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

Optimal. Leaf size=262 \[ -\frac {\left (-30 a^2 b c^2 e+12 a^2 c^3 d+20 a b^3 c e-12 a b^2 c^2 d-3 b^5 e+2 b^4 c d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}-\frac {\left (2 a c e-3 b^2 e+2 b c d\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac {x^2 \left (8 a c e-3 b^2 e+2 b c d\right )}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {x \left (11 a b c e-6 a c^2 d-3 b^3 e+2 b^2 c d\right )}{c^3 \left (b^2-4 a c\right )} \]

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Rubi [A]  time = 0.64, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {818, 800, 634, 618, 206, 628} \begin {gather*} -\frac {\left (-30 a^2 b c^2 e+12 a^2 c^3 d-12 a b^2 c^2 d+20 a b^3 c e+2 b^4 c d-3 b^5 e\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}-\frac {x^2 \left (8 a c e-3 b^2 e+2 b c d\right )}{2 c^2 \left (b^2-4 a c\right )}-\frac {\left (2 a c e-3 b^2 e+2 b c d\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac {x \left (11 a b c e-6 a c^2 d+2 b^2 c d-3 b^3 e\right )}{c^3 \left (b^2-4 a c\right )}+\frac {x^3 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*b^2*c*d - 6*a*c^2*d - 3*b^3*e + 11*a*b*c*e)*x)/(c^3*(b^2 - 4*a*c)) - ((2*b*c*d - 3*b^2*e + 8*a*c*e)*x^2)/(
2*c^2*(b^2 - 4*a*c)) + (x^3*(a*(2*c*d - b*e) + (b*c*d - b^2*e + 2*a*c*e)*x))/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2
)) - ((2*b^4*c*d - 12*a*b^2*c^2*d + 12*a^2*c^3*d - 3*b^5*e + 20*a*b^3*c*e - 30*a^2*b*c^2*e)*ArcTanh[(b + 2*c*x
)/Sqrt[b^2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(3/2)) - ((2*b*c*d - 3*b^2*e + 2*a*c*e)*Log[a + b*x + c*x^2])/(2*c^4)

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 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 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

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

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), 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] && IntegerQ[m]

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rubi steps

\begin {align*} \int \frac {x^4 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx &=\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {x^2 \left (-3 a (2 c d-b e)-\left (2 b c d-3 b^2 e+8 a c e\right ) x\right )}{a+b x+c x^2} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \left (\frac {2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e}{c^2}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x}{c}-\frac {a \left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right )+\left (b^2-4 a c\right ) \left (2 b c d-3 b^2 e+2 a c e\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right ) x}{c^3 \left (b^2-4 a c\right )}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {a \left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right )+\left (b^2-4 a c\right ) \left (2 b c d-3 b^2 e+2 a c e\right ) x}{a+b x+c x^2} \, dx}{c^3 \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right ) x}{c^3 \left (b^2-4 a c\right )}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (2 b c d-3 b^2 e+2 a c e\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}+\frac {\left (2 b^4 c d-12 a b^2 c^2 d+12 a^2 c^3 d-3 b^5 e+20 a b^3 c e-30 a^2 b c^2 e\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^4 \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right ) x}{c^3 \left (b^2-4 a c\right )}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (2 b c d-3 b^2 e+2 a c e\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac {\left (2 b^4 c d-12 a b^2 c^2 d+12 a^2 c^3 d-3 b^5 e+20 a b^3 c e-30 a^2 b c^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^4 \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right ) x}{c^3 \left (b^2-4 a c\right )}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (2 b^4 c d-12 a b^2 c^2 d+12 a^2 c^3 d-3 b^5 e+20 a b^3 c e-30 a^2 b c^2 e\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}-\frac {\left (2 b c d-3 b^2 e+2 a c e\right ) \log \left (a+b x+c x^2\right )}{2 c^4}\\ \end {align*}

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Mathematica [A]  time = 0.36, size = 249, normalized size = 0.95 \begin {gather*} \frac {\frac {2 \left (30 a^2 b c^2 e-12 a^2 c^3 d-20 a b^3 c e+12 a b^2 c^2 d+3 b^5 e-2 b^4 c d\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac {2 \left (2 a^3 c^2 e+a^2 c \left (-4 b^2 e+b c (3 d+5 e x)-2 c^2 d x\right )+a b^2 \left (b^2 e-b c (d+5 e x)+4 c^2 d x\right )+b^4 x (b e-c d)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\left (-2 a c e+3 b^2 e-2 b c d\right ) \log (a+x (b+c x))+2 c x (c d-2 b e)+c^2 e x^2}{2 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c*(c*d - 2*b*e)*x + c^2*e*x^2 + (2*(2*a^3*c^2*e + b^4*(-(c*d) + b*e)*x + a*b^2*(b^2*e + 4*c^2*d*x - b*c*(d
+ 5*e*x)) + a^2*c*(-4*b^2*e - 2*c^2*d*x + b*c*(3*d + 5*e*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*(-2*b^4*
c*d + 12*a*b^2*c^2*d - 12*a^2*c^3*d + 3*b^5*e - 20*a*b^3*c*e + 30*a^2*b*c^2*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 +
4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (-2*b*c*d + 3*b^2*e - 2*a*c*e)*Log[a + x*(b + c*x)])/(2*c^4)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.48, size = 1696, normalized size = 6.47

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e*x^4 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d - 3*(b^5*c^2 - 8*
a*b^3*c^3 + 16*a^2*b*c^4)*e)*x^3 + (2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d - (4*b^6*c - 33*a*b^4*c^2 + 72*
a^2*b^2*c^3 - 16*a^3*c^4)*e)*x^2 + ((2*(b^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*d - (3*b^5*c - 20*a*b^3*c^2 + 30*a^
2*b*c^3)*e)*x^2 + 2*(a*b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*d - (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*e + (2*(
b^5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*d - (3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*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)) - 2*(a*b^5*c - 7*a^2*b^3*c^2
+ 12*a^3*b*c^3)*d + 2*(a*b^6 - 8*a^2*b^4*c + 18*a^3*b^2*c^2 - 8*a^4*c^3)*e - 2*((b^6*c - 9*a*b^4*c^2 + 26*a^2
*b^2*c^3 - 24*a^3*c^4)*d - (b^7 - 11*a*b^5*c + 41*a^2*b^3*c^2 - 52*a^3*b*c^3)*e)*x - ((2*(b^5*c^2 - 8*a*b^3*c^
3 + 16*a^2*b*c^4)*d - (3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*e)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c
^2 + 16*a^3*b*c^3)*d - (3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*e + (2*(b^6*c - 8*a*b^4*c^2 + 16
*a^2*b^2*c^3)*d - (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*e)*x)*log(c*x^2 + b*x + a))/(a*b^4*c^4
- 8*a^2*b^2*c^5 + 16*a^3*c^6 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^2 + (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^
6)*x), 1/2*((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e*x^4 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d - 3*(b^5*c^
2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e)*x^3 + (2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d - (4*b^6*c - 33*a*b^4*c^2
+ 72*a^2*b^2*c^3 - 16*a^3*c^4)*e)*x^2 - 2*((2*(b^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*d - (3*b^5*c - 20*a*b^3*c^2
+ 30*a^2*b*c^3)*e)*x^2 + 2*(a*b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*d - (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*
e + (2*(b^5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*d - (3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*e)*x)*sqrt(-b^2 + 4*a*c)*
arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - 2*(a*b^5*c - 7*a^2*b^3*c^2 + 12*a^3*b*c^3)*d + 2*(a*b^
6 - 8*a^2*b^4*c + 18*a^3*b^2*c^2 - 8*a^4*c^3)*e - 2*((b^6*c - 9*a*b^4*c^2 + 26*a^2*b^2*c^3 - 24*a^3*c^4)*d - (
b^7 - 11*a*b^5*c + 41*a^2*b^3*c^2 - 52*a^3*b*c^3)*e)*x - ((2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d - (3*b^6
*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*e)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d - (3*a*
b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*e + (2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d - (3*b^7 - 2
6*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*e)*x)*log(c*x^2 + b*x + a))/(a*b^4*c^4 - 8*a^2*b^2*c^5 + 16*a^3*c^6
+ (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^2 + (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x)]

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giac [A]  time = 0.16, size = 297, normalized size = 1.13 \begin {gather*} \frac {{\left (2 \, b^{4} c d - 12 \, a b^{2} c^{2} d + 12 \, a^{2} c^{3} d - 3 \, b^{5} e + 20 \, a b^{3} c e - 30 \, a^{2} b c^{2} e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {{\left (2 \, b c d - 3 \, b^{2} e + 2 \, a c e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac {c^{2} x^{2} e + 2 \, c^{2} d x - 4 \, b c x e}{2 \, c^{4}} - \frac {a b^{3} c d - 3 \, a^{2} b c^{2} d - a b^{4} e + 4 \, a^{2} b^{2} c e - 2 \, a^{3} c^{2} e + {\left (b^{4} c d - 4 \, a b^{2} c^{2} d + 2 \, a^{2} c^{3} d - b^{5} e + 5 \, a b^{3} c e - 5 \, a^{2} b c^{2} e\right )} x}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b^4*c*d - 12*a*b^2*c^2*d + 12*a^2*c^3*d - 3*b^5*e + 20*a*b^3*c*e - 30*a^2*b*c^2*e)*arctan((2*c*x + b)/sqrt(
-b^2 + 4*a*c))/((b^2*c^4 - 4*a*c^5)*sqrt(-b^2 + 4*a*c)) - 1/2*(2*b*c*d - 3*b^2*e + 2*a*c*e)*log(c*x^2 + b*x +
a)/c^4 + 1/2*(c^2*x^2*e + 2*c^2*d*x - 4*b*c*x*e)/c^4 - (a*b^3*c*d - 3*a^2*b*c^2*d - a*b^4*e + 4*a^2*b^2*c*e -
2*a^3*c^2*e + (b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*b*c^2*e)*x)/((c*x^2 + b*x +
a)*(b^2 - 4*a*c)*c^4)

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maple [B]  time = 0.06, size = 809, normalized size = 3.09 \begin {gather*} -\frac {5 a^{2} b e x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{2}}+\frac {30 a^{2} b e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c^{2}}+\frac {2 a^{2} d x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c}-\frac {12 a^{2} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c}+\frac {5 a \,b^{3} e x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{3}}-\frac {20 a \,b^{3} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c^{3}}-\frac {4 a \,b^{2} d x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{2}}+\frac {12 a \,b^{2} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c^{2}}-\frac {b^{5} e x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{4}}+\frac {3 b^{5} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c^{4}}+\frac {b^{4} d x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{3}}-\frac {2 b^{4} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c^{3}}-\frac {2 a^{3} e}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{2}}+\frac {4 a^{2} b^{2} e}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{3}}-\frac {3 a^{2} b d}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{2}}-\frac {4 a^{2} e \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) c^{2}}-\frac {a \,b^{4} e}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{4}}+\frac {a \,b^{3} d}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{3}}+\frac {7 a \,b^{2} e \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) c^{3}}-\frac {4 a b d \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) c^{2}}-\frac {3 b^{4} e \ln \left (c \,x^{2}+b x +a \right )}{2 \left (4 a c -b^{2}\right ) c^{4}}+\frac {b^{3} d \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) c^{3}}+\frac {e \,x^{2}}{2 c^{2}}-\frac {2 b e x}{c^{3}}+\frac {d x}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x+d)/(c*x^2+b*x+a)^2,x)

[Out]

-5/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a^2*b*e+5/c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a*b^3*e-4/c^2/(c*x^2+b*x+a)/(4*a*
c-b^2)*x*a*b^2*d-20/c^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^3*e+12/c^2/(4*a*c-b^2)^(3/2)
*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2*d+7/c^3/(4*a*c-b^2)*ln(c*x^2+b*x+a)*a*b^2*e-4/c^2/(4*a*c-b^2)*ln(c*
x^2+b*x+a)*a*b*d+30/c^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b*e+1/c^3/(c*x^2+b*x+a)*a/(4
*a*c-b^2)*b^3*d-3/c^2/(c*x^2+b*x+a)*a^2/(4*a*c-b^2)*b*d-1/c^4/(c*x^2+b*x+a)*a/(4*a*c-b^2)*b^4*e-1/c^4/(c*x^2+b
*x+a)/(4*a*c-b^2)*x*b^5*e+1/c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*x*b^4*d+4/c^3/(c*x^2+b*x+a)*a^2/(4*a*c-b^2)*b^2*e+2/
c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a^2*d-4/c^2/(4*a*c-b^2)*ln(c*x^2+b*x+a)*e*a^2-3/2/c^4/(4*a*c-b^2)*ln(c*x^2+b*x+a
)*b^4*e+1/c^3/(4*a*c-b^2)*ln(c*x^2+b*x+a)*b^3*d-12/c/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2
*d+1/2/c^2*e*x^2+1/c^2*d*x+3/c^4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^5*e-2/c^3/(4*a*c-b^2)
^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4*d-2/c^2/(c*x^2+b*x+a)*a^3/(4*a*c-b^2)*e-2/c^3*b*e*x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)/(c*x^2+b*x+a)^2,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 = 1.98, size = 427, normalized size = 1.63 \begin {gather*} x\,\left (\frac {d}{c^2}-\frac {2\,b\,e}{c^3}\right )-\frac {\frac {a\,\left (2\,e\,a^2\,c^2-4\,e\,a\,b^2\,c+3\,d\,a\,b\,c^2+e\,b^4-d\,b^3\,c\right )}{c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (5\,e\,a^2\,b\,c^2-2\,d\,a^2\,c^3-5\,e\,a\,b^3\,c+4\,d\,a\,b^2\,c^2+e\,b^5-d\,b^4\,c\right )}{c\,\left (4\,a\,c-b^2\right )}}{c^4\,x^2+b\,c^3\,x+a\,c^3}+\frac {e\,x^2}{2\,c^2}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (128\,e\,a^4\,c^4-288\,e\,a^3\,b^2\,c^3+128\,d\,a^3\,b\,c^4+168\,e\,a^2\,b^4\,c^2-96\,d\,a^2\,b^3\,c^3-38\,e\,a\,b^6\,c+24\,d\,a\,b^5\,c^2+3\,e\,b^8-2\,d\,b^7\,c\right )}{2\,\left (64\,a^3\,c^7-48\,a^2\,b^2\,c^6+12\,a\,b^4\,c^5-b^6\,c^4\right )}+\frac {\mathrm {atan}\left (\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}-\frac {b^3\,c^3-4\,a\,b\,c^4}{c^3\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (30\,e\,a^2\,b\,c^2-12\,d\,a^2\,c^3-20\,e\,a\,b^3\,c+12\,d\,a\,b^2\,c^2+3\,e\,b^5-2\,d\,b^4\,c\right )}{c^4\,{\left (4\,a\,c-b^2\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(d + e*x))/(a + b*x + c*x^2)^2,x)

[Out]

x*(d/c^2 - (2*b*e)/c^3) - ((a*(b^4*e + 2*a^2*c^2*e - b^3*c*d + 3*a*b*c^2*d - 4*a*b^2*c*e))/(c*(4*a*c - b^2)) +
(x*(b^5*e - 2*a^2*c^3*d - b^4*c*d - 5*a*b^3*c*e + 4*a*b^2*c^2*d + 5*a^2*b*c^2*e))/(c*(4*a*c - b^2)))/(a*c^3 +
c^4*x^2 + b*c^3*x) + (e*x^2)/(2*c^2) - (log(a + b*x + c*x^2)*(3*b^8*e + 128*a^4*c^4*e - 2*b^7*c*d - 96*a^2*b^
3*c^3*d + 168*a^2*b^4*c^2*e - 288*a^3*b^2*c^3*e - 38*a*b^6*c*e + 24*a*b^5*c^2*d + 128*a^3*b*c^4*d))/(2*(64*a^3
*c^7 - b^6*c^4 + 12*a*b^4*c^5 - 48*a^2*b^2*c^6)) + (atan((2*c*x)/(4*a*c - b^2)^(1/2) - (b^3*c^3 - 4*a*b*c^4)/(
c^3*(4*a*c - b^2)^(3/2)))*(3*b^5*e - 12*a^2*c^3*d - 2*b^4*c*d - 20*a*b^3*c*e + 12*a*b^2*c^2*d + 30*a^2*b*c^2*e
))/(c^4*(4*a*c - b^2)^(3/2))

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sympy [B]  time = 7.71, size = 1572, normalized size = 6.00

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x+d)/(c*x**2+b*x+a)**2,x)

[Out]

x*(-2*b*e/c**3 + d/c**2) + (-sqrt(-(4*a*c - b**2)**3)*(30*a**2*b*c**2*e - 12*a**2*c**3*d - 20*a*b**3*c*e + 12*
a*b**2*c**2*d + 3*b**5*e - 2*b**4*c*d)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a
*c*e - 3*b**2*e + 2*b*c*d)/(2*c**4))*log(x + (16*a**3*c**2*e - 17*a**2*b**2*c*e + 10*a**2*b*c**2*d + 16*a**2*c
**5*(-sqrt(-(4*a*c - b**2)**3)*(30*a**2*b*c**2*e - 12*a**2*c**3*d - 20*a*b**3*c*e + 12*a*b**2*c**2*d + 3*b**5*
e - 2*b**4*c*d)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c*e - 3*b**2*e + 2*b*c
*d)/(2*c**4)) + 3*a*b**4*e - 2*a*b**3*c*d - 8*a*b**2*c**4*(-sqrt(-(4*a*c - b**2)**3)*(30*a**2*b*c**2*e - 12*a*
*2*c**3*d - 20*a*b**3*c*e + 12*a*b**2*c**2*d + 3*b**5*e - 2*b**4*c*d)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**
2 + 12*a*b**4*c - b**6)) - (2*a*c*e - 3*b**2*e + 2*b*c*d)/(2*c**4)) + b**4*c**3*(-sqrt(-(4*a*c - b**2)**3)*(30
*a**2*b*c**2*e - 12*a**2*c**3*d - 20*a*b**3*c*e + 12*a*b**2*c**2*d + 3*b**5*e - 2*b**4*c*d)/(2*c**4*(64*a**3*c
**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c*e - 3*b**2*e + 2*b*c*d)/(2*c**4)))/(30*a**2*b*c**2*e -
12*a**2*c**3*d - 20*a*b**3*c*e + 12*a*b**2*c**2*d + 3*b**5*e - 2*b**4*c*d)) + (sqrt(-(4*a*c - b**2)**3)*(30*a
**2*b*c**2*e - 12*a**2*c**3*d - 20*a*b**3*c*e + 12*a*b**2*c**2*d + 3*b**5*e - 2*b**4*c*d)/(2*c**4*(64*a**3*c**
3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c*e - 3*b**2*e + 2*b*c*d)/(2*c**4))*log(x + (16*a**3*c**2*
e - 17*a**2*b**2*c*e + 10*a**2*b*c**2*d + 16*a**2*c**5*(sqrt(-(4*a*c - b**2)**3)*(30*a**2*b*c**2*e - 12*a**2*c
**3*d - 20*a*b**3*c*e + 12*a*b**2*c**2*d + 3*b**5*e - 2*b**4*c*d)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 +
12*a*b**4*c - b**6)) - (2*a*c*e - 3*b**2*e + 2*b*c*d)/(2*c**4)) + 3*a*b**4*e - 2*a*b**3*c*d - 8*a*b**2*c**4*(s
qrt(-(4*a*c - b**2)**3)*(30*a**2*b*c**2*e - 12*a**2*c**3*d - 20*a*b**3*c*e + 12*a*b**2*c**2*d + 3*b**5*e - 2*b
**4*c*d)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c*e - 3*b**2*e + 2*b*c*d)/(2*
c**4)) + b**4*c**3*(sqrt(-(4*a*c - b**2)**3)*(30*a**2*b*c**2*e - 12*a**2*c**3*d - 20*a*b**3*c*e + 12*a*b**2*c*
*2*d + 3*b**5*e - 2*b**4*c*d)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c*e - 3*
b**2*e + 2*b*c*d)/(2*c**4)))/(30*a**2*b*c**2*e - 12*a**2*c**3*d - 20*a*b**3*c*e + 12*a*b**2*c**2*d + 3*b**5*e
- 2*b**4*c*d)) + (-2*a**3*c**2*e + 4*a**2*b**2*c*e - 3*a**2*b*c**2*d - a*b**4*e + a*b**3*c*d + x*(-5*a**2*b*c*
*2*e + 2*a**2*c**3*d + 5*a*b**3*c*e - 4*a*b**2*c**2*d - b**5*e + b**4*c*d))/(4*a**2*c**5 - a*b**2*c**4 + x**2*
(4*a*c**6 - b**2*c**5) + x*(4*a*b*c**5 - b**3*c**4)) + e*x**2/(2*c**2)

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