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

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

[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)

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Rubi [A]  time = 0.0963824, antiderivative size = 219, normalized size of antiderivative = 1., 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 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]

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}{\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*}

Mathematica [A]  time = 0.30673, 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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Maple [B]  time = 0.157, size = 496, normalized size = 2.3 \begin{align*}{\frac{bd-2\,ae+ \left ( -be+2\,cd \right ) x}{ \left ( 16\,ac-4\,{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{4}}}-{\frac{7\,bcxe}{6\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{3}}}+{\frac{7\,{c}^{2}xd}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{3}}}-{\frac{7\,{b}^{2}e}{12\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{3}}}+{\frac{7\,bcd}{6\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{3}}}-{\frac{35\,{c}^{2}xbe}{6\, \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+{\frac{35\,x{c}^{3}d}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{35\,{b}^{2}ce}{12\, \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+{\frac{35\,b{c}^{2}d}{6\, \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-35\,{\frac{{c}^{3}xbe}{ \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) }}+70\,{\frac{{c}^{4}xd}{ \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) }}-{\frac{35\,{b}^{2}{c}^{2}e}{2\, \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) }}+35\,{\frac{db{c}^{3}}{ \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) }}-70\,{\frac{b{c}^{3}e}{ \left ( 4\,ac-{b}^{2} \right ) ^{9/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+140\,{\frac{{c}^{4}d}{ \left ( 4\,ac-{b}^{2} \right ) ^{9/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \end{align*}

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*(b*d-2*a*e+(-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*x*c*b*e+7/3/(4*a*
c-b^2)^2/(c*x^2+b*x+a)^3*x*c^2*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., 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)/(c*x^2+b*x+a)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.2368, size = 6942, normalized size = 31.7 \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)/(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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Sympy [B]  time = 8.58564, size = 1564, normalized size = 7.14 \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)/(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 + 17
4*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 + 875*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 + 696*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))/(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*(184
32*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 + 4608*a**5*
b**5*c**2 - 768*a**4*b**7*c + 48*a**3*b**9))

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Giac [B]  time = 1.14646, size = 826, normalized size = 3.77 \begin{align*} \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}} \end{align*}

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)