*e**2 + 60*b*c**3*d**2*e - 40*c**4*d**3)) - (32*a**4*c*e**3
+ 22*a**3*b**2*e**3 - 156*a**3*b*c*d*e**2 + 192*a**3*c**2*d**2*e - 6*a**2*b**3*d*e**2 + 54*a**2*b**2*c*d**2*e
- 132*a**2*b*c**2*d**3 - 3*a*b**4*d**2*e + 26*a*b**3*c*d**3 - 2*b**5*d**3 + x**5*(36*a*b*c**3*e**3 - 72*a*c**
4*d*e**2 + 6*b**3*c**2*e**3 - 72*b**2*c**3*d*e**2 + 180*b*c**4*d**2*e - 120*c**5*d**3) + x**4*(90*a*b**2*c**2*
e**3 - 180*a*b*c**3*d*e**2 + 15*b**4*c*e**3 - 180*b**3*c**2*d*e**2 + 450*b**2*c**3*d**2*e - 300*b*c**4*d**3) +
x**3*(96*a**2*b*c**2*e**3 - 192*a**2*c**3*d*e**2 + 82*a*b**3*c*e**3 - 324*a*b**2*c**2*d*e**2 + 480*a*b*c**3*d
**2*e - 320*a*c**4*d**3 + 11*b**5*e**3 - 132*b**4*c*d*e**2 + 330*b**3*c**2*d**2*e - 220*b**2*c**3*d**3) + x**2
*(96*a**3*c**2*e**3 + 72*a**2*b**2*c*e**3 - 288*a**2*b*c**2*d*e**2 + 51*a*b**4*e**3 - 306*a*b**3*c*d*e**2 + 72
0*a*b**2*c**2*d**2*e - 480*a*b*c**3*d**3 - 18*b**5*d*e**2 + 45*b**4*c*d**2*e - 30*b**3*c**2*d**3) + x*(60*a**3
*b*c*e**3 + 72*a**3*c**2*d*e**2 + 60*a**2*b**3*e**3 - 396*a**2*b**2*c*d*e**2 + 396*a**2*b*c**2*d**2*e - 264*a*
*2*c**3*d**3 - 18*a*b**4*d*e**2 + 162*a*b**3*c*d**2*e - 108*a*b**2*c**2*d**3 - 9*b**5*d**2*e + 6*b**4*c*d**3))
/(384*a**6*c**3 - 288*a**5*b**2*c**2 + 72*a**4*b**4*c - 6*a**3*b**6 + x**6*(384*a**3*c**6 - 288*a**2*b**2*c**5
+ 72*a*b**4*c**4 - 6*b**6*c**3) + x**5*(1152*a**3*b*c**5 - 864*a**2*b**3*c**4 + 216*a*b**5*c**3 - 18*b**7*c**
2) + x**4*(1152*a**4*c**5 + 288*a**3*b**2*c**4 - 648*a**2*b**4*c**3 + 198*a*b**6*c**2 - 18*b**8*c) + x**3*(230
4*a**4*b*c**4 - 1344*a**3*b**3*c**3 + 144*a**2*b**5*c**2 + 36*a*b**7*c - 6*b**9) + x**2*(1152*a**5*c**4 + 288*
a**4*b**2*c**3 - 648*a**3*b**4*c**2 + 198*a**2*b**6*c - 18*a*b**8) + x*(1152*a**5*b*c**3 - 864*a**4*b**3*c**2
+ 216*a**3*b**5*c - 18*a**2*b**7))

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Giac [B]  time = 1.13938, size = 1118, normalized size = 3.65 \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)^4,x, algorithm="giac")

[Out]

-2*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 + 12*a*c^2*d*e^2 - b^3*e^3 - 6*a*b*c*e^3)*arctan((2*c*x + b)/
sqrt(-b^2 + 4*a*c))/((b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(-b^2 + 4*a*c)) - 1/6*(120*c^5*d^3*x
^5 - 180*b*c^4*d^2*x^5*e + 300*b*c^4*d^3*x^4 + 72*b^2*c^3*d*x^5*e^2 + 72*a*c^4*d*x^5*e^2 - 450*b^2*c^3*d^2*x^4
*e + 220*b^2*c^3*d^3*x^3 + 320*a*c^4*d^3*x^3 - 6*b^3*c^2*x^5*e^3 - 36*a*b*c^3*x^5*e^3 + 180*b^3*c^2*d*x^4*e^2
+ 180*a*b*c^3*d*x^4*e^2 - 330*b^3*c^2*d^2*x^3*e - 480*a*b*c^3*d^2*x^3*e + 30*b^3*c^2*d^3*x^2 + 480*a*b*c^3*d^3
*x^2 - 15*b^4*c*x^4*e^3 - 90*a*b^2*c^2*x^4*e^3 + 132*b^4*c*d*x^3*e^2 + 324*a*b^2*c^2*d*x^3*e^2 + 192*a^2*c^3*d
*x^3*e^2 - 45*b^4*c*d^2*x^2*e - 720*a*b^2*c^2*d^2*x^2*e - 6*b^4*c*d^3*x + 108*a*b^2*c^2*d^3*x + 264*a^2*c^3*d^
3*x - 11*b^5*x^3*e^3 - 82*a*b^3*c*x^3*e^3 - 96*a^2*b*c^2*x^3*e^3 + 18*b^5*d*x^2*e^2 + 306*a*b^3*c*d*x^2*e^2 +
288*a^2*b*c^2*d*x^2*e^2 + 9*b^5*d^2*x*e - 162*a*b^3*c*d^2*x*e - 396*a^2*b*c^2*d^2*x*e + 2*b^5*d^3 - 26*a*b^3*c
*d^3 + 132*a^2*b*c^2*d^3 - 51*a*b^4*x^2*e^3 - 72*a^2*b^2*c*x^2*e^3 - 96*a^3*c^2*x^2*e^3 + 18*a*b^4*d*x*e^2 + 3
96*a^2*b^2*c*d*x*e^2 - 72*a^3*c^2*d*x*e^2 + 3*a*b^4*d^2*e - 54*a^2*b^2*c*d^2*e - 192*a^3*c^2*d^2*e - 60*a^2*b^
3*x*e^3 - 60*a^3*b*c*x*e^3 + 6*a^2*b^3*d*e^2 + 156*a^3*b*c*d*e^2 - 22*a^3*b^2*e^3 - 32*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)^3)