3.2202 \(\int \frac{(d+e x)^4}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=169 \[ \frac{3 (d+e x) \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{12 \left (a e^2-b d e+c d^2\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.122729, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {722, 618, 206} \[ \frac{3 (d+e x) \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{12 \left (a e^2-b d e+c d^2\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 722

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

Rule 618

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

Rule 206

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

Rubi steps

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

Mathematica [B]  time = 0.598989, size = 413, normalized size = 2.44 \[ \frac{1}{2} \left (\frac{b c \left (-3 a^2 e^4+6 a c d e^2 (d+2 e x)+c^2 d^3 (d-4 e x)\right )+2 c^2 \left (a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)+c^2 d^4 x\right )+2 b^2 c e^2 \left (3 c d^2 x-2 a e (d+e x)\right )+b^3 e^3 (a e-4 c d x)+b^4 e^4 x}{c^3 \left (4 a c-b^2\right ) (a+x (b+c x))^2}+\frac{2 b c^2 \left (11 a^2 e^4+6 a c d e^2 (d-2 e x)+3 c^2 d^3 (d-4 e x)\right )+4 c^3 \left (-a^2 e^3 (16 d+5 e x)+6 a c d^2 e^2 x+3 c^2 d^4 x\right )+4 b^2 c^2 e \left (a e^2 (5 d+4 e x)-3 c d^2 (d-e x)\right )+2 b^3 c e^2 \left (3 c d^2-4 a e^2\right )-2 b^4 c e^3 (2 d+e x)+b^5 e^4}{c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{24 \left (e (a e-b d)+c d^2\right )^2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b^4*e^4*x + b^3*e^3*(a*e - 4*c*d*x) + 2*b^2*c*e^2*(3*c*d^2*x - 2*a*e*(d + e*x)) + b*c*(-3*a^2*e^4 + c^2*d^3*
(d - 4*e*x) + 6*a*c*d*e^2*(d + 2*e*x)) + 2*c^2*(c^2*d^4*x + a^2*e^3*(4*d + e*x) - 2*a*c*d^2*e*(2*d + 3*e*x)))/
(c^3*(-b^2 + 4*a*c)*(a + x*(b + c*x))^2) + (b^5*e^4 + 2*b^3*c*e^2*(3*c*d^2 - 4*a*e^2) - 2*b^4*c*e^3*(2*d + e*x
) + 2*b*c^2*(11*a^2*e^4 + 3*c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d - 2*e*x)) + 4*b^2*c^2*e*(-3*c*d^2*(d - e*x) +
a*e^2*(5*d + 4*e*x)) + 4*c^3*(3*c^2*d^4*x + 6*a*c*d^2*e^2*x - a^2*e^3*(16*d + 5*e*x)))/(c^3*(b^2 - 4*a*c)^2*(
a + x*(b + c*x))) + (24*(c*d^2 + e*(-(b*d) + a*e))^2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5
/2))/2

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Maple [B]  time = 0.162, size = 932, normalized size = 5.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-(10*a^2*c^2*e^4-8*a*b^2*c*e^4+12*a*b*c^2*d*e^3-12*a*c^3*d^2*e^2+b^4*e^4-6*b^2*c^2*d^2*e^2+12*b*c^3*d^3*e-6*c
^4*d^4)/c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+1/2*(2*a^2*b*c^2*e^4-64*a^2*c^3*d*e^3+8*a*b^3*c*e^4-4*a*b^2*c^2*d*e^3
+36*a*b*c^3*d^2*e^2-b^5*e^4-4*b^4*c*d*e^3+18*b^3*c^2*d^2*e^2-36*b^2*c^3*d^3*e+18*b*c^4*d^4)/(16*a^2*c^2-8*a*b^
2*c+b^4)/c^2*x^2-(6*a^3*c^2*e^4-10*a^2*b^2*c*e^4+20*a^2*b*c^2*d*e^3+12*a^2*c^3*d^2*e^2+a*b^4*e^4+4*a*b^3*c*d*e
^3-30*a*b^2*c^2*d^2*e^2+20*a*b*c^3*d^3*e-10*a*c^4*d^4+4*b^3*c^2*d^3*e-2*b^2*c^3*d^4)/(16*a^2*c^2-8*a*b^2*c+b^4
)/c^2*x+1/2/c^2*(10*a^3*b*c*e^4-32*a^3*c^2*d*e^3-a^2*b^3*e^4-4*a^2*b^2*c*d*e^3+36*a^2*b*c^2*d^2*e^2-32*a^2*c^3
*d^3*e-4*a*b^2*c^2*d^3*e+10*a*b*c^3*d^4-b^3*c^2*d^4)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2+b*x+a)^2+12/(16*a^2*c^
2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*e^4-24/(16*a^2*c^2-8*a*b^2*c+b^4)/(
4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d*e^3*a*b+24/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)
*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c*d^2*e^2+12/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x
+b)/(4*a*c-b^2)^(1/2))*b^2*d^2*e^2-24/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2
)^(1/2))*b*c*d^3*e+12/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^2*d^4

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.34423, size = 5349, normalized size = 31.65 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*((b^5*c^2 - 14*a*b^3*c^3 + 40*a^2*b*c^4)*d^4 + 4*(a*b^4*c^2 + 4*a^2*b^2*c^3 - 32*a^3*c^4)*d^3*e - 36*(a^
2*b^3*c^2 - 4*a^3*b*c^3)*d^2*e^2 + 4*(a^2*b^4*c + 4*a^3*b^2*c^2 - 32*a^4*c^3)*d*e^3 + (a^2*b^5 - 14*a^3*b^3*c
+ 40*a^4*b*c^2)*e^4 - 2*(6*(b^2*c^5 - 4*a*c^6)*d^4 - 12*(b^3*c^4 - 4*a*b*c^5)*d^3*e + 6*(b^4*c^3 - 2*a*b^2*c^4
- 8*a^2*c^5)*d^2*e^2 - 12*(a*b^3*c^3 - 4*a^2*b*c^4)*d*e^3 - (b^6*c - 12*a*b^4*c^2 + 42*a^2*b^2*c^3 - 40*a^3*c
^4)*e^4)*x^3 - (18*(b^3*c^4 - 4*a*b*c^5)*d^4 - 36*(b^4*c^3 - 4*a*b^2*c^4)*d^3*e + 18*(b^5*c^2 - 2*a*b^3*c^3 -
8*a^2*b*c^4)*d^2*e^2 - 4*(b^6*c - 3*a*b^4*c^2 + 12*a^2*b^2*c^3 - 64*a^3*c^4)*d*e^3 - (b^7 - 12*a*b^5*c + 30*a^
2*b^3*c^2 + 8*a^3*b*c^3)*e^4)*x^2 - 12*(a^2*c^4*d^4 - 2*a^2*b*c^3*d^3*e - 2*a^3*b*c^2*d*e^3 + a^4*c^2*e^4 + (a
^2*b^2*c^2 + 2*a^3*c^3)*d^2*e^2 + (c^6*d^4 - 2*b*c^5*d^3*e - 2*a*b*c^4*d*e^3 + a^2*c^4*e^4 + (b^2*c^4 + 2*a*c^
5)*d^2*e^2)*x^4 + 2*(b*c^5*d^4 - 2*b^2*c^4*d^3*e - 2*a*b^2*c^3*d*e^3 + a^2*b*c^3*e^4 + (b^3*c^3 + 2*a*b*c^4)*d
^2*e^2)*x^3 + ((b^2*c^4 + 2*a*c^5)*d^4 - 2*(b^3*c^3 + 2*a*b*c^4)*d^3*e + (b^4*c^2 + 4*a*b^2*c^3 + 4*a^2*c^4)*d
^2*e^2 - 2*(a*b^3*c^2 + 2*a^2*b*c^3)*d*e^3 + (a^2*b^2*c^2 + 2*a^3*c^3)*e^4)*x^2 + 2*(a*b*c^4*d^4 - 2*a*b^2*c^3
*d^3*e - 2*a^2*b^2*c^2*d*e^3 + a^3*b*c^2*e^4 + (a*b^3*c^2 + 2*a^2*b*c^3)*d^2*e^2)*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*(2*(b^4*c^3 + a*b^2*c^
4 - 20*a^2*c^5)*d^4 - 4*(b^5*c^2 + a*b^3*c^3 - 20*a^2*b*c^4)*d^3*e + 6*(5*a*b^4*c^2 - 22*a^2*b^2*c^3 + 8*a^3*c
^4)*d^2*e^2 - 4*(a*b^5*c + a^2*b^3*c^2 - 20*a^3*b*c^3)*d*e^3 - (a*b^6 - 14*a^2*b^4*c + 46*a^3*b^2*c^2 - 24*a^4
*c^3)*e^4)*x)/(a^2*b^6*c^2 - 12*a^3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5 + (b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b
^2*c^6 - 64*a^3*c^7)*x^4 + 2*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^3 + (b^8*c^2 - 10*a*b^
6*c^3 + 24*a^2*b^4*c^4 + 32*a^3*b^2*c^5 - 128*a^4*c^6)*x^2 + 2*(a*b^7*c^2 - 12*a^2*b^5*c^3 + 48*a^3*b^3*c^4 -
64*a^4*b*c^5)*x), -1/2*((b^5*c^2 - 14*a*b^3*c^3 + 40*a^2*b*c^4)*d^4 + 4*(a*b^4*c^2 + 4*a^2*b^2*c^3 - 32*a^3*c^
4)*d^3*e - 36*(a^2*b^3*c^2 - 4*a^3*b*c^3)*d^2*e^2 + 4*(a^2*b^4*c + 4*a^3*b^2*c^2 - 32*a^4*c^3)*d*e^3 + (a^2*b^
5 - 14*a^3*b^3*c + 40*a^4*b*c^2)*e^4 - 2*(6*(b^2*c^5 - 4*a*c^6)*d^4 - 12*(b^3*c^4 - 4*a*b*c^5)*d^3*e + 6*(b^4*
c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d^2*e^2 - 12*(a*b^3*c^3 - 4*a^2*b*c^4)*d*e^3 - (b^6*c - 12*a*b^4*c^2 + 42*a^2*b
^2*c^3 - 40*a^3*c^4)*e^4)*x^3 - (18*(b^3*c^4 - 4*a*b*c^5)*d^4 - 36*(b^4*c^3 - 4*a*b^2*c^4)*d^3*e + 18*(b^5*c^2
- 2*a*b^3*c^3 - 8*a^2*b*c^4)*d^2*e^2 - 4*(b^6*c - 3*a*b^4*c^2 + 12*a^2*b^2*c^3 - 64*a^3*c^4)*d*e^3 - (b^7 - 1
2*a*b^5*c + 30*a^2*b^3*c^2 + 8*a^3*b*c^3)*e^4)*x^2 + 24*(a^2*c^4*d^4 - 2*a^2*b*c^3*d^3*e - 2*a^3*b*c^2*d*e^3 +
a^4*c^2*e^4 + (a^2*b^2*c^2 + 2*a^3*c^3)*d^2*e^2 + (c^6*d^4 - 2*b*c^5*d^3*e - 2*a*b*c^4*d*e^3 + a^2*c^4*e^4 +
(b^2*c^4 + 2*a*c^5)*d^2*e^2)*x^4 + 2*(b*c^5*d^4 - 2*b^2*c^4*d^3*e - 2*a*b^2*c^3*d*e^3 + a^2*b*c^3*e^4 + (b^3*c
^3 + 2*a*b*c^4)*d^2*e^2)*x^3 + ((b^2*c^4 + 2*a*c^5)*d^4 - 2*(b^3*c^3 + 2*a*b*c^4)*d^3*e + (b^4*c^2 + 4*a*b^2*c
^3 + 4*a^2*c^4)*d^2*e^2 - 2*(a*b^3*c^2 + 2*a^2*b*c^3)*d*e^3 + (a^2*b^2*c^2 + 2*a^3*c^3)*e^4)*x^2 + 2*(a*b*c^4*
d^4 - 2*a*b^2*c^3*d^3*e - 2*a^2*b^2*c^2*d*e^3 + a^3*b*c^2*e^4 + (a*b^3*c^2 + 2*a^2*b*c^3)*d^2*e^2)*x)*sqrt(-b^
2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - 2*(2*(b^4*c^3 + a*b^2*c^4 - 20*a^2*c^5)*d^4
- 4*(b^5*c^2 + a*b^3*c^3 - 20*a^2*b*c^4)*d^3*e + 6*(5*a*b^4*c^2 - 22*a^2*b^2*c^3 + 8*a^3*c^4)*d^2*e^2 - 4*(a*
b^5*c + a^2*b^3*c^2 - 20*a^3*b*c^3)*d*e^3 - (a*b^6 - 14*a^2*b^4*c + 46*a^3*b^2*c^2 - 24*a^4*c^3)*e^4)*x)/(a^2*
b^6*c^2 - 12*a^3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5 + (b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7
)*x^4 + 2*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^3 + (b^8*c^2 - 10*a*b^6*c^3 + 24*a^2*b^4*
c^4 + 32*a^3*b^2*c^5 - 128*a^4*c^6)*x^2 + 2*(a*b^7*c^2 - 12*a^2*b^5*c^3 + 48*a^3*b^3*c^4 - 64*a^4*b*c^5)*x)]

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Sympy [B]  time = 38.2636, size = 1355, normalized size = 8.02 \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)**4/(c*x**2+b*x+a)**3,x)

[Out]

-6*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2*log(x + (-384*a**3*c**3*sqrt(-1/(4*a*c - b**2)**5)*
(a*e**2 - b*d*e + c*d**2)**2 + 288*a**2*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 + 6*
a**2*b*e**4 - 72*a*b**4*c*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 - 12*a*b**2*d*e**3 + 12*a*b*
c*d**2*e**2 + 6*b**6*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 + 6*b**3*d**2*e**2 - 12*b**2*c*d*
*3*e + 6*b*c**2*d**4)/(12*a**2*c*e**4 - 24*a*b*c*d*e**3 + 24*a*c**2*d**2*e**2 + 12*b**2*c*d**2*e**2 - 24*b*c**
2*d**3*e + 12*c**3*d**4)) + 6*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2*log(x + (384*a**3*c**3*s
qrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 - 288*a**2*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2
- b*d*e + c*d**2)**2 + 6*a**2*b*e**4 + 72*a*b**4*c*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 -
12*a*b**2*d*e**3 + 12*a*b*c*d**2*e**2 - 6*b**6*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 + 6*b**
3*d**2*e**2 - 12*b**2*c*d**3*e + 6*b*c**2*d**4)/(12*a**2*c*e**4 - 24*a*b*c*d*e**3 + 24*a*c**2*d**2*e**2 + 12*b
**2*c*d**2*e**2 - 24*b*c**2*d**3*e + 12*c**3*d**4)) - (-10*a**3*b*c*e**4 + 32*a**3*c**2*d*e**3 + a**2*b**3*e**
4 + 4*a**2*b**2*c*d*e**3 - 36*a**2*b*c**2*d**2*e**2 + 32*a**2*c**3*d**3*e + 4*a*b**2*c**2*d**3*e - 10*a*b*c**3
*d**4 + b**3*c**2*d**4 + x**3*(20*a**2*c**3*e**4 - 16*a*b**2*c**2*e**4 + 24*a*b*c**3*d*e**3 - 24*a*c**4*d**2*e
**2 + 2*b**4*c*e**4 - 12*b**2*c**3*d**2*e**2 + 24*b*c**4*d**3*e - 12*c**5*d**4) + x**2*(-2*a**2*b*c**2*e**4 +
64*a**2*c**3*d*e**3 - 8*a*b**3*c*e**4 + 4*a*b**2*c**2*d*e**3 - 36*a*b*c**3*d**2*e**2 + b**5*e**4 + 4*b**4*c*d*
e**3 - 18*b**3*c**2*d**2*e**2 + 36*b**2*c**3*d**3*e - 18*b*c**4*d**4) + x*(12*a**3*c**2*e**4 - 20*a**2*b**2*c*
e**4 + 40*a**2*b*c**2*d*e**3 + 24*a**2*c**3*d**2*e**2 + 2*a*b**4*e**4 + 8*a*b**3*c*d*e**3 - 60*a*b**2*c**2*d**
2*e**2 + 40*a*b*c**3*d**3*e - 20*a*c**4*d**4 + 8*b**3*c**2*d**3*e - 4*b**2*c**3*d**4))/(32*a**4*c**4 - 16*a**3
*b**2*c**3 + 2*a**2*b**4*c**2 + x**4*(32*a**2*c**6 - 16*a*b**2*c**5 + 2*b**4*c**4) + x**3*(64*a**2*b*c**5 - 32
*a*b**3*c**4 + 4*b**5*c**3) + x**2*(64*a**3*c**5 - 12*a*b**4*c**3 + 2*b**6*c**2) + x*(64*a**3*b*c**4 - 32*a**2
*b**3*c**3 + 4*a*b**5*c**2))

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Giac [B]  time = 1.13065, size = 845, normalized size = 5. \begin{align*} \frac{12 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{12 \, c^{5} d^{4} x^{3} - 24 \, b c^{4} d^{3} x^{3} e + 18 \, b c^{4} d^{4} x^{2} + 12 \, b^{2} c^{3} d^{2} x^{3} e^{2} + 24 \, a c^{4} d^{2} x^{3} e^{2} - 36 \, b^{2} c^{3} d^{3} x^{2} e + 4 \, b^{2} c^{3} d^{4} x + 20 \, a c^{4} d^{4} x - 24 \, a b c^{3} d x^{3} e^{3} + 18 \, b^{3} c^{2} d^{2} x^{2} e^{2} + 36 \, a b c^{3} d^{2} x^{2} e^{2} - 8 \, b^{3} c^{2} d^{3} x e - 40 \, a b c^{3} d^{3} x e - b^{3} c^{2} d^{4} + 10 \, a b c^{3} d^{4} - 2 \, b^{4} c x^{3} e^{4} + 16 \, a b^{2} c^{2} x^{3} e^{4} - 20 \, a^{2} c^{3} x^{3} e^{4} - 4 \, b^{4} c d x^{2} e^{3} - 4 \, a b^{2} c^{2} d x^{2} e^{3} - 64 \, a^{2} c^{3} d x^{2} e^{3} + 60 \, a b^{2} c^{2} d^{2} x e^{2} - 24 \, a^{2} c^{3} d^{2} x e^{2} - 4 \, a b^{2} c^{2} d^{3} e - 32 \, a^{2} c^{3} d^{3} e - b^{5} x^{2} e^{4} + 8 \, a b^{3} c x^{2} e^{4} + 2 \, a^{2} b c^{2} x^{2} e^{4} - 8 \, a b^{3} c d x e^{3} - 40 \, a^{2} b c^{2} d x e^{3} + 36 \, a^{2} b c^{2} d^{2} e^{2} - 2 \, a b^{4} x e^{4} + 20 \, a^{2} b^{2} c x e^{4} - 12 \, a^{3} c^{2} x e^{4} - 4 \, a^{2} b^{2} c d e^{3} - 32 \, a^{3} c^{2} d e^{3} - a^{2} b^{3} e^{4} + 10 \, a^{3} b c e^{4}}{2 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )}{\left (c x^{2} + b x + a\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

12*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*arctan((2*c*x + b)/sqrt(-b^2
+ 4*a*c))/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(-b^2 + 4*a*c)) + 1/2*(12*c^5*d^4*x^3 - 24*b*c^4*d^3*x^3*e + 18*
b*c^4*d^4*x^2 + 12*b^2*c^3*d^2*x^3*e^2 + 24*a*c^4*d^2*x^3*e^2 - 36*b^2*c^3*d^3*x^2*e + 4*b^2*c^3*d^4*x + 20*a*
c^4*d^4*x - 24*a*b*c^3*d*x^3*e^3 + 18*b^3*c^2*d^2*x^2*e^2 + 36*a*b*c^3*d^2*x^2*e^2 - 8*b^3*c^2*d^3*x*e - 40*a*
b*c^3*d^3*x*e - b^3*c^2*d^4 + 10*a*b*c^3*d^4 - 2*b^4*c*x^3*e^4 + 16*a*b^2*c^2*x^3*e^4 - 20*a^2*c^3*x^3*e^4 - 4
*b^4*c*d*x^2*e^3 - 4*a*b^2*c^2*d*x^2*e^3 - 64*a^2*c^3*d*x^2*e^3 + 60*a*b^2*c^2*d^2*x*e^2 - 24*a^2*c^3*d^2*x*e^
2 - 4*a*b^2*c^2*d^3*e - 32*a^2*c^3*d^3*e - b^5*x^2*e^4 + 8*a*b^3*c*x^2*e^4 + 2*a^2*b*c^2*x^2*e^4 - 8*a*b^3*c*d
*x*e^3 - 40*a^2*b*c^2*d*x*e^3 + 36*a^2*b*c^2*d^2*e^2 - 2*a*b^4*x*e^4 + 20*a^2*b^2*c*x*e^4 - 12*a^3*c^2*x*e^4 -
4*a^2*b^2*c*d*e^3 - 32*a^3*c^2*d*e^3 - a^2*b^3*e^4 + 10*a^3*b*c*e^4)/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*(c
*x^2 + b*x + a)^2)