3.7.51 \(\int \frac {1}{(d f+e f x)^2 (a+b (d+e x)^2+c (d+e x)^4)^2} \, dx\) [651]

Optimal. Leaf size=360 \[ -\frac {3 b^2-10 a c}{2 a^2 \left (b^2-4 a c\right ) e f^2 (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\sqrt {c} \left (3 b^3-16 a b c+\left (3 b^2-10 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} e f^2}+\frac {\sqrt {c} \left (3 b^3-16 a b c-\left (3 b^2-10 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} e f^2} \]

[Out]

1/2*(10*a*c-3*b^2)/a^2/(-4*a*c+b^2)/e/f^2/(e*x+d)+1/2*(b^2-2*a*c+b*c*(e*x+d)^2)/a/(-4*a*c+b^2)/e/f^2/(e*x+d)/(
a+b*(e*x+d)^2+c*(e*x+d)^4)-1/4*arctan((e*x+d)*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(3*b^3-16*
a*b*c+(-10*a*c+3*b^2)*(-4*a*c+b^2)^(1/2))/a^2/(-4*a*c+b^2)^(3/2)/e/f^2*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/
4*arctan((e*x+d)*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(3*b^3-16*a*b*c-(-10*a*c+3*b^2)*(-4*a*c
+b^2)^(1/2))/a^2/(-4*a*c+b^2)^(3/2)/e/f^2*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]
time = 1.13, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1156, 1135, 1295, 1180, 211} \begin {gather*} -\frac {\sqrt {c} \left (\left (3 b^2-10 a c\right ) \sqrt {b^2-4 a c}-16 a b c+3 b^3\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 e f^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (-\left (3 b^2-10 a c\right ) \sqrt {b^2-4 a c}-16 a b c+3 b^3\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 \sqrt {2} a^2 e f^2 \left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {3 b^2-10 a c}{2 a^2 e f^2 \left (b^2-4 a c\right ) (d+e x)}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a e f^2 \left (b^2-4 a c\right ) (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d*f + e*f*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]

[Out]

-1/2*(3*b^2 - 10*a*c)/(a^2*(b^2 - 4*a*c)*e*f^2*(d + e*x)) + (b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)
*e*f^2*(d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (Sqrt[c]*(3*b^3 - 16*a*b*c + (3*b^2 - 10*a*c)*Sqrt[b^2
- 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^2*(b^2 - 4*a*c)^(3/2)
*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e*f^2) + (Sqrt[c]*(3*b^3 - 16*a*b*c - (3*b^2 - 10*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[
(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^2*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2
- 4*a*c]]*e*f^2)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 1135

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-(d*x)^(m + 1))*(b^2 - 2*
a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*d*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*
a*c)), Int[(d*x)^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[b^2*(m + 2*p + 3) - 2*a*c*(m + 4*p + 5) + b*c*(m + 4*p + 7
)*x^2, x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p] && (Integ
erQ[p] || IntegerQ[m])

Rule 1156

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1295

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d*(f
*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1)/(a*f*(m + 1))), x] + Dist[1/(a*f^2*(m + 1)), Int[(f*x)^(m + 2)*(a + b
*x^2 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rubi steps

\begin {align*} \int \frac {1}{(d f+e f x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx,x,d+e x\right )}{e f^2}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\text {Subst}\left (\int \frac {-3 b^2+10 a c-3 b c x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx,x,d+e x\right )}{2 a \left (b^2-4 a c\right ) e f^2}\\ &=-\frac {3 b^2-10 a c}{2 a^2 \left (b^2-4 a c\right ) e f^2 (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\text {Subst}\left (\int \frac {-b \left (3 b^2-13 a c\right )-c \left (3 b^2-10 a c\right ) x^2}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{2 a^2 \left (b^2-4 a c\right ) e f^2}\\ &=-\frac {3 b^2-10 a c}{2 a^2 \left (b^2-4 a c\right ) e f^2 (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\left (c \left (3 b^2-10 a c+\frac {3 b^3}{\sqrt {b^2-4 a c}}-\frac {16 a b c}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,d+e x\right )}{4 a^2 \left (b^2-4 a c\right ) e f^2}-\frac {\left (c \left (3 b^2-10 a c-\frac {3 b^3}{\sqrt {b^2-4 a c}}+\frac {16 a b c}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,d+e x\right )}{4 a^2 \left (b^2-4 a c\right ) e f^2}\\ &=-\frac {3 b^2-10 a c}{2 a^2 \left (b^2-4 a c\right ) e f^2 (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\sqrt {c} \left (3 b^2-10 a c+\frac {3 b^3}{\sqrt {b^2-4 a c}}-\frac {16 a b c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}} e f^2}-\frac {\sqrt {c} \left (3 b^2-10 a c-\frac {3 b^3}{\sqrt {b^2-4 a c}}+\frac {16 a b c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}} e f^2}\\ \end {align*}

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Mathematica [A]
time = 1.03, size = 342, normalized size = 0.95 \begin {gather*} \frac {-\frac {4}{d+e x}+\frac {2 (d+e x) \left (b^3-3 a b c+b^2 c (d+e x)^2-2 a c^2 (d+e x)^2\right )}{\left (-b^2+4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\sqrt {2} \sqrt {c} \left (-3 b^3+16 a b c-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (3 b^3-16 a b c-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{4 a^2 e f^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d*f + e*f*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]

[Out]

(-4/(d + e*x) + (2*(d + e*x)*(b^3 - 3*a*b*c + b^2*c*(d + e*x)^2 - 2*a*c^2*(d + e*x)^2))/((-b^2 + 4*a*c)*(a + b
*(d + e*x)^2 + c*(d + e*x)^4)) + (Sqrt[2]*Sqrt[c]*(-3*b^3 + 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b
^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sq
rt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(3*b^3 - 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c])*A
rcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c
]]))/(4*a^2*e*f^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.21, size = 445, normalized size = 1.24

method	result	size
default	\(\frac {-\frac {1}{a^{2} e \left (e x +d \right )}-\frac {\frac {\frac {c \,e^{2} \left (2 a c -b^{2}\right ) x^{3}}{8 a c -2 b^{2}}+\frac {3 d c e \left (2 a c -b^{2}\right ) x^{2}}{2 \left (4 a c -b^{2}\right )}+\frac {\left (6 a \,c^{2} d^{2}-3 b^{2} c \,d^{2}+3 a b c -b^{3}\right ) x}{8 a c -2 b^{2}}+\frac {d \left (2 a \,c^{2} d^{2}-b^{2} c \,d^{2}+3 a b c -b^{3}\right )}{2 e \left (4 a c -b^{2}\right )}}{c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 d e b x +d^{2} b +a}+\frac {\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (c \,e^{2} \left (10 a c -3 b^{2}\right ) \textit {\_R}^{2}+2 c d e \left (10 a c -3 b^{2}\right ) \textit {\_R} +10 a \,c^{2} d^{2}-3 b^{2} c \,d^{2}+13 a b c -3 b^{3}\right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}}{4 \left (4 a c -b^{2}\right ) e}}{a^{2}}}{f^{2}}\)	\(445\)
risch	\(\frac {-\frac {e^{3} c \left (10 a c -3 b^{2}\right ) x^{4}}{2 a^{2} \left (4 a c -b^{2}\right )}-\frac {2 d \,e^{2} c \left (10 a c -3 b^{2}\right ) x^{3}}{a^{2} \left (4 a c -b^{2}\right )}-\frac {\left (60 a \,c^{2} d^{2}-18 b^{2} c \,d^{2}+11 a b c -3 b^{3}\right ) e \,x^{2}}{2 a^{2} \left (4 a c -b^{2}\right )}-\frac {d \left (20 a \,c^{2} d^{2}-6 b^{2} c \,d^{2}+11 a b c -3 b^{3}\right ) x}{a^{2} \left (4 a c -b^{2}\right )}-\frac {10 a \,c^{2} d^{4}-3 b^{2} c \,d^{4}+11 a b c \,d^{2}-3 b^{3} d^{2}+8 a^{2} c -2 a \,b^{2}}{2 e \,a^{2} \left (4 a c -b^{2}\right )}}{f^{2} \left (e x +d \right ) \left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 d e b x +d^{2} b +a \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (4096 a^{11} c^{6} e^{4} f^{8}-6144 a^{10} b^{2} c^{5} e^{4} f^{8}+3840 a^{9} b^{4} c^{4} e^{4} f^{8}-1280 a^{8} b^{6} c^{3} e^{4} f^{8}+240 a^{7} b^{8} c^{2} e^{4} f^{8}-24 a^{6} b^{10} c \,e^{4} f^{8}+a^{5} b^{12} e^{4} f^{8}\right ) \textit {\_Z}^{4}+\left (26880 a^{6} b \,c^{6} e^{2} f^{4}-44800 a^{5} b^{3} c^{5} e^{2} f^{4}+30240 a^{4} b^{5} c^{4} e^{2} f^{4}-10656 a^{3} b^{7} c^{3} e^{2} f^{4}+2077 a^{2} b^{9} c^{2} e^{2} f^{4}-213 a \,b^{11} c \,e^{2} f^{4}+9 b^{13} e^{2} f^{4}\right ) \textit {\_Z}^{2}+10000 a^{2} c^{7}-4200 a \,b^{2} c^{6}+441 b^{4} c^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (10240 a^{11} c^{6} e^{5} f^{8}-15872 a^{10} b^{2} c^{5} e^{5} f^{8}+10240 a^{9} b^{4} c^{4} e^{5} f^{8}-3520 a^{8} b^{6} c^{3} e^{5} f^{8}+680 a^{7} b^{8} c^{2} e^{5} f^{8}-70 a^{6} b^{10} c \,e^{5} f^{8}+3 a^{5} b^{12} e^{5} f^{8}\right ) \textit {\_R}^{4}+\left (57280 a^{6} b \,c^{6} e^{3} f^{4}-92752 a^{5} b^{3} c^{5} e^{3} f^{4}+61540 a^{4} b^{5} c^{4} e^{3} f^{4}-21471 a^{3} b^{7} c^{3} e^{3} f^{4}+4163 a^{2} b^{9} c^{2} e^{3} f^{4}-426 a \,b^{11} c \,e^{3} f^{4}+18 b^{13} e^{3} f^{4}\right ) \textit {\_R}^{2}+20000 e \,a^{2} c^{7}-8400 a \,b^{2} c^{6} e +882 b^{4} c^{5} e \right ) x +\left (10240 a^{11} c^{6} d \,e^{4} f^{8}-15872 a^{10} b^{2} c^{5} d \,e^{4} f^{8}+10240 a^{9} b^{4} c^{4} d \,e^{4} f^{8}-3520 a^{8} b^{6} c^{3} d \,e^{4} f^{8}+680 a^{7} b^{8} c^{2} d \,e^{4} f^{8}-70 a^{6} b^{10} c d \,e^{4} f^{8}+3 a^{5} b^{12} d \,e^{4} f^{8}\right ) \textit {\_R}^{4}+\left (2560 a^{9} c^{6} e^{3} f^{6}-6656 a^{8} b^{2} c^{5} e^{3} f^{6}+5824 a^{7} b^{4} c^{4} e^{3} f^{6}-2464 a^{6} b^{6} c^{3} e^{3} f^{6}+554 a^{5} b^{8} c^{2} e^{3} f^{6}-64 a^{4} b^{10} c \,e^{3} f^{6}+3 a^{3} b^{12} e^{3} f^{6}\right ) \textit {\_R}^{3}+\left (57280 a^{6} b \,c^{6} d \,e^{2} f^{4}-92752 a^{5} b^{3} c^{5} d \,e^{2} f^{4}+61540 a^{4} b^{5} c^{4} d \,e^{2} f^{4}-21471 a^{3} b^{7} c^{3} d \,e^{2} f^{4}+4163 a^{2} b^{9} c^{2} d \,e^{2} f^{4}-426 a \,b^{11} c d \,e^{2} f^{4}+18 b^{13} d \,e^{2} f^{4}\right ) \textit {\_R}^{2}+\left (1200 a^{4} b \,c^{6} e \,f^{2}-552 a^{3} b^{3} c^{5} e \,f^{2}+63 a^{2} b^{5} c^{4} e \,f^{2}\right ) \textit {\_R} +20000 a^{2} c^{7} d -8400 a \,b^{2} c^{6} d +882 b^{4} c^{5} d \right )\right )}{4}\)	\(1268\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*f*x+d*f)^2/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x,method=_RETURNVERBOSE)

[Out]

1/f^2*(-1/a^2/e/(e*x+d)-1/a^2*((1/2*c*e^2*(2*a*c-b^2)/(4*a*c-b^2)*x^3+3/2*d*c*e*(2*a*c-b^2)/(4*a*c-b^2)*x^2+1/
2*(6*a*c^2*d^2-3*b^2*c*d^2+3*a*b*c-b^3)/(4*a*c-b^2)*x+1/2*d/e*(2*a*c^2*d^2-b^2*c*d^2+3*a*b*c-b^3)/(4*a*c-b^2))
/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*e*x+b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+a)+1/4/(4*a*c-b^2)/e*sum
((c*e^2*(10*a*c-3*b^2)*_R^2+2*c*d*e*(10*a*c-3*b^2)*_R+10*a*c^2*d^2-3*b^2*c*d^2+13*a*b*c-3*b^3)/(2*_R^3*c*e^3+6
*_R^2*c*d*e^2+6*_R*c*d^2*e+2*c*d^3+_R*b*e+b*d)*ln(x-_R),_R=RootOf(e^4*c*_Z^4+4*d*e^3*c*_Z^3+(6*c*d^2*e^2+b*e^2
)*_Z^2+(4*c*d^3*e+2*b*d*e)*_Z+d^4*c+d^2*b+a))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*f*x+d*f)^2/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x, algorithm="maxima")

[Out]

-1/2*((3*b^2*c - 10*a*c^2)*d^4 + 4*(3*b^2*c*e^3 - 10*a*c^2*e^3)*d*x^3 + (3*b^2*c*e^4 - 10*a*c^2*e^4)*x^4 + 2*a
*b^2 - 8*a^2*c + (3*b^3 - 11*a*b*c)*d^2 + (3*b^3*e^2 - 11*a*b*c*e^2 + 6*(3*b^2*c*e^2 - 10*a*c^2*e^2)*d^2)*x^2
+ 2*(2*(3*b^2*c*e - 10*a*c^2*e)*d^3 + (3*b^3*e - 11*a*b*c*e)*d)*x)/(5*(a^2*b^2*c*e^5 - 4*a^3*c^2*e^5)*d*f^2*x^
4 + (a^2*b^2*c*e^6 - 4*a^3*c^2*e^6)*f^2*x^5 + (a^2*b^3*e^4 - 4*a^3*b*c*e^4 + 10*(a^2*b^2*c*e^4 - 4*a^3*c^2*e^4
)*d^2)*f^2*x^3 + (10*(a^2*b^2*c*e^3 - 4*a^3*c^2*e^3)*d^3 + 3*(a^2*b^3*e^3 - 4*a^3*b*c*e^3)*d)*f^2*x^2 + (a^3*b
^2*e^2 - 4*a^4*c*e^2 + 5*(a^2*b^2*c*e^2 - 4*a^3*c^2*e^2)*d^4 + 3*(a^2*b^3*e^2 - 4*a^3*b*c*e^2)*d^2)*f^2*x + ((
a^2*b^2*c*e - 4*a^3*c^2*e)*d^5 + (a^2*b^3*e - 4*a^3*b*c*e)*d^3 + (a^3*b^2*e - 4*a^4*c*e)*d)*f^2) - 1/2*integra
te((3*b^3 - 13*a*b*c + (3*b^2*c - 10*a*c^2)*d^2 + 2*(3*b^2*c*e - 10*a*c^2*e)*d*x + (3*b^2*c*e^2 - 10*a*c^2*e^2
)*x^2)/((b^2*c - 4*a*c^2)*d^4 + 4*(b^2*c*e^3 - 4*a*c^2*e^3)*d*x^3 + (b^2*c*e^4 - 4*a*c^2*e^4)*x^4 + a*b^2 - 4*
a^2*c + (b^3 - 4*a*b*c)*d^2 + (b^3*e^2 - 4*a*b*c*e^2 + 6*(b^2*c*e^2 - 4*a*c^2*e^2)*d^2)*x^2 + 2*(2*(b^2*c*e -
4*a*c^2*e)*d^3 + (b^3*e - 4*a*b*c*e)*d)*x), x)/(a^2*f^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4442 vs. \(2 (315) = 630\).
time = 0.54, size = 4442, normalized size = 12.34 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*f*x+d*f)^2/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x, algorithm="fricas")

[Out]

-1/4*(2*(3*b^2*c - 10*a*c^2)*x^4*e^4 + 8*(3*b^2*c - 10*a*c^2)*d*x^3*e^3 + 2*(3*b^2*c - 10*a*c^2)*d^4 + 2*(3*b^
3 - 11*a*b*c + 6*(3*b^2*c - 10*a*c^2)*d^2)*x^2*e^2 + 4*a*b^2 - 16*a^2*c + 2*(3*b^3 - 11*a*b*c)*d^2 + 4*(2*(3*b
^2*c - 10*a*c^2)*d^3 + (3*b^3 - 11*a*b*c)*d)*x*e + sqrt(1/2)*((a^2*b^2*c - 4*a^3*c^2)*f^2*x^5*e^6 + 5*(a^2*b^2
*c - 4*a^3*c^2)*d*f^2*x^4*e^5 + (a^2*b^3 - 4*a^3*b*c + 10*(a^2*b^2*c - 4*a^3*c^2)*d^2)*f^2*x^3*e^4 + (10*(a^2*
b^2*c - 4*a^3*c^2)*d^3 + 3*(a^2*b^3 - 4*a^3*b*c)*d)*f^2*x^2*e^3 + (a^3*b^2 - 4*a^4*c + 5*(a^2*b^2*c - 4*a^3*c^
2)*d^4 + 3*(a^2*b^3 - 4*a^3*b*c)*d^2)*f^2*x*e^2 + ((a^2*b^2*c - 4*a^3*c^2)*d^5 + (a^2*b^3 - 4*a^3*b*c)*d^3 + (
a^3*b^2 - 4*a^4*c)*d)*f^2*e)*sqrt(-(9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3 + (a^5*b^6 - 12*a^6*
b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*f^4*sqrt((81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 62
5*a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*f^8)))*e^(-2)/((a^5*b^6 - 12*a^6*b^4*c
+ 48*a^7*b^2*c^2 - 64*a^8*c^3)*f^4))*log(-(189*b^6*c^3 - 1971*a*b^4*c^4 + 5625*a^2*b^2*c^5 - 2500*a^3*c^6)*x*e
- (189*b^6*c^3 - 1971*a*b^4*c^4 + 5625*a^2*b^2*c^5 - 2500*a^3*c^6)*d + 1/2*sqrt(1/2)*((3*a^5*b^10 - 55*a^6*b^
8*c + 392*a^7*b^6*c^2 - 1344*a^8*b^4*c^3 + 2176*a^9*b^2*c^4 - 1280*a^10*c^5)*f^6*sqrt((81*b^8 - 918*a*b^6*c +
3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)
*f^8))*e - (27*b^11 - 486*a*b^9*c + 3330*a^2*b^7*c^2 - 10549*a^3*b^5*c^3 + 14408*a^4*b^3*c^4 - 5200*a^5*b*c^5)
*f^2*e)*sqrt(-(9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3 + (a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^
2 - 64*a^8*c^3)*f^4*sqrt((81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)/((a^10*b^6
- 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*f^8)))*e^(-2)/((a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64
*a^8*c^3)*f^4))) - sqrt(1/2)*((a^2*b^2*c - 4*a^3*c^2)*f^2*x^5*e^6 + 5*(a^2*b^2*c - 4*a^3*c^2)*d*f^2*x^4*e^5 +
(a^2*b^3 - 4*a^3*b*c + 10*(a^2*b^2*c - 4*a^3*c^2)*d^2)*f^2*x^3*e^4 + (10*(a^2*b^2*c - 4*a^3*c^2)*d^3 + 3*(a^2*
b^3 - 4*a^3*b*c)*d)*f^2*x^2*e^3 + (a^3*b^2 - 4*a^4*c + 5*(a^2*b^2*c - 4*a^3*c^2)*d^4 + 3*(a^2*b^3 - 4*a^3*b*c)
*d^2)*f^2*x*e^2 + ((a^2*b^2*c - 4*a^3*c^2)*d^5 + (a^2*b^3 - 4*a^3*b*c)*d^3 + (a^3*b^2 - 4*a^4*c)*d)*f^2*e)*sqr
t(-(9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3 + (a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*
c^3)*f^4*sqrt((81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)/((a^10*b^6 - 12*a^11*
b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*f^8)))*e^(-2)/((a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*f
^4))*log(-(189*b^6*c^3 - 1971*a*b^4*c^4 + 5625*a^2*b^2*c^5 - 2500*a^3*c^6)*x*e - (189*b^6*c^3 - 1971*a*b^4*c^4
+ 5625*a^2*b^2*c^5 - 2500*a^3*c^6)*d - 1/2*sqrt(1/2)*((3*a^5*b^10 - 55*a^6*b^8*c + 392*a^7*b^6*c^2 - 1344*a^8
*b^4*c^3 + 2176*a^9*b^2*c^4 - 1280*a^10*c^5)*f^6*sqrt((81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*
c^3 + 625*a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*f^8))*e - (27*b^11 - 486*a*b^9*
c + 3330*a^2*b^7*c^2 - 10549*a^3*b^5*c^3 + 14408*a^4*b^3*c^4 - 5200*a^5*b*c^5)*f^2*e)*sqrt(-(9*b^7 - 105*a*b^5
*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3 + (a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*f^4*sqrt((81*b^8
- 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c
^2 - 64*a^13*c^3)*f^8)))*e^(-2)/((a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*f^4))) - sqrt(1/2)*((a
^2*b^2*c - 4*a^3*c^2)*f^2*x^5*e^6 + 5*(a^2*b^2*c - 4*a^3*c^2)*d*f^2*x^4*e^5 + (a^2*b^3 - 4*a^3*b*c + 10*(a^2*b
^2*c - 4*a^3*c^2)*d^2)*f^2*x^3*e^4 + (10*(a^2*b^2*c - 4*a^3*c^2)*d^3 + 3*(a^2*b^3 - 4*a^3*b*c)*d)*f^2*x^2*e^3
+ (a^3*b^2 - 4*a^4*c + 5*(a^2*b^2*c - 4*a^3*c^2)*d^4 + 3*(a^2*b^3 - 4*a^3*b*c)*d^2)*f^2*x*e^2 + ((a^2*b^2*c -
4*a^3*c^2)*d^5 + (a^2*b^3 - 4*a^3*b*c)*d^3 + (a^3*b^2 - 4*a^4*c)*d)*f^2*e)*sqrt(-(9*b^7 - 105*a*b^5*c + 385*a^
2*b^3*c^2 - 420*a^3*b*c^3 - (a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*f^4*sqrt((81*b^8 - 918*a*b^
6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^1
3*c^3)*f^8)))*e^(-2)/((a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*f^4))*log(-(189*b^6*c^3 - 1971*a*
b^4*c^4 + 5625*a^2*b^2*c^5 - 2500*a^3*c^6)*x*e - (189*b^6*c^3 - 1971*a*b^4*c^4 + 5625*a^2*b^2*c^5 - 2500*a^3*c
^6)*d + 1/2*sqrt(1/2)*((3*a^5*b^10 - 55*a^6*b^8*c + 392*a^7*b^6*c^2 - 1344*a^8*b^4*c^3 + 2176*a^9*b^2*c^4 - 12
80*a^10*c^5)*f^6*sqrt((81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)/((a^10*b^6 -
12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*f^8))*e + (27*b^11 - 486*a*b^9*c + 3330*a^2*b^7*c^2 - 10549*a^3
*b^5*c^3 + 14408*a^4*b^3*c^4 - 5200*a^5*b*c^5)*f^2*e)*sqrt(-(9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b
*c^3 - (a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*f^4*sqrt((81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^
2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)/((a^10*b^6 ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*f*x+d*f)**2/(a+b*(e*x+d)**2+c*(e*x+d)**4)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 999 vs. \(2 (315) = 630\).
time = 4.02, size = 999, normalized size = 2.78 \begin {gather*} -\frac {\frac {b^{2} c e^{\left (-1\right )}}{{\left (f x e + d f\right )} f} - \frac {2 \, a c^{2} e^{\left (-1\right )}}{{\left (f x e + d f\right )} f} + \frac {b^{3} f e^{\left (-1\right )}}{{\left (f x e + d f\right )}^{3}} - \frac {3 \, a b c f e^{\left (-1\right )}}{{\left (f x e + d f\right )}^{3}}}{2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} {\left (c + \frac {b f^{2}}{{\left (f x e + d f\right )}^{2}} + \frac {a f^{4}}{{\left (f x e + d f\right )}^{4}}\right )}} - \frac {e^{\left (-1\right )}}{{\left (f x e + d f\right )} a^{2} f} + \frac {{\left ({\left (3 \, a^{4} b^{7} - 31 \, a^{5} b^{5} c + 96 \, a^{6} b^{3} c^{2} - 80 \, a^{7} b c^{3}\right )} \sqrt {2 \, a b + 2 \, \sqrt {b^{2} - 4 \, a c} a} f^{8} e^{4} + 2 \, {\left (3 \, a^{3} b^{2} c - 10 \, a^{4} c^{2}\right )} \sqrt {2 \, a b + 2 \, \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} f^{4} {\left | a^{2} b^{2} f^{4} e^{2} - 4 \, a^{3} c f^{4} e^{2} \right |} e^{2} - {\left (a^{2} b^{2} f^{4} e^{2} - 4 \, a^{3} c f^{4} e^{2}\right )}^{2} {\left (3 \, b^{3} - 13 \, a b c\right )} \sqrt {2 \, a b + 2 \, \sqrt {b^{2} - 4 \, a c} a}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} e^{\left (-1\right )}}{{\left (f x e + d f\right )} f \sqrt {\frac {a^{2} b^{3} f^{4} e^{2} - 4 \, a^{3} b c f^{4} e^{2} + \sqrt {{\left (a^{2} b^{3} f^{4} e^{2} - 4 \, a^{3} b c f^{4} e^{2}\right )}^{2} - 4 \, {\left (a^{3} b^{2} f^{8} e^{4} - 4 \, a^{4} c f^{8} e^{4}\right )} {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )}}}{a^{3} b^{2} f^{8} e^{4} - 4 \, a^{4} c f^{8} e^{4}}}}\right ) e^{\left (-3\right )}}{16 \, {\left (a^{5} b^{2} c - 4 \, a^{6} c^{2}\right )} \sqrt {b^{2} - 4 \, a c} f^{6} {\left | a^{2} b^{2} f^{4} e^{2} - 4 \, a^{3} c f^{4} e^{2} \right |} {\left | a \right |}} - \frac {{\left ({\left (3 \, a^{4} b^{7} - 31 \, a^{5} b^{5} c + 96 \, a^{6} b^{3} c^{2} - 80 \, a^{7} b c^{3}\right )} \sqrt {2 \, a b - 2 \, \sqrt {b^{2} - 4 \, a c} a} f^{8} e^{4} - 2 \, {\left (3 \, a^{3} b^{2} c - 10 \, a^{4} c^{2}\right )} \sqrt {2 \, a b - 2 \, \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} f^{4} {\left | a^{2} b^{2} f^{4} e^{2} - 4 \, a^{3} c f^{4} e^{2} \right |} e^{2} - {\left (a^{2} b^{2} f^{4} e^{2} - 4 \, a^{3} c f^{4} e^{2}\right )}^{2} {\left (3 \, b^{3} - 13 \, a b c\right )} \sqrt {2 \, a b - 2 \, \sqrt {b^{2} - 4 \, a c} a}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} e^{\left (-1\right )}}{{\left (f x e + d f\right )} f \sqrt {\frac {a^{2} b^{3} f^{4} e^{2} - 4 \, a^{3} b c f^{4} e^{2} - \sqrt {{\left (a^{2} b^{3} f^{4} e^{2} - 4 \, a^{3} b c f^{4} e^{2}\right )}^{2} - 4 \, {\left (a^{3} b^{2} f^{8} e^{4} - 4 \, a^{4} c f^{8} e^{4}\right )} {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )}}}{a^{3} b^{2} f^{8} e^{4} - 4 \, a^{4} c f^{8} e^{4}}}}\right ) e^{\left (-3\right )}}{16 \, {\left (a^{5} b^{2} c - 4 \, a^{6} c^{2}\right )} \sqrt {b^{2} - 4 \, a c} f^{6} {\left | a^{2} b^{2} f^{4} e^{2} - 4 \, a^{3} c f^{4} e^{2} \right |} {\left | a \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*f*x+d*f)^2/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x, algorithm="giac")

[Out]

-1/2*(b^2*c*e^(-1)/((f*x*e + d*f)*f) - 2*a*c^2*e^(-1)/((f*x*e + d*f)*f) + b^3*f*e^(-1)/(f*x*e + d*f)^3 - 3*a*b
*c*f*e^(-1)/(f*x*e + d*f)^3)/((a^2*b^2 - 4*a^3*c)*(c + b*f^2/(f*x*e + d*f)^2 + a*f^4/(f*x*e + d*f)^4)) - e^(-1
)/((f*x*e + d*f)*a^2*f) + 1/16*((3*a^4*b^7 - 31*a^5*b^5*c + 96*a^6*b^3*c^2 - 80*a^7*b*c^3)*sqrt(2*a*b + 2*sqrt
(b^2 - 4*a*c)*a)*f^8*e^4 + 2*(3*a^3*b^2*c - 10*a^4*c^2)*sqrt(2*a*b + 2*sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*
f^4*abs(a^2*b^2*f^4*e^2 - 4*a^3*c*f^4*e^2)*e^2 - (a^2*b^2*f^4*e^2 - 4*a^3*c*f^4*e^2)^2*(3*b^3 - 13*a*b*c)*sqrt
(2*a*b + 2*sqrt(b^2 - 4*a*c)*a))*arctan(2*sqrt(1/2)*e^(-1)/((f*x*e + d*f)*f*sqrt((a^2*b^3*f^4*e^2 - 4*a^3*b*c*
f^4*e^2 + sqrt((a^2*b^3*f^4*e^2 - 4*a^3*b*c*f^4*e^2)^2 - 4*(a^3*b^2*f^8*e^4 - 4*a^4*c*f^8*e^4)*(a^2*b^2*c - 4*
a^3*c^2)))/(a^3*b^2*f^8*e^4 - 4*a^4*c*f^8*e^4))))*e^(-3)/((a^5*b^2*c - 4*a^6*c^2)*sqrt(b^2 - 4*a*c)*f^6*abs(a^
2*b^2*f^4*e^2 - 4*a^3*c*f^4*e^2)*abs(a)) - 1/16*((3*a^4*b^7 - 31*a^5*b^5*c + 96*a^6*b^3*c^2 - 80*a^7*b*c^3)*sq
rt(2*a*b - 2*sqrt(b^2 - 4*a*c)*a)*f^8*e^4 - 2*(3*a^3*b^2*c - 10*a^4*c^2)*sqrt(2*a*b - 2*sqrt(b^2 - 4*a*c)*a)*s
qrt(b^2 - 4*a*c)*f^4*abs(a^2*b^2*f^4*e^2 - 4*a^3*c*f^4*e^2)*e^2 - (a^2*b^2*f^4*e^2 - 4*a^3*c*f^4*e^2)^2*(3*b^3
- 13*a*b*c)*sqrt(2*a*b - 2*sqrt(b^2 - 4*a*c)*a))*arctan(2*sqrt(1/2)*e^(-1)/((f*x*e + d*f)*f*sqrt((a^2*b^3*f^4
*e^2 - 4*a^3*b*c*f^4*e^2 - sqrt((a^2*b^3*f^4*e^2 - 4*a^3*b*c*f^4*e^2)^2 - 4*(a^3*b^2*f^8*e^4 - 4*a^4*c*f^8*e^4
)*(a^2*b^2*c - 4*a^3*c^2)))/(a^3*b^2*f^8*e^4 - 4*a^4*c*f^8*e^4))))*e^(-3)/((a^5*b^2*c - 4*a^6*c^2)*sqrt(b^2 -
4*a*c)*f^6*abs(a^2*b^2*f^4*e^2 - 4*a^3*c*f^4*e^2)*abs(a))

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Mupad [B]
time = 7.29, size = 2500, normalized size = 6.94 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d*f + e*f*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x)

[Out]

- atan(((-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 +
30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 - 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*b^2*c*(-(4*a
*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2*f^4 + 4096*a^11*c^6*e^2*f^4 + 240*a^7*b^8*c^2*e^2*f^4 - 1280*a^8*b^6*c^3
*e^2*f^4 + 3840*a^9*b^4*c^4*e^2*f^4 - 6144*a^10*b^2*c^5*e^2*f^4 - 24*a^6*b^10*c*e^2*f^4)))^(1/2)*((-(9*b^13 -
9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 -
44800*a^5*b^3*c^5 - 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/
(32*(a^5*b^12*e^2*f^4 + 4096*a^11*c^6*e^2*f^4 + 240*a^7*b^8*c^2*e^2*f^4 - 1280*a^8*b^6*c^3*e^2*f^4 + 3840*a^9*
b^4*c^4*e^2*f^4 - 6144*a^10*b^2*c^5*e^2*f^4 - 24*a^6*b^10*c*e^2*f^4)))^(1/2)*((-(9*b^13 - 9*b^4*(-(4*a*c - b^2
)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 -
25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2*f^
4 + 4096*a^11*c^6*e^2*f^4 + 240*a^7*b^8*c^2*e^2*f^4 - 1280*a^8*b^6*c^3*e^2*f^4 + 3840*a^9*b^4*c^4*e^2*f^4 - 61
44*a^10*b^2*c^5*e^2*f^4 - 24*a^6*b^10*c*e^2*f^4)))^(1/2)*(x*(256*a^10*b^13*c^2*e^14*f^10 - 6144*a^11*b^11*c^3*
e^14*f^10 + 61440*a^12*b^9*c^4*e^14*f^10 - 327680*a^13*b^7*c^5*e^14*f^10 + 983040*a^14*b^5*c^6*e^14*f^10 - 157
2864*a^15*b^3*c^7*e^14*f^10 + 1048576*a^16*b*c^8*e^14*f^10) + 1048576*a^16*b*c^8*d*e^13*f^10 + 256*a^10*b^13*c
^2*d*e^13*f^10 - 6144*a^11*b^11*c^3*d*e^13*f^10 + 61440*a^12*b^9*c^4*d*e^13*f^10 - 327680*a^13*b^7*c^5*d*e^13*
f^10 + 983040*a^14*b^5*c^6*d*e^13*f^10 - 1572864*a^15*b^3*c^7*d*e^13*f^10) - 192*a^8*b^13*c^2*e^12*f^8 + 4672*
a^9*b^11*c^3*e^12*f^8 - 47360*a^10*b^9*c^4*e^12*f^8 + 256000*a^11*b^7*c^5*e^12*f^8 - 778240*a^12*b^5*c^6*e^12*
f^8 + 1261568*a^13*b^3*c^7*e^12*f^8 - 851968*a^14*b*c^8*e^12*f^8) + x*(204800*a^12*c^9*e^12*f^6 + 144*a^6*b^12
*c^3*e^12*f^6 - 3264*a^7*b^10*c^4*e^12*f^6 + 30112*a^8*b^8*c^5*e^12*f^6 - 143360*a^9*b^6*c^6*e^12*f^6 + 365568
*a^10*b^4*c^7*e^12*f^6 - 458752*a^11*b^2*c^8*e^12*f^6) + 204800*a^12*c^9*d*e^11*f^6 + 144*a^6*b^12*c^3*d*e^11*
f^6 - 3264*a^7*b^10*c^4*d*e^11*f^6 + 30112*a^8*b^8*c^5*d*e^11*f^6 - 143360*a^9*b^6*c^6*d*e^11*f^6 + 365568*a^1
0*b^4*c^7*d*e^11*f^6 - 458752*a^11*b^2*c^8*d*e^11*f^6)*1i + (-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880
*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 - 25*a^2*c^2*(-(4*a*
c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2*f^4 + 4096*a^11*c^6*
e^2*f^4 + 240*a^7*b^8*c^2*e^2*f^4 - 1280*a^8*b^6*c^3*e^2*f^4 + 3840*a^9*b^4*c^4*e^2*f^4 - 6144*a^10*b^2*c^5*e^
2*f^4 - 24*a^6*b^10*c*e^2*f^4)))^(1/2)*((-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^
2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 - 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) -
213*a*b^11*c + 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2*f^4 + 4096*a^11*c^6*e^2*f^4 + 240*a^7*b^
8*c^2*e^2*f^4 - 1280*a^8*b^6*c^3*e^2*f^4 + 3840*a^9*b^4*c^4*e^2*f^4 - 6144*a^10*b^2*c^5*e^2*f^4 - 24*a^6*b^10*
c*e^2*f^4)))^(1/2)*((-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^
3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 - 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*
b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2*f^4 + 4096*a^11*c^6*e^2*f^4 + 240*a^7*b^8*c^2*e^2*f^4 - 1280
*a^8*b^6*c^3*e^2*f^4 + 3840*a^9*b^4*c^4*e^2*f^4 - 6144*a^10*b^2*c^5*e^2*f^4 - 24*a^6*b^10*c*e^2*f^4)))^(1/2)*(
x*(256*a^10*b^13*c^2*e^14*f^10 - 6144*a^11*b^11*c^3*e^14*f^10 + 61440*a^12*b^9*c^4*e^14*f^10 - 327680*a^13*b^7
*c^5*e^14*f^10 + 983040*a^14*b^5*c^6*e^14*f^10 - 1572864*a^15*b^3*c^7*e^14*f^10 + 1048576*a^16*b*c^8*e^14*f^10
) + 1048576*a^16*b*c^8*d*e^13*f^10 + 256*a^10*b^13*c^2*d*e^13*f^10 - 6144*a^11*b^11*c^3*d*e^13*f^10 + 61440*a^
12*b^9*c^4*d*e^13*f^10 - 327680*a^13*b^7*c^5*d*e^13*f^10 + 983040*a^14*b^5*c^6*d*e^13*f^10 - 1572864*a^15*b^3*
c^7*d*e^13*f^10) + 192*a^8*b^13*c^2*e^12*f^8 - 4672*a^9*b^11*c^3*e^12*f^8 + 47360*a^10*b^9*c^4*e^12*f^8 - 2560
00*a^11*b^7*c^5*e^12*f^8 + 778240*a^12*b^5*c^6*e^12*f^8 - 1261568*a^13*b^3*c^7*e^12*f^8 + 851968*a^14*b*c^8*e^
12*f^8) + x*(204800*a^12*c^9*e^12*f^6 + 144*a^6*b^12*c^3*e^12*f^6 - 3264*a^7*b^10*c^4*e^12*f^6 + 30112*a^8*b^8
*c^5*e^12*f^6 - 143360*a^9*b^6*c^6*e^12*f^6 + 365568*a^10*b^4*c^7*e^12*f^6 - 458752*a^11*b^2*c^8*e^12*f^6) + 2
04800*a^12*c^9*d*e^11*f^6 + 144*a^6*b^12*c^3*d*e^11*f^6 - 3264*a^7*b^10*c^4*d*e^11*f^6 + 30112*a^8*b^8*c^5*d*e
^11*f^6 - 143360*a^9*b^6*c^6*d*e^11*f^6 + 365568*a^10*b^4*c^7*d*e^11*f^6 - 458752*a^11*b^2*c^8*d*e^11*f^6)*1i)
/((-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*
a^4*b^5*c^4 - 44800*a^5*b^3*c^5 - 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*b^2*c*(-(4*a*c - b
^2)^9)^(1/2))/(32*(a^5*b^12*e^2*f^4 + 4096*a^11...

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