3.7.43 \(\int \frac {1}{(d f+e f x)^2 (a+b (d+e x)^2+c (d+e x)^4)} \, dx\) [643]

Optimal. Leaf size=204 \[ -\frac {1}{a e f^2 (d+e x)}-\frac {\sqrt {c} \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b-\sqrt {b^2-4 a c}} e f^2}-\frac {\sqrt {c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b+\sqrt {b^2-4 a c}} e f^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.21, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1156, 1137, 1180, 211} \begin {gather*} -\frac {\sqrt {c} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a e f^2 \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} a e f^2 \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {1}{a e f^2 (d+e x)} \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)),x]

[Out]

-(1/(a*e*f^2*(d + e*x))) - (Sqrt[c]*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt
[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e*f^2) - (Sqrt[c]*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sq
rt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*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 1137

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

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

________________________________________________________________________________________

Mathematica [A]
time = 0.23, size = 209, normalized size = 1.02 \begin {gather*} -\frac {\frac {2}{d+e x}+\frac {\sqrt {2} \sqrt {c} \left (b+\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 )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-b+\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 )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{2 a 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)),x]

[Out]

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.20, size = 172, normalized size = 0.84

method	result	size
default	\(\frac {-\frac {1}{a e \left (e x +d \right )}+\frac {\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (-\textit {\_R}^{2} c \,e^{2}-2 \textit {\_R} c d e -c \,d^{2}-b \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}}{2 a e}}{f^{2}}\)	\(172\)
risch	\(-\frac {1}{a e \,f^{2} \left (e x +d \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (16 f^{8} e^{4} c^{2} a^{5}-8 b^{2} f^{8} e^{4} c \,a^{4}+b^{4} f^{8} e^{4} a^{3}\right ) \textit {\_Z}^{4}+\left (12 a^{2} b \,c^{2} e^{2} f^{4}-7 a \,b^{3} c \,e^{2} f^{4}+b^{5} e^{2} f^{4}\right ) \textit {\_Z}^{2}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (40 a^{5} c^{2} e^{5} f^{8}-22 a^{4} b^{2} c \,e^{5} f^{8}+3 a^{3} b^{4} e^{5} f^{8}\right ) \textit {\_R}^{4}+\left (25 a^{2} b \,c^{2} e^{3} f^{4}-14 a \,b^{3} c \,e^{3} f^{4}+2 b^{5} e^{3} f^{4}\right ) \textit {\_R}^{2}+2 c^{3} e \right ) x +\left (40 a^{5} c^{2} d \,e^{4} f^{8}-22 a^{4} b^{2} c d \,e^{4} f^{8}+3 a^{3} b^{4} d \,e^{4} f^{8}\right ) \textit {\_R}^{4}+\left (4 a^{4} c^{2} e^{3} f^{6}-5 a^{3} b^{2} c \,e^{3} f^{6}+a^{2} b^{4} e^{3} f^{6}\right ) \textit {\_R}^{3}+\left (25 a^{2} b \,c^{2} d \,e^{2} f^{4}-14 a \,b^{3} c d \,e^{2} f^{4}+2 b^{5} d \,e^{2} f^{4}\right ) \textit {\_R}^{2}+2 c^{3} d \right )\right )}{2}\)	\(376\)

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),x,method=_RETURNVERBOSE)

[Out]

1/f^2*(-1/a/e/(e*x+d)+1/2/a/e*sum((-_R^2*c*e^2-2*_R*c*d*e-c*d^2-b)/(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)))

________________________________________________________________________________________

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),x, algorithm="maxima")

[Out]

-1/(a*f^2*x*e^2 + a*d*f^2*e) - integrate((c*x^2*e^2 + 2*c*d*x*e + c*d^2 + b)/(c*x^4*e^4 + 4*c*d*x^3*e^3 + c*d^
4 + b*d^2 + (6*c*d^2*e^2 + b*e^2)*x^2 + 2*(2*c*d^3*e + b*d*e)*x + a), x)/(a*f^2)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1413 vs. \(2 (167) = 334\).
time = 0.37, size = 1413, normalized size = 6.93 \begin {gather*} -\frac {\sqrt {\frac {1}{2}} {\left (a f^{2} x e^{2} + a d f^{2} e\right )} \sqrt {\frac {{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f^{4} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} f^{8}}} - b^{3} + 3 \, a b c}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f^{4}}} e^{\left (-1\right )} \log \left (-2 \, {\left (b^{2} c^{2} - a c^{3}\right )} x e + \sqrt {\frac {1}{2}} {\left ({\left (a^{3} b^{4} - 6 \, a^{4} b^{2} c + 8 \, a^{5} c^{2}\right )} f^{6} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} f^{8}}} e + {\left (b^{5} - 5 \, a b^{3} c + 4 \, a^{2} b c^{2}\right )} f^{2} e\right )} \sqrt {\frac {{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f^{4} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} f^{8}}} - b^{3} + 3 \, a b c}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f^{4}}} e^{\left (-1\right )} - 2 \, {\left (b^{2} c^{2} - a c^{3}\right )} d\right ) - \sqrt {\frac {1}{2}} {\left (a f^{2} x e^{2} + a d f^{2} e\right )} \sqrt {\frac {{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f^{4} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} f^{8}}} - b^{3} + 3 \, a b c}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f^{4}}} e^{\left (-1\right )} \log \left (-2 \, {\left (b^{2} c^{2} - a c^{3}\right )} x e - \sqrt {\frac {1}{2}} {\left ({\left (a^{3} b^{4} - 6 \, a^{4} b^{2} c + 8 \, a^{5} c^{2}\right )} f^{6} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} f^{8}}} e + {\left (b^{5} - 5 \, a b^{3} c + 4 \, a^{2} b c^{2}\right )} f^{2} e\right )} \sqrt {\frac {{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f^{4} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} f^{8}}} - b^{3} + 3 \, a b c}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f^{4}}} e^{\left (-1\right )} - 2 \, {\left (b^{2} c^{2} - a c^{3}\right )} d\right ) - \sqrt {\frac {1}{2}} {\left (a f^{2} x e^{2} + a d f^{2} e\right )} \sqrt {-\frac {{\left ({\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f^{4} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} f^{8}}} + b^{3} - 3 \, a b c\right )} e^{\left (-2\right )}}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f^{4}}} \log \left (-2 \, {\left (b^{2} c^{2} - a c^{3}\right )} x e - 2 \, {\left (b^{2} c^{2} - a c^{3}\right )} d + \sqrt {\frac {1}{2}} {\left ({\left (a^{3} b^{4} - 6 \, a^{4} b^{2} c + 8 \, a^{5} c^{2}\right )} f^{6} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} f^{8}}} e - {\left (b^{5} - 5 \, a b^{3} c + 4 \, a^{2} b c^{2}\right )} f^{2} e\right )} \sqrt {-\frac {{\left ({\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f^{4} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} f^{8}}} + b^{3} - 3 \, a b c\right )} e^{\left (-2\right )}}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f^{4}}}\right ) + \sqrt {\frac {1}{2}} {\left (a f^{2} x e^{2} + a d f^{2} e\right )} \sqrt {-\frac {{\left ({\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f^{4} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} f^{8}}} + b^{3} - 3 \, a b c\right )} e^{\left (-2\right )}}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f^{4}}} \log \left (-2 \, {\left (b^{2} c^{2} - a c^{3}\right )} x e - 2 \, {\left (b^{2} c^{2} - a c^{3}\right )} d - \sqrt {\frac {1}{2}} {\left ({\left (a^{3} b^{4} - 6 \, a^{4} b^{2} c + 8 \, a^{5} c^{2}\right )} f^{6} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} f^{8}}} e - {\left (b^{5} - 5 \, a b^{3} c + 4 \, a^{2} b c^{2}\right )} f^{2} e\right )} \sqrt {-\frac {{\left ({\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f^{4} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} f^{8}}} + b^{3} - 3 \, a b c\right )} e^{\left (-2\right )}}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f^{4}}}\right ) + 2}{2 \, {\left (a f^{2} x e^{2} + a d f^{2} e\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),x, algorithm="fricas")

[Out]

-1/2*(sqrt(1/2)*(a*f^2*x*e^2 + a*d*f^2*e)*sqrt(((a^3*b^2 - 4*a^4*c)*f^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6
*b^2 - 4*a^7*c)*f^8)) - b^3 + 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*f^4))*e^(-1)*log(-2*(b^2*c^2 - a*c^3)*x*e + sqrt(1
/2)*((a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*f^6*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*f^8))*e + (
b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*f^2*e)*sqrt(((a^3*b^2 - 4*a^4*c)*f^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2
- 4*a^7*c)*f^8)) - b^3 + 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*f^4))*e^(-1) - 2*(b^2*c^2 - a*c^3)*d) - sqrt(1/2)*(a*f
^2*x*e^2 + a*d*f^2*e)*sqrt(((a^3*b^2 - 4*a^4*c)*f^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*f^8)
) - b^3 + 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*f^4))*e^(-1)*log(-2*(b^2*c^2 - a*c^3)*x*e - sqrt(1/2)*((a^3*b^4 - 6*a^
4*b^2*c + 8*a^5*c^2)*f^6*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*f^8))*e + (b^5 - 5*a*b^3*c + 4*
a^2*b*c^2)*f^2*e)*sqrt(((a^3*b^2 - 4*a^4*c)*f^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*f^8)) -
b^3 + 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*f^4))*e^(-1) - 2*(b^2*c^2 - a*c^3)*d) - sqrt(1/2)*(a*f^2*x*e^2 + a*d*f^2*e
)*sqrt(-((a^3*b^2 - 4*a^4*c)*f^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*f^8)) + b^3 - 3*a*b*c)*
e^(-2)/((a^3*b^2 - 4*a^4*c)*f^4))*log(-2*(b^2*c^2 - a*c^3)*x*e - 2*(b^2*c^2 - a*c^3)*d + sqrt(1/2)*((a^3*b^4 -
6*a^4*b^2*c + 8*a^5*c^2)*f^6*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*f^8))*e - (b^5 - 5*a*b^3*c
+ 4*a^2*b*c^2)*f^2*e)*sqrt(-((a^3*b^2 - 4*a^4*c)*f^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*f^
8)) + b^3 - 3*a*b*c)*e^(-2)/((a^3*b^2 - 4*a^4*c)*f^4))) + sqrt(1/2)*(a*f^2*x*e^2 + a*d*f^2*e)*sqrt(-((a^3*b^2
- 4*a^4*c)*f^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*f^8)) + b^3 - 3*a*b*c)*e^(-2)/((a^3*b^2 -
4*a^4*c)*f^4))*log(-2*(b^2*c^2 - a*c^3)*x*e - 2*(b^2*c^2 - a*c^3)*d - sqrt(1/2)*((a^3*b^4 - 6*a^4*b^2*c + 8*a
^5*c^2)*f^6*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*f^8))*e - (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*f^
2*e)*sqrt(-((a^3*b^2 - 4*a^4*c)*f^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*f^8)) + b^3 - 3*a*b*
c)*e^(-2)/((a^3*b^2 - 4*a^4*c)*f^4))) + 2)/(a*f^2*x*e^2 + a*d*f^2*e)

________________________________________________________________________________________

Sympy [A]
time = 3.25, size = 258, normalized size = 1.26 \begin {gather*} \operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{5} c^{2} e^{4} f^{8} - 128 a^{4} b^{2} c e^{4} f^{8} + 16 a^{3} b^{4} e^{4} f^{8}\right ) + t^{2} \cdot \left (48 a^{2} b c^{2} e^{2} f^{4} - 28 a b^{3} c e^{2} f^{4} + 4 b^{5} e^{2} f^{4}\right ) + c^{3}, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{5} c^{2} e^{3} f^{6} + 48 t^{3} a^{4} b^{2} c e^{3} f^{6} - 8 t^{3} a^{3} b^{4} e^{3} f^{6} - 10 t a^{2} b c^{2} e f^{2} + 10 t a b^{3} c e f^{2} - 2 t b^{5} e f^{2} + a c^{3} d - b^{2} c^{2} d}{a c^{3} e - b^{2} c^{2} e} \right )} \right )\right )} - \frac {1}{a d e f^{2} + a e^{2} f^{2} x} \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),x)

[Out]

RootSum(_t**4*(256*a**5*c**2*e**4*f**8 - 128*a**4*b**2*c*e**4*f**8 + 16*a**3*b**4*e**4*f**8) + _t**2*(48*a**2*
b*c**2*e**2*f**4 - 28*a*b**3*c*e**2*f**4 + 4*b**5*e**2*f**4) + c**3, Lambda(_t, _t*log(x + (-64*_t**3*a**5*c**
2*e**3*f**6 + 48*_t**3*a**4*b**2*c*e**3*f**6 - 8*_t**3*a**3*b**4*e**3*f**6 - 10*_t*a**2*b*c**2*e*f**2 + 10*_t*
a*b**3*c*e*f**2 - 2*_t*b**5*e*f**2 + a*c**3*d - b**2*c**2*d)/(a*c**3*e - b**2*c**2*e)))) - 1/(a*d*e*f**2 + a*e
**2*f**2*x)

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error index.cc index_gcd Error: Bad Argument ValueError index.cc index_gcd Error: Bad Argument ValueDone

________________________________________________________________________________________

Mupad [B]
time = 3.87, size = 2500, normalized size = 12.25 \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)),x)

[Out]

- atan(((-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a
^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2)*(x*(4*a^4*c^4*e^12*f^6 - 2*a^3*b^2*c^3*e^12
*f^6) - ((x*(8*a^5*b^3*c^2*e^14*f^10 - 32*a^6*b*c^3*e^14*f^10) - 32*a^6*b*c^3*d*e^13*f^10 + 8*a^5*b^3*c^2*d*e^
13*f^10)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(
a^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2) - 4*a^4*b^3*c^2*e^12*f^8 + 16*a^5*b*c^3*e^
12*f^8)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a
^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2) + 4*a^4*c^4*d*e^11*f^6 - 2*a^3*b^2*c^3*d*e^
11*f^6)*1i + (-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/
(8*(a^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2)*(x*(4*a^4*c^4*e^12*f^6 - 2*a^3*b^2*c^3
*e^12*f^6) - ((x*(8*a^5*b^3*c^2*e^14*f^10 - 32*a^6*b*c^3*e^14*f^10) - 32*a^6*b*c^3*d*e^13*f^10 + 8*a^5*b^3*c^2
*d*e^13*f^10)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))
/(8*(a^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2) + 4*a^4*b^3*c^2*e^12*f^8 - 16*a^5*b*c
^3*e^12*f^8)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/
(8*(a^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2) + 4*a^4*c^4*d*e^11*f^6 - 2*a^3*b^2*c^3
*d*e^11*f^6)*1i)/((-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1
/2))/(8*(a^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2)*(x*(4*a^4*c^4*e^12*f^6 - 2*a^3*b^
2*c^3*e^12*f^6) - ((x*(8*a^5*b^3*c^2*e^14*f^10 - 32*a^6*b*c^3*e^14*f^10) - 32*a^6*b*c^3*d*e^13*f^10 + 8*a^5*b^
3*c^2*d*e^13*f^10)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(
1/2))/(8*(a^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2) + 4*a^4*b^3*c^2*e^12*f^8 - 16*a^
5*b*c^3*e^12*f^8)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1
/2))/(8*(a^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2) + 4*a^4*c^4*d*e^11*f^6 - 2*a^3*b^
2*c^3*d*e^11*f^6) - (-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^
(1/2))/(8*(a^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2)*(x*(4*a^4*c^4*e^12*f^6 - 2*a^3*
b^2*c^3*e^12*f^6) - ((x*(8*a^5*b^3*c^2*e^14*f^10 - 32*a^6*b*c^3*e^14*f^10) - 32*a^6*b*c^3*d*e^13*f^10 + 8*a^5*
b^3*c^2*d*e^13*f^10)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)
^(1/2))/(8*(a^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2) - 4*a^4*b^3*c^2*e^12*f^8 + 16*
a^5*b*c^3*e^12*f^8)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^
(1/2))/(8*(a^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2) + 4*a^4*c^4*d*e^11*f^6 - 2*a^3*
b^2*c^3*d*e^11*f^6) + 2*a^3*c^4*e^10*f^4))*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c -
a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2)*2i - ata
n(((-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^
4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2)*(x*(4*a^4*c^4*e^12*f^6 - 2*a^3*b^2*c^3*e^12*f^6)
- ((x*(8*a^5*b^3*c^2*e^14*f^10 - 32*a^6*b*c^3*e^14*f^10) - 32*a^6*b*c^3*d*e^13*f^10 + 8*a^5*b^3*c^2*d*e^13*f^
10)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b
^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2) - 4*a^4*b^3*c^2*e^12*f^8 + 16*a^5*b*c^3*e^12*f^
8)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^
4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2) + 4*a^4*c^4*d*e^11*f^6 - 2*a^3*b^2*c^3*d*e^11*f^
6)*1i + (-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a
^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2)*(x*(4*a^4*c^4*e^12*f^6 - 2*a^3*b^2*c^3*e^12
*f^6) - ((x*(8*a^5*b^3*c^2*e^14*f^10 - 32*a^6*b*c^3*e^14*f^10) - 32*a^6*b*c^3*d*e^13*f^10 + 8*a^5*b^3*c^2*d*e^
13*f^10)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(
a^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2) + 4*a^4*b^3*c^2*e^12*f^8 - 16*a^5*b*c^3*e^
12*f^8)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a
^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4)))^(1/2) + 4*a^4*c^4*d*e^11*f^6 - 2*a^3*b^2*c^3*d*e^
11*f^6)*1i)/((-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/
(8*(a^3*b^4*e^2*f^4 + 16*a^5*c^2*e^2*f^4 - 8*a^...

________________________________________________________________________________________