3.22.90 \(\int \frac {(d+e x)^5}{(a+b x+c x^2)^2} \, dx\) [2190]

Optimal. Leaf size=374 \[ \frac {e^2 \left (12 c^3 d^3-3 b^3 e^3-10 c^2 d e (b d+3 a e)+b c e^2 (10 b d+11 a e)\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac {e^3 \left (16 c^2 d^2+3 b^2 e^2-2 c e (5 b d+4 a e)\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {e^4 (2 c d-b e) x^3}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {(2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 c^4} \]

[Out]

e^2*(12*c^3*d^3-3*b^3*e^3-10*c^2*d*e*(3*a*e+b*d)+b*c*e^2*(11*a*e+10*b*d))*x/c^3/(-4*a*c+b^2)+1/2*e^3*(16*c^2*d
^2+3*b^2*e^2-2*c*e*(4*a*e+5*b*d))*x^2/c^2/(-4*a*c+b^2)+e^4*(-b*e+2*c*d)*x^3/c/(-4*a*c+b^2)-(e*x+d)^4*(b*d-2*a*
e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)+(-b*e+2*c*d)*(2*c^4*d^4-3*b^4*e^4-4*c^3*d^2*e*(-5*a*e+b*d)+4*b^2*
c*e^3*(5*a*e+b*d)-2*c^2*e^2*(15*a^2*e^2+10*a*b*d*e+b^2*d^2))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^4/(-4*a*c
+b^2)^(3/2)+1/2*e^3*(10*c^2*d^2+3*b^2*e^2-2*c*e*(a*e+5*b*d))*ln(c*x^2+b*x+a)/c^4

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Rubi [A]
time = 0.46, antiderivative size = 374, 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 c d-b e) \left (-2 c^2 e^2 \left (15 a^2 e^2+10 a b d e+b^2 d^2\right )+4 b^2 c e^3 (5 a e+b d)-4 c^3 d^2 e (b d-5 a e)-3 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^4 \left (b^2-4 a c\right )^{3/2}}+\frac {e^3 x^2 \left (-2 c e (4 a e+5 b d)+3 b^2 e^2+16 c^2 d^2\right )}{2 c^2 \left (b^2-4 a c\right )}+\frac {e^3 \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac {e^4 x^3 (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (-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^2 x \left (-10 c^2 d e (3 a e+b d)+b c e^2 (11 a e+10 b d)-3 b^3 e^3+12 c^3 d^3\right )}{c^3 \left (b^2-4 a c\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5/(a + b*x + c*x^2)^2,x]

[Out]

(e^2*(12*c^3*d^3 - 3*b^3*e^3 - 10*c^2*d*e*(b*d + 3*a*e) + b*c*e^2*(10*b*d + 11*a*e))*x)/(c^3*(b^2 - 4*a*c)) +
(e^3*(16*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + 4*a*e))*x^2)/(2*c^2*(b^2 - 4*a*c)) + (e^4*(2*c*d - b*e)*x^3)/(c*
(b^2 - 4*a*c)) - ((d + e*x)^4*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + ((2*c*d - b
*e)*(2*c^4*d^4 - 3*b^4*e^4 - 4*c^3*d^2*e*(b*d - 5*a*e) + 4*b^2*c*e^3*(b*d + 5*a*e) - 2*c^2*e^2*(b^2*d^2 + 10*a
*b*d*e + 15*a^2*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(3/2)) + (e^3*(10*c^2*d^2 + 3
*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[a + b*x + c*x^2])/(2*c^4)

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)^5}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {(d+e x)^4 (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)^3 \left (2 c d^2-e (5 b d-8 a e)-3 e (2 c d-b e) x\right )}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=-\frac {(d+e x)^4 (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 {e^2 \left (12 c^3 d^3-3 b^3 e^3-10 c^2 d e (b d+3 a e)+b c e^2 (10 b d+11 a e)\right )}{c^3}-\frac {e^3 \left (16 c^2 d^2+3 b^2 e^2-2 c e (5 b d+4 a e)\right ) x}{c^2}-\frac {3 e^4 (2 c d-b e) x^2}{c}+\frac {2 c^4 d^5-3 a b^3 e^5-5 c^3 d^3 e (b d-4 a e)-10 a c^2 d e^3 (b d+3 a e)+a b c e^4 (10 b d+11 a e)-\left (b^2-4 a c\right ) e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx}{-b^2+4 a c}\\ &=\frac {e^2 \left (12 c^3 d^3-3 b^3 e^3-10 c^2 d e (b d+3 a e)+b c e^2 (10 b d+11 a e)\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac {e^3 \left (16 c^2 d^2+3 b^2 e^2-2 c e (5 b d+4 a e)\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {e^4 (2 c d-b e) x^3}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (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 {2 c^4 d^5-3 a b^3 e^5-5 c^3 d^3 e (b d-4 a e)-10 a c^2 d e^3 (b d+3 a e)+a b c e^4 (10 b d+11 a e)-\left (b^2-4 a c\right ) e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) x}{a+b x+c x^2} \, dx}{c^3 \left (b^2-4 a c\right )}\\ &=\frac {e^2 \left (12 c^3 d^3-3 b^3 e^3-10 c^2 d e (b d+3 a e)+b c e^2 (10 b d+11 a e)\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac {e^3 \left (16 c^2 d^2+3 b^2 e^2-2 c e (5 b d+4 a e)\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {e^4 (2 c d-b e) x^3}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (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 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right )\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}-\frac {\left ((2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^4 \left (b^2-4 a c\right )}\\ &=\frac {e^2 \left (12 c^3 d^3-3 b^3 e^3-10 c^2 d e (b d+3 a e)+b c e^2 (10 b d+11 a e)\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac {e^3 \left (16 c^2 d^2+3 b^2 e^2-2 c e (5 b d+4 a e)\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {e^4 (2 c d-b e) x^3}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (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 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac {\left ((2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\right )\right ) \text {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 {e^2 \left (12 c^3 d^3-3 b^3 e^3-10 c^2 d e (b d+3 a e)+b c e^2 (10 b d+11 a e)\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac {e^3 \left (16 c^2 d^2+3 b^2 e^2-2 c e (5 b d+4 a e)\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {e^4 (2 c d-b e) x^3}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (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 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\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 {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 c^4}\\ \end {align*}

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Mathematica [A]
time = 0.42, size = 422, normalized size = 1.13 \begin {gather*} \frac {2 c e^4 (5 c d-2 b e) x+c^2 e^5 x^2+\frac {2 \left (b^5 e^5 x+b^4 e^4 (a e-5 c d x)-5 b^3 c e^3 \left (-2 c d^2 x+a e (d+e x)\right )-2 b^2 c e^2 \left (2 a^2 e^3+5 c^2 d^3 x-5 a c d e (d+2 e x)\right )+2 c^2 \left (a^3 e^5-c^3 d^5 x-5 a^2 c d e^3 (2 d+e x)+5 a c^2 d^3 e (d+2 e x)\right )+b c^2 \left (-c^2 d^4 (d-5 e x)+5 a^2 e^4 (3 d+e x)-10 a c d^2 e^2 (d+3 e x)\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 (-2 c d+b e) \left (-2 c^4 d^4+3 b^4 e^4+4 c^3 d^2 e (b d-5 a e)-4 b^2 c e^3 (b d+5 a e)+2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\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 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (a+x (b+c x))}{2 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5/(a + b*x + c*x^2)^2,x]

[Out]

(2*c*e^4*(5*c*d - 2*b*e)*x + c^2*e^5*x^2 + (2*(b^5*e^5*x + b^4*e^4*(a*e - 5*c*d*x) - 5*b^3*c*e^3*(-2*c*d^2*x +
a*e*(d + e*x)) - 2*b^2*c*e^2*(2*a^2*e^3 + 5*c^2*d^3*x - 5*a*c*d*e*(d + 2*e*x)) + 2*c^2*(a^3*e^5 - c^3*d^5*x -
5*a^2*c*d*e^3*(2*d + e*x) + 5*a*c^2*d^3*e*(d + 2*e*x)) + b*c^2*(-(c^2*d^4*(d - 5*e*x)) + 5*a^2*e^4*(3*d + e*x
) - 10*a*c*d^2*e^2*(d + 3*e*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*(-2*c*d + b*e)*(-2*c^4*d^4 + 3*b^4*e^
4 + 4*c^3*d^2*e*(b*d - 5*a*e) - 4*b^2*c*e^3*(b*d + 5*a*e) + 2*c^2*e^2*(b^2*d^2 + 10*a*b*d*e + 15*a^2*e^2))*Arc
Tan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))
*Log[a + x*(b + c*x)])/(2*c^4)

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Maple [A]
time = 0.93, size = 645, normalized size = 1.72

method	result	size
default	\(-\frac {e^{4} \left (-\frac {1}{2} c e \,x^{2}+2 b e x -5 c d x \right )}{c^{3}}+\frac {\frac {-\frac {\left (5 a^{2} b \,c^{2} e^{5}-10 a^{2} c^{3} d \,e^{4}-5 a \,b^{3} c \,e^{5}+20 a \,b^{2} c^{2} d \,e^{4}-30 a b \,c^{3} d^{2} e^{3}+20 a \,c^{4} d^{3} e^{2}+b^{5} e^{5}-5 b^{4} c d \,e^{4}+10 b^{3} c^{2} d^{2} e^{3}-10 b^{2} c^{3} d^{3} e^{2}+5 b \,c^{4} d^{4} e -2 c^{5} d^{5}\right ) x}{c \left (4 a c -b^{2}\right )}-\frac {2 a^{3} c^{2} e^{5}-4 a^{2} b^{2} c \,e^{5}+15 a^{2} b \,c^{2} d \,e^{4}-20 a^{2} c^{3} d^{2} e^{3}+a \,b^{4} e^{5}-5 a \,b^{3} c d \,e^{4}+10 a \,b^{2} c^{2} d^{2} e^{3}-10 a b \,c^{3} d^{3} e^{2}+10 a \,c^{4} d^{4} e -b \,c^{4} d^{5}}{c \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {\frac {\left (-8 a^{2} c^{2} e^{5}+14 a \,b^{2} e^{5} c -40 a b \,c^{2} d \,e^{4}+40 e^{3} c^{3} d^{2} a -3 b^{4} e^{5}+10 b^{3} c d \,e^{4}-10 b^{2} c^{2} d^{2} e^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (11 a^{2} b \,e^{5} c -30 d \,e^{4} a^{2} c^{2}-3 a \,b^{3} e^{5}+10 a \,b^{2} c d \,e^{4}-10 a b \,c^{2} d^{2} e^{3}+20 a \,c^{3} d^{3} e^{2}-5 b \,c^{3} d^{4} e +2 c^{4} d^{5}-\frac {\left (-8 a^{2} c^{2} e^{5}+14 a \,b^{2} e^{5} c -40 a b \,c^{2} d \,e^{4}+40 e^{3} c^{3} d^{2} a -3 b^{4} e^{5}+10 b^{3} c d \,e^{4}-10 b^{2} c^{2} d^{2} e^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{c^{3}}\)	\(645\)
risch	\(\text {Expression too large to display}\)	\(11237\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

-e^4/c^3*(-1/2*c*e*x^2+2*b*e*x-5*c*d*x)+1/c^3*((-(5*a^2*b*c^2*e^5-10*a^2*c^3*d*e^4-5*a*b^3*c*e^5+20*a*b^2*c^2*
d*e^4-30*a*b*c^3*d^2*e^3+20*a*c^4*d^3*e^2+b^5*e^5-5*b^4*c*d*e^4+10*b^3*c^2*d^2*e^3-10*b^2*c^3*d^3*e^2+5*b*c^4*
d^4*e-2*c^5*d^5)/c/(4*a*c-b^2)*x-(2*a^3*c^2*e^5-4*a^2*b^2*c*e^5+15*a^2*b*c^2*d*e^4-20*a^2*c^3*d^2*e^3+a*b^4*e^
5-5*a*b^3*c*d*e^4+10*a*b^2*c^2*d^2*e^3-10*a*b*c^3*d^3*e^2+10*a*c^4*d^4*e-b*c^4*d^5)/c/(4*a*c-b^2))/(c*x^2+b*x+
a)+1/(4*a*c-b^2)*(1/2*(-8*a^2*c^2*e^5+14*a*b^2*c*e^5-40*a*b*c^2*d*e^4+40*a*c^3*d^2*e^3-3*b^4*e^5+10*b^3*c*d*e^
4-10*b^2*c^2*d^2*e^3)/c*ln(c*x^2+b*x+a)+2*(11*a^2*b*e^5*c-30*d*e^4*a^2*c^2-3*a*b^3*e^5+10*a*b^2*c*d*e^4-10*a*b
*c^2*d^2*e^3+20*a*c^3*d^3*e^2-5*b*c^3*d^4*e+2*c^4*d^5-1/2*(-8*a^2*c^2*e^5+14*a*b^2*c*e^5-40*a*b*c^2*d*e^4+40*a
*c^3*d^2*e^3-3*b^4*e^5+10*b^3*c*d*e^4-10*b^2*c^2*d^2*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)^5/(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. 1321 vs. \(2 (370) = 740\).
time = 2.52, size = 2662, normalized size = 7.12 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

[-1/2*(4*(b^2*c^5 - 4*a*c^6)*d^5*x + 2*(b^3*c^4 - 4*a*b*c^5)*d^5 + (4*c^6*d^5*x^2 + 4*b*c^5*d^5*x + 4*a*c^5*d^
5 + (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2 + (3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*x^2 + (3*b^6 - 20*a*b^4*c
+ 30*a^2*b^2*c^2)*x)*e^5 - 10*((b^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*d*x^2 + (b^5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3
)*d*x + (a*b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*d)*e^4 + 10*((b^3*c^3 - 6*a*b*c^4)*d^2*x^2 + (b^4*c^2 - 6*a*b^2*
c^3)*d^2*x + (a*b^3*c^2 - 6*a^2*b*c^3)*d^2)*e^3 + 40*(a*c^5*d^3*x^2 + a*b*c^4*d^3*x + a^2*c^4*d^3)*e^2 - 10*(b
*c^5*d^4*x^2 + b^2*c^4*d^4*x + a*b*c^4*d^4)*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)) - (2*a*b^6 - 16*a^2*b^4*c + 36*a^3*b^2*c^2 - 16*a^4*c^3 + (b^4*c
^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 - 3*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 - (4*b^6*c - 33*a*b^4*c^2 +
72*a^2*b^2*c^3 - 16*a^3*c^4)*x^2 + 2*(b^7 - 11*a*b^5*c + 41*a^2*b^3*c^2 - 52*a^3*b*c^3)*x)*e^5 - 10*((b^4*c^3
- 8*a*b^2*c^4 + 16*a^2*c^5)*d*x^3 + (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*x^2 - (b^6*c - 9*a*b^4*c^2 + 26*a
^2*b^2*c^3 - 24*a^3*c^4)*d*x - (a*b^5*c - 7*a^2*b^3*c^2 + 12*a^3*b*c^3)*d)*e^4 - 20*((b^5*c^2 - 7*a*b^3*c^3 +
12*a^2*b*c^4)*d^2*x + (a*b^4*c^2 - 6*a^2*b^2*c^3 + 8*a^3*c^4)*d^2)*e^3 + 20*((b^4*c^3 - 6*a*b^2*c^4 + 8*a^2*c^
5)*d^3*x + (a*b^3*c^3 - 4*a^2*b*c^4)*d^3)*e^2 - 10*((b^3*c^4 - 4*a*b*c^5)*d^4*x + 2*(a*b^2*c^4 - 4*a^2*c^5)*d^
4)*e - ((3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3 + (3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*
a^3*c^4)*x^2 + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*x)*e^5 - 10*((b^5*c^2 - 8*a*b^3*c^3 + 16*a
^2*b*c^4)*d*x^2 + (b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d*x + (a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d)*e^4
+ 10*((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^2*x^2 + (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*x + (a*b^4*c^
2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^2)*e^3)*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*(4*(b^2*c^5 - 4*a*c^6)*
d^5*x + 2*(b^3*c^4 - 4*a*b*c^5)*d^5 - 2*(4*c^6*d^5*x^2 + 4*b*c^5*d^5*x + 4*a*c^5*d^5 + (3*a*b^5 - 20*a^2*b^3*c
+ 30*a^3*b*c^2 + (3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*x^2 + (3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*x)*e^5 -
10*((b^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*d*x^2 + (b^5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*d*x + (a*b^4*c - 6*a^2*b^
2*c^2 + 6*a^3*c^3)*d)*e^4 + 10*((b^3*c^3 - 6*a*b*c^4)*d^2*x^2 + (b^4*c^2 - 6*a*b^2*c^3)*d^2*x + (a*b^3*c^2 - 6
*a^2*b*c^3)*d^2)*e^3 + 40*(a*c^5*d^3*x^2 + a*b*c^4*d^3*x + a^2*c^4*d^3)*e^2 - 10*(b*c^5*d^4*x^2 + b^2*c^4*d^4*
x + a*b*c^4*d^4)*e)*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^6 - 16*a
^2*b^4*c + 36*a^3*b^2*c^2 - 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 - 3*(b^5*c^2 - 8*a*b^3*c^3 +
16*a^2*b*c^4)*x^3 - (4*b^6*c - 33*a*b^4*c^2 + 72*a^2*b^2*c^3 - 16*a^3*c^4)*x^2 + 2*(b^7 - 11*a*b^5*c + 41*a^2
*b^3*c^2 - 52*a^3*b*c^3)*x)*e^5 - 10*((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d*x^3 + (b^5*c^2 - 8*a*b^3*c^3 + 16
*a^2*b*c^4)*d*x^2 - (b^6*c - 9*a*b^4*c^2 + 26*a^2*b^2*c^3 - 24*a^3*c^4)*d*x - (a*b^5*c - 7*a^2*b^3*c^2 + 12*a^
3*b*c^3)*d)*e^4 - 20*((b^5*c^2 - 7*a*b^3*c^3 + 12*a^2*b*c^4)*d^2*x + (a*b^4*c^2 - 6*a^2*b^2*c^3 + 8*a^3*c^4)*d
^2)*e^3 + 20*((b^4*c^3 - 6*a*b^2*c^4 + 8*a^2*c^5)*d^3*x + (a*b^3*c^3 - 4*a^2*b*c^4)*d^3)*e^2 - 10*((b^3*c^4 -
4*a*b*c^5)*d^4*x + 2*(a*b^2*c^4 - 4*a^2*c^5)*d^4)*e - ((3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3 +
(3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*x^2 + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*
c^3)*x)*e^5 - 10*((b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*x^2 + (b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d*x +
(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d)*e^4 + 10*((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^2*x^2 + (b^5*c^2
- 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*x + (a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^2)*e^3)*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)]

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2671 vs. \(2 (371) = 742\).
time = 48.78, size = 2671, normalized size = 7.14 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(c*x**2+b*x+a)**2,x)

[Out]

x*(-2*b*e**5/c**3 + 5*d*e**4/c**2) + (-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) -
sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*
d**2*e**2 + 3*b**4*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(64
*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (16*a**3*c**2*e**5 - 17*a**2*b**2*c*e**5 + 50*a
**2*b*c**2*d*e**4 + 16*a**2*c**5*(-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) - sqrt
(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2
*e**2 + 3*b**4*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(64*a**
3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 80*a**2*c**3*d**2*e**3 + 3*a*b**4*e**5 - 10*a*b**3*c*d*e*
*4 - 8*a*b**2*c**4*(-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) - sqrt(-(4*a*c - b**
2)**3)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4
*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a*
*2*b**2*c**2 + 12*a*b**4*c - b**6))) + 10*a*b**2*c**2*d**2*e**3 + 20*a*b*c**3*d**3*e**2 + b**4*c**3*(-e**3*(2*
a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) - sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(30*a**2
*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e**4 - 4*b**3*c*d*e**3 + 2*b
**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c -
b**6))) - 5*b**2*c**3*d**4*e + 2*b*c**4*d**5)/(30*a**2*b*c**2*e**5 - 60*a**2*c**3*d*e**4 - 20*a*b**3*c*e**5 +
60*a*b**2*c**2*d*e**4 - 60*a*b*c**3*d**2*e**3 + 40*a*c**4*d**3*e**2 + 3*b**5*e**5 - 10*b**4*c*d*e**4 + 10*b**3
*c**2*d**2*e**3 - 10*b*c**4*d**4*e + 4*c**5*d**5)) + (-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d
**2)/(2*c**4) + sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e
**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d
**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (16*a**3*c**2*e**5 - 17*a**2*b*
*2*c*e**5 + 50*a**2*b*c**2*d*e**4 + 16*a**2*c**5*(-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)
/(2*c**4) + sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3
- 20*a*c**3*d**2*e**2 + 3*b**4*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)
/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 80*a**2*c**3*d**2*e**3 + 3*a*b**4*e**5 -
10*a*b**3*c*d*e**4 - 8*a*b**2*c**4*(-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) + sq
rt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d*
*2*e**2 + 3*b**4*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(64*a
**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 10*a*b**2*c**2*d**2*e**3 + 20*a*b*c**3*d**3*e**2 + b**4
*c**3*(-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) + sqrt(-(4*a*c - b**2)**3)*(b*e -
2*c*d)*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e**4 - 4*b**
3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2
+ 12*a*b**4*c - b**6))) - 5*b**2*c**3*d**4*e + 2*b*c**4*d**5)/(30*a**2*b*c**2*e**5 - 60*a**2*c**3*d*e**4 - 20*
a*b**3*c*e**5 + 60*a*b**2*c**2*d*e**4 - 60*a*b*c**3*d**2*e**3 + 40*a*c**4*d**3*e**2 + 3*b**5*e**5 - 10*b**4*c*
d*e**4 + 10*b**3*c**2*d**2*e**3 - 10*b*c**4*d**4*e + 4*c**5*d**5)) + (-2*a**3*c**2*e**5 + 4*a**2*b**2*c*e**5 -
15*a**2*b*c**2*d*e**4 + 20*a**2*c**3*d**2*e**3 - a*b**4*e**5 + 5*a*b**3*c*d*e**4 - 10*a*b**2*c**2*d**2*e**3 +
10*a*b*c**3*d**3*e**2 - 10*a*c**4*d**4*e + b*c**4*d**5 + x*(-5*a**2*b*c**2*e**5 + 10*a**2*c**3*d*e**4 + 5*a*b
**3*c*e**5 - 20*a*b**2*c**2*d*e**4 + 30*a*b*c**3*d**2*e**3 - 20*a*c**4*d**3*e**2 - b**5*e**5 + 5*b**4*c*d*e**4
- 10*b**3*c**2*d**2*e**3 + 10*b**2*c**3*d**3*e**2 - 5*b*c**4*d**4*e + 2*c**5*d**5))/(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**5*x**2/(2*c**2)

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Giac [A]
time = 1.01, size = 511, normalized size = 1.37 \begin {gather*} -\frac {{\left (4 \, c^{5} d^{5} - 10 \, b c^{4} d^{4} e + 40 \, a c^{4} d^{3} e^{2} + 10 \, b^{3} c^{2} d^{2} e^{3} - 60 \, a b c^{3} d^{2} e^{3} - 10 \, b^{4} c d e^{4} + 60 \, a b^{2} c^{2} d e^{4} - 60 \, a^{2} c^{3} d e^{4} + 3 \, b^{5} e^{5} - 20 \, a b^{3} c e^{5} + 30 \, a^{2} b c^{2} e^{5}\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 (10 \, c^{2} d^{2} e^{3} - 10 \, b c d e^{4} + 3 \, b^{2} e^{5} - 2 \, a c e^{5}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac {c^{2} x^{2} e^{5} + 10 \, c^{2} d x e^{4} - 4 \, b c x e^{5}}{2 \, c^{4}} - \frac {b c^{4} d^{5} - 10 \, a c^{4} d^{4} e + 10 \, a b c^{3} d^{3} e^{2} - 10 \, a b^{2} c^{2} d^{2} e^{3} + 20 \, a^{2} c^{3} d^{2} e^{3} + 5 \, a b^{3} c d e^{4} - 15 \, a^{2} b c^{2} d e^{4} - a b^{4} e^{5} + 4 \, a^{2} b^{2} c e^{5} - 2 \, a^{3} c^{2} e^{5} + {\left (2 \, c^{5} d^{5} - 5 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - 20 \, a c^{4} d^{3} e^{2} - 10 \, b^{3} c^{2} d^{2} e^{3} + 30 \, a b c^{3} d^{2} e^{3} + 5 \, b^{4} c d e^{4} - 20 \, a b^{2} c^{2} d e^{4} + 10 \, a^{2} c^{3} d e^{4} - b^{5} e^{5} + 5 \, a b^{3} c e^{5} - 5 \, a^{2} b c^{2} e^{5}\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((e*x+d)^5/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

-(4*c^5*d^5 - 10*b*c^4*d^4*e + 40*a*c^4*d^3*e^2 + 10*b^3*c^2*d^2*e^3 - 60*a*b*c^3*d^2*e^3 - 10*b^4*c*d*e^4 + 6
0*a*b^2*c^2*d*e^4 - 60*a^2*c^3*d*e^4 + 3*b^5*e^5 - 20*a*b^3*c*e^5 + 30*a^2*b*c^2*e^5)*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*(10*c^2*d^2*e^3 - 10*b*c*d*e^4 + 3*b^2*e^5 - 2*a
*c*e^5)*log(c*x^2 + b*x + a)/c^4 + 1/2*(c^2*x^2*e^5 + 10*c^2*d*x*e^4 - 4*b*c*x*e^5)/c^4 - (b*c^4*d^5 - 10*a*c^
4*d^4*e + 10*a*b*c^3*d^3*e^2 - 10*a*b^2*c^2*d^2*e^3 + 20*a^2*c^3*d^2*e^3 + 5*a*b^3*c*d*e^4 - 15*a^2*b*c^2*d*e^
4 - a*b^4*e^5 + 4*a^2*b^2*c*e^5 - 2*a^3*c^2*e^5 + (2*c^5*d^5 - 5*b*c^4*d^4*e + 10*b^2*c^3*d^3*e^2 - 20*a*c^4*d
^3*e^2 - 10*b^3*c^2*d^2*e^3 + 30*a*b*c^3*d^2*e^3 + 5*b^4*c*d*e^4 - 20*a*b^2*c^2*d*e^4 + 10*a^2*c^3*d*e^4 - b^5
*e^5 + 5*a*b^3*c*e^5 - 5*a^2*b*c^2*e^5)*x)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*c^4)

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Mupad [B]
time = 2.12, size = 1083, normalized size = 2.90 \begin {gather*} \frac {e^5\,x^2}{2\,c^2}-x\,\left (\frac {2\,b\,e^5}{c^3}-\frac {5\,d\,e^4}{c^2}\right )-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (128\,a^4\,c^4\,e^5-288\,a^3\,b^2\,c^3\,e^5+640\,a^3\,b\,c^4\,d\,e^4-640\,a^3\,c^5\,d^2\,e^3+168\,a^2\,b^4\,c^2\,e^5-480\,a^2\,b^3\,c^3\,d\,e^4+480\,a^2\,b^2\,c^4\,d^2\,e^3-38\,a\,b^6\,c\,e^5+120\,a\,b^5\,c^2\,d\,e^4-120\,a\,b^4\,c^3\,d^2\,e^3+3\,b^8\,e^5-10\,b^7\,c\,d\,e^4+10\,b^6\,c^2\,d^2\,e^3\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 {\frac {2\,a^3\,c^2\,e^5-4\,a^2\,b^2\,c\,e^5+15\,a^2\,b\,c^2\,d\,e^4-20\,a^2\,c^3\,d^2\,e^3+a\,b^4\,e^5-5\,a\,b^3\,c\,d\,e^4+10\,a\,b^2\,c^2\,d^2\,e^3-10\,a\,b\,c^3\,d^3\,e^2+10\,a\,c^4\,d^4\,e-b\,c^4\,d^5}{c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (5\,a^2\,b\,c^2\,e^5-10\,a^2\,c^3\,d\,e^4-5\,a\,b^3\,c\,e^5+20\,a\,b^2\,c^2\,d\,e^4-30\,a\,b\,c^3\,d^2\,e^3+20\,a\,c^4\,d^3\,e^2+b^5\,e^5-5\,b^4\,c\,d\,e^4+10\,b^3\,c^2\,d^2\,e^3-10\,b^2\,c^3\,d^3\,e^2+5\,b\,c^4\,d^4\,e-2\,c^5\,d^5\right )}{c\,\left (4\,a\,c-b^2\right )}}{c^4\,x^2+b\,c^3\,x+a\,c^3}-\frac {\mathrm {atan}\left (\frac {c^4\,\left (\frac {\left (b^3\,c^3-4\,a\,b\,c^4\right )\,\left (b\,e-2\,c\,d\right )\,\left (30\,a^2\,c^2\,e^4-20\,a\,b^2\,c\,e^4+20\,a\,b\,c^2\,d\,e^3-20\,a\,c^3\,d^2\,e^2+3\,b^4\,e^4-4\,b^3\,c\,d\,e^3+2\,b^2\,c^2\,d^2\,e^2+4\,b\,c^3\,d^3\,e-2\,c^4\,d^4\right )}{c^7\,{\left (4\,a\,c-b^2\right )}^4}-\frac {2\,x\,\left (b\,e-2\,c\,d\right )\,\left (30\,a^2\,c^2\,e^4-20\,a\,b^2\,c\,e^4+20\,a\,b\,c^2\,d\,e^3-20\,a\,c^3\,d^2\,e^2+3\,b^4\,e^4-4\,b^3\,c\,d\,e^3+2\,b^2\,c^2\,d^2\,e^2+4\,b\,c^3\,d^3\,e-2\,c^4\,d^4\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^3}\right )\,{\left (4\,a\,c-b^2\right )}^{5/2}}{30\,a^2\,b\,c^2\,e^5-60\,a^2\,c^3\,d\,e^4-20\,a\,b^3\,c\,e^5+60\,a\,b^2\,c^2\,d\,e^4-60\,a\,b\,c^3\,d^2\,e^3+40\,a\,c^4\,d^3\,e^2+3\,b^5\,e^5-10\,b^4\,c\,d\,e^4+10\,b^3\,c^2\,d^2\,e^3-10\,b\,c^4\,d^4\,e+4\,c^5\,d^5}\right )\,\left (b\,e-2\,c\,d\right )\,\left (30\,a^2\,c^2\,e^4-20\,a\,b^2\,c\,e^4+20\,a\,b\,c^2\,d\,e^3-20\,a\,c^3\,d^2\,e^2+3\,b^4\,e^4-4\,b^3\,c\,d\,e^3+2\,b^2\,c^2\,d^2\,e^2+4\,b\,c^3\,d^3\,e-2\,c^4\,d^4\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((d + e*x)^5/(a + b*x + c*x^2)^2,x)

[Out]

(e^5*x^2)/(2*c^2) - x*((2*b*e^5)/c^3 - (5*d*e^4)/c^2) - (log(a + b*x + c*x^2)*(3*b^8*e^5 + 128*a^4*c^4*e^5 + 1
68*a^2*b^4*c^2*e^5 - 288*a^3*b^2*c^3*e^5 - 640*a^3*c^5*d^2*e^3 + 10*b^6*c^2*d^2*e^3 - 38*a*b^6*c*e^5 - 10*b^7*
c*d*e^4 + 480*a^2*b^2*c^4*d^2*e^3 + 120*a*b^5*c^2*d*e^4 + 640*a^3*b*c^4*d*e^4 - 120*a*b^4*c^3*d^2*e^3 - 480*a^
2*b^3*c^3*d*e^4))/(2*(64*a^3*c^7 - b^6*c^4 + 12*a*b^4*c^5 - 48*a^2*b^2*c^6)) - ((a*b^4*e^5 - b*c^4*d^5 + 2*a^3
*c^2*e^5 - 4*a^2*b^2*c*e^5 - 20*a^2*c^3*d^2*e^3 + 10*a*c^4*d^4*e - 5*a*b^3*c*d*e^4 - 10*a*b*c^3*d^3*e^2 + 15*a
^2*b*c^2*d*e^4 + 10*a*b^2*c^2*d^2*e^3)/(c*(4*a*c - b^2)) + (x*(b^5*e^5 - 2*c^5*d^5 + 5*a^2*b*c^2*e^5 + 20*a*c^
4*d^3*e^2 - 10*a^2*c^3*d*e^4 - 10*b^2*c^3*d^3*e^2 + 10*b^3*c^2*d^2*e^3 - 5*a*b^3*c*e^5 + 5*b*c^4*d^4*e - 5*b^4
*c*d*e^4 - 30*a*b*c^3*d^2*e^3 + 20*a*b^2*c^2*d*e^4))/(c*(4*a*c - b^2)))/(a*c^3 + c^4*x^2 + b*c^3*x) - (atan((c
^4*(((b^3*c^3 - 4*a*b*c^4)*(b*e - 2*c*d)*(3*b^4*e^4 - 2*c^4*d^4 + 30*a^2*c^2*e^4 - 20*a*c^3*d^2*e^2 + 2*b^2*c^
2*d^2*e^2 - 20*a*b^2*c*e^4 + 4*b*c^3*d^3*e - 4*b^3*c*d*e^3 + 20*a*b*c^2*d*e^3))/(c^7*(4*a*c - b^2)^4) - (2*x*(
b*e - 2*c*d)*(3*b^4*e^4 - 2*c^4*d^4 + 30*a^2*c^2*e^4 - 20*a*c^3*d^2*e^2 + 2*b^2*c^2*d^2*e^2 - 20*a*b^2*c*e^4 +
4*b*c^3*d^3*e - 4*b^3*c*d*e^3 + 20*a*b*c^2*d*e^3))/(c^3*(4*a*c - b^2)^3))*(4*a*c - b^2)^(5/2))/(3*b^5*e^5 + 4
*c^5*d^5 + 30*a^2*b*c^2*e^5 + 40*a*c^4*d^3*e^2 - 60*a^2*c^3*d*e^4 + 10*b^3*c^2*d^2*e^3 - 20*a*b^3*c*e^5 - 10*b
*c^4*d^4*e - 10*b^4*c*d*e^4 - 60*a*b*c^3*d^2*e^3 + 60*a*b^2*c^2*d*e^4))*(b*e - 2*c*d)*(3*b^4*e^4 - 2*c^4*d^4 +
30*a^2*c^2*e^4 - 20*a*c^3*d^2*e^2 + 2*b^2*c^2*d^2*e^2 - 20*a*b^2*c*e^4 + 4*b*c^3*d^3*e - 4*b^3*c*d*e^3 + 20*a
*b*c^2*d*e^3))/(c^4*(4*a*c - b^2)^(3/2))

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