3.2191 \(\int \frac{(d+e x)^4}{(a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=260 \[ \frac{2 e^2 x \left (-c e (3 a e+2 b d)+b^2 e^2+3 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}+\frac{2 \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{e^3 x^2 (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{c^3} \]

[Out]

(2*e^2*(3*c^2*d^2 + b^2*e^2 - c*e*(2*b*d + 3*a*e))*x)/(c^2*(b^2 - 4*a*c)) + (e^3*(2*c*d - b*e)*x^2)/(c*(b^2 -
4*a*c)) - ((d + e*x)^3*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + (2*(2*c^4*d^4 - b^
4*e^4 - 4*c^3*d^2*e*(b*d - 3*a*e) - 6*a*c^2*e^3*(2*b*d + a*e) + 2*b^2*c*e^3*(b*d + 3*a*e))*ArcTanh[(b + 2*c*x)
/Sqrt[b^2 - 4*a*c]])/(c^3*(b^2 - 4*a*c)^(3/2)) + (e^3*(2*c*d - b*e)*Log[a + b*x + c*x^2])/c^3

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Rubi [A]  time = 0.567281, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {738, 800, 634, 618, 206, 628} \[ \frac{2 e^2 x \left (-c e (3 a e+2 b d)+b^2 e^2+3 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}+\frac{2 \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{e^3 x^2 (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e^2*(3*c^2*d^2 + b^2*e^2 - c*e*(2*b*d + 3*a*e))*x)/(c^2*(b^2 - 4*a*c)) + (e^3*(2*c*d - b*e)*x^2)/(c*(b^2 -
4*a*c)) - ((d + e*x)^3*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + (2*(2*c^4*d^4 - b^
4*e^4 - 4*c^3*d^2*e*(b*d - 3*a*e) - 6*a*c^2*e^3*(2*b*d + a*e) + 2*b^2*c*e^3*(b*d + 3*a*e))*ArcTanh[(b + 2*c*x)
/Sqrt[b^2 - 4*a*c]])/(c^3*(b^2 - 4*a*c)^(3/2)) + (e^3*(2*c*d - b*e)*Log[a + b*x + c*x^2])/c^3

Rule 738

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[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, 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, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

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 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 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])

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]

Rubi steps

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

Mathematica [A]  time = 0.495343, size = 298, normalized size = 1.15 \[ \frac{\frac{-b c \left (-3 a^2 e^4+6 a c d e^2 (d+2 e x)+c^2 d^3 (d-4 e x)\right )-2 c^2 \left (a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)+c^2 d^4 x\right )+2 b^2 c e^2 \left (2 a e (d+e x)-3 c d^2 x\right )+b^3 e^3 (4 c d x-a e)-b^4 e^4 x}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac{2 \left (-2 b^2 c e^3 (3 a e+b d)+4 c^3 d^2 e (b d-3 a e)+6 a c^2 e^3 (a e+2 b d)+b^4 e^4-2 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+e^3 (2 c d-b e) \log (a+x (b+c x))+c e^4 x}{c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.161, size = 1037, normalized size = 4. \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)^2,x)

[Out]

6/c/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b*d^2*e^2-4/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a*b^2*e^4-4/c^2/(c*x^2+b*x+a)/(4*a
*c-b^2)*x*b^3*d*e^3-4/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^2*d*e^3-24/c/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c
-b^2)^(1/2))*a*b*d*e^3-2/c^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4*e^4+24/(4*a*c-b^2)^(3/2
)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*d^2*e^2+12/c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a*b*d*e^3+e^4*x/c^2-8/(4*a*c-
b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*d^3*e+2*c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*d^4+4*c/(4*a*c-b^2)^(3/
2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d^4+1/(c*x^2+b*x+a)/(4*a*c-b^2)*d^4*b+6/c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*b
^2*d^2*e^2-3/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*b*e^4-4/c^2/(4*a*c-b^2)*ln(c*x^2+b*x+a)*a*b*e^4+4/c^2/(4*a*c-b^
2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*d*e^3-2/c^2/(4*a*c-b^2)*ln(c*x^2+b*x+a)*b^2*d*e^3+8/c/(4*a*c-
b^2)*ln(c*x^2+b*x+a)*a*d*e^3-12/c/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*e^4+1/c^3/(4*a*c-b
^2)*ln(c*x^2+b*x+a)*b^3*e^4-8/(c*x^2+b*x+a)/(4*a*c-b^2)*a*d^3*e+1/c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^3*e^4+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*e^4+1/c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*x*b^4*e^4+2
/c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a^2*e^4-12/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a*d^2*e^2-4/(c*x^2+b*x+a)/(4*a*c-b^2)*x*
b*d^3*e+8/c/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*d*e^3

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.73383, size = 3791, normalized size = 14.58 \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)^2,x, algorithm="fricas")

[Out]

[((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^4*x^3 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e^4*x^2 - (b^3*c^3 - 4*a
*b*c^4)*d^4 + 8*(a*b^2*c^3 - 4*a^2*c^4)*d^3*e - 6*(a*b^3*c^2 - 4*a^2*b*c^3)*d^2*e^2 + 4*(a*b^4*c - 6*a^2*b^2*c
^2 + 8*a^3*c^3)*d*e^3 - (a*b^5 - 7*a^2*b^3*c + 12*a^3*b*c^2)*e^4 + (2*a*c^4*d^4 - 4*a*b*c^3*d^3*e + 12*a^2*c^3
*d^2*e^2 + 2*(a*b^3*c - 6*a^2*b*c^2)*d*e^3 - (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*e^4 + (2*c^5*d^4 - 4*b*c^4*d^3*
e + 12*a*c^4*d^2*e^2 + 2*(b^3*c^2 - 6*a*b*c^3)*d*e^3 - (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*e^4)*x^2 + (2*b*c^4*d
^4 - 4*b^2*c^3*d^3*e + 12*a*b*c^3*d^2*e^2 + 2*(b^4*c - 6*a*b^2*c^2)*d*e^3 - (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*e^
4)*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*(b^2*c^4 - 4*a*c^5)*d^4 - 4*(b^3*c^3 - 4*a*b*c^4)*d^3*e + 6*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2*
e^2 - 4*(b^5*c - 7*a*b^3*c^2 + 12*a^2*b*c^3)*d*e^3 + (b^6 - 9*a*b^4*c + 26*a^2*b^2*c^2 - 24*a^3*c^3)*e^4)*x +
(2*(a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*d*e^3 - (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e^4 + (2*(b^4*c^2 - 8*a
*b^2*c^3 + 16*a^2*c^4)*d*e^3 - (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e^4)*x^2 + (2*(b^5*c - 8*a*b^3*c^2 + 16*a^
2*b*c^3)*d*e^3 - (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*e^4)*x)*log(c*x^2 + b*x + a))/(a*b^4*c^3 - 8*a^2*b^2*c^4 +
16*a^3*c^5 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^2 + (b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x), ((b^4*c^2
- 8*a*b^2*c^3 + 16*a^2*c^4)*e^4*x^3 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e^4*x^2 - (b^3*c^3 - 4*a*b*c^4)*d^4
+ 8*(a*b^2*c^3 - 4*a^2*c^4)*d^3*e - 6*(a*b^3*c^2 - 4*a^2*b*c^3)*d^2*e^2 + 4*(a*b^4*c - 6*a^2*b^2*c^2 + 8*a^3*
c^3)*d*e^3 - (a*b^5 - 7*a^2*b^3*c + 12*a^3*b*c^2)*e^4 + 2*(2*a*c^4*d^4 - 4*a*b*c^3*d^3*e + 12*a^2*c^3*d^2*e^2
+ 2*(a*b^3*c - 6*a^2*b*c^2)*d*e^3 - (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*e^4 + (2*c^5*d^4 - 4*b*c^4*d^3*e + 12*a*
c^4*d^2*e^2 + 2*(b^3*c^2 - 6*a*b*c^3)*d*e^3 - (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*e^4)*x^2 + (2*b*c^4*d^4 - 4*b^
2*c^3*d^3*e + 12*a*b*c^3*d^2*e^2 + 2*(b^4*c - 6*a*b^2*c^2)*d*e^3 - (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*e^4)*x)*sqr
t(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (2*(b^2*c^4 - 4*a*c^5)*d^4 - 4*(b^3*c^
3 - 4*a*b*c^4)*d^3*e + 6*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^2 - 4*(b^5*c - 7*a*b^3*c^2 + 12*a^2*b*c^3)*
d*e^3 + (b^6 - 9*a*b^4*c + 26*a^2*b^2*c^2 - 24*a^3*c^3)*e^4)*x + (2*(a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*d*e
^3 - (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e^4 + (2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*e^3 - (b^5*c - 8*a*b
^3*c^2 + 16*a^2*b*c^3)*e^4)*x^2 + (2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e^3 - (b^6 - 8*a*b^4*c + 16*a^2*b^
2*c^2)*e^4)*x)*log(c*x^2 + b*x + a))/(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2
*c^6)*x^2 + (b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x)]

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Sympy [B]  time = 13.0234, size = 1924, normalized size = 7.4 \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)**2,x)

[Out]

(-e**3*(b*e - 2*c*d)/c**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2*e**4 - 6*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 - 2*b**3*c*d*e**3 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(c**3*(64*a**3*c**3 - 48*
a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-10*a**2*b*c*e**4 - 16*a**2*c**4*(-e**3*(b*e - 2*c*d)/c**3 - s
qrt(-(4*a*c - b**2)**3)*(6*a**2*c**2*e**4 - 6*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 - 2*b**3*c*d*e**3 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c
- b**6))) + 32*a**2*c**2*d*e**3 + 2*a*b**3*e**4 + 8*a*b**2*c**3*(-e**3*(b*e - 2*c*d)/c**3 - sqrt(-(4*a*c - b**
2)**3)*(6*a**2*c**2*e**4 - 6*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 - 2*b**3*c*d
*e**3 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 4*a*b
**2*c*d*e**3 - 12*a*b*c**2*d**2*e**2 - b**4*c**2*(-e**3*(b*e - 2*c*d)/c**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*
c**2*e**4 - 6*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 - 2*b**3*c*d*e**3 + 4*b*c**
3*d**3*e - 2*c**4*d**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 4*b**2*c**2*d**3*e -
2*b*c**3*d**4)/(12*a**2*c**2*e**4 - 12*a*b**2*c*e**4 + 24*a*b*c**2*d*e**3 - 24*a*c**3*d**2*e**2 + 2*b**4*e**4
- 4*b**3*c*d*e**3 + 8*b*c**3*d**3*e - 4*c**4*d**4)) + (-e**3*(b*e - 2*c*d)/c**3 + sqrt(-(4*a*c - b**2)**3)*(6
*a**2*c**2*e**4 - 6*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 - 2*b**3*c*d*e**3 + 4
*b*c**3*d**3*e - 2*c**4*d**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-10*a**
2*b*c*e**4 - 16*a**2*c**4*(-e**3*(b*e - 2*c*d)/c**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2*e**4 - 6*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 - 2*b**3*c*d*e**3 + 4*b*c**3*d**3*e - 2*c**4*d**4)
/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 32*a**2*c**2*d*e**3 + 2*a*b**3*e**4 + 8*a*b
**2*c**3*(-e**3*(b*e - 2*c*d)/c**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2*e**4 - 6*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 - 2*b**3*c*d*e**3 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(c**3*(64*a**3*c
**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 4*a*b**2*c*d*e**3 - 12*a*b*c**2*d**2*e**2 - b**4*c**2*(-e**3
*(b*e - 2*c*d)/c**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2*e**4 - 6*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 - 2*b**3*c*d*e**3 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(c**3*(64*a**3*c**3 - 48*a**2*b
**2*c**2 + 12*a*b**4*c - b**6))) + 4*b**2*c**2*d**3*e - 2*b*c**3*d**4)/(12*a**2*c**2*e**4 - 12*a*b**2*c*e**4 +
24*a*b*c**2*d*e**3 - 24*a*c**3*d**2*e**2 + 2*b**4*e**4 - 4*b**3*c*d*e**3 + 8*b*c**3*d**3*e - 4*c**4*d**4)) +
(-3*a**2*b*c*e**4 + 8*a**2*c**2*d*e**3 + a*b**3*e**4 - 4*a*b**2*c*d*e**3 + 6*a*b*c**2*d**2*e**2 - 8*a*c**3*d**
3*e + b*c**3*d**4 + x*(2*a**2*c**2*e**4 - 4*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 - 4*b**3*c*d*e**3 + 6*b**2*c**2*d**2*e**2 - 4*b*c**3*d**3*e + 2*c**4*d**4))/(4*a**2*c**4 - a*b**2*c**3 + x*
*2*(4*a*c**5 - b**2*c**4) + x*(4*a*b*c**4 - b**3*c**3)) + e**4*x/c**2

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Giac [A]  time = 1.13373, size = 479, normalized size = 1.84 \begin{align*} -\frac{2 \,{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 12 \, a c^{3} d^{2} e^{2} + 2 \, b^{3} c d e^{3} - 12 \, a b c^{2} d e^{3} - b^{4} e^{4} + 6 \, a b^{2} c e^{4} - 6 \, a^{2} c^{2} e^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{x e^{4}}{c^{2}} + \frac{{\left (2 \, c d e^{3} - b e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{c^{3}} - \frac{\frac{{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 12 \, a c^{3} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + 12 \, a b c^{2} d e^{3} + b^{4} e^{4} - 4 \, a b^{2} c e^{4} + 2 \, a^{2} c^{2} e^{4}\right )} x}{c} + \frac{b c^{3} d^{4} - 8 \, a c^{3} d^{3} e + 6 \, a b c^{2} d^{2} e^{2} - 4 \, a b^{2} c d e^{3} + 8 \, a^{2} c^{2} d e^{3} + a b^{3} e^{4} - 3 \, a^{2} b c e^{4}}{c}}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} c^{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)^2,x, algorithm="giac")

[Out]

-2*(2*c^4*d^4 - 4*b*c^3*d^3*e + 12*a*c^3*d^2*e^2 + 2*b^3*c*d*e^3 - 12*a*b*c^2*d*e^3 - b^4*e^4 + 6*a*b^2*c*e^4
- 6*a^2*c^2*e^4)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^3 - 4*a*c^4)*sqrt(-b^2 + 4*a*c)) + x*e^4/c^2 +
(2*c*d*e^3 - b*e^4)*log(c*x^2 + b*x + a)/c^3 - ((2*c^4*d^4 - 4*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 - 12*a*c^3*d^2
*e^2 - 4*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3 + b^4*e^4 - 4*a*b^2*c*e^4 + 2*a^2*c^2*e^4)*x/c + (b*c^3*d^4 - 8*a*c^3*
d^3*e + 6*a*b*c^2*d^2*e^2 - 4*a*b^2*c*d*e^3 + 8*a^2*c^2*d*e^3 + a*b^3*e^4 - 3*a^2*b*c*e^4)/c)/((c*x^2 + b*x +
a)*(b^2 - 4*a*c)*c^2)