3.644 \(\int \frac {1}{(d f+e f x)^3 (a+b (d+e x)^2+c (d+e x)^4)} \, dx\)

Optimal. Leaf size=133 \[ -\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 e f^3 \sqrt {b^2-4 a c}}+\frac {b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f^3}-\frac {b \log (d+e x)}{a^2 e f^3}-\frac {1}{2 a e f^3 (d+e x)^2} \]

[Out]

-1/2/a/e/f^3/(e*x+d)^2-b*ln(e*x+d)/a^2/e/f^3+1/4*b*ln(a+b*(e*x+d)^2+c*(e*x+d)^4)/a^2/e/f^3-1/2*(-2*a*c+b^2)*ar
ctanh((b+2*c*(e*x+d)^2)/(-4*a*c+b^2)^(1/2))/a^2/e/f^3/(-4*a*c+b^2)^(1/2)

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Rubi [A]  time = 0.20, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {1142, 1114, 709, 800, 634, 618, 206, 628} \[ -\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 e f^3 \sqrt {b^2-4 a c}}+\frac {b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f^3}-\frac {b \log (d+e x)}{a^2 e f^3}-\frac {1}{2 a e f^3 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]

[Out]

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

Rule 206

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

Rule 618

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 628

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 634

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 709

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1))/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d - b*e - c
*e*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, m}, 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[m, -1]

Rule 800

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]

Rule 1114

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

Rule 1142

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]

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 157, normalized size = 1.18 \[ \frac {\frac {\left (b \sqrt {b^2-4 a c}-2 a c+b^2\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c (d+e x)^2\right )}{\sqrt {b^2-4 a c}}+\frac {\left (b \sqrt {b^2-4 a c}+2 a c-b^2\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c (d+e x)^2\right )}{\sqrt {b^2-4 a c}}-\frac {2 a}{(d+e x)^2}-4 b \log (d+e x)}{4 a^2 e f^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]

[Out]

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

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fricas [B]  time = 1.22, size = 828, normalized size = 6.23 \[ \left [-\frac {2 \, a b^{2} - 8 \, a^{2} c + {\left ({\left (b^{2} - 2 \, a c\right )} e^{2} x^{2} + 2 \, {\left (b^{2} - 2 \, a c\right )} d e x + {\left (b^{2} - 2 \, a c\right )} d^{2}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} e^{4} x^{4} + 8 \, c^{2} d e^{3} x^{3} + 2 \, c^{2} d^{4} + 2 \, {\left (6 \, c^{2} d^{2} + b c\right )} e^{2} x^{2} + 2 \, b c d^{2} + 4 \, {\left (2 \, c^{2} d^{3} + b c d\right )} e x + b^{2} - 2 \, a c + {\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a}\right ) - {\left ({\left (b^{3} - 4 \, a b c\right )} e^{2} x^{2} + 2 \, {\left (b^{3} - 4 \, a b c\right )} d e x + {\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} \log \left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a\right ) + 4 \, {\left ({\left (b^{3} - 4 \, a b c\right )} e^{2} x^{2} + 2 \, {\left (b^{3} - 4 \, a b c\right )} d e x + {\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} \log \left (e x + d\right )}{4 \, {\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e^{3} f^{3} x^{2} + 2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d e^{2} f^{3} x + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{2} e f^{3}\right )}}, -\frac {2 \, a b^{2} - 8 \, a^{2} c + 2 \, {\left ({\left (b^{2} - 2 \, a c\right )} e^{2} x^{2} + 2 \, {\left (b^{2} - 2 \, a c\right )} d e x + {\left (b^{2} - 2 \, a c\right )} d^{2}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left ({\left (b^{3} - 4 \, a b c\right )} e^{2} x^{2} + 2 \, {\left (b^{3} - 4 \, a b c\right )} d e x + {\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} \log \left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a\right ) + 4 \, {\left ({\left (b^{3} - 4 \, a b c\right )} e^{2} x^{2} + 2 \, {\left (b^{3} - 4 \, a b c\right )} d e x + {\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} \log \left (e x + d\right )}{4 \, {\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e^{3} f^{3} x^{2} + 2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d e^{2} f^{3} x + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{2} e f^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*f*x+d*f)^3/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="fricas")

[Out]

[-1/4*(2*a*b^2 - 8*a^2*c + ((b^2 - 2*a*c)*e^2*x^2 + 2*(b^2 - 2*a*c)*d*e*x + (b^2 - 2*a*c)*d^2)*sqrt(b^2 - 4*a*
c)*log((2*c^2*e^4*x^4 + 8*c^2*d*e^3*x^3 + 2*c^2*d^4 + 2*(6*c^2*d^2 + b*c)*e^2*x^2 + 2*b*c*d^2 + 4*(2*c^2*d^3 +
b*c*d)*e*x + b^2 - 2*a*c + (2*c*e^2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqrt(b^2 - 4*a*c))/(c*e^4*x^4 + 4*c*d*e^3*
x^3 + c*d^4 + (6*c*d^2 + b)*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a)) - ((b^3 - 4*a*b*c)*e^2*x^2 + 2*(b^3
- 4*a*b*c)*d*e*x + (b^3 - 4*a*b*c)*d^2)*log(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^2*x^2 + b*d^2
+ 2*(2*c*d^3 + b*d)*e*x + a) + 4*((b^3 - 4*a*b*c)*e^2*x^2 + 2*(b^3 - 4*a*b*c)*d*e*x + (b^3 - 4*a*b*c)*d^2)*log
(e*x + d))/((a^2*b^2 - 4*a^3*c)*e^3*f^3*x^2 + 2*(a^2*b^2 - 4*a^3*c)*d*e^2*f^3*x + (a^2*b^2 - 4*a^3*c)*d^2*e*f^
3), -1/4*(2*a*b^2 - 8*a^2*c + 2*((b^2 - 2*a*c)*e^2*x^2 + 2*(b^2 - 2*a*c)*d*e*x + (b^2 - 2*a*c)*d^2)*sqrt(-b^2
+ 4*a*c)*arctan(-(2*c*e^2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - ((b^3 - 4*a*b*c)*
e^2*x^2 + 2*(b^3 - 4*a*b*c)*d*e*x + (b^3 - 4*a*b*c)*d^2)*log(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)
*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a) + 4*((b^3 - 4*a*b*c)*e^2*x^2 + 2*(b^3 - 4*a*b*c)*d*e*x + (b^3 -
4*a*b*c)*d^2)*log(e*x + d))/((a^2*b^2 - 4*a^3*c)*e^3*f^3*x^2 + 2*(a^2*b^2 - 4*a^3*c)*d*e^2*f^3*x + (a^2*b^2 -
4*a^3*c)*d^2*e*f^3)]

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giac [B]  time = 1.11, size = 348, normalized size = 2.62 \[ \frac {b e^{\left (-1\right )} \log \left ({\left | c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + 6 \, c d^{2} x^{2} e^{2} + 4 \, c d^{3} x e + c d^{4} + b x^{2} e^{2} + 2 \, b d x e + b d^{2} + a \right |}\right )}{4 \, a^{2} f^{3}} - \frac {b e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\right )}{a^{2} f^{3}} - \frac {e^{\left (-1\right )}}{2 \, {\left (x e + d\right )}^{2} a f^{3}} + \frac {{\left (\frac {{\left (a^{2} b^{2} c f^{3} e^{3} - 2 \, a^{3} c^{2} f^{3} e^{3}\right )} \log \left ({\left | b x^{2} e^{2} + 2 \, b d x e + \sqrt {b^{2} - 4 \, a c} x^{2} e^{2} + 2 \, \sqrt {b^{2} - 4 \, a c} d x e + b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} + 2 \, a \right |}\right )}{\sqrt {b^{2} - 4 \, a c}} - \frac {{\left (a^{2} b^{2} c f^{3} e^{3} - 2 \, a^{3} c^{2} f^{3} e^{3}\right )} \log \left ({\left | -b x^{2} e^{2} - 2 \, b d x e + \sqrt {b^{2} - 4 \, a c} x^{2} e^{2} + 2 \, \sqrt {b^{2} - 4 \, a c} d x e - b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} - 2 \, a \right |}\right )}{\sqrt {b^{2} - 4 \, a c}}\right )} e^{\left (-4\right )}}{4 \, a^{4} c f^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*f*x+d*f)^3/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.01, size = 222, normalized size = 1.67 \[ -\frac {b \ln \left (e x +d \right )}{a^{2} e \,f^{3}}-\frac {1}{2 \left (e x +d \right )^{2} a e \,f^{3}}+\frac {\left (\RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{3} b c \,e^{3}+3 \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{2} b c d \,e^{2}+b c \,d^{3}-a c d +b^{2} d +\left (3 b c \,d^{2}-a c +b^{2}\right ) \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right ) e \right ) \ln \left (-\RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )+x \right )}{2 a^{2} e \,f^{3} \left (2 c \,e^{3} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{3}+6 c d \,e^{2} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{2}+6 e c \,d^{2} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )+2 c \,d^{3}+b e \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )+b d \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*f*x+d*f)^3/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="maxima")

[Out]

Timed out

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mupad [B]  time = 6.98, size = 5947, normalized size = 44.71 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((16*a^6*f^9*x*(((3*b^4 + a^2*c^2 - 9*a*b^2*c)*(((((2*b^3*e*f^3 - 8*a*b*c*e*f^3)*((2*(20*a^3*c^4*d*e^17*f
^6 + 2*a^2*b^2*c^3*d*e^17*f^6))/(a^3*f^9) + ((40*a^4*b*c^3*d*e^18*f^9 - 12*a^3*b^3*c^2*d*e^18*f^9)*(2*b^3*e*f^
3 - 8*a*b*c*e*f^3))/(a^3*f^9*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6))))/(2*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^
6)) + (12*b*c^4*d*e^16)/(a^2*f^6))*(2*b^3*e*f^3 - 8*a*b*c*e*f^3))/(2*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6)) +
(2*c^5*d*e^15)/(a^3*f^9) - (((((2*(20*a^3*c^4*d*e^17*f^6 + 2*a^2*b^2*c^3*d*e^17*f^6))/(a^3*f^9) + ((40*a^4*b*
c^3*d*e^18*f^9 - 12*a^3*b^3*c^2*d*e^18*f^9)*(2*b^3*e*f^3 - 8*a*b*c*e*f^3))/(a^3*f^9*(16*a^3*c*e^2*f^6 - 4*a^2*
b^2*e^2*f^6)))*(2*a*c - b^2))/(4*a^2*e*f^3*(4*a*c - b^2)^(1/2)) + ((40*a^4*b*c^3*d*e^18*f^9 - 12*a^3*b^3*c^2*d
*e^18*f^9)*(2*b^3*e*f^3 - 8*a*b*c*e*f^3)*(2*a*c - b^2))/(4*a^5*e*f^12*(4*a*c - b^2)^(1/2)*(16*a^3*c*e^2*f^6 -
4*a^2*b^2*e^2*f^6)))*(2*a*c - b^2))/(4*a^2*e*f^3*(4*a*c - b^2)^(1/2)) - ((40*a^4*b*c^3*d*e^18*f^9 - 12*a^3*b^3
*c^2*d*e^18*f^9)*(2*b^3*e*f^3 - 8*a*b*c*e*f^3)*(2*a*c - b^2)^2)/(16*a^7*e^2*f^15*(4*a*c - b^2)*(16*a^3*c*e^2*f
^6 - 4*a^2*b^2*e^2*f^6))))/(8*a^3*c^2*(a^2*c^2 - 6*b^4 + 24*a*b^2*c)) + ((3*b^5 + 13*a^2*b*c^2 - 15*a*b^3*c)*(
(((((2*(20*a^3*c^4*d*e^17*f^6 + 2*a^2*b^2*c^3*d*e^17*f^6))/(a^3*f^9) + ((40*a^4*b*c^3*d*e^18*f^9 - 12*a^3*b^3*
c^2*d*e^18*f^9)*(2*b^3*e*f^3 - 8*a*b*c*e*f^3))/(a^3*f^9*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6)))*(2*a*c - b^2)
)/(4*a^2*e*f^3*(4*a*c - b^2)^(1/2)) + ((40*a^4*b*c^3*d*e^18*f^9 - 12*a^3*b^3*c^2*d*e^18*f^9)*(2*b^3*e*f^3 - 8*
a*b*c*e*f^3)*(2*a*c - b^2))/(4*a^5*e*f^12*(4*a*c - b^2)^(1/2)*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6)))*(2*b^3*
e*f^3 - 8*a*b*c*e*f^3))/(2*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6)) + ((((2*b^3*e*f^3 - 8*a*b*c*e*f^3)*((2*(20*
a^3*c^4*d*e^17*f^6 + 2*a^2*b^2*c^3*d*e^17*f^6))/(a^3*f^9) + ((40*a^4*b*c^3*d*e^18*f^9 - 12*a^3*b^3*c^2*d*e^18*
f^9)*(2*b^3*e*f^3 - 8*a*b*c*e*f^3))/(a^3*f^9*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6))))/(2*(16*a^3*c*e^2*f^6 -
4*a^2*b^2*e^2*f^6)) + (12*b*c^4*d*e^16)/(a^2*f^6))*(2*a*c - b^2))/(4*a^2*e*f^3*(4*a*c - b^2)^(1/2)) - ((40*a^4
*b*c^3*d*e^18*f^9 - 12*a^3*b^3*c^2*d*e^18*f^9)*(2*a*c - b^2)^3)/(32*a^9*e^3*f^18*(4*a*c - b^2)^(3/2))))/(8*a^3
*c^2*(4*a*c - b^2)^(1/2)*(a^2*c^2 - 6*b^4 + 24*a*b^2*c)))*(4*a*c - b^2)^(3/2))/(4*a^2*c^4*e^14 + b^4*c^2*e^14
- 4*a*b^2*c^3*e^14) + (16*a^6*f^9*x^2*(((3*b^4 + a^2*c^2 - 9*a*b^2*c)*((((((20*a^3*c^4*e^18*f^6 + 2*a^2*b^2*c^
3*e^18*f^6)/(a^3*f^9) - ((2*b^3*e*f^3 - 8*a*b*c*e*f^3)*(12*a^3*b^3*c^2*e^19*f^9 - 40*a^4*b*c^3*e^19*f^9))/(2*a
^3*f^9*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6)))*(2*b^3*e*f^3 - 8*a*b*c*e*f^3))/(2*(16*a^3*c*e^2*f^6 - 4*a^2*b^
2*e^2*f^6)) + (6*b*c^4*e^17)/(a^2*f^6))*(2*b^3*e*f^3 - 8*a*b*c*e*f^3))/(2*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^
6)) + (c^5*e^16)/(a^3*f^9) - ((2*a*c - b^2)*((((20*a^3*c^4*e^18*f^6 + 2*a^2*b^2*c^3*e^18*f^6)/(a^3*f^9) - ((2*
b^3*e*f^3 - 8*a*b*c*e*f^3)*(12*a^3*b^3*c^2*e^19*f^9 - 40*a^4*b*c^3*e^19*f^9))/(2*a^3*f^9*(16*a^3*c*e^2*f^6 - 4
*a^2*b^2*e^2*f^6)))*(2*a*c - b^2))/(4*a^2*e*f^3*(4*a*c - b^2)^(1/2)) - ((2*b^3*e*f^3 - 8*a*b*c*e*f^3)*(12*a^3*
b^3*c^2*e^19*f^9 - 40*a^4*b*c^3*e^19*f^9)*(2*a*c - b^2))/(8*a^5*e*f^12*(4*a*c - b^2)^(1/2)*(16*a^3*c*e^2*f^6 -
4*a^2*b^2*e^2*f^6))))/(4*a^2*e*f^3*(4*a*c - b^2)^(1/2)) + ((2*b^3*e*f^3 - 8*a*b*c*e*f^3)*(12*a^3*b^3*c^2*e^19
*f^9 - 40*a^4*b*c^3*e^19*f^9)*(2*a*c - b^2)^2)/(32*a^7*e^2*f^15*(4*a*c - b^2)*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^
2*f^6))))/(8*a^3*c^2*(a^2*c^2 - 6*b^4 + 24*a*b^2*c)) + ((3*b^5 + 13*a^2*b*c^2 - 15*a*b^3*c)*(((2*b^3*e*f^3 - 8
*a*b*c*e*f^3)*((((20*a^3*c^4*e^18*f^6 + 2*a^2*b^2*c^3*e^18*f^6)/(a^3*f^9) - ((2*b^3*e*f^3 - 8*a*b*c*e*f^3)*(12
*a^3*b^3*c^2*e^19*f^9 - 40*a^4*b*c^3*e^19*f^9))/(2*a^3*f^9*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6)))*(2*a*c - b
^2))/(4*a^2*e*f^3*(4*a*c - b^2)^(1/2)) - ((2*b^3*e*f^3 - 8*a*b*c*e*f^3)*(12*a^3*b^3*c^2*e^19*f^9 - 40*a^4*b*c^
3*e^19*f^9)*(2*a*c - b^2))/(8*a^5*e*f^12*(4*a*c - b^2)^(1/2)*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6))))/(2*(16*
a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6)) + ((12*a^3*b^3*c^2*e^19*f^9 - 40*a^4*b*c^3*e^19*f^9)*(2*a*c - b^2)^3)/(64*
a^9*e^3*f^18*(4*a*c - b^2)^(3/2)) + (((((20*a^3*c^4*e^18*f^6 + 2*a^2*b^2*c^3*e^18*f^6)/(a^3*f^9) - ((2*b^3*e*f
^3 - 8*a*b*c*e*f^3)*(12*a^3*b^3*c^2*e^19*f^9 - 40*a^4*b*c^3*e^19*f^9))/(2*a^3*f^9*(16*a^3*c*e^2*f^6 - 4*a^2*b^
2*e^2*f^6)))*(2*b^3*e*f^3 - 8*a*b*c*e*f^3))/(2*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6)) + (6*b*c^4*e^17)/(a^2*f
^6))*(2*a*c - b^2))/(4*a^2*e*f^3*(4*a*c - b^2)^(1/2))))/(8*a^3*c^2*(4*a*c - b^2)^(1/2)*(a^2*c^2 - 6*b^4 + 24*a
*b^2*c)))*(4*a*c - b^2)^(3/2))/(4*a^2*c^4*e^14 + b^4*c^2*e^14 - 4*a*b^2*c^3*e^14) + (2*a^3*f^9*(4*a*c - b^2)^(
3/2)*(3*b^4 + a^2*c^2 - 9*a*b^2*c)*((b*c^4*e^14 + c^5*d^2*e^14)/(a^3*f^9) + (((((4*a^2*b^3*c^2*e^16*f^6 + 20*a
^3*c^4*d^2*e^16*f^6 - 4*a^3*b*c^3*e^16*f^6 + 2*a^2*b^2*c^3*d^2*e^16*f^6)/(a^3*f^9) - ((2*b^3*e*f^3 - 8*a*b*c*e
*f^3)*(4*a^4*b^2*c^2*e^17*f^9 - 40*a^4*b*c^3*d^2*e^17*f^9 + 12*a^3*b^3*c^2*d^2*e^17*f^9))/(2*a^3*f^9*(16*a^3*c
*e^2*f^6 - 4*a^2*b^2*e^2*f^6)))*(2*b^3*e*f^3 - 8*a*b*c*e*f^3))/(2*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6)) + (4
*a*b^2*c^3*e^15*f^3 - a^2*c^4*e^15*f^3 + 6*a*b*c^4*d^2*e^15*f^3)/(a^3*f^9))*(2*b^3*e*f^3 - 8*a*b*c*e*f^3))/(2*
(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6)) - ((2*a*c - b^2)*((((4*a^2*b^3*c^2*e^16*f^6 + 20*a^3*c^4*d^2*e^16*f^6
- 4*a^3*b*c^3*e^16*f^6 + 2*a^2*b^2*c^3*d^2*e^16*f^6)/(a^3*f^9) - ((2*b^3*e*f^3 - 8*a*b*c*e*f^3)*(4*a^4*b^2*c^2
*e^17*f^9 - 40*a^4*b*c^3*d^2*e^17*f^9 + 12*a^3*b^3*c^2*d^2*e^17*f^9))/(2*a^3*f^9*(16*a^3*c*e^2*f^6 - 4*a^2*b^2
*e^2*f^6)))*(2*a*c - b^2))/(4*a^2*e*f^3*(4*a*c - b^2)^(1/2)) - ((2*b^3*e*f^3 - 8*a*b*c*e*f^3)*(2*a*c - b^2)*(4
*a^4*b^2*c^2*e^17*f^9 - 40*a^4*b*c^3*d^2*e^17*f^9 + 12*a^3*b^3*c^2*d^2*e^17*f^9))/(8*a^5*e*f^12*(4*a*c - b^2)^
(1/2)*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6))))/(4*a^2*e*f^3*(4*a*c - b^2)^(1/2)) + ((2*b^3*e*f^3 - 8*a*b*c*e*
f^3)*(2*a*c - b^2)^2*(4*a^4*b^2*c^2*e^17*f^9 - 40*a^4*b*c^3*d^2*e^17*f^9 + 12*a^3*b^3*c^2*d^2*e^17*f^9))/(32*a
^7*e^2*f^15*(4*a*c - b^2)*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6))))/(c^2*(a^2*c^2 - 6*b^4 + 24*a*b^2*c)*(4*a^2
*c^4*e^14 + b^4*c^2*e^14 - 4*a*b^2*c^3*e^14)) + (2*a^3*f^9*(4*a*c - b^2)*(((2*b^3*e*f^3 - 8*a*b*c*e*f^3)*((((4
*a^2*b^3*c^2*e^16*f^6 + 20*a^3*c^4*d^2*e^16*f^6 - 4*a^3*b*c^3*e^16*f^6 + 2*a^2*b^2*c^3*d^2*e^16*f^6)/(a^3*f^9)
- ((2*b^3*e*f^3 - 8*a*b*c*e*f^3)*(4*a^4*b^2*c^2*e^17*f^9 - 40*a^4*b*c^3*d^2*e^17*f^9 + 12*a^3*b^3*c^2*d^2*e^1
7*f^9))/(2*a^3*f^9*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6)))*(2*a*c - b^2))/(4*a^2*e*f^3*(4*a*c - b^2)^(1/2)) -
((2*b^3*e*f^3 - 8*a*b*c*e*f^3)*(2*a*c - b^2)*(4*a^4*b^2*c^2*e^17*f^9 - 40*a^4*b*c^3*d^2*e^17*f^9 + 12*a^3*b^3
*c^2*d^2*e^17*f^9))/(8*a^5*e*f^12*(4*a*c - b^2)^(1/2)*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6))))/(2*(16*a^3*c*e
^2*f^6 - 4*a^2*b^2*e^2*f^6)) + (((((4*a^2*b^3*c^2*e^16*f^6 + 20*a^3*c^4*d^2*e^16*f^6 - 4*a^3*b*c^3*e^16*f^6 +
2*a^2*b^2*c^3*d^2*e^16*f^6)/(a^3*f^9) - ((2*b^3*e*f^3 - 8*a*b*c*e*f^3)*(4*a^4*b^2*c^2*e^17*f^9 - 40*a^4*b*c^3*
d^2*e^17*f^9 + 12*a^3*b^3*c^2*d^2*e^17*f^9))/(2*a^3*f^9*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6)))*(2*b^3*e*f^3
- 8*a*b*c*e*f^3))/(2*(16*a^3*c*e^2*f^6 - 4*a^2*b^2*e^2*f^6)) + (4*a*b^2*c^3*e^15*f^3 - a^2*c^4*e^15*f^3 + 6*a*
b*c^4*d^2*e^15*f^3)/(a^3*f^9))*(2*a*c - b^2))/(4*a^2*e*f^3*(4*a*c - b^2)^(1/2)) + ((2*a*c - b^2)^3*(4*a^4*b^2*
c^2*e^17*f^9 - 40*a^4*b*c^3*d^2*e^17*f^9 + 12*a^3*b^3*c^2*d^2*e^17*f^9))/(64*a^9*e^3*f^18*(4*a*c - b^2)^(3/2))
)*(3*b^5 + 13*a^2*b*c^2 - 15*a*b^3*c))/(c^2*(a^2*c^2 - 6*b^4 + 24*a*b^2*c)*(4*a^2*c^4*e^14 + b^4*c^2*e^14 - 4*
a*b^2*c^3*e^14)))*(2*a*c - b^2))/(2*a^2*e*f^3*(4*a*c - b^2)^(1/2)) - 1/(2*a*e*(d^2*f^3 + e^2*f^3*x^2 + 2*d*e*f
^3*x)) - (b*log(d + e*x))/(a^2*e*f^3) - (log(((c^5*e^16*x^2)/(a^3*f^9) - ((b + a^2*e*f^3*(-(2*a*c - b^2)^2/(a^
4*e^2*f^6*(4*a*c - b^2)))^(1/2))*((c^3*e^15*(4*b^2 - a*c + 6*b*c*d^2))/(a^2*f^6) - ((b + a^2*e*f^3*(-(2*a*c -
b^2)^2/(a^4*e^2*f^6*(4*a*c - b^2)))^(1/2))*((2*c^2*e^16*(2*b^3 + 10*a*c^2*d^2 + b^2*c*d^2 - 2*a*b*c))/(a*f^3)
+ (2*c^3*e^18*x^2*(10*a*c + b^2))/(a*f^3) + (b*c^2*e^16*(b + a^2*e*f^3*(-(2*a*c - b^2)^2/(a^4*e^2*f^6*(4*a*c -
b^2)))^(1/2))*(a*b + 3*b^2*d^2 + 3*b^2*e^2*x^2 - 10*a*c*d^2 + 6*b^2*d*e*x - 10*a*c*e^2*x^2 - 20*a*c*d*e*x))/(
a^2*f^3) + (4*c^3*d*e^17*x*(10*a*c + b^2))/(a*f^3)))/(4*a^2*e*f^3) + (6*b*c^4*e^17*x^2)/(a^2*f^6) + (12*b*c^4*
d*e^16*x)/(a^2*f^6)))/(4*a^2*e*f^3) + (c^4*e^14*(b + c*d^2))/(a^3*f^9) + (2*c^5*d*e^15*x)/(a^3*f^9))*((c^5*e^1
6*x^2)/(a^3*f^9) - ((b - a^2*e*f^3*(-(2*a*c - b^2)^2/(a^4*e^2*f^6*(4*a*c - b^2)))^(1/2))*((c^3*e^15*(4*b^2 - a
*c + 6*b*c*d^2))/(a^2*f^6) - ((b - a^2*e*f^3*(-(2*a*c - b^2)^2/(a^4*e^2*f^6*(4*a*c - b^2)))^(1/2))*((2*c^2*e^1
6*(2*b^3 + 10*a*c^2*d^2 + b^2*c*d^2 - 2*a*b*c))/(a*f^3) + (2*c^3*e^18*x^2*(10*a*c + b^2))/(a*f^3) + (b*c^2*e^1
6*(b - a^2*e*f^3*(-(2*a*c - b^2)^2/(a^4*e^2*f^6*(4*a*c - b^2)))^(1/2))*(a*b + 3*b^2*d^2 + 3*b^2*e^2*x^2 - 10*a
*c*d^2 + 6*b^2*d*e*x - 10*a*c*e^2*x^2 - 20*a*c*d*e*x))/(a^2*f^3) + (4*c^3*d*e^17*x*(10*a*c + b^2))/(a*f^3)))/(
4*a^2*e*f^3) + (6*b*c^4*e^17*x^2)/(a^2*f^6) + (12*b*c^4*d*e^16*x)/(a^2*f^6)))/(4*a^2*e*f^3) + (c^4*e^14*(b + c
*d^2))/(a^3*f^9) + (2*c^5*d*e^15*x)/(a^3*f^9)))*(2*b^3*e*f^3 - 8*a*b*c*e*f^3))/(2*(16*a^3*c*e^2*f^6 - 4*a^2*b^
2*e^2*f^6))

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sympy [B]  time = 156.63, size = 532, normalized size = 4.00 \[ \left (\frac {b}{4 a^{2} e f^{3}} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e f^{3} \left (4 a c - b^{2}\right )}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a^{3} c e f^{3} \left (\frac {b}{4 a^{2} e f^{3}} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e f^{3} \left (4 a c - b^{2}\right )}\right ) + 2 a^{2} b^{2} e f^{3} \left (\frac {b}{4 a^{2} e f^{3}} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e f^{3} \left (4 a c - b^{2}\right )}\right ) + 3 a b c + 2 a c^{2} d^{2} - b^{3} - b^{2} c d^{2}}{2 a c^{2} e^{2} - b^{2} c e^{2}} \right )} + \left (\frac {b}{4 a^{2} e f^{3}} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e f^{3} \left (4 a c - b^{2}\right )}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a^{3} c e f^{3} \left (\frac {b}{4 a^{2} e f^{3}} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e f^{3} \left (4 a c - b^{2}\right )}\right ) + 2 a^{2} b^{2} e f^{3} \left (\frac {b}{4 a^{2} e f^{3}} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e f^{3} \left (4 a c - b^{2}\right )}\right ) + 3 a b c + 2 a c^{2} d^{2} - b^{3} - b^{2} c d^{2}}{2 a c^{2} e^{2} - b^{2} c e^{2}} \right )} - \frac {1}{2 a d^{2} e f^{3} + 4 a d e^{2} f^{3} x + 2 a e^{3} f^{3} x^{2}} - \frac {b \log {\left (\frac {d}{e} + x \right )}}{a^{2} e f^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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