\(\int \frac {(d+e x)^3}{(a+b x+c x^2)^5} \, dx\) [2223]

Optimal result

Integrand size = 20, antiderivative size = 378 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^5} \, dx=-\frac {(b+2 c x) (d+e x)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^2 \left (14 b c d-3 b^2 e-16 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {3 b^3 d e^2-32 a c e \left (7 c d^2+a e^2\right )+2 b c d \left (35 c d^2+99 a e^2\right )-b^2 \left (49 c d^2 e+27 a e^3\right )+2 (2 c d-b e) \left (35 c^2 d^2+12 b^2 e^2-c e (35 b d+13 a e)\right ) x}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (b+2 c x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {10 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}} \]

[Out]

-1/4*(2*c*x+b)*(e*x+d)^3/(-4*a*c+b^2)/(c*x^2+b*x+a)^4+1/12*(e*x+d)^2*(14*b*c*d-3*b^2*e-16*a*c*e+14*c*(-b*e+2*c
*d)*x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^3+1/12*(-3*b^3*d*e^2+32*a*c*e*(a*e^2+7*c*d^2)-2*b*c*d*(99*a*e^2+35*c*d^2)+
b^2*(27*a*e^3+49*c*d^2*e)-2*(-b*e+2*c*d)*(35*c^2*d^2+12*b^2*e^2-c*e*(13*a*e+35*b*d))*x)/(-4*a*c+b^2)^3/(c*x^2+
b*x+a)^2+5/2*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(2*c*x+b)/(-4*a*c+b^2)^4/(c*x^2+b*x+a)-10*c*(
-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(9/2)

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {750, 834, 791, 628, 632, 212} \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^5} \, dx=-\frac {10 c (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{\left (b^2-4 a c\right )^{9/2}}+\frac {5 (b+2 c x) (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {(b+2 c x) (d+e x)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^2 \left (-16 a c e-3 b^2 e+14 c x (2 c d-b e)+14 b c d\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {2 x (2 c d-b e) \left (-c e (13 a e+35 b d)+12 b^2 e^2+35 c^2 d^2\right )-b^2 \left (27 a e^3+49 c d^2 e\right )+2 b c d \left (99 a e^2+35 c d^2\right )-32 a c e \left (a e^2+7 c d^2\right )+3 b^3 d e^2}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2} \]

[In]

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

[Out]

-1/4*((b + 2*c*x)*(d + e*x)^3)/((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 212

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

Rule 628

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 632

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 750

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 791

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] &&
NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 834

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*} \text {integral}& = -\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 ) \text {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] (verified)

Time = 0.77 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.24 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^5} \, dx=\frac {1}{12} \left (\frac {5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2+c e (-7 b d+3 a e)\right ) (b+2 c x)}{c \left (-b^2+4 a c\right )^3 (a+x (b+c x))^2}+\frac {30 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2+c e (-7 b d+3 a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^4 (a+x (b+c x))}+\frac {-3 b^4 e^3+b^3 c e^2 (9 d-2 e x)+4 c^2 \left (-8 a^2 e^3+7 c^2 d^3 x+3 a c d e^2 x\right )+b^2 c e \left (13 a e^2-3 c d (7 d-6 e x)\right )+2 b c^2 \left (7 c d^2 (d-3 e x)+3 a e^2 (d-e x)\right )}{c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^3}+\frac {3 \left (-b^3 e^3 x+b^2 e^2 (-a e+3 c d x)+2 c \left (a^2 e^3+c^2 d^3 x-3 a c d e (d+e x)\right )+b c \left (c d^2 (d-3 e x)+3 a e^2 (d+e x)\right )\right )}{c^2 \left (-b^2+4 a c\right ) (a+x (b+c x))^4}+\frac {120 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2+c e (-7 b d+3 a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{9/2}}\right ) \]

[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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1364\) vs. \(2(365)=730\).

Time = 17.14 (sec) , antiderivative size = 1365, normalized size of antiderivative = 3.61

method	result	size
default	\(\text {Expression too large to display}\)	\(1365\)
risch	\(\text {Expression too large to display}\)	\(2719\)

[In]

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

[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)^4-10*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)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2899 vs. \(2 (366) = 732\).

Time = 0.38 (sec) , antiderivative size = 5818, normalized size of antiderivative = 15.39 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^5} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2994 vs. \(2 (379) = 758\).

Time = 54.78 (sec) , antiderivative size = 2994, normalized size of antiderivative = 7.92 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^5} \, dx=\text {Too large to display} \]

[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 - 11
52*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 - 1
260*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 +
1890*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
+ 7700*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
- 6132*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 - 3
348*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*(12288*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 + 15
36*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 - 12288*a**6*b**3*c**3 + 4608*a**5*b**5*c**2 - 768*a**4*b**7*c + 48*a**3*b**9))

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^5} \, dx=\text {Exception raised: ValueError} \]

[In]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1431 vs. \(2 (366) = 732\).

Time = 0.28 (sec) , antiderivative size = 1431, normalized size of antiderivative = 3.79 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^5} \, dx=\text {Too large to display} \]

[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*e*x^7 + 540*b^2*c^5*d*e^2*x^7 + 360*a*c^6*d*e^2*x^7 - 60*b^3*c^4*e
^3*x^7 - 180*a*b*c^5*e^3*x^7 + 2940*b*c^6*d^3*x^6 - 4410*b^2*c^5*d^2*e*x^6 + 1890*b^3*c^4*d*e^2*x^6 + 1260*a*b
*c^5*d*e^2*x^6 - 210*b^4*c^3*e^3*x^6 - 630*a*b^2*c^4*e^3*x^6 + 3640*b^2*c^5*d^3*x^5 + 3080*a*c^6*d^3*x^5 - 546
0*b^3*c^4*d^2*e*x^5 - 4620*a*b*c^5*d^2*e*x^5 + 2340*b^4*c^3*d*e^2*x^5 + 3540*a*b^2*c^4*d*e^2*x^5 + 1320*a^2*c^
5*d*e^2*x^5 - 260*b^5*c^2*e^3*x^5 - 1000*a*b^3*c^3*e^3*x^5 - 660*a^2*b*c^4*e^3*x^5 + 1750*b^3*c^4*d^3*x^4 + 77
00*a*b*c^5*d^3*x^4 - 2625*b^4*c^3*d^2*e*x^4 - 11550*a*b^2*c^4*d^2*e*x^4 + 1125*b^5*c^2*d*e^2*x^4 + 5700*a*b^3*
c^3*d*e^2*x^4 + 3300*a^2*b*c^4*d*e^2*x^4 - 125*b^6*c*e^3*x^4 - 925*a*b^4*c^2*e^3*x^4 - 1650*a^2*b^2*c^3*e^3*x^
4 + 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 - 252*b^5*c^2*d^2*e*x^3 - 8484*a*b^3*c
^3*d^2*e*x^3 - 6132*a^2*b*c^4*d^2*e*x^3 + 108*b^6*c*d*e^2*x^3 + 3708*a*b^4*c^2*d*e^2*x^3 + 5052*a^2*b^2*c^3*d*
e^2*x^3 + 1752*a^3*c^4*d*e^2*x^3 - 12*b^7*e^3*x^3 - 440*a*b^5*c*e^3*x^3 - 1504*a^2*b^3*c^2*e^3*x^3 - 876*a^3*b
*c^3*e^3*x^3 - 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 + 42*b^6*c*d^2*e*x^2 - 1176
*a*b^4*c^2*d^2*e*x^2 - 9198*a^2*b^2*c^3*d^2*e*x^2 - 18*b^7*d*e^2*x^2 + 492*a*b^5*c*d*e^2*x^2 + 4278*a^2*b^3*c^
2*d*e^2*x^2 + 2628*a^3*b*c^3*d*e^2*x^2 - 18*a*b^6*e^3*x^2 - 798*a^2*b^4*c*e^3*x^2 - 802*a^3*b^2*c^2*e^3*x^2 -
512*a^4*c^3*e^3*x^2 + 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*d^2*e*x + 228*a*b^5*c*d^2*e*x - 2088*a^2*b^3*c^2*d^2*e*x - 3348*a^3*b*c^3*d^2*e*x - 12*a*b^6*d*e^2*x + 336*
a^2*b^4*c*d*e^2*x + 3348*a^3*b^2*c^2*d*e^2*x - 360*a^4*c^3*d*e^2*x - 12*a^2*b^5*e^3*x - 604*a^3*b^3*c*e^3*x -
332*a^4*b*c^2*e^3*x - 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 - 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 - 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)

Mupad [B] (verification not implemented)

Time = 11.40 (sec) , antiderivative size = 1711, normalized size of antiderivative = 4.53 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^5} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(10*c*atan((((10*c^2*x*(b*e - 2*c*d)*(b^2*e^2 + 7*c^2*d^2 + 3*a*c*e^2 - 7*b*c*d*e))/(4*a*c - b^2)^(9/2) + (5*c
*(b*e - 2*c*d)*(b^2*e^2 + 7*c^2*d^2 + 3*a*c*e^2 - 7*b*c*d*e)*(b^9 + 256*a^4*b*c^4 + 96*a^2*b^5*c^2 - 256*a^3*b
^3*c^3 - 16*a*b^7*c))/((4*a*c - b^2)^(9/2)*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)
))*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c))/(70*c^4*d^3 - 5*b^3*c*e^3 + 45*b^2*c^2
*d*e^2 - 15*a*b*c^2*e^3 + 30*a*c^3*d*e^2 - 105*b*c^3*d^2*e))*(b*e - 2*c*d)*(b^2*e^2 + 7*c^2*d^2 + 3*a*c*e^2 -
7*b*c*d*e))/(4*a*c - b^2)^(9/2) - ((3*b^7*d^3 + 3*a^3*b^4*e^3 + 128*a^5*c^2*e^3 - 1116*a^3*b*c^3*d^3 + 166*a^4
*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 1152*a^4*c^3*d^2*e + 326*a^2*b^3*c^2*d^3 - 50*a*b^5*c*d^3 + 3*a*b^6*d^2*e - 57*
a^2*b^4*c*d^2*e - 84*a^3*b^3*c*d*e^2 - 972*a^4*b*c^2*d*e^2 + 522*a^3*b^2*c^2*d^2*e)/(12*(b^8 + 256*a^4*c^4 + 9
6*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) + (x*(3*b^7*d^2*e - 2*b^6*c*d^3 + 3*a^2*b^5*e^3 - 558*a^3*c^4*d
^3 + 38*a*b^4*c^2*d^3 + 151*a^3*b^3*c*e^3 + 83*a^4*b*c^2*e^3 + 90*a^4*c^3*d*e^2 - 348*a^2*b^2*c^3*d^3 + 3*a*b^
6*d*e^2 - 57*a*b^5*c*d^2*e - 84*a^2*b^4*c*d*e^2 + 837*a^3*b*c^3*d^2*e + 522*a^2*b^3*c^2*d^2*e - 837*a^3*b^2*c^
2*d*e^2))/(3*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) + (x^2*(9*a*b^6*e^3 + 9*b^7*
d*e^2 + 256*a^4*c^3*e^3 + 14*b^5*c^2*d^3 - 392*a*b^3*c^3*d^3 - 3066*a^2*b*c^4*d^3 + 399*a^2*b^4*c*e^3 + 401*a^
3*b^2*c^2*e^3 - 21*b^6*c*d^2*e - 246*a*b^5*c*d*e^2 + 588*a*b^4*c^2*d^2*e - 1314*a^3*b*c^3*d*e^2 + 4599*a^2*b^2
*c^3*d^2*e - 2139*a^2*b^3*c^2*d*e^2))/(6*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c))
+ (5*c^4*x^7*(b^3*e^3 - 14*c^3*d^3 + 3*a*b*c*e^3 - 6*a*c^2*d*e^2 + 21*b*c^2*d^2*e - 9*b^2*c*d*e^2))/(b^8 + 256
*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c) + (5*x^5*(11*a*c^3 + 13*b^2*c^2)*(b^3*e^3 - 14*c^3*d
^3 + 3*a*b*c*e^3 - 6*a*c^2*d*e^2 + 21*b*c^2*d^2*e - 9*b^2*c*d*e^2))/(3*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 2
56*a^3*b^2*c^3 - 16*a*b^6*c)) + (25*x^4*(5*b^3*c + 22*a*b*c^2)*(b^3*e^3 - 14*c^3*d^3 + 3*a*b*c*e^3 - 6*a*c^2*d
*e^2 + 21*b*c^2*d^2*e - 9*b^2*c*d*e^2))/(12*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c
)) + (x^3*(3*b^4 + 73*a^2*c^2 + 101*a*b^2*c)*(b^3*e^3 - 14*c^3*d^3 + 3*a*b*c*e^3 - 6*a*c^2*d*e^2 + 21*b*c^2*d^
2*e - 9*b^2*c*d*e^2))/(3*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) + (35*b*c^3*x^6*
(b^3*e^3 - 14*c^3*d^3 + 3*a*b*c*e^3 - 6*a*c^2*d*e^2 + 21*b*c^2*d^2*e - 9*b^2*c*d*e^2))/(2*(b^8 + 256*a^4*c^4 +
96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)))/(x^4*(b^4 + 6*a^2*c^2 + 12*a*b^2*c) + a^4 + c^4*x^8 + x^2*(4
*a^3*c + 6*a^2*b^2) + x^6*(4*a*c^3 + 6*b^2*c^2) + x^3*(4*a*b^3 + 12*a^2*b*c) + x^5*(4*b^3*c + 12*a*b*c^2) + 4*
b*c^3*x^7 + 4*a^3*b*x)