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

Optimal. Leaf size=291 \[ \frac {\left (2 a^3 c d^2-a^2 \left (b^2 d^2+6 b c d e+2 c^2 e^2\right )+2 a b^2 e (b d+2 c e)+b^4 \left (-e^2\right )\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {(a d-b e) \left (a b d+2 a c e+b^2 (-e)\right ) \log \left (a x^2+b x+c\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )^2}-\frac {e^3}{d^2 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac {e^3 \log (d+e x) \left (4 a d^2-e (3 b d-2 c e)\right )}{d^3 \left (a d^2-e (b d-c e)\right )^2}-\frac {\log (x) (b d+2 c e)}{c^2 d^3}-\frac {1}{c d^2 x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.56, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1569, 893, 634, 618, 206, 628} \begin {gather*} \frac {\left (-a^2 \left (b^2 d^2+6 b c d e+2 c^2 e^2\right )+2 a^3 c d^2+2 a b^2 e (b d+2 c e)+b^4 \left (-e^2\right )\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {(a d-b e) \left (a b d+2 a c e+b^2 (-e)\right ) \log \left (a x^2+b x+c\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )^2}-\frac {e^3}{d^2 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac {e^3 \log (d+e x) \left (4 a d^2-e (3 b d-2 c e)\right )}{d^3 \left (a d^2-e (b d-c e)\right )^2}-\frac {\log (x) (b d+2 c e)}{c^2 d^3}-\frac {1}{c d^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(c*d^2*x)) - e^3/(d^2*(a*d^2 - e*(b*d - c*e))*(d + e*x)) + ((2*a^3*c*d^2 - b^4*e^2 + 2*a*b^2*e*(b*d + 2*c*
e) - a^2*(b^2*d^2 + 6*b*c*d*e + 2*c^2*e^2))*ArcTanh[(b + 2*a*x)/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]*(a*
d^2 - e*(b*d - c*e))^2) - ((b*d + 2*c*e)*Log[x])/(c^2*d^3) + (e^3*(4*a*d^2 - e*(3*b*d - 2*c*e))*Log[d + e*x])/
(d^3*(a*d^2 - e*(b*d - c*e))^2) + ((a*d - b*e)*(a*b*d - b^2*e + 2*a*c*e)*Log[c + b*x + a*x^2])/(2*c^2*(a*d^2 -
e*(b*d - c*e))^2)

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 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 1569

Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^(mn_.) + (c_.)*(x_)^(mn2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbo
l] :> Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, m, n, q}, x] && E
qQ[mn, -n] && EqQ[mn2, 2*mn] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) x^4 (d+e x)^2} \, dx &=\int \frac {1}{x^2 (d+e x)^2 \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac {1}{c d^2 x^2}+\frac {-b d-2 c e}{c^2 d^3 x}+\frac {e^4}{d^2 \left (a d^2-e (b d-c e)\right ) (d+e x)^2}+\frac {e^4 \left (4 a d^2-e (3 b d-2 c e)\right )}{d^3 \left (a d^2-e (b d-c e)\right )^2 (d+e x)}+\frac {-a^3 c d^2+b^4 e^2-a b^2 e (2 b d+3 c e)+a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )+a (a d-b e) \left (a b d-b^2 e+2 a c e\right ) x}{c^2 \left (a d^2-e (b d-c e)\right )^2 \left (c+b x+a x^2\right )}\right ) \, dx\\ &=-\frac {1}{c d^2 x}-\frac {e^3}{d^2 \left (a d^2-e (b d-c e)\right ) (d+e x)}-\frac {(b d+2 c e) \log (x)}{c^2 d^3}+\frac {e^3 \left (4 a d^2-e (3 b d-2 c e)\right ) \log (d+e x)}{d^3 \left (a d^2-e (b d-c e)\right )^2}+\frac {\int \frac {-a^3 c d^2+b^4 e^2-a b^2 e (2 b d+3 c e)+a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )+a (a d-b e) \left (a b d-b^2 e+2 a c e\right ) x}{c+b x+a x^2} \, dx}{c^2 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac {1}{c d^2 x}-\frac {e^3}{d^2 \left (a d^2-e (b d-c e)\right ) (d+e x)}-\frac {(b d+2 c e) \log (x)}{c^2 d^3}+\frac {e^3 \left (4 a d^2-e (3 b d-2 c e)\right ) \log (d+e x)}{d^3 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left ((a d-b e) \left (a b d-b^2 e+2 a c e\right )\right ) \int \frac {b+2 a x}{c+b x+a x^2} \, dx}{2 c^2 \left (a d^2-e (b d-c e)\right )^2}-\frac {\left (2 a^3 c d^2-b^4 e^2+2 a b^2 e (b d+2 c e)-a^2 \left (b^2 d^2+6 b c d e+2 c^2 e^2\right )\right ) \int \frac {1}{c+b x+a x^2} \, dx}{2 c^2 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac {1}{c d^2 x}-\frac {e^3}{d^2 \left (a d^2-e (b d-c e)\right ) (d+e x)}-\frac {(b d+2 c e) \log (x)}{c^2 d^3}+\frac {e^3 \left (4 a d^2-e (3 b d-2 c e)\right ) \log (d+e x)}{d^3 \left (a d^2-e (b d-c e)\right )^2}+\frac {(a d-b e) \left (a b d-b^2 e+2 a c e\right ) \log \left (c+b x+a x^2\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (2 a^3 c d^2-b^4 e^2+2 a b^2 e (b d+2 c e)-a^2 \left (b^2 d^2+6 b c d e+2 c^2 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )}{c^2 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac {1}{c d^2 x}-\frac {e^3}{d^2 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {\left (2 a^3 c d^2-b^4 e^2+2 a b^2 e (b d+2 c e)-a^2 \left (b^2 d^2+6 b c d e+2 c^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}-\frac {(b d+2 c e) \log (x)}{c^2 d^3}+\frac {e^3 \left (4 a d^2-e (3 b d-2 c e)\right ) \log (d+e x)}{d^3 \left (a d^2-e (b d-c e)\right )^2}+\frac {(a d-b e) \left (a b d-b^2 e+2 a c e\right ) \log \left (c+b x+a x^2\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.34, size = 287, normalized size = 0.99 \begin {gather*} \frac {\left (-2 a^3 c d^2+a^2 \left (b^2 d^2+6 b c d e+2 c^2 e^2\right )-2 a b^2 e (b d+2 c e)+b^4 e^2\right ) \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{c^2 \sqrt {4 a c-b^2} \left (a d^2+e (c e-b d)\right )^2}+\frac {(a d-b e) \left (a b d+2 a c e+b^2 (-e)\right ) \log (x (a x+b)+c)}{2 c^2 \left (a d^2+e (c e-b d)\right )^2}-\frac {e^3}{d^2 (d+e x) \left (a d^2+e (c e-b d)\right )}+\frac {e^3 \log (d+e x) \left (4 a d^2+e (2 c e-3 b d)\right )}{d^3 \left (a d^2+e (c e-b d)\right )^2}-\frac {\log (x) (b d+2 c e)}{c^2 d^3}-\frac {1}{c d^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) x^4 (d+e x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F(-1)]  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/(a+c/x^2+b/x)/x^4/(e*x+d)^2,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [A]  time = 0.36, size = 487, normalized size = 1.67 \begin {gather*} -\frac {{\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} c d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 6 \, a^{2} b c d e^{3} + b^{4} e^{4} - 4 \, a b^{2} c e^{4} + 2 \, a^{2} c^{2} e^{4}\right )} \arctan \left (-\frac {{\left (2 \, a d - \frac {2 \, a d^{2}}{x e + d} - b e + \frac {2 \, b d e}{x e + d} - \frac {2 \, c e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt {-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (a^{2} c^{2} d^{4} - 2 \, a b c^{2} d^{3} e + b^{2} c^{2} d^{2} e^{2} + 2 \, a c^{3} d^{2} e^{2} - 2 \, b c^{3} d e^{3} + c^{4} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (a^{2} b d^{2} - 2 \, a b^{2} d e + 2 \, a^{2} c d e + b^{3} e^{2} - 2 \, a b c e^{2}\right )} \log \left (-a + \frac {2 \, a d}{x e + d} - \frac {a d^{2}}{{\left (x e + d\right )}^{2}} - \frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {c e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \, {\left (a^{2} c^{2} d^{4} - 2 \, a b c^{2} d^{3} e + b^{2} c^{2} d^{2} e^{2} + 2 \, a c^{3} d^{2} e^{2} - 2 \, b c^{3} d e^{3} + c^{4} e^{4}\right )}} - \frac {e^{7}}{{\left (a d^{4} e^{4} - b d^{3} e^{5} + c d^{2} e^{6}\right )} {\left (x e + d\right )}} - \frac {{\left (b d e + 2 \, c e^{2}\right )} e^{\left (-1\right )} \log \left ({\left | -\frac {d}{x e + d} + 1 \right |}\right )}{c^{2} d^{3}} + \frac {e}{c d^{3} {\left (\frac {d}{x e + d} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.01, size = 791, normalized size = 2.72 \begin {gather*} -\frac {2 a^{3} d^{2} \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} \sqrt {4 a c -b^{2}}\, c}+\frac {a^{2} b^{2} d^{2} \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} \sqrt {4 a c -b^{2}}\, c^{2}}+\frac {6 a^{2} b d e \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} \sqrt {4 a c -b^{2}}\, c}+\frac {2 a^{2} e^{2} \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} \sqrt {4 a c -b^{2}}}-\frac {2 a \,b^{3} d e \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} \sqrt {4 a c -b^{2}}\, c^{2}}-\frac {4 a \,b^{2} e^{2} \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} \sqrt {4 a c -b^{2}}\, c}+\frac {b^{4} e^{2} \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} \sqrt {4 a c -b^{2}}\, c^{2}}+\frac {a^{2} b \,d^{2} \ln \left (a \,x^{2}+b x +c \right )}{2 \left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} c^{2}}+\frac {a^{2} d e \ln \left (a \,x^{2}+b x +c \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} c}-\frac {a \,b^{2} d e \ln \left (a \,x^{2}+b x +c \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} c^{2}}-\frac {a b \,e^{2} \ln \left (a \,x^{2}+b x +c \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} c}+\frac {4 a \,e^{3} \ln \left (e x +d \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} d}+\frac {b^{3} e^{2} \ln \left (a \,x^{2}+b x +c \right )}{2 \left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} c^{2}}-\frac {3 b \,e^{4} \ln \left (e x +d \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} d^{2}}+\frac {2 c \,e^{5} \ln \left (e x +d \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} d^{3}}-\frac {e^{3}}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) \left (e x +d \right ) d^{2}}-\frac {b \ln \relax (x )}{c^{2} d^{2}}-\frac {2 e \ln \relax (x )}{c \,d^{3}}-\frac {1}{c \,d^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/c^2/(a*d^2-b*d*e+c*e^2)^2*a^2*ln(a*x^2+b*x+c)*b*d^2+1/c/(a*d^2-b*d*e+c*e^2)^2*a^2*ln(a*x^2+b*x+c)*d*e-1/c^
2/(a*d^2-b*d*e+c*e^2)^2*a*ln(a*x^2+b*x+c)*b^2*d*e-1/c/(a*d^2-b*d*e+c*e^2)^2*a*ln(a*x^2+b*x+c)*b*e^2+1/2/c^2/(a
*d^2-b*d*e+c*e^2)^2*ln(a*x^2+b*x+c)*b^3*e^2-2/c/(a*d^2-b*d*e+c*e^2)^2/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*
c-b^2)^(1/2))*a^3*d^2+1/c^2/(a*d^2-b*d*e+c*e^2)^2/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*a^2*b^
2*d^2+6/c/(a*d^2-b*d*e+c*e^2)^2/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*a^2*b*d*e+2/(a*d^2-b*d*e
+c*e^2)^2/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*a^2*e^2-2/c^2/(a*d^2-b*d*e+c*e^2)^2/(4*a*c-b^2
)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*a*b^3*d*e-4/c/(a*d^2-b*d*e+c*e^2)^2/(4*a*c-b^2)^(1/2)*arctan((2*a*
x+b)/(4*a*c-b^2)^(1/2))*a*b^2*e^2+1/c^2/(a*d^2-b*d*e+c*e^2)^2/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(
1/2))*b^4*e^2-1/c/d^2/x-1/c^2/d^2*ln(x)*b-2/c/d^3*ln(x)*e-e^3/(a*d^2-b*d*e+c*e^2)/d^2/(e*x+d)+4*e^3/(a*d^2-b*d
*e+c*e^2)^2/d*ln(e*x+d)*a-3*e^4/(a*d^2-b*d*e+c*e^2)^2/d^2*ln(e*x+d)*b+2*e^5/(a*d^2-b*d*e+c*e^2)^2/d^3*ln(e*x+d
)*c

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

________________________________________________________________________________________

mupad [B]  time = 31.16, size = 4948, normalized size = 17.00

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(d + e*x)*(2*c*e^5 + 4*a*d^2*e^3 - 3*b*d*e^4))/(a^2*d^7 + b^2*d^5*e^2 + c^2*d^3*e^4 - 2*a*b*d^6*e + 2*a*c*
d^5*e^2 - 2*b*c*d^4*e^3) - (1/(c*d) + (x*(2*c*e^3 + a*d^2*e - b*d*e^2))/(c*d^2*(a*d^2 + c*e^2 - b*d*e)))/(d*x
+ e*x^2) - (log((((a*e*(a^5*b*d^8 + 4*b^3*c^3*e^8 + b^6*d^3*e^5 - 2*a*b^5*d^4*e^4 - 2*a^4*b^2*d^7*e + 16*a^2*c
^4*d*e^7 - 4*b^4*c^2*d*e^7 - b^5*c*d^2*e^6 + a^2*b^4*d^5*e^3 + a^3*b^3*d^6*e^2 + 16*a^3*c^3*d^3*e^5 + a^4*c^2*
d^5*e^3 - 12*a*b*c^4*e^8 + 2*a^5*c*d^7*e - 16*a^2*b^2*c^2*d^3*e^5 + 4*a*b^2*c^3*d*e^7 - 2*a^4*b*c*d^6*e^2 + 13
*a*b^3*c^2*d^2*e^6 - 20*a^2*b*c^3*d^2*e^6 + a^2*b^3*c*d^4*e^4 + 8*a^3*b*c^2*d^4*e^4))/(c^2*d^4*(a*d^2 + c*e^2
- b*d*e)^2) - (((a*e*(a^4*c*d^6 + 8*a*c^4*e^6 - a^3*b^2*d^6 - 2*b^2*c^3*e^6 + b^5*d^3*e^3 - 3*a*b^4*d^4*e^2 +
3*a^2*b^3*d^5*e + b^3*c^2*d*e^5 + b^4*c*d^2*e^4 + 8*a^2*c^3*d^2*e^4 - 7*a^3*c^2*d^4*e^2 - 4*a*b*c^3*d*e^5 - 7*
a^3*b*c*d^5*e - 7*a*b^3*c*d^3*e^3 - 6*a*b^2*c^2*d^2*e^4 + 12*a^2*b*c^2*d^3*e^3 + 12*a^2*b^2*c*d^4*e^2))/(c*d^2
*(a*d^2 + c*e^2 - b*d*e)) + (a*e*(b^5*e^2 + b^4*e^2*(b^2 - 4*a*c)^(1/2) + a^2*b^3*d^2 + 8*a^2*b*c^2*e^2 + a^2*
b^2*d^2*(b^2 - 4*a*c)^(1/2) + 2*a^2*c^2*e^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^4*d*e - 4*a^3*b*c*d^2 - 6*a*b^3*c*e^2
- 8*a^3*c^2*d*e - 2*a^3*c*d^2*(b^2 - 4*a*c)^(1/2) + 10*a^2*b^2*c*d*e - 4*a*b^2*c*e^2*(b^2 - 4*a*c)^(1/2) - 2*a
*b^3*d*e*(b^2 - 4*a*c)^(1/2) + 6*a^2*b*c*d*e*(b^2 - 4*a*c)^(1/2))*(4*a^2*c^2*d^3*e + b^2*c^2*d*e^3 + b^3*c*d^2
*e^2 + 2*a^2*b^2*d^4*x + 2*b^2*c^2*e^4*x + 2*b^4*d^2*e^2*x + a^2*b*c*d^4 - 4*a*c^3*d*e^3 - 6*a^3*c*d^4*x - 8*a
*c^3*e^4*x - 2*a*b^2*c*d^3*e - 4*a*b^3*d^3*e*x - 2*b^3*c*d*e^3*x - 3*a*b*c^2*d^2*e^2 - 6*a^2*c^2*d^2*e^2*x + 8
*a*b*c^2*d*e^3*x + 14*a^2*b*c*d^3*e*x - 6*a*b^2*c*d^2*e^2*x))/(2*c^2*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e)^2)
- (2*a*e*x*(a*d - b*e)*(a^3*b*d^5 + 8*a*c^3*e^5 - 2*b^2*c^2*e^5 + b^4*d^2*e^3 - a*b^3*d^3*e^2 - a^2*b^2*d^4*e
+ 16*a^2*c^2*d^2*e^3 + 2*a^3*c*d^4*e + 2*b^3*c*d*e^4 - 8*a*b*c^2*d*e^4 - 8*a*b^2*c*d^2*e^3 + 4*a^2*b*c*d^3*e^2
))/(c*d^2*(a*d^2 + c*e^2 - b*d*e)))*(b^5*e^2 + b^4*e^2*(b^2 - 4*a*c)^(1/2) + a^2*b^3*d^2 + 8*a^2*b*c^2*e^2 + a
^2*b^2*d^2*(b^2 - 4*a*c)^(1/2) + 2*a^2*c^2*e^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^4*d*e - 4*a^3*b*c*d^2 - 6*a*b^3*c*e
^2 - 8*a^3*c^2*d*e - 2*a^3*c*d^2*(b^2 - 4*a*c)^(1/2) + 10*a^2*b^2*c*d*e - 4*a*b^2*c*e^2*(b^2 - 4*a*c)^(1/2) -
2*a*b^3*d*e*(b^2 - 4*a*c)^(1/2) + 6*a^2*b*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*c^2*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*
d*e)^2) + (a*e*x*(a^6*d^8 + 8*a^2*c^4*e^8 + 4*b^4*c^2*e^8 + b^6*d^2*e^6 - 16*a*b^2*c^3*e^8 - 2*a*b^5*d^3*e^5 +
2*a^5*c*d^6*e^2 + a^2*b^4*d^4*e^4 + a^4*b^2*d^6*e^2 + 8*a^3*c^3*d^2*e^6 + 18*a^4*c^2*d^4*e^4 - 2*a^5*b*d^7*e
- 4*b^5*c*d*e^7 - 26*a^2*b^2*c^2*d^2*e^6 + 8*a*b^3*c^2*d*e^7 + 4*a*b^4*c*d^2*e^6 + 16*a^2*b*c^3*d*e^7 + 6*a^4*
b*c*d^5*e^3 + 10*a^2*b^3*c*d^3*e^5 - 18*a^3*b^2*c*d^4*e^4))/(c^2*d^4*(a*d^2 + c*e^2 - b*d*e)^2))*(b^5*e^2 + b^
4*e^2*(b^2 - 4*a*c)^(1/2) + a^2*b^3*d^2 + 8*a^2*b*c^2*e^2 + a^2*b^2*d^2*(b^2 - 4*a*c)^(1/2) + 2*a^2*c^2*e^2*(b
^2 - 4*a*c)^(1/2) - 2*a*b^4*d*e - 4*a^3*b*c*d^2 - 6*a*b^3*c*e^2 - 8*a^3*c^2*d*e - 2*a^3*c*d^2*(b^2 - 4*a*c)^(1
/2) + 10*a^2*b^2*c*d*e - 4*a*b^2*c*e^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^3*d*e*(b^2 - 4*a*c)^(1/2) + 6*a^2*b*c*d*e*(
b^2 - 4*a*c)^(1/2)))/(2*c^2*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e)^2) + (a^4*e^4*(b*d + 2*c*e)*(3*a*d^2 + 2*c*e
^2 - 3*b*d*e))/(c^2*d^4*(a*d^2 + c*e^2 - b*d*e)^2) + (4*a^5*e^4*x*(a*d - b*e))/(c^2*d^2*(a*d^2 + c*e^2 - b*d*e
)^2))*(b^5*e^2 + b^4*e^2*(b^2 - 4*a*c)^(1/2) + a^2*b^3*d^2 + 8*a^2*b*c^2*e^2 + a^2*b^2*d^2*(b^2 - 4*a*c)^(1/2)
+ 2*a^2*c^2*e^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^4*d*e - 4*a^3*b*c*d^2 - 6*a*b^3*c*e^2 - 8*a^3*c^2*d*e - 2*a^3*c*d
^2*(b^2 - 4*a*c)^(1/2) + 10*a^2*b^2*c*d*e - 4*a*b^2*c*e^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^3*d*e*(b^2 - 4*a*c)^(1/2
) + 6*a^2*b*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*(4*a*c^5*e^4 + 4*a^3*c^3*d^4 - b^2*c^4*e^4 + 2*b^3*c^3*d*e^3 - a^2*
b^2*c^2*d^4 + 8*a^2*c^4*d^2*e^2 - b^4*c^2*d^2*e^2 - 8*a*b*c^4*d*e^3 + 2*a*b^3*c^2*d^3*e - 8*a^2*b*c^3*d^3*e +
2*a*b^2*c^3*d^2*e^2)) + (log((a^4*e^4*(b*d + 2*c*e)*(3*a*d^2 + 2*c*e^2 - 3*b*d*e))/(c^2*d^4*(a*d^2 + c*e^2 - b
*d*e)^2) - (((a*e*(a^5*b*d^8 + 4*b^3*c^3*e^8 + b^6*d^3*e^5 - 2*a*b^5*d^4*e^4 - 2*a^4*b^2*d^7*e + 16*a^2*c^4*d*
e^7 - 4*b^4*c^2*d*e^7 - b^5*c*d^2*e^6 + a^2*b^4*d^5*e^3 + a^3*b^3*d^6*e^2 + 16*a^3*c^3*d^3*e^5 + a^4*c^2*d^5*e
^3 - 12*a*b*c^4*e^8 + 2*a^5*c*d^7*e - 16*a^2*b^2*c^2*d^3*e^5 + 4*a*b^2*c^3*d*e^7 - 2*a^4*b*c*d^6*e^2 + 13*a*b^
3*c^2*d^2*e^6 - 20*a^2*b*c^3*d^2*e^6 + a^2*b^3*c*d^4*e^4 + 8*a^3*b*c^2*d^4*e^4))/(c^2*d^4*(a*d^2 + c*e^2 - b*d
*e)^2) - (((a*e*(b^4*e^2*(b^2 - 4*a*c)^(1/2) - b^5*e^2 - a^2*b^3*d^2 - 8*a^2*b*c^2*e^2 + a^2*b^2*d^2*(b^2 - 4*
a*c)^(1/2) + 2*a^2*c^2*e^2*(b^2 - 4*a*c)^(1/2) + 2*a*b^4*d*e + 4*a^3*b*c*d^2 + 6*a*b^3*c*e^2 + 8*a^3*c^2*d*e -
2*a^3*c*d^2*(b^2 - 4*a*c)^(1/2) - 10*a^2*b^2*c*d*e - 4*a*b^2*c*e^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^3*d*e*(b^2 - 4
*a*c)^(1/2) + 6*a^2*b*c*d*e*(b^2 - 4*a*c)^(1/2))*(4*a^2*c^2*d^3*e + b^2*c^2*d*e^3 + b^3*c*d^2*e^2 + 2*a^2*b^2*
d^4*x + 2*b^2*c^2*e^4*x + 2*b^4*d^2*e^2*x + a^2*b*c*d^4 - 4*a*c^3*d*e^3 - 6*a^3*c*d^4*x - 8*a*c^3*e^4*x - 2*a*
b^2*c*d^3*e - 4*a*b^3*d^3*e*x - 2*b^3*c*d*e^3*x - 3*a*b*c^2*d^2*e^2 - 6*a^2*c^2*d^2*e^2*x + 8*a*b*c^2*d*e^3*x
+ 14*a^2*b*c*d^3*e*x - 6*a*b^2*c*d^2*e^2*x))/(2*c^2*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e)^2) - (a*e*(a^4*c*d^6
+ 8*a*c^4*e^6 - a^3*b^2*d^6 - 2*b^2*c^3*e^6 + b^5*d^3*e^3 - 3*a*b^4*d^4*e^2 + 3*a^2*b^3*d^5*e + b^3*c^2*d*e^5
+ b^4*c*d^2*e^4 + 8*a^2*c^3*d^2*e^4 - 7*a^3*c^2*d^4*e^2 - 4*a*b*c^3*d*e^5 - 7*a^3*b*c*d^5*e - 7*a*b^3*c*d^3*e
^3 - 6*a*b^2*c^2*d^2*e^4 + 12*a^2*b*c^2*d^3*e^3 + 12*a^2*b^2*c*d^4*e^2))/(c*d^2*(a*d^2 + c*e^2 - b*d*e)) + (2*
a*e*x*(a*d - b*e)*(a^3*b*d^5 + 8*a*c^3*e^5 - 2*b^2*c^2*e^5 + b^4*d^2*e^3 - a*b^3*d^3*e^2 - a^2*b^2*d^4*e + 16*
a^2*c^2*d^2*e^3 + 2*a^3*c*d^4*e + 2*b^3*c*d*e^4 - 8*a*b*c^2*d*e^4 - 8*a*b^2*c*d^2*e^3 + 4*a^2*b*c*d^3*e^2))/(c
*d^2*(a*d^2 + c*e^2 - b*d*e)))*(b^4*e^2*(b^2 - 4*a*c)^(1/2) - b^5*e^2 - a^2*b^3*d^2 - 8*a^2*b*c^2*e^2 + a^2*b^
2*d^2*(b^2 - 4*a*c)^(1/2) + 2*a^2*c^2*e^2*(b^2 - 4*a*c)^(1/2) + 2*a*b^4*d*e + 4*a^3*b*c*d^2 + 6*a*b^3*c*e^2 +
8*a^3*c^2*d*e - 2*a^3*c*d^2*(b^2 - 4*a*c)^(1/2) - 10*a^2*b^2*c*d*e - 4*a*b^2*c*e^2*(b^2 - 4*a*c)^(1/2) - 2*a*b
^3*d*e*(b^2 - 4*a*c)^(1/2) + 6*a^2*b*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*c^2*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e)^
2) + (a*e*x*(a^6*d^8 + 8*a^2*c^4*e^8 + 4*b^4*c^2*e^8 + b^6*d^2*e^6 - 16*a*b^2*c^3*e^8 - 2*a*b^5*d^3*e^5 + 2*a^
5*c*d^6*e^2 + a^2*b^4*d^4*e^4 + a^4*b^2*d^6*e^2 + 8*a^3*c^3*d^2*e^6 + 18*a^4*c^2*d^4*e^4 - 2*a^5*b*d^7*e - 4*b
^5*c*d*e^7 - 26*a^2*b^2*c^2*d^2*e^6 + 8*a*b^3*c^2*d*e^7 + 4*a*b^4*c*d^2*e^6 + 16*a^2*b*c^3*d*e^7 + 6*a^4*b*c*d
^5*e^3 + 10*a^2*b^3*c*d^3*e^5 - 18*a^3*b^2*c*d^4*e^4))/(c^2*d^4*(a*d^2 + c*e^2 - b*d*e)^2))*(b^4*e^2*(b^2 - 4*
a*c)^(1/2) - b^5*e^2 - a^2*b^3*d^2 - 8*a^2*b*c^2*e^2 + a^2*b^2*d^2*(b^2 - 4*a*c)^(1/2) + 2*a^2*c^2*e^2*(b^2 -
4*a*c)^(1/2) + 2*a*b^4*d*e + 4*a^3*b*c*d^2 + 6*a*b^3*c*e^2 + 8*a^3*c^2*d*e - 2*a^3*c*d^2*(b^2 - 4*a*c)^(1/2) -
10*a^2*b^2*c*d*e - 4*a*b^2*c*e^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^3*d*e*(b^2 - 4*a*c)^(1/2) + 6*a^2*b*c*d*e*(b^2 -
4*a*c)^(1/2)))/(2*c^2*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e)^2) + (4*a^5*e^4*x*(a*d - b*e))/(c^2*d^2*(a*d^2 +
c*e^2 - b*d*e)^2))*(b^4*e^2*(b^2 - 4*a*c)^(1/2) - b^5*e^2 - a^2*b^3*d^2 - 8*a^2*b*c^2*e^2 + a^2*b^2*d^2*(b^2 -
4*a*c)^(1/2) + 2*a^2*c^2*e^2*(b^2 - 4*a*c)^(1/2) + 2*a*b^4*d*e + 4*a^3*b*c*d^2 + 6*a*b^3*c*e^2 + 8*a^3*c^2*d*
e - 2*a^3*c*d^2*(b^2 - 4*a*c)^(1/2) - 10*a^2*b^2*c*d*e - 4*a*b^2*c*e^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^3*d*e*(b^2
- 4*a*c)^(1/2) + 6*a^2*b*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*(4*a*c^5*e^4 + 4*a^3*c^3*d^4 - b^2*c^4*e^4 + 2*b^3*c^3
*d*e^3 - a^2*b^2*c^2*d^4 + 8*a^2*c^4*d^2*e^2 - b^4*c^2*d^2*e^2 - 8*a*b*c^4*d*e^3 + 2*a*b^3*c^2*d^3*e - 8*a^2*b
*c^3*d^3*e + 2*a*b^2*c^3*d^2*e^2)) - (log(x)*(b*d + 2*c*e))/(c^2*d^3)

________________________________________________________________________________________

sympy [F(-1)]  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/(a+c/x**2+b/x)/x**4/(e*x+d)**2,x)

[Out]

Timed out

________________________________________________________________________________________