3.2225 \(\int \frac {d+e x}{(a+b x+c x^2)^5} \, dx\)

Optimal. Leaf size=219 \[ -\frac {70 c^3 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}+\frac {35 c^2 (b+2 c x) (2 c d-b e)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {35 c (b+2 c x) (2 c d-b e)}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {7 (b+2 c x) (2 c d-b e)}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {-2 a e+x (2 c d-b e)+b d}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4} \]

[Out]

1/4*(-b*d+2*a*e-(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^4+7/12*(-b*e+2*c*d)*(2*c*x+b)/(-4*a*c+b^2)^2/(c*x^2
+b*x+a)^3-35/12*c*(-b*e+2*c*d)*(2*c*x+b)/(-4*a*c+b^2)^3/(c*x^2+b*x+a)^2+35/2*c^2*(-b*e+2*c*d)*(2*c*x+b)/(-4*a*
c+b^2)^4/(c*x^2+b*x+a)-70*c^3*(-b*e+2*c*d)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(9/2)

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Rubi [A]  time = 0.10, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {638, 614, 618, 206} \[ \frac {35 c^2 (b+2 c x) (2 c d-b e)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {70 c^3 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}-\frac {35 c (b+2 c x) (2 c d-b e)}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {7 (b+2 c x) (2 c d-b e)}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {-2 a e+x (2 c d-b e)+b d}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(a + b*x + c*x^2)^5,x]

[Out]

-(b*d - 2*a*e + (2*c*d - b*e)*x)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + (7*(2*c*d - b*e)*(b + 2*c*x))/(12*(b^
2 - 4*a*c)^2*(a + b*x + c*x^2)^3) - (35*c*(2*c*d - b*e)*(b + 2*c*x))/(12*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)^2)
+ (35*c^2*(2*c*d - b*e)*(b + 2*c*x))/(2*(b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (70*c^3*(2*c*d - b*e)*ArcTanh[(b
+ 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(9/2)

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])

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 638

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

Rubi steps

\begin {align*} \int \frac {d+e x}{\left (a+b x+c x^2\right )^5} \, dx &=-\frac {b d-2 a e+(2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}-\frac {(7 (2 c d-b e)) \int \frac {1}{\left (a+b x+c x^2\right )^4} \, dx}{4 \left (b^2-4 a c\right )}\\ &=-\frac {b d-2 a e+(2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {7 (2 c d-b e) (b+2 c x)}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac {(35 c (2 c d-b e)) \int \frac {1}{\left (a+b x+c x^2\right )^3} \, dx}{6 \left (b^2-4 a c\right )^2}\\ &=-\frac {b d-2 a e+(2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {7 (2 c d-b e) (b+2 c x)}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {35 c (2 c d-b e) (b+2 c x)}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac {\left (35 c^2 (2 c d-b e)\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 d-2 a e+(2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {7 (2 c d-b e) (b+2 c x)}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {35 c (2 c d-b e) (b+2 c x)}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {35 c^2 (2 c d-b e) (b+2 c x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}+\frac {\left (35 c^3 (2 c d-b e)\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^4}\\ &=-\frac {b d-2 a e+(2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {7 (2 c d-b e) (b+2 c x)}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {35 c (2 c d-b e) (b+2 c x)}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {35 c^2 (2 c d-b e) (b+2 c x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {\left (70 c^3 (2 c d-b e)\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 d-2 a e+(2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {7 (2 c d-b e) (b+2 c x)}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {35 c (2 c d-b e) (b+2 c x)}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {35 c^2 (2 c d-b e) (b+2 c x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {70 c^3 (2 c d-b e) \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*}

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Mathematica [A]  time = 0.31, size = 209, normalized size = 0.95 \[ \frac {-\frac {840 c^3 (b e-2 c d) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+\frac {35 c \left (b^2-4 a c\right ) (b+2 c x) (b e-2 c d)}{(a+x (b+c x))^2}-\frac {7 \left (b^2-4 a c\right )^2 (b+2 c x) (b e-2 c d)}{(a+x (b+c x))^3}+\frac {3 \left (b^2-4 a c\right )^3 (2 a e-b d+b e x-2 c d x)}{(a+x (b+c x))^4}+\frac {210 c^2 (b+2 c x) (2 c d-b e)}{a+x (b+c x)}}{12 \left (b^2-4 a c\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*(b^2 - 4*a*c)^3*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x))/(a + x*(b + c*x))^4 - (7*(b^2 - 4*a*c)^2*(-2*c*d + b*e
)*(b + 2*c*x))/(a + x*(b + c*x))^3 + (35*c*(b^2 - 4*a*c)*(-2*c*d + b*e)*(b + 2*c*x))/(a + x*(b + c*x))^2 + (21
0*c^2*(2*c*d - b*e)*(b + 2*c*x))/(a + x*(b + c*x)) - (840*c^3*(-2*c*d + b*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*
a*c]])/Sqrt[-b^2 + 4*a*c])/(12*(b^2 - 4*a*c)^4)

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fricas [B]  time = 0.87, size = 3202, normalized size = 14.62 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(420*(2*(b^2*c^7 - 4*a*c^8)*d - (b^3*c^6 - 4*a*b*c^7)*e)*x^7 + 1470*(2*(b^3*c^6 - 4*a*b*c^7)*d - (b^4*c^
5 - 4*a*b^2*c^6)*e)*x^6 + 140*(2*(13*b^4*c^5 - 41*a*b^2*c^6 - 44*a^2*c^7)*d - (13*b^5*c^4 - 41*a*b^3*c^5 - 44*
a^2*b*c^6)*e)*x^5 + 175*(2*(5*b^5*c^4 + 2*a*b^3*c^5 - 88*a^2*b*c^6)*d - (5*b^6*c^3 + 2*a*b^4*c^4 - 88*a^2*b^2*
c^5)*e)*x^4 + 28*(2*(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*b^7*c^2 + 89*a*b^5*c^3 -
331*a^2*b^3*c^4 - 292*a^3*b*c^5)*e)*x^3 - 14*(2*(b^7*c^2 - 32*a*b^5*c^3 - 107*a^2*b^3*c^4 + 876*a^3*b*c^5)*d
- (b^8*c - 32*a*b^6*c^2 - 107*a^2*b^4*c^3 + 876*a^3*b^2*c^4)*e)*x^2 - 420*(2*a^4*c^4*d - a^4*b*c^3*e + (2*c^8*
d - b*c^7*e)*x^8 + 4*(2*b*c^7*d - b^2*c^6*e)*x^7 + 2*(2*(3*b^2*c^6 + 2*a*c^7)*d - (3*b^3*c^5 + 2*a*b*c^6)*e)*x
^6 + 4*(2*(b^3*c^5 + 3*a*b*c^6)*d - (b^4*c^4 + 3*a*b^2*c^5)*e)*x^5 + (2*(b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*d
- (b^5*c^3 + 12*a*b^3*c^4 + 6*a^2*b*c^5)*e)*x^4 + 4*(2*(a*b^3*c^4 + 3*a^2*b*c^5)*d - (a*b^4*c^3 + 3*a^2*b^2*c
^4)*e)*x^3 + 2*(2*(3*a^2*b^2*c^4 + 2*a^3*c^5)*d - (3*a^2*b^3*c^3 + 2*a^3*b*c^4)*e)*x^2 + 4*(2*a^3*b*c^4*d - a^
3*b^2*c^3*e)*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)) - (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 - (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)*e + 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 - (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)*e
)*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), 1/12*(420*(2*(b^2*c^7 - 4*a*c^8)*d - (b^3*c^6 - 4*a*
b*c^7)*e)*x^7 + 1470*(2*(b^3*c^6 - 4*a*b*c^7)*d - (b^4*c^5 - 4*a*b^2*c^6)*e)*x^6 + 140*(2*(13*b^4*c^5 - 41*a*b
^2*c^6 - 44*a^2*c^7)*d - (13*b^5*c^4 - 41*a*b^3*c^5 - 44*a^2*b*c^6)*e)*x^5 + 175*(2*(5*b^5*c^4 + 2*a*b^3*c^5 -
88*a^2*b*c^6)*d - (5*b^6*c^3 + 2*a*b^4*c^4 - 88*a^2*b^2*c^5)*e)*x^4 + 28*(2*(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*b^7*c^2 + 89*a*b^5*c^3 - 331*a^2*b^3*c^4 - 292*a^3*b*c^5)*e)*x^3 - 14*(2*(b^7
*c^2 - 32*a*b^5*c^3 - 107*a^2*b^3*c^4 + 876*a^3*b*c^5)*d - (b^8*c - 32*a*b^6*c^2 - 107*a^2*b^4*c^3 + 876*a^3*b
^2*c^4)*e)*x^2 - 840*(2*a^4*c^4*d - a^4*b*c^3*e + (2*c^8*d - b*c^7*e)*x^8 + 4*(2*b*c^7*d - b^2*c^6*e)*x^7 + 2*
(2*(3*b^2*c^6 + 2*a*c^7)*d - (3*b^3*c^5 + 2*a*b*c^6)*e)*x^6 + 4*(2*(b^3*c^5 + 3*a*b*c^6)*d - (b^4*c^4 + 3*a*b^
2*c^5)*e)*x^5 + (2*(b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*d - (b^5*c^3 + 12*a*b^3*c^4 + 6*a^2*b*c^5)*e)*x^4 + 4*
(2*(a*b^3*c^4 + 3*a^2*b*c^5)*d - (a*b^4*c^3 + 3*a^2*b^2*c^4)*e)*x^3 + 2*(2*(3*a^2*b^2*c^4 + 2*a^3*c^5)*d - (3*
a^2*b^3*c^3 + 2*a^3*b*c^4)*e)*x^2 + 4*(2*a^3*b*c^4*d - a^3*b^2*c^3*e)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2
+ 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (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 - (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)*e + 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 - (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)*e)*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 - 10
24*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 - 204
8*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 + 104
96*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)]

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giac [B]  time = 0.19, size = 612, normalized size = 2.79 \[ \frac {70 \, {\left (2 \, c^{4} d - b c^{3} e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {840 \, c^{7} d x^{7} - 420 \, b c^{6} x^{7} e + 2940 \, b c^{6} d x^{6} - 1470 \, b^{2} c^{5} x^{6} e + 3640 \, b^{2} c^{5} d x^{5} + 3080 \, a c^{6} d x^{5} - 1820 \, b^{3} c^{4} x^{5} e - 1540 \, a b c^{5} x^{5} e + 1750 \, b^{3} c^{4} d x^{4} + 7700 \, a b c^{5} d x^{4} - 875 \, b^{4} c^{3} x^{4} e - 3850 \, a b^{2} c^{4} x^{4} e + 168 \, b^{4} c^{3} d x^{3} + 5656 \, a b^{2} c^{4} d x^{3} + 4088 \, a^{2} c^{5} d x^{3} - 84 \, b^{5} c^{2} x^{3} e - 2828 \, a b^{3} c^{3} x^{3} e - 2044 \, a^{2} b c^{4} x^{3} e - 28 \, b^{5} c^{2} d x^{2} + 784 \, a b^{3} c^{3} d x^{2} + 6132 \, a^{2} b c^{4} d x^{2} + 14 \, b^{6} c x^{2} e - 392 \, a b^{4} c^{2} x^{2} e - 3066 \, a^{2} b^{2} c^{3} x^{2} e + 8 \, b^{6} c d x - 152 \, a b^{4} c^{2} d x + 1392 \, a^{2} b^{2} c^{3} d x + 2232 \, a^{3} c^{4} d x - 4 \, b^{7} x e + 76 \, a b^{5} c x e - 696 \, a^{2} b^{3} c^{2} x e - 1116 \, a^{3} b c^{3} x e - 3 \, b^{7} d + 50 \, a b^{5} c d - 326 \, a^{2} b^{3} c^{2} d + 1116 \, a^{3} b c^{3} d - a b^{6} e + 19 \, a^{2} b^{4} c e - 174 \, a^{3} b^{2} c^{2} e - 384 \, a^{4} c^{3} e}{12 \, {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} {\left (c x^{2} + b x + a\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

70*(2*c^4*d - b*c^3*e)*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*x^7 - 420*b*c^6*x^7*e + 2940*b*c^6*d*x^6 - 1470*b^2
*c^5*x^6*e + 3640*b^2*c^5*d*x^5 + 3080*a*c^6*d*x^5 - 1820*b^3*c^4*x^5*e - 1540*a*b*c^5*x^5*e + 1750*b^3*c^4*d*
x^4 + 7700*a*b*c^5*d*x^4 - 875*b^4*c^3*x^4*e - 3850*a*b^2*c^4*x^4*e + 168*b^4*c^3*d*x^3 + 5656*a*b^2*c^4*d*x^3
+ 4088*a^2*c^5*d*x^3 - 84*b^5*c^2*x^3*e - 2828*a*b^3*c^3*x^3*e - 2044*a^2*b*c^4*x^3*e - 28*b^5*c^2*d*x^2 + 78
4*a*b^3*c^3*d*x^2 + 6132*a^2*b*c^4*d*x^2 + 14*b^6*c*x^2*e - 392*a*b^4*c^2*x^2*e - 3066*a^2*b^2*c^3*x^2*e + 8*b
^6*c*d*x - 152*a*b^4*c^2*d*x + 1392*a^2*b^2*c^3*d*x + 2232*a^3*c^4*d*x - 4*b^7*x*e + 76*a*b^5*c*x*e - 696*a^2*
b^3*c^2*x*e - 1116*a^3*b*c^3*x*e - 3*b^7*d + 50*a*b^5*c*d - 326*a^2*b^3*c^2*d + 1116*a^3*b*c^3*d - a*b^6*e + 1
9*a^2*b^4*c*e - 174*a^3*b^2*c^2*e - 384*a^4*c^3*e)/((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)*(c*x^2 + b*x + a)^4)

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maple [B]  time = 0.06, size = 496, normalized size = 2.26 \[ -\frac {35 b \,c^{3} e x}{\left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )}-\frac {70 b \,c^{3} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {9}{2}}}+\frac {70 c^{4} d x}{\left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )}+\frac {140 c^{4} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {9}{2}}}-\frac {35 b^{2} c^{2} e}{2 \left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )}+\frac {35 b \,c^{3} d}{\left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )}-\frac {35 b \,c^{2} e x}{6 \left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )^{2}}+\frac {35 c^{3} d x}{3 \left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )^{2}}-\frac {35 b^{2} c e}{12 \left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )^{2}}+\frac {35 b \,c^{2} d}{6 \left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )^{2}}-\frac {7 b c e x}{6 \left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{3}}+\frac {7 c^{2} d x}{3 \left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{3}}-\frac {7 b^{2} e}{12 \left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{3}}+\frac {7 b c d}{6 \left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{3}}+\frac {-2 a e +b d +\left (-b e +2 c d \right ) x}{4 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(-2*a*e+b*d+(-b*e+2*c*d)*x)/(4*a*c-b^2)/(c*x^2+b*x+a)^4-7/6/(4*a*c-b^2)^2/(c*x^2+b*x+a)^3*c*x*b*e+7/3/(4*a
*c-b^2)^2/(c*x^2+b*x+a)^3*c^2*x*d-7/12/(4*a*c-b^2)^2/(c*x^2+b*x+a)^3*b^2*e+7/6/(4*a*c-b^2)^2/(c*x^2+b*x+a)^3*b
*c*d-35/6/(4*a*c-b^2)^3*c^2/(c*x^2+b*x+a)^2*x*b*e+35/3/(4*a*c-b^2)^3*c^3/(c*x^2+b*x+a)^2*x*d-35/12/(4*a*c-b^2)
^3*c/(c*x^2+b*x+a)^2*b^2*e+35/6/(4*a*c-b^2)^3*c^2/(c*x^2+b*x+a)^2*b*d-35/(4*a*c-b^2)^4*c^3/(c*x^2+b*x+a)*x*b*e
+70/(4*a*c-b^2)^4*c^4/(c*x^2+b*x+a)*x*d-35/2/(4*a*c-b^2)^4*c^2/(c*x^2+b*x+a)*b^2*e+35/(4*a*c-b^2)^4*c^3/(c*x^2
+b*x+a)*b*d-70/(4*a*c-b^2)^(9/2)*c^3*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*e+140/(4*a*c-b^2)^(9/2)*c^4*arctan(
(2*c*x+b)/(4*a*c-b^2)^(1/2))*d

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(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 details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 0.84, size = 992, normalized size = 4.53 \[ \frac {70\,c^3\,\mathrm {atan}\left (\frac {\left (\frac {70\,c^4\,x\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{9/2}}+\frac {35\,c^3\,\left (b\,e-2\,c\,d\right )\,\left (256\,a^4\,b\,c^4-256\,a^3\,b^3\,c^3+96\,a^2\,b^5\,c^2-16\,a\,b^7\,c+b^9\right )}{{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (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\right )}\right )\,\left (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\right )}{70\,c^4\,d-35\,b\,c^3\,e}\right )\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{9/2}}-\frac {\frac {384\,e\,a^4\,c^3+174\,e\,a^3\,b^2\,c^2-1116\,d\,a^3\,b\,c^3-19\,e\,a^2\,b^4\,c+326\,d\,a^2\,b^3\,c^2+e\,a\,b^6-50\,d\,a\,b^5\,c+3\,d\,b^7}{12\,\left (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\right )}+\frac {35\,c^6\,x^7\,\left (b\,e-2\,c\,d\right )}{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}+\frac {x\,\left (b\,e-2\,c\,d\right )\,\left (279\,a^3\,c^3+174\,a^2\,b^2\,c^2-19\,a\,b^4\,c+b^6\right )}{3\,\left (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\right )}+\frac {7\,c\,x^2\,\left (b\,e-2\,c\,d\right )\,\left (219\,a^2\,b\,c^2+28\,a\,b^3\,c-b^5\right )}{6\,\left (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\right )}+\frac {245\,b\,c^5\,x^6\,\left (b\,e-2\,c\,d\right )}{2\,\left (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\right )}+\frac {7\,c\,x^3\,\left (b\,e-2\,c\,d\right )\,\left (73\,a^2\,c^3+101\,a\,b^2\,c^2+3\,b^4\,c\right )}{3\,\left (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\right )}+\frac {35\,c^2\,x^5\,\left (b\,e-2\,c\,d\right )\,\left (13\,b^2\,c^2+11\,a\,c^3\right )}{3\,\left (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\right )}+\frac {175\,c\,x^4\,\left (5\,b^3\,c^2+22\,a\,b\,c^3\right )\,\left (b\,e-2\,c\,d\right )}{12\,\left (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\right )}}{x^4\,\left (6\,a^2\,c^2+12\,a\,b^2\,c+b^4\right )+a^4+c^4\,x^8+x^2\,\left (4\,c\,a^3+6\,a^2\,b^2\right )+x^6\,\left (6\,b^2\,c^2+4\,a\,c^3\right )+x^3\,\left (12\,c\,a^2\,b+4\,a\,b^3\right )+x^5\,\left (4\,b^3\,c+12\,a\,b\,c^2\right )+4\,b\,c^3\,x^7+4\,a^3\,b\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(a + b*x + c*x^2)^5,x)

[Out]

(70*c^3*atan((((70*c^4*x*(b*e - 2*c*d))/(4*a*c - b^2)^(9/2) + (35*c^3*(b*e - 2*c*d)*(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 - 35*
b*c^3*e))*(b*e - 2*c*d))/(4*a*c - b^2)^(9/2) - ((3*b^7*d + 384*a^4*c^3*e + a*b^6*e + 326*a^2*b^3*c^2*d + 174*a
^3*b^2*c^2*e - 50*a*b^5*c*d - 1116*a^3*b*c^3*d - 19*a^2*b^4*c*e)/(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)) + (35*c^6*x^7*(b*e - 2*c*d))/(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*(b*e - 2*c*d)*(b^6 + 279*a^3*c^3 + 174*a^2*b^2*c^2 - 19*a*b^4*c))/(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)) + (7*c*x^2*(b*e - 2*c*d)*(219*a^2*b*c^2 - b^5 + 28*a*b^3*c))/(
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)) + (245*b*c^5*x^6*(b*e - 2*c*d))/(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)) + (7*c*x^3*(b*e - 2*c*d)*(3*b^4*c + 73*a^2*c^
3 + 101*a*b^2*c^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*c^2*x^5*(b*e
- 2*c*d)*(11*a*c^3 + 13*b^2*c^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)) + (
175*c*x^4*(5*b^3*c^2 + 22*a*b*c^3)*(b*e - 2*c*d))/(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^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)

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sympy [B]  time = 5.28, size = 1564, normalized size = 7.14 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35*c**3*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*log(x + (-35840*a**5*c**8*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2
*c*d) + 44800*a**4*b**2*c**7*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d) - 22400*a**3*b**4*c**6*sqrt(-1/(4*a*c -
b**2)**9)*(b*e - 2*c*d) + 5600*a**2*b**6*c**5*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d) - 700*a*b**8*c**4*sqrt(
-1/(4*a*c - b**2)**9)*(b*e - 2*c*d) + 35*b**10*c**3*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d) + 35*b**2*c**3*e
- 70*b*c**4*d)/(70*b*c**4*e - 140*c**5*d)) - 35*c**3*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*log(x + (35840*a
**5*c**8*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d) - 44800*a**4*b**2*c**7*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c
*d) + 22400*a**3*b**4*c**6*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d) - 5600*a**2*b**6*c**5*sqrt(-1/(4*a*c - b**
2)**9)*(b*e - 2*c*d) + 700*a*b**8*c**4*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d) - 35*b**10*c**3*sqrt(-1/(4*a*c
- b**2)**9)*(b*e - 2*c*d) + 35*b**2*c**3*e - 70*b*c**4*d)/(70*b*c**4*e - 140*c**5*d)) + (-384*a**4*c**3*e - 1
74*a**3*b**2*c**2*e + 1116*a**3*b*c**3*d + 19*a**2*b**4*c*e - 326*a**2*b**3*c**2*d - a*b**6*e + 50*a*b**5*c*d
- 3*b**7*d + x**7*(-420*b*c**6*e + 840*c**7*d) + x**6*(-1470*b**2*c**5*e + 2940*b*c**6*d) + x**5*(-1540*a*b*c*
*5*e + 3080*a*c**6*d - 1820*b**3*c**4*e + 3640*b**2*c**5*d) + x**4*(-3850*a*b**2*c**4*e + 7700*a*b*c**5*d - 87
5*b**4*c**3*e + 1750*b**3*c**4*d) + x**3*(-2044*a**2*b*c**4*e + 4088*a**2*c**5*d - 2828*a*b**3*c**3*e + 5656*a
*b**2*c**4*d - 84*b**5*c**2*e + 168*b**4*c**3*d) + x**2*(-3066*a**2*b**2*c**3*e + 6132*a**2*b*c**4*d - 392*a*b
**4*c**2*e + 784*a*b**3*c**3*d + 14*b**6*c*e - 28*b**5*c**2*d) + x*(-1116*a**3*b*c**3*e + 2232*a**3*c**4*d - 6
96*a**2*b**3*c**2*e + 1392*a**2*b**2*c**3*d + 76*a*b**5*c*e - 152*a*b**4*c**2*d - 4*b**7*e + 8*b**6*c*d))/(307
2*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 - 12
288*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 - 12288*a**6*b**3*c**3 + 46
08*a**5*b**5*c**2 - 768*a**4*b**7*c + 48*a**3*b**9))

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