3.22.91 \(\int \frac {(d+e x)^4}{(a+b x+c x^2)^2} \, dx\) [2191]

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

[Out]

2*e^2*(3*c^2*d^2+b^2*e^2-c*e*(3*a*e+2*b*d))*x/c^2/(-4*a*c+b^2)+e^3*(-b*e+2*c*d)*x^2/c/(-4*a*c+b^2)-(e*x+d)^3*(
b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)+2*(2*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(-3*a*e+b*d)-6*a*c^2*e^3
*(a*e+2*b*d)+2*b^2*c*e^3*(3*a*e+b*d))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^3/(-4*a*c+b^2)^(3/2)+e^3*(-b*e+2
*c*d)*ln(c*x^2+b*x+a)/c^3

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Rubi [A]
time = 0.38, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {752, 814, 648, 632, 212, 642} \begin {gather*} \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 {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 {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 (2 c d-b e) \log \left (a+b x+c x^2\right )}{c^3} \end {gather*}

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 212

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

Rule 632

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 642

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 648

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 752

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 814

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]

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 ) \text {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*}

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Mathematica [A]
time = 0.29, size = 298, normalized size = 1.15 \begin {gather*} \frac {c e^4 x+\frac {-b^4 e^4 x+b^3 e^3 (-a e+4 c d x)+2 b^2 c e^2 \left (-3 c d^2 x+2 a e (d+e x)\right )-b c \left (-3 a^2 e^4+c^2 d^3 (d-4 e x)+6 a c d e^2 (d+2 e x)\right )-2 c^2 \left (c^2 d^4 x+a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\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 ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+e^3 (2 c d-b e) \log (a+x (b+c x))}{c^3} \end {gather*}

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 [A]
time = 0.97, size = 436, normalized size = 1.68

method	result	size
default	\(\frac {e^{4} x}{c^{2}}-\frac {\frac {-\frac {\left (2 e^{4} a^{2} c^{2}-4 a \,b^{2} c \,e^{4}+12 a b \,c^{2} d \,e^{3}-12 d^{2} e^{2} c^{3} a +b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +2 c^{4} d^{4}\right ) x}{c \left (4 a c -b^{2}\right )}+\frac {3 a^{2} b c \,e^{4}-8 a^{2} c^{2} d \,e^{3}-a \,b^{3} e^{4}+4 a \,b^{2} c d \,e^{3}-6 a b \,c^{2} d^{2} e^{2}+8 a \,c^{3} d^{3} e -d^{4} b \,c^{3}}{c \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {\frac {\left (4 a b c \,e^{4}-8 d \,e^{3} c^{2} a -b^{3} e^{4}+2 b^{2} d \,e^{3} c \right ) \ln \left (c \,x^{2}+b x +a \right )}{c}+\frac {4 \left (3 e^{4} a^{2} c -a \,b^{2} e^{4}+2 a b c d \,e^{3}-6 d^{2} e^{2} c^{2} a +2 d^{3} e b \,c^{2}-d^{4} c^{3}-\frac {\left (4 a b c \,e^{4}-8 d \,e^{3} c^{2} a -b^{3} e^{4}+2 b^{2} d \,e^{3} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{c^{2}}\)	\(436\)
risch	\(\text {Expression too large to display}\)	\(5644\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((e*x+d)^4/(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 deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 878 vs. \(2 (260) = 520\).
time = 2.23, size = 1776, normalized size = 6.83 \begin {gather*} \left [-\frac {2 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{4} x + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{4} - {\left (2 \, c^{5} d^{4} x^{2} + 2 \, b c^{4} d^{4} x + 2 \, a c^{4} d^{4} - {\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2} + {\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x\right )} e^{4} + 2 \, {\left ({\left (b^{3} c^{2} - 6 \, a b c^{3}\right )} d x^{2} + {\left (b^{4} c - 6 \, a b^{2} c^{2}\right )} d x + {\left (a b^{3} c - 6 \, a^{2} b c^{2}\right )} d\right )} e^{3} + 12 \, {\left (a c^{4} d^{2} x^{2} + a b c^{3} d^{2} x + a^{2} c^{3} d^{2}\right )} e^{2} - 4 \, {\left (b c^{4} d^{3} x^{2} + b^{2} c^{3} d^{3} x + a b c^{3} d^{3}\right )} e\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + {\left (a b^{5} - 7 \, a^{2} b^{3} c + 12 \, a^{3} b c^{2} - {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{3} - {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2} + {\left (b^{6} - 9 \, a b^{4} c + 26 \, a^{2} b^{2} c^{2} - 24 \, a^{3} c^{3}\right )} x\right )} e^{4} - 4 \, {\left ({\left (b^{5} c - 7 \, a b^{3} c^{2} + 12 \, a^{2} b c^{3}\right )} d x + {\left (a b^{4} c - 6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} c^{3}\right )} d\right )} e^{3} + 6 \, {\left ({\left (b^{4} c^{2} - 6 \, a b^{2} c^{3} + 8 \, a^{2} c^{4}\right )} d^{2} x + {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} d^{2}\right )} e^{2} - 4 \, {\left ({\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{3} x + 2 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4}\right )} d^{3}\right )} e + {\left ({\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x\right )} e^{4} - 2 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x + {\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} d\right )} e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{a b^{4} c^{3} - 8 \, a^{2} b^{2} c^{4} + 16 \, a^{3} c^{5} + {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} x^{2} + {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} x}, -\frac {2 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{4} x + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{4} - 2 \, {\left (2 \, c^{5} d^{4} x^{2} + 2 \, b c^{4} d^{4} x + 2 \, a c^{4} d^{4} - {\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2} + {\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x\right )} e^{4} + 2 \, {\left ({\left (b^{3} c^{2} - 6 \, a b c^{3}\right )} d x^{2} + {\left (b^{4} c - 6 \, a b^{2} c^{2}\right )} d x + {\left (a b^{3} c - 6 \, a^{2} b c^{2}\right )} d\right )} e^{3} + 12 \, {\left (a c^{4} d^{2} x^{2} + a b c^{3} d^{2} x + a^{2} c^{3} d^{2}\right )} e^{2} - 4 \, {\left (b c^{4} d^{3} x^{2} + b^{2} c^{3} d^{3} x + a b c^{3} d^{3}\right )} e\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left (a b^{5} - 7 \, a^{2} b^{3} c + 12 \, a^{3} b c^{2} - {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{3} - {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2} + {\left (b^{6} - 9 \, a b^{4} c + 26 \, a^{2} b^{2} c^{2} - 24 \, a^{3} c^{3}\right )} x\right )} e^{4} - 4 \, {\left ({\left (b^{5} c - 7 \, a b^{3} c^{2} + 12 \, a^{2} b c^{3}\right )} d x + {\left (a b^{4} c - 6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} c^{3}\right )} d\right )} e^{3} + 6 \, {\left ({\left (b^{4} c^{2} - 6 \, a b^{2} c^{3} + 8 \, a^{2} c^{4}\right )} d^{2} x + {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} d^{2}\right )} e^{2} - 4 \, {\left ({\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{3} x + 2 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4}\right )} d^{3}\right )} e + {\left ({\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x\right )} e^{4} - 2 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x + {\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} d\right )} e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{a b^{4} c^{3} - 8 \, a^{2} b^{2} c^{4} + 16 \, a^{3} c^{5} + {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} x^{2} + {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} x}\right ] \end {gather*}

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]

[-(2*(b^2*c^4 - 4*a*c^5)*d^4*x + (b^3*c^3 - 4*a*b*c^4)*d^4 - (2*c^5*d^4*x^2 + 2*b*c^4*d^4*x + 2*a*c^4*d^4 - (a
*b^4 - 6*a^2*b^2*c + 6*a^3*c^2 + (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*x^2 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*x)*e^
4 + 2*((b^3*c^2 - 6*a*b*c^3)*d*x^2 + (b^4*c - 6*a*b^2*c^2)*d*x + (a*b^3*c - 6*a^2*b*c^2)*d)*e^3 + 12*(a*c^4*d^
2*x^2 + a*b*c^3*d^2*x + a^2*c^3*d^2)*e^2 - 4*(b*c^4*d^3*x^2 + b^2*c^3*d^3*x + a*b*c^3*d^3)*e)*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)) + (a*b^5 - 7*a^2*
b^3*c + 12*a^3*b*c^2 - (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^3 - (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^2 + (
b^6 - 9*a*b^4*c + 26*a^2*b^2*c^2 - 24*a^3*c^3)*x)*e^4 - 4*((b^5*c - 7*a*b^3*c^2 + 12*a^2*b*c^3)*d*x + (a*b^4*c
- 6*a^2*b^2*c^2 + 8*a^3*c^3)*d)*e^3 + 6*((b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2*x + (a*b^3*c^2 - 4*a^2*b*c^3
)*d^2)*e^2 - 4*((b^3*c^3 - 4*a*b*c^4)*d^3*x + 2*(a*b^2*c^3 - 4*a^2*c^4)*d^3)*e + ((a*b^5 - 8*a^2*b^3*c + 16*a^
3*b*c^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^2 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*x)*e^4 - 2*((b^4*c^2 -
8*a*b^2*c^3 + 16*a^2*c^4)*d*x^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*x + (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^
3*c^3)*d)*e^3)*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), -(2*(b^2*c^4 - 4*a*c^5)*d^4*x + (b^3*c^3 - 4*a*b*c^4)*d
^4 - 2*(2*c^5*d^4*x^2 + 2*b*c^4*d^4*x + 2*a*c^4*d^4 - (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2 + (b^4*c - 6*a*b^2*c^2
+ 6*a^2*c^3)*x^2 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*x)*e^4 + 2*((b^3*c^2 - 6*a*b*c^3)*d*x^2 + (b^4*c - 6*a*b^2*
c^2)*d*x + (a*b^3*c - 6*a^2*b*c^2)*d)*e^3 + 12*(a*c^4*d^2*x^2 + a*b*c^3*d^2*x + a^2*c^3*d^2)*e^2 - 4*(b*c^4*d^
3*x^2 + b^2*c^3*d^3*x + a*b*c^3*d^3)*e)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c
)) + (a*b^5 - 7*a^2*b^3*c + 12*a^3*b*c^2 - (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^3 - (b^5*c - 8*a*b^3*c^2 + 1
6*a^2*b*c^3)*x^2 + (b^6 - 9*a*b^4*c + 26*a^2*b^2*c^2 - 24*a^3*c^3)*x)*e^4 - 4*((b^5*c - 7*a*b^3*c^2 + 12*a^2*b
*c^3)*d*x + (a*b^4*c - 6*a^2*b^2*c^2 + 8*a^3*c^3)*d)*e^3 + 6*((b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2*x + (a*b
^3*c^2 - 4*a^2*b*c^3)*d^2)*e^2 - 4*((b^3*c^3 - 4*a*b*c^4)*d^3*x + 2*(a*b^2*c^3 - 4*a^2*c^4)*d^3)*e + ((a*b^5 -
8*a^2*b^3*c + 16*a^3*b*c^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^2 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*x)
*e^4 - 2*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*x^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*x + (a*b^4*c - 8
*a^2*b^2*c^2 + 16*a^3*c^3)*d)*e^3)*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)]

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1924 vs. \(2 (252) = 504\).
time = 12.53, size = 1924, normalized size = 7.40 \begin {gather*} \left (- \frac {e^{3} \left (b e - 2 c d\right )}{c^{3}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (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}\right )}{c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) \log {\left (x + \frac {- 10 a^{2} b c e^{4} - 16 a^{2} c^{4} \left (- \frac {e^{3} \left (b e - 2 c d\right )}{c^{3}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (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}\right )}{c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 32 a^{2} c^{2} d e^{3} + 2 a b^{3} e^{4} + 8 a b^{2} c^{3} \left (- \frac {e^{3} \left (b e - 2 c d\right )}{c^{3}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (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}\right )}{c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) - 4 a b^{2} c d e^{3} - 12 a b c^{2} d^{2} e^{2} - b^{4} c^{2} \left (- \frac {e^{3} \left (b e - 2 c d\right )}{c^{3}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (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}\right )}{c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 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}} \right )} + \left (- \frac {e^{3} \left (b e - 2 c d\right )}{c^{3}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (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}\right )}{c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) \log {\left (x + \frac {- 10 a^{2} b c e^{4} - 16 a^{2} c^{4} \left (- \frac {e^{3} \left (b e - 2 c d\right )}{c^{3}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (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}\right )}{c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 32 a^{2} c^{2} d e^{3} + 2 a b^{3} e^{4} + 8 a b^{2} c^{3} \left (- \frac {e^{3} \left (b e - 2 c d\right )}{c^{3}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (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}\right )}{c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) - 4 a b^{2} c d e^{3} - 12 a b c^{2} d^{2} e^{2} - b^{4} c^{2} \left (- \frac {e^{3} \left (b e - 2 c d\right )}{c^{3}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (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}\right )}{c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 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}} \right )} + \frac {- 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 \left (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}\right )}{4 a^{2} c^{4} - a b^{2} c^{3} + x^{2} \cdot \left (4 a c^{5} - b^{2} c^{4}\right ) + x \left (4 a b c^{4} - b^{3} c^{3}\right )} + \frac {e^{4} x}{c^{2}} \end {gather*}

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 = 0.99, size = 355, normalized size = 1.37 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 1.87, size = 525, normalized size = 2.02 \begin {gather*} \frac {\frac {-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}{c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (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\right )}{c\,\left (4\,a\,c-b^2\right )}}{c^3\,x^2+b\,c^2\,x+a\,c^2}+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-128\,a^3\,b\,c^3\,e^4+256\,d\,a^3\,c^4\,e^3+96\,a^2\,b^3\,c^2\,e^4-192\,d\,a^2\,b^2\,c^3\,e^3-24\,a\,b^5\,c\,e^4+48\,d\,a\,b^4\,c^2\,e^3+2\,b^7\,e^4-4\,d\,b^6\,c\,e^3\right )}{2\,\left (64\,a^3\,c^6-48\,a^2\,b^2\,c^5+12\,a\,b^4\,c^4-b^6\,c^3\right )}+\frac {e^4\,x}{c^2}-\frac {2\,\mathrm {atan}\left (\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}-\frac {b^3\,c^2-4\,a\,b\,c^3}{c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (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\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*b^3*e^4 + b*c^3*d^4 + 8*a^2*c^2*d*e^3 - 3*a^2*b*c*e^4 - 8*a*c^3*d^3*e - 4*a*b^2*c*d*e^3 + 6*a*b*c^2*d^2*e^
2)/(c*(4*a*c - b^2)) + (x*(b^4*e^4 + 2*c^4*d^4 + 2*a^2*c^2*e^4 - 12*a*c^3*d^2*e^2 + 6*b^2*c^2*d^2*e^2 - 4*a*b^
2*c*e^4 - 4*b*c^3*d^3*e - 4*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3))/(c*(4*a*c - b^2)))/(a*c^2 + c^3*x^2 + b*c^2*x) +
(log(a + b*x + c*x^2)*(2*b^7*e^4 - 128*a^3*b*c^3*e^4 + 256*a^3*c^4*d*e^3 + 96*a^2*b^3*c^2*e^4 - 24*a*b^5*c*e^4
- 4*b^6*c*d*e^3 + 48*a*b^4*c^2*d*e^3 - 192*a^2*b^2*c^3*d*e^3))/(2*(64*a^3*c^6 - b^6*c^3 + 12*a*b^4*c^4 - 48*a
^2*b^2*c^5)) + (e^4*x)/c^2 - (2*atan((2*c*x)/(4*a*c - b^2)^(1/2) - (b^3*c^2 - 4*a*b*c^3)/(c^2*(4*a*c - b^2)^(3
/2)))*(b^4*e^4 - 2*c^4*d^4 + 6*a^2*c^2*e^4 - 12*a*c^3*d^2*e^2 - 6*a*b^2*c*e^4 + 4*b*c^3*d^3*e - 2*b^3*c*d*e^3
+ 12*a*b*c^2*d*e^3))/(c^3*(4*a*c - b^2)^(3/2))

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