3.23.16 \(\int \frac {(d+e x)^2}{(a+b x+c x^2)^4} \, dx\) [2216]

Optimal. Leaf size=260 \[ -\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {3 b^2 d e+8 a c d e-5 b \left (c d^2+a e^2\right )-2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}} \]

[Out]

-1/3*(e*x+d)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^3+1/3*(-3*b^2*d*e-8*a*c*d*e+5*b*(a*e^2+c*d^
2)+2*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^2-2*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b
*d))*(2*c*x+b)/(-4*a*c+b^2)^3/(c*x^2+b*x+a)+8*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*arctanh((2*c*x+b)/(-4*a*c
+b^2)^(1/2))/(-4*a*c+b^2)^(7/2)

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Rubi [A]
time = 0.16, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {752, 652, 628, 632, 212} \begin {gather*} -\frac {2 (b+2 c x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {-2 x \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-5 b \left (a e^2+c d^2\right )+8 a c d e+3 b^2 d e}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {8 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*((d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)^3) - (3*b^2*d*e + 8*a*c*d*e
- 5*b*(c*d^2 + a*e^2) - 2*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*x)/(3*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2)
- (2*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(b + 2*c*x))/((b^2 - 4*a*c)^3*(a + b*x + c*x^2)) + (8*c*(5*c^2
*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(7/2)

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 628

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1
)*(b^2 - 4*a*c))), x] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

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 652

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

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]

Rubi steps

\begin {align*} \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^4} \, dx &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {\int \frac {2 \left (5 c d^2-e (3 b d-a e)\right )+4 e (2 c d-b e) x}{\left (a+b x+c x^2\right )^3} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {3 b^2 d e+8 a c d e-5 b \left (c d^2+a e^2\right )-2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {\left (2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) \int \frac {1}{\left (a+b x+c x^2\right )^2} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {3 b^2 d e+8 a c d e-5 b \left (c d^2+a e^2\right )-2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {\left (4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3}\\ &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {3 b^2 d e+8 a c d e-5 b \left (c d^2+a e^2\right )-2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {\left (8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) \text {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 )^3}\\ &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {3 b^2 d e+8 a c d e-5 b \left (c d^2+a e^2\right )-2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.34, size = 258, normalized size = 0.99 \begin {gather*} \frac {1}{3} \left (\frac {\left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right ) (b+2 c x)}{c \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac {6 \left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 (a+x (b+c x))}+\frac {a b e^2+2 c^2 d^2 x+b^2 e^2 x+b c d (d-2 e x)-2 a c e (2 d+e x)}{c \left (-b^2+4 a c\right ) (a+x (b+c x))^3}+\frac {24 c \left (5 c^2 d^2+b^2 e^2+c e (-5 b d+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 )^{7/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*(b + 2*c*x))/(c*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (6*(5*c^2*
d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*(b + 2*c*x))/((b^2 - 4*a*c)^3*(a + x*(b + c*x))) + (a*b*e^2 + 2*c^2*d^2*x
+ b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))^3) + (24*c*(5*c^2*d
^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2))/3

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(648\) vs. \(2(252)=504\).
time = 0.94, size = 649, normalized size = 2.50

method	result	size
default	\(\frac {\frac {4 c^{3} \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) x^{5}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {10 c^{2} \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) b \,x^{4}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {2 \left (16 a c +11 b^{2}\right ) c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) x^{3}}{3 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {b \left (16 a c +b^{2}\right ) \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) x^{2}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {\left (4 a^{3} c^{2} e^{2}-22 a^{2} b^{2} c \,e^{2}+44 a^{2} b \,c^{2} d e -44 a^{2} c^{3} d^{2}-a \,b^{4} e^{2}+18 a \,b^{3} c d e -18 a \,c^{2} d^{2} b^{2}-b^{5} d e +b^{4} c \,d^{2}\right ) x}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {26 a^{3} b c \,e^{2}-64 a^{3} c^{2} d e +a^{2} b^{3} e^{2}-18 a^{2} b^{2} c d e +66 a^{2} b \,c^{2} d^{2}+a \,b^{4} d e -13 a \,b^{3} c \,d^{2}+b^{5} d^{2}}{192 a^{3} c^{3}-144 a^{2} b^{2} c^{2}+36 a \,b^{4} c -3 b^{6}}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {8 c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {4 a c -b^{2}}}\)	\(649\)
risch	\(\frac {\frac {4 c^{3} \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) x^{5}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {10 c^{2} \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) b \,x^{4}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {2 \left (16 a c +11 b^{2}\right ) c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) x^{3}}{3 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {b \left (16 a c +b^{2}\right ) \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) x^{2}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {\left (4 a^{3} c^{2} e^{2}-22 a^{2} b^{2} c \,e^{2}+44 a^{2} b \,c^{2} d e -44 a^{2} c^{3} d^{2}-a \,b^{4} e^{2}+18 a \,b^{3} c d e -18 a \,c^{2} d^{2} b^{2}-b^{5} d e +b^{4} c \,d^{2}\right ) x}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {26 a^{3} b c \,e^{2}-64 a^{3} c^{2} d e +a^{2} b^{3} e^{2}-18 a^{2} b^{2} c d e +66 a^{2} b \,c^{2} d^{2}+a \,b^{4} d e -13 a \,b^{3} c \,d^{2}+b^{5} d^{2}}{192 a^{3} c^{3}-144 a^{2} b^{2} c^{2}+36 a \,b^{4} c -3 b^{6}}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {4 c^{2} \ln \left (\left (-128 a^{3} c^{4}+96 a^{2} b^{2} c^{3}-24 a \,b^{4} c^{2}+2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}-64 a^{3} b \,c^{3}+48 a^{2} b^{3} c^{2}-12 a \,b^{5} c +b^{7}\right ) a \,e^{2}}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}+\frac {4 c \ln \left (\left (-128 a^{3} c^{4}+96 a^{2} b^{2} c^{3}-24 a \,b^{4} c^{2}+2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}-64 a^{3} b \,c^{3}+48 a^{2} b^{3} c^{2}-12 a \,b^{5} c +b^{7}\right ) b^{2} e^{2}}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}-\frac {20 c^{2} \ln \left (\left (-128 a^{3} c^{4}+96 a^{2} b^{2} c^{3}-24 a \,b^{4} c^{2}+2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}-64 a^{3} b \,c^{3}+48 a^{2} b^{3} c^{2}-12 a \,b^{5} c +b^{7}\right ) b d e}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}+\frac {20 c^{3} \ln \left (\left (-128 a^{3} c^{4}+96 a^{2} b^{2} c^{3}-24 a \,b^{4} c^{2}+2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}-64 a^{3} b \,c^{3}+48 a^{2} b^{3} c^{2}-12 a \,b^{5} c +b^{7}\right ) d^{2}}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}-\frac {4 c^{2} \ln \left (\left (128 a^{3} c^{4}-96 a^{2} b^{2} c^{3}+24 a \,b^{4} c^{2}-2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}+64 a^{3} b \,c^{3}-48 a^{2} b^{3} c^{2}+12 a \,b^{5} c -b^{7}\right ) a \,e^{2}}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}-\frac {4 c \ln \left (\left (128 a^{3} c^{4}-96 a^{2} b^{2} c^{3}+24 a \,b^{4} c^{2}-2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}+64 a^{3} b \,c^{3}-48 a^{2} b^{3} c^{2}+12 a \,b^{5} c -b^{7}\right ) b^{2} e^{2}}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}+\frac {20 c^{2} \ln \left (\left (128 a^{3} c^{4}-96 a^{2} b^{2} c^{3}+24 a \,b^{4} c^{2}-2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}+64 a^{3} b \,c^{3}-48 a^{2} b^{3} c^{2}+12 a \,b^{5} c -b^{7}\right ) b d e}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}-\frac {20 c^{3} \ln \left (\left (128 a^{3} c^{4}-96 a^{2} b^{2} c^{3}+24 a \,b^{4} c^{2}-2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}+64 a^{3} b \,c^{3}-48 a^{2} b^{3} c^{2}+12 a \,b^{5} c -b^{7}\right ) d^{2}}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}\)	\(1340\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*c^3*(a*c*e^2+b^2*e^2-5*b*c*d*e+5*c^2*d^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^5+10*c^2*(a*c*e^2+b^
2*e^2-5*b*c*d*e+5*c^2*d^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*b*x^4+2/3*(16*a*c+11*b^2)*c*(a*c*e^2+b^2
*e^2-5*b*c*d*e+5*c^2*d^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3+b*(16*a*c+b^2)*(a*c*e^2+b^2*e^2-5*b*c
*d*e+5*c^2*d^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2-(4*a^3*c^2*e^2-22*a^2*b^2*c*e^2+44*a^2*b*c^2*d*
e-44*a^2*c^3*d^2-a*b^4*e^2+18*a*b^3*c*d*e-18*a*b^2*c^2*d^2-b^5*d*e+b^4*c*d^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*
b^4*c-b^6)*x+1/3*(26*a^3*b*c*e^2-64*a^3*c^2*d*e+a^2*b^3*e^2-18*a^2*b^2*c*d*e+66*a^2*b*c^2*d^2+a*b^4*d*e-13*a*b
^3*c*d^2+b^5*d^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6))/(c*x^2+b*x+a)^3+8*c*(a*c*e^2+b^2*e^2-5*b*c*d*e+5
*c^2*d^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(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)^2/(c*x^2+b*x+a)^4,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. 1366 vs. \(2 (259) = 518\).
time = 2.06, size = 2753, normalized size = 10.59 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/3*(60*(b^2*c^5 - 4*a*c^6)*d^2*x^5 + 150*(b^3*c^4 - 4*a*b*c^5)*d^2*x^4 + 10*(11*b^4*c^3 - 28*a*b^2*c^4 - 64
*a^2*c^5)*d^2*x^3 + 15*(b^5*c^2 + 12*a*b^3*c^3 - 64*a^2*b*c^4)*d^2*x^2 - 3*(b^6*c - 22*a*b^4*c^2 + 28*a^2*b^2*
c^3 + 176*a^3*c^4)*d^2*x + (b^7 - 17*a*b^5*c + 118*a^2*b^3*c^2 - 264*a^3*b*c^3)*d^2 + 12*(5*c^6*d^2*x^6 + 15*b
*c^5*d^2*x^5 + 15*a^2*b*c^3*d^2*x + 5*a^3*c^3*d^2 + 15*(b^2*c^4 + a*c^5)*d^2*x^4 + 5*(b^3*c^3 + 6*a*b*c^4)*d^2
*x^3 + 15*(a*b^2*c^3 + a^2*c^4)*d^2*x^2 + ((b^2*c^4 + a*c^5)*x^6 + a^3*b^2*c + a^4*c^2 + 3*(b^3*c^3 + a*b*c^4)
*x^5 + 3*(b^4*c^2 + 2*a*b^2*c^3 + a^2*c^4)*x^4 + (b^5*c + 7*a*b^3*c^2 + 6*a^2*b*c^3)*x^3 + 3*(a*b^4*c + 2*a^2*
b^2*c^2 + a^3*c^3)*x^2 + 3*(a^2*b^3*c + a^3*b*c^2)*x)*e^2 - 5*(b*c^5*d*x^6 + 3*b^2*c^4*d*x^5 + 3*a^2*b^2*c^2*d
*x + a^3*b*c^2*d + 3*(b^3*c^3 + a*b*c^4)*d*x^4 + (b^4*c^2 + 6*a*b^2*c^3)*d*x^3 + 3*(a*b^3*c^2 + a^2*b*c^3)*d*x
^2)*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^2*b^5 + 22*a^3*b^3*c - 104*a^4*b*c^2 + 12*(b^4*c^3 - 3*a*b^2*c^4 - 4*a^2*c^5)*x^5 + 30*(b^5*c^2 - 3
*a*b^3*c^3 - 4*a^2*b*c^4)*x^4 + 2*(11*b^6*c - 17*a*b^4*c^2 - 92*a^2*b^2*c^3 - 64*a^3*c^4)*x^3 + 3*(b^7 + 13*a*
b^5*c - 52*a^2*b^3*c^2 - 64*a^3*b*c^3)*x^2 + 3*(a*b^6 + 18*a^2*b^4*c - 92*a^3*b^2*c^2 + 16*a^4*c^3)*x)*e^2 - (
60*(b^3*c^4 - 4*a*b*c^5)*d*x^5 + 150*(b^4*c^3 - 4*a*b^2*c^4)*d*x^4 + 10*(11*b^5*c^2 - 28*a*b^3*c^3 - 64*a^2*b*
c^4)*d*x^3 + 15*(b^6*c + 12*a*b^4*c^2 - 64*a^2*b^2*c^3)*d*x^2 - 3*(b^7 - 22*a*b^5*c + 28*a^2*b^3*c^2 + 176*a^3
*b*c^3)*d*x - (a*b^6 - 22*a^2*b^4*c + 8*a^3*b^2*c^2 + 256*a^4*c^3)*d)*e)/(a^3*b^8 - 16*a^4*b^6*c + 96*a^5*b^4*
c^2 - 256*a^6*b^2*c^3 + 256*a^7*c^4 + (b^8*c^3 - 16*a*b^6*c^4 + 96*a^2*b^4*c^5 - 256*a^3*b^2*c^6 + 256*a^4*c^7
)*x^6 + 3*(b^9*c^2 - 16*a*b^7*c^3 + 96*a^2*b^5*c^4 - 256*a^3*b^3*c^5 + 256*a^4*b*c^6)*x^5 + 3*(b^10*c - 15*a*b
^8*c^2 + 80*a^2*b^6*c^3 - 160*a^3*b^4*c^4 + 256*a^5*c^6)*x^4 + (b^11 - 10*a*b^9*c + 320*a^3*b^5*c^3 - 1280*a^4
*b^3*c^4 + 1536*a^5*b*c^5)*x^3 + 3*(a*b^10 - 15*a^2*b^8*c + 80*a^3*b^6*c^2 - 160*a^4*b^4*c^3 + 256*a^6*c^5)*x^
2 + 3*(a^2*b^9 - 16*a^3*b^7*c + 96*a^4*b^5*c^2 - 256*a^5*b^3*c^3 + 256*a^6*b*c^4)*x), -1/3*(60*(b^2*c^5 - 4*a*
c^6)*d^2*x^5 + 150*(b^3*c^4 - 4*a*b*c^5)*d^2*x^4 + 10*(11*b^4*c^3 - 28*a*b^2*c^4 - 64*a^2*c^5)*d^2*x^3 + 15*(b
^5*c^2 + 12*a*b^3*c^3 - 64*a^2*b*c^4)*d^2*x^2 - 3*(b^6*c - 22*a*b^4*c^2 + 28*a^2*b^2*c^3 + 176*a^3*c^4)*d^2*x
+ (b^7 - 17*a*b^5*c + 118*a^2*b^3*c^2 - 264*a^3*b*c^3)*d^2 - 24*(5*c^6*d^2*x^6 + 15*b*c^5*d^2*x^5 + 15*a^2*b*c
^3*d^2*x + 5*a^3*c^3*d^2 + 15*(b^2*c^4 + a*c^5)*d^2*x^4 + 5*(b^3*c^3 + 6*a*b*c^4)*d^2*x^3 + 15*(a*b^2*c^3 + a^
2*c^4)*d^2*x^2 + ((b^2*c^4 + a*c^5)*x^6 + a^3*b^2*c + a^4*c^2 + 3*(b^3*c^3 + a*b*c^4)*x^5 + 3*(b^4*c^2 + 2*a*b
^2*c^3 + a^2*c^4)*x^4 + (b^5*c + 7*a*b^3*c^2 + 6*a^2*b*c^3)*x^3 + 3*(a*b^4*c + 2*a^2*b^2*c^2 + a^3*c^3)*x^2 +
3*(a^2*b^3*c + a^3*b*c^2)*x)*e^2 - 5*(b*c^5*d*x^6 + 3*b^2*c^4*d*x^5 + 3*a^2*b^2*c^2*d*x + a^3*b*c^2*d + 3*(b^3
*c^3 + a*b*c^4)*d*x^4 + (b^4*c^2 + 6*a*b^2*c^3)*d*x^3 + 3*(a*b^3*c^2 + a^2*b*c^3)*d*x^2)*e)*sqrt(-b^2 + 4*a*c)
*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (a^2*b^5 + 22*a^3*b^3*c - 104*a^4*b*c^2 + 12*(b^4*c^3
- 3*a*b^2*c^4 - 4*a^2*c^5)*x^5 + 30*(b^5*c^2 - 3*a*b^3*c^3 - 4*a^2*b*c^4)*x^4 + 2*(11*b^6*c - 17*a*b^4*c^2 -
92*a^2*b^2*c^3 - 64*a^3*c^4)*x^3 + 3*(b^7 + 13*a*b^5*c - 52*a^2*b^3*c^2 - 64*a^3*b*c^3)*x^2 + 3*(a*b^6 + 18*a^
2*b^4*c - 92*a^3*b^2*c^2 + 16*a^4*c^3)*x)*e^2 - (60*(b^3*c^4 - 4*a*b*c^5)*d*x^5 + 150*(b^4*c^3 - 4*a*b^2*c^4)*
d*x^4 + 10*(11*b^5*c^2 - 28*a*b^3*c^3 - 64*a^2*b*c^4)*d*x^3 + 15*(b^6*c + 12*a*b^4*c^2 - 64*a^2*b^2*c^3)*d*x^2
- 3*(b^7 - 22*a*b^5*c + 28*a^2*b^3*c^2 + 176*a^3*b*c^3)*d*x - (a*b^6 - 22*a^2*b^4*c + 8*a^3*b^2*c^2 + 256*a^4
*c^3)*d)*e)/(a^3*b^8 - 16*a^4*b^6*c + 96*a^5*b^4*c^2 - 256*a^6*b^2*c^3 + 256*a^7*c^4 + (b^8*c^3 - 16*a*b^6*c^4
+ 96*a^2*b^4*c^5 - 256*a^3*b^2*c^6 + 256*a^4*c^7)*x^6 + 3*(b^9*c^2 - 16*a*b^7*c^3 + 96*a^2*b^5*c^4 - 256*a^3*
b^3*c^5 + 256*a^4*b*c^6)*x^5 + 3*(b^10*c - 15*a*b^8*c^2 + 80*a^2*b^6*c^3 - 160*a^3*b^4*c^4 + 256*a^5*c^6)*x^4
+ (b^11 - 10*a*b^9*c + 320*a^3*b^5*c^3 - 1280*a^4*b^3*c^4 + 1536*a^5*b*c^5)*x^3 + 3*(a*b^10 - 15*a^2*b^8*c + 8
0*a^3*b^6*c^2 - 160*a^4*b^4*c^3 + 256*a^6*c^5)*x^2 + 3*(a^2*b^9 - 16*a^3*b^7*c + 96*a^4*b^5*c^2 - 256*a^5*b^3*
c^3 + 256*a^6*b*c^4)*x)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1635 vs. \(2 (257) = 514\).
time = 3.82, size = 1635, normalized size = 6.29 \begin {gather*} - 4 c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log {\left (x + \frac {- 1024 a^{4} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) + 1024 a^{3} b^{2} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) - 384 a^{2} b^{4} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) + 64 a b^{6} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) + 4 a b c^{2} e^{2} - 4 b^{8} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) + 4 b^{3} c e^{2} - 20 b^{2} c^{2} d e + 20 b c^{3} d^{2}}{8 a c^{3} e^{2} + 8 b^{2} c^{2} e^{2} - 40 b c^{3} d e + 40 c^{4} d^{2}} \right )} + 4 c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log {\left (x + \frac {1024 a^{4} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) - 1024 a^{3} b^{2} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) + 384 a^{2} b^{4} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) - 64 a b^{6} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) + 4 a b c^{2} e^{2} + 4 b^{8} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) + 4 b^{3} c e^{2} - 20 b^{2} c^{2} d e + 20 b c^{3} d^{2}}{8 a c^{3} e^{2} + 8 b^{2} c^{2} e^{2} - 40 b c^{3} d e + 40 c^{4} d^{2}} \right )} + \frac {26 a^{3} b c e^{2} - 64 a^{3} c^{2} d e + a^{2} b^{3} e^{2} - 18 a^{2} b^{2} c d e + 66 a^{2} b c^{2} d^{2} + a b^{4} d e - 13 a b^{3} c d^{2} + b^{5} d^{2} + x^{5} \cdot \left (12 a c^{4} e^{2} + 12 b^{2} c^{3} e^{2} - 60 b c^{4} d e + 60 c^{5} d^{2}\right ) + x^{4} \cdot \left (30 a b c^{3} e^{2} + 30 b^{3} c^{2} e^{2} - 150 b^{2} c^{3} d e + 150 b c^{4} d^{2}\right ) + x^{3} \cdot \left (32 a^{2} c^{3} e^{2} + 54 a b^{2} c^{2} e^{2} - 160 a b c^{3} d e + 160 a c^{4} d^{2} + 22 b^{4} c e^{2} - 110 b^{3} c^{2} d e + 110 b^{2} c^{3} d^{2}\right ) + x^{2} \cdot \left (48 a^{2} b c^{2} e^{2} + 51 a b^{3} c e^{2} - 240 a b^{2} c^{2} d e + 240 a b c^{3} d^{2} + 3 b^{5} e^{2} - 15 b^{4} c d e + 15 b^{3} c^{2} d^{2}\right ) + x \left (- 12 a^{3} c^{2} e^{2} + 66 a^{2} b^{2} c e^{2} - 132 a^{2} b c^{2} d e + 132 a^{2} c^{3} d^{2} + 3 a b^{4} e^{2} - 54 a b^{3} c d e + 54 a b^{2} c^{2} d^{2} + 3 b^{5} d e - 3 b^{4} c d^{2}\right )}{192 a^{6} c^{3} - 144 a^{5} b^{2} c^{2} + 36 a^{4} b^{4} c - 3 a^{3} b^{6} + x^{6} \cdot \left (192 a^{3} c^{6} - 144 a^{2} b^{2} c^{5} + 36 a b^{4} c^{4} - 3 b^{6} c^{3}\right ) + x^{5} \cdot \left (576 a^{3} b c^{5} - 432 a^{2} b^{3} c^{4} + 108 a b^{5} c^{3} - 9 b^{7} c^{2}\right ) + x^{4} \cdot \left (576 a^{4} c^{5} + 144 a^{3} b^{2} c^{4} - 324 a^{2} b^{4} c^{3} + 99 a b^{6} c^{2} - 9 b^{8} c\right ) + x^{3} \cdot \left (1152 a^{4} b c^{4} - 672 a^{3} b^{3} c^{3} + 72 a^{2} b^{5} c^{2} + 18 a b^{7} c - 3 b^{9}\right ) + x^{2} \cdot \left (576 a^{5} c^{4} + 144 a^{4} b^{2} c^{3} - 324 a^{3} b^{4} c^{2} + 99 a^{2} b^{6} c - 9 a b^{8}\right ) + x \left (576 a^{5} b c^{3} - 432 a^{4} b^{3} c^{2} + 108 a^{3} b^{5} c - 9 a^{2} b^{7}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*c*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)*log(x + (-1024*a**4*c**5*sqrt
(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) + 1024*a**3*b**2*c**4*sqrt(-1/(4*a*c -
b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) - 384*a**2*b**4*c**3*sqrt(-1/(4*a*c - b**2)**7)*(a
*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) + 64*a*b**6*c**2*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e*
*2 - 5*b*c*d*e + 5*c**2*d**2) + 4*a*b*c**2*e**2 - 4*b**8*c*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 -
5*b*c*d*e + 5*c**2*d**2) + 4*b**3*c*e**2 - 20*b**2*c**2*d*e + 20*b*c**3*d**2)/(8*a*c**3*e**2 + 8*b**2*c**2*e**
2 - 40*b*c**3*d*e + 40*c**4*d**2)) + 4*c*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2
*d**2)*log(x + (1024*a**4*c**5*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) - 1
024*a**3*b**2*c**4*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) + 384*a**2*b**4
*c**3*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) - 64*a*b**6*c**2*sqrt(-1/(4*
a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) + 4*a*b*c**2*e**2 + 4*b**8*c*sqrt(-1/(4*a*c -
b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) + 4*b**3*c*e**2 - 20*b**2*c**2*d*e + 20*b*c**3*d**
2)/(8*a*c**3*e**2 + 8*b**2*c**2*e**2 - 40*b*c**3*d*e + 40*c**4*d**2)) + (26*a**3*b*c*e**2 - 64*a**3*c**2*d*e +
a**2*b**3*e**2 - 18*a**2*b**2*c*d*e + 66*a**2*b*c**2*d**2 + a*b**4*d*e - 13*a*b**3*c*d**2 + b**5*d**2 + x**5*
(12*a*c**4*e**2 + 12*b**2*c**3*e**2 - 60*b*c**4*d*e + 60*c**5*d**2) + x**4*(30*a*b*c**3*e**2 + 30*b**3*c**2*e*
*2 - 150*b**2*c**3*d*e + 150*b*c**4*d**2) + x**3*(32*a**2*c**3*e**2 + 54*a*b**2*c**2*e**2 - 160*a*b*c**3*d*e +
160*a*c**4*d**2 + 22*b**4*c*e**2 - 110*b**3*c**2*d*e + 110*b**2*c**3*d**2) + x**2*(48*a**2*b*c**2*e**2 + 51*a
*b**3*c*e**2 - 240*a*b**2*c**2*d*e + 240*a*b*c**3*d**2 + 3*b**5*e**2 - 15*b**4*c*d*e + 15*b**3*c**2*d**2) + x*
(-12*a**3*c**2*e**2 + 66*a**2*b**2*c*e**2 - 132*a**2*b*c**2*d*e + 132*a**2*c**3*d**2 + 3*a*b**4*e**2 - 54*a*b*
*3*c*d*e + 54*a*b**2*c**2*d**2 + 3*b**5*d*e - 3*b**4*c*d**2))/(192*a**6*c**3 - 144*a**5*b**2*c**2 + 36*a**4*b*
*4*c - 3*a**3*b**6 + x**6*(192*a**3*c**6 - 144*a**2*b**2*c**5 + 36*a*b**4*c**4 - 3*b**6*c**3) + x**5*(576*a**3
*b*c**5 - 432*a**2*b**3*c**4 + 108*a*b**5*c**3 - 9*b**7*c**2) + x**4*(576*a**4*c**5 + 144*a**3*b**2*c**4 - 324
*a**2*b**4*c**3 + 99*a*b**6*c**2 - 9*b**8*c) + x**3*(1152*a**4*b*c**4 - 672*a**3*b**3*c**3 + 72*a**2*b**5*c**2
+ 18*a*b**7*c - 3*b**9) + x**2*(576*a**5*c**4 + 144*a**4*b**2*c**3 - 324*a**3*b**4*c**2 + 99*a**2*b**6*c - 9*
a*b**8) + x*(576*a**5*b*c**3 - 432*a**4*b**3*c**2 + 108*a**3*b**5*c - 9*a**2*b**7))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (259) = 518\).
time = 0.84, size = 603, normalized size = 2.32 \begin {gather*} -\frac {8 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2} + a c^{2} e^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {60 \, c^{5} d^{2} x^{5} - 60 \, b c^{4} d x^{5} e + 150 \, b c^{4} d^{2} x^{4} + 12 \, b^{2} c^{3} x^{5} e^{2} + 12 \, a c^{4} x^{5} e^{2} - 150 \, b^{2} c^{3} d x^{4} e + 110 \, b^{2} c^{3} d^{2} x^{3} + 160 \, a c^{4} d^{2} x^{3} + 30 \, b^{3} c^{2} x^{4} e^{2} + 30 \, a b c^{3} x^{4} e^{2} - 110 \, b^{3} c^{2} d x^{3} e - 160 \, a b c^{3} d x^{3} e + 15 \, b^{3} c^{2} d^{2} x^{2} + 240 \, a b c^{3} d^{2} x^{2} + 22 \, b^{4} c x^{3} e^{2} + 54 \, a b^{2} c^{2} x^{3} e^{2} + 32 \, a^{2} c^{3} x^{3} e^{2} - 15 \, b^{4} c d x^{2} e - 240 \, a b^{2} c^{2} d x^{2} e - 3 \, b^{4} c d^{2} x + 54 \, a b^{2} c^{2} d^{2} x + 132 \, a^{2} c^{3} d^{2} x + 3 \, b^{5} x^{2} e^{2} + 51 \, a b^{3} c x^{2} e^{2} + 48 \, a^{2} b c^{2} x^{2} e^{2} + 3 \, b^{5} d x e - 54 \, a b^{3} c d x e - 132 \, a^{2} b c^{2} d x e + b^{5} d^{2} - 13 \, a b^{3} c d^{2} + 66 \, a^{2} b c^{2} d^{2} + 3 \, a b^{4} x e^{2} + 66 \, a^{2} b^{2} c x e^{2} - 12 \, a^{3} c^{2} x e^{2} + a b^{4} d e - 18 \, a^{2} b^{2} c d e - 64 \, a^{3} c^{2} d e + a^{2} b^{3} e^{2} + 26 \, a^{3} b c e^{2}}{3 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} {\left (c x^{2} + b x + a\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*c*e^2 + a*c^2*e^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^6 - 12*a*b^4*c
+ 48*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(-b^2 + 4*a*c)) - 1/3*(60*c^5*d^2*x^5 - 60*b*c^4*d*x^5*e + 150*b*c^4*d^2*x
^4 + 12*b^2*c^3*x^5*e^2 + 12*a*c^4*x^5*e^2 - 150*b^2*c^3*d*x^4*e + 110*b^2*c^3*d^2*x^3 + 160*a*c^4*d^2*x^3 + 3
0*b^3*c^2*x^4*e^2 + 30*a*b*c^3*x^4*e^2 - 110*b^3*c^2*d*x^3*e - 160*a*b*c^3*d*x^3*e + 15*b^3*c^2*d^2*x^2 + 240*
a*b*c^3*d^2*x^2 + 22*b^4*c*x^3*e^2 + 54*a*b^2*c^2*x^3*e^2 + 32*a^2*c^3*x^3*e^2 - 15*b^4*c*d*x^2*e - 240*a*b^2*
c^2*d*x^2*e - 3*b^4*c*d^2*x + 54*a*b^2*c^2*d^2*x + 132*a^2*c^3*d^2*x + 3*b^5*x^2*e^2 + 51*a*b^3*c*x^2*e^2 + 48
*a^2*b*c^2*x^2*e^2 + 3*b^5*d*x*e - 54*a*b^3*c*d*x*e - 132*a^2*b*c^2*d*x*e + b^5*d^2 - 13*a*b^3*c*d^2 + 66*a^2*
b*c^2*d^2 + 3*a*b^4*x*e^2 + 66*a^2*b^2*c*x*e^2 - 12*a^3*c^2*x*e^2 + a*b^4*d*e - 18*a^2*b^2*c*d*e - 64*a^3*c^2*
d*e + a^2*b^3*e^2 + 26*a^3*b*c*e^2)/((b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*(c*x^2 + b*x + a)^3)

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Mupad [B]
time = 1.37, size = 872, normalized size = 3.35 \begin {gather*} -\frac {\frac {26\,a^3\,b\,c\,e^2-64\,a^3\,c^2\,d\,e+a^2\,b^3\,e^2-18\,a^2\,b^2\,c\,d\,e+66\,a^2\,b\,c^2\,d^2+a\,b^4\,d\,e-13\,a\,b^3\,c\,d^2+b^5\,d^2}{3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x\,\left (-4\,a^3\,c^2\,e^2+22\,a^2\,b^2\,c\,e^2-44\,a^2\,b\,c^2\,d\,e+44\,a^2\,c^3\,d^2+a\,b^4\,e^2-18\,a\,b^3\,c\,d\,e+18\,a\,b^2\,c^2\,d^2+b^5\,d\,e-b^4\,c\,d^2\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {2\,x^3\,\left (11\,b^2\,c+16\,a\,c^2\right )\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x^2\,\left (b^3+16\,a\,c\,b\right )\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {4\,c^3\,x^5\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {10\,b\,c^2\,x^4\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}}{x^2\,\left (3\,c\,a^2+3\,a\,b^2\right )+x^4\,\left (3\,b^2\,c+3\,a\,c^2\right )+a^3+x^3\,\left (b^3+6\,a\,c\,b\right )+c^3\,x^6+3\,b\,c^2\,x^5+3\,a^2\,b\,x}-\frac {8\,c\,\mathrm {atan}\left (\frac {\left (\frac {8\,c^2\,x\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}}+\frac {4\,c\,\left (-64\,a^3\,b\,c^3+48\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{4\,b^2\,c\,e^2-20\,b\,c^2\,d\,e+20\,c^3\,d^2+4\,a\,c^2\,e^2}\right )\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((b^5*d^2 + a^2*b^3*e^2 + 66*a^2*b*c^2*d^2 + a*b^4*d*e - 13*a*b^3*c*d^2 + 26*a^3*b*c*e^2 - 64*a^3*c^2*d*e -
18*a^2*b^2*c*d*e)/(3*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (x*(a*b^4*e^2 - b^4*c*d^2 + 44*a^2*c^
3*d^2 - 4*a^3*c^2*e^2 + b^5*d*e + 18*a*b^2*c^2*d^2 + 22*a^2*b^2*c*e^2 - 44*a^2*b*c^2*d*e - 18*a*b^3*c*d*e))/(b
^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c) + (2*x^3*(16*a*c^2 + 11*b^2*c)*(b^2*e^2 + 5*c^2*d^2 + a*c*e^2 -
5*b*c*d*e))/(3*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (x^2*(b^3 + 16*a*b*c)*(b^2*e^2 + 5*c^2*d^2
+ a*c*e^2 - 5*b*c*d*e))/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c) + (4*c^3*x^5*(b^2*e^2 + 5*c^2*d^2 +
a*c*e^2 - 5*b*c*d*e))/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c) + (10*b*c^2*x^4*(b^2*e^2 + 5*c^2*d^2 +
a*c*e^2 - 5*b*c*d*e))/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c))/(x^2*(3*a*b^2 + 3*a^2*c) + x^4*(3*a*c^
2 + 3*b^2*c) + a^3 + x^3*(b^3 + 6*a*b*c) + c^3*x^6 + 3*b*c^2*x^5 + 3*a^2*b*x) - (8*c*atan((((8*c^2*x*(b^2*e^2
+ 5*c^2*d^2 + a*c*e^2 - 5*b*c*d*e))/(4*a*c - b^2)^(7/2) + (4*c*(b^7 - 64*a^3*b*c^3 + 48*a^2*b^3*c^2 - 12*a*b^5
*c)*(b^2*e^2 + 5*c^2*d^2 + a*c*e^2 - 5*b*c*d*e))/((4*a*c - b^2)^(7/2)*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*
a*b^4*c)))*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c))/(20*c^3*d^2 + 4*a*c^2*e^2 + 4*b^2*c*e^2 - 20*b*c^
2*d*e))*(b^2*e^2 + 5*c^2*d^2 + a*c*e^2 - 5*b*c*d*e))/(4*a*c - b^2)^(7/2)

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