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

Optimal. Leaf size=248 \[ -\frac {\left (a^2 d^2-a e (2 b d+c e)+b^2 e^2\right ) \log \left (a x^2+b x+c\right )}{2 c \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (a^2 d (b d+4 c e)-a b e (2 b d+3 c e)+b^3 e^2\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {e^2}{d (d+e x) \left (a d^2-b d e+c e^2\right )}-\frac {e^2 \log (d+e x) \left (3 a d^2-e (2 b d-c e)\right )}{d^2 \left (a d^2-e (b d-c e)\right )^2}+\frac {\log (x)}{c d^2} \]

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Rubi [A]  time = 0.41, antiderivative size = 249, 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 d^2-a e (2 b d+c e)+b^2 e^2\right ) \log \left (a x^2+b x+c\right )}{2 c \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (a^2 d (b d+4 c e)-a b e (2 b d+3 c e)+b^3 e^2\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {e^2}{d (d+e x) \left (a d^2-e (b d-c e)\right )}-\frac {e^2 \log (d+e x) \left (3 a d^2-e (2 b d-c e)\right )}{d^2 \left (a d^2-e (b d-c e)\right )^2}+\frac {\log (x)}{c d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.25, size = 246, normalized size = 0.99 \begin {gather*} \frac {\left (-a^2 d^2+a e (2 b d+c e)-b^2 e^2\right ) \log (x (a x+b)+c)}{2 c \left (a d^2+e (c e-b d)\right )^2}-\frac {\left (a^2 d (b d+4 c e)-a b e (2 b d+3 c e)+b^3 e^2\right ) \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{c \sqrt {4 a c-b^2} \left (a d^2+e (c e-b d)\right )^2}+\frac {e^2}{d (d+e x) \left (a d^2+e (c e-b d)\right )}-\frac {e^2 \log (d+e x) \left (3 a d^2+e (c e-2 b d)\right )}{\left (a d^3+d e (c e-b d)\right )^2}+\frac {\log (x)}{c d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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^3 (d+e x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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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^3/(e*x+d)^2,x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.41, size = 391, normalized size = 1.58 \begin {gather*} -\frac {{\left (a^{2} b d^{2} e^{2} - 2 \, a b^{2} d e^{3} + 4 \, a^{2} c d e^{3} + b^{3} e^{4} - 3 \, a b c 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 d^{4} - 2 \, a b c d^{3} e + b^{2} c d^{2} e^{2} + 2 \, a c^{2} d^{2} e^{2} - 2 \, b c^{2} d e^{3} + c^{3} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {{\left (a^{2} d^{2} - 2 \, a b d e + b^{2} e^{2} - a 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 d^{4} - 2 \, a b c d^{3} e + b^{2} c d^{2} e^{2} + 2 \, a c^{2} d^{2} e^{2} - 2 \, b c^{2} d e^{3} + c^{3} e^{4}\right )}} + \frac {e^{5}}{{\left (a d^{3} e^{3} - b d^{2} e^{4} + c d e^{5}\right )} {\left (x e + d\right )}} + \frac {\log \left ({\left | -\frac {d}{x e + d} + 1 \right |}\right )}{c d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.01, size = 589, normalized size = 2.38 \begin {gather*} -\frac {a^{2} b \,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 {4 a^{2} 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}}}+\frac {2 a \,b^{2} 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 {3 a b \,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 {b^{3} 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 {a^{2} 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}+\frac {a b 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 {3 a \,e^{2} \ln \left (e x +d \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2}}+\frac {a \,e^{2} \ln \left (a \,x^{2}+b x +c \right )}{2 \left (a \,d^{2}-d e b +c \,e^{2}\right )^{2}}-\frac {b^{2} 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}+\frac {2 b \,e^{3} \ln \left (e x +d \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} d}-\frac {c \,e^{4} \ln \left (e x +d \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} d^{2}}+\frac {e^{2}}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) \left (e x +d \right ) d}+\frac {\ln \relax (x )}{c \,d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+c/x^2+b/x)/x^3/(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?

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mupad [B]  time = 25.28, size = 3510, normalized size = 14.15

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((a^4*e^4)/(d*(a*d^2 + c*e^2 - b*d*e)^2) + (a^4*e^5*x)/(d^2*(a*d^2 + c*e^2 - b*d*e)^2) - (((a*e^3*(3*a^3*b
*d^4 + b^3*c*e^4 - b^4*d*e^3 + 5*a*b^3*d^2*e^2 - 7*a^2*b^2*d^3*e + 8*a^2*c^2*d*e^3 - 3*a*b*c^2*e^4 + 9*a^3*c*d
^3*e - a*b^2*c*d*e^3 - 8*a^2*b*c*d^2*e^2))/(d^2*(a*d^2 + c*e^2 - b*d*e)^2) + (((a*e*(a^3*b*d^5 - 4*a*c^3*e^5 +
b^2*c^2*e^5 - b^4*d^2*e^3 + 3*a*b^3*d^3*e^2 - 3*a^2*b^2*d^4*e - 8*a^2*c^2*d^2*e^3 + 4*a^3*c*d^4*e - b^3*c*d*e
^4 + 4*a*b*c^2*d*e^4 + 6*a*b^2*c*d^2*e^3 - 9*a^2*b*c*d^3*e^2))/(a*d^3 - b*d^2*e + c*d*e^2) + (a*e*x*(3*a^4*d^5
+ 2*b^3*c*e^5 - 4*b^4*d*e^4 + 9*a*b^3*d^2*e^3 + 4*a^2*c^2*d*e^4 + 19*a^3*c*d^3*e^2 - 3*a^2*b^2*d^3*e^2 - 8*a*
b*c^2*e^5 - 5*a^3*b*d^4*e + 15*a*b^2*c*d*e^4 - 36*a^2*b*c*d^2*e^3))/(a*d^3 - b*d^2*e + c*d*e^2) - (a*e*(b^4*e^
2 - 4*a^3*c*d^2 + b^3*e^2*(b^2 - 4*a*c)^(1/2) + a^2*b^2*d^2 + 4*a^2*c^2*e^2 - 2*a*b^3*d*e - 5*a*b^2*c*e^2 + a^
2*b*d^2*(b^2 - 4*a*c)^(1/2) + 8*a^2*b*c*d*e - 3*a*b*c*e^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^2*d*e*(b^2 - 4*a*c)^(1/2
) + 4*a^2*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*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e)^2))*(b^4*e^2 - 4*a^3*c*d^2 + b^3*e
^2*(b^2 - 4*a*c)^(1/2) + a^2*b^2*d^2 + 4*a^2*c^2*e^2 - 2*a*b^3*d*e - 5*a*b^2*c*e^2 + a^2*b*d^2*(b^2 - 4*a*c)^(
1/2) + 8*a^2*b*c*d*e - 3*a*b*c*e^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^2*d*e*(b^2 - 4*a*c)^(1/2) + 4*a^2*c*d*e*(b^2 -
4*a*c)^(1/2)))/(2*c*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e)^2) + (a*e^3*x*(9*a^4*d^4 + b^4*e^4 + 2*a^2*c^2*e^4 -
6*a^3*c*d^2*e^2 + 8*a^2*b^2*d^2*e^2 - 4*a*b^2*c*e^4 - 4*a*b^3*d*e^3 - 12*a^3*b*d^3*e + 10*a^2*b*c*d*e^3))/(d^
2*(a*d^2 + c*e^2 - b*d*e)^2))*(b^4*e^2 - 4*a^3*c*d^2 + b^3*e^2*(b^2 - 4*a*c)^(1/2) + a^2*b^2*d^2 + 4*a^2*c^2*e
^2 - 2*a*b^3*d*e - 5*a*b^2*c*e^2 + a^2*b*d^2*(b^2 - 4*a*c)^(1/2) + 8*a^2*b*c*d*e - 3*a*b*c*e^2*(b^2 - 4*a*c)^(
1/2) - 2*a*b^2*d*e*(b^2 - 4*a*c)^(1/2) + 4*a^2*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*c*(4*a*c - b^2)*(a*d^2 + c*e^2 -
b*d*e)^2))*(b^4*e^2 - 4*a^3*c*d^2 + b^3*e^2*(b^2 - 4*a*c)^(1/2) + a^2*b^2*d^2 + 4*a^2*c^2*e^2 - 2*a*b^3*d*e -
5*a*b^2*c*e^2 + a^2*b*d^2*(b^2 - 4*a*c)^(1/2) + 8*a^2*b*c*d*e - 3*a*b*c*e^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^2*d*e
*(b^2 - 4*a*c)^(1/2) + 4*a^2*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*(4*a*c^4*e^4 + 4*a^3*c^2*d^4 - b^2*c^3*e^4 - a^2*b
^2*c*d^4 + 2*b^3*c^2*d*e^3 - b^4*c*d^2*e^2 + 8*a^2*c^3*d^2*e^2 - 8*a*b*c^3*d*e^3 + 2*a*b^3*c*d^3*e - 8*a^2*b*c
^2*d^3*e + 2*a*b^2*c^2*d^2*e^2)) + (log((a^4*e^4)/(d*(a*d^2 + c*e^2 - b*d*e)^2) + (a^4*e^5*x)/(d^2*(a*d^2 + c*
e^2 - b*d*e)^2) - (((a*e^3*(3*a^3*b*d^4 + b^3*c*e^4 - b^4*d*e^3 + 5*a*b^3*d^2*e^2 - 7*a^2*b^2*d^3*e + 8*a^2*c^
2*d*e^3 - 3*a*b*c^2*e^4 + 9*a^3*c*d^3*e - a*b^2*c*d*e^3 - 8*a^2*b*c*d^2*e^2))/(d^2*(a*d^2 + c*e^2 - b*d*e)^2)
+ (((a*e*(a^3*b*d^5 - 4*a*c^3*e^5 + b^2*c^2*e^5 - b^4*d^2*e^3 + 3*a*b^3*d^3*e^2 - 3*a^2*b^2*d^4*e - 8*a^2*c^2*
d^2*e^3 + 4*a^3*c*d^4*e - b^3*c*d*e^4 + 4*a*b*c^2*d*e^4 + 6*a*b^2*c*d^2*e^3 - 9*a^2*b*c*d^3*e^2))/(a*d^3 - b*d
^2*e + c*d*e^2) + (a*e*x*(3*a^4*d^5 + 2*b^3*c*e^5 - 4*b^4*d*e^4 + 9*a*b^3*d^2*e^3 + 4*a^2*c^2*d*e^4 + 19*a^3*c
*d^3*e^2 - 3*a^2*b^2*d^3*e^2 - 8*a*b*c^2*e^5 - 5*a^3*b*d^4*e + 15*a*b^2*c*d*e^4 - 36*a^2*b*c*d^2*e^3))/(a*d^3
- b*d^2*e + c*d*e^2) - (a*e*(b^4*e^2 - 4*a^3*c*d^2 - b^3*e^2*(b^2 - 4*a*c)^(1/2) + a^2*b^2*d^2 + 4*a^2*c^2*e^2
- 2*a*b^3*d*e - 5*a*b^2*c*e^2 - a^2*b*d^2*(b^2 - 4*a*c)^(1/2) + 8*a^2*b*c*d*e + 3*a*b*c*e^2*(b^2 - 4*a*c)^(1/
2) + 2*a*b^2*d*e*(b^2 - 4*a*c)^(1/2) - 4*a^2*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*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e)
^2))*(b^4*e^2 - 4*a^3*c*d^2 - b^3*e^2*(b^2 - 4*a*c)^(1/2) + a^2*b^2*d^2 + 4*a^2*c^2*e^2 - 2*a*b^3*d*e - 5*a*b^
2*c*e^2 - a^2*b*d^2*(b^2 - 4*a*c)^(1/2) + 8*a^2*b*c*d*e + 3*a*b*c*e^2*(b^2 - 4*a*c)^(1/2) + 2*a*b^2*d*e*(b^2 -
4*a*c)^(1/2) - 4*a^2*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*c*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e)^2) + (a*e^3*x*(9*
a^4*d^4 + b^4*e^4 + 2*a^2*c^2*e^4 - 6*a^3*c*d^2*e^2 + 8*a^2*b^2*d^2*e^2 - 4*a*b^2*c*e^4 - 4*a*b^3*d*e^3 - 12*a
^3*b*d^3*e + 10*a^2*b*c*d*e^3))/(d^2*(a*d^2 + c*e^2 - b*d*e)^2))*(b^4*e^2 - 4*a^3*c*d^2 - b^3*e^2*(b^2 - 4*a*c
)^(1/2) + a^2*b^2*d^2 + 4*a^2*c^2*e^2 - 2*a*b^3*d*e - 5*a*b^2*c*e^2 - a^2*b*d^2*(b^2 - 4*a*c)^(1/2) + 8*a^2*b*
c*d*e + 3*a*b*c*e^2*(b^2 - 4*a*c)^(1/2) + 2*a*b^2*d*e*(b^2 - 4*a*c)^(1/2) - 4*a^2*c*d*e*(b^2 - 4*a*c)^(1/2)))/
(2*c*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e)^2))*(b^4*e^2 - 4*a^3*c*d^2 - b^3*e^2*(b^2 - 4*a*c)^(1/2) + a^2*b^2*
d^2 + 4*a^2*c^2*e^2 - 2*a*b^3*d*e - 5*a*b^2*c*e^2 - a^2*b*d^2*(b^2 - 4*a*c)^(1/2) + 8*a^2*b*c*d*e + 3*a*b*c*e^
2*(b^2 - 4*a*c)^(1/2) + 2*a*b^2*d*e*(b^2 - 4*a*c)^(1/2) - 4*a^2*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*(4*a*c^4*e^4 +
4*a^3*c^2*d^4 - b^2*c^3*e^4 - a^2*b^2*c*d^4 + 2*b^3*c^2*d*e^3 - b^4*c*d^2*e^2 + 8*a^2*c^3*d^2*e^2 - 8*a*b*c^3*
d*e^3 + 2*a*b^3*c*d^3*e - 8*a^2*b*c^2*d^3*e + 2*a*b^2*c^2*d^2*e^2)) - (log(d + e*x)*(c*e^4 + 3*a*d^2*e^2 - 2*b
*d*e^3))/(a^2*d^6 + b^2*d^4*e^2 + c^2*d^2*e^4 - 2*a*b*d^5*e + 2*a*c*d^4*e^2 - 2*b*c*d^3*e^3) + log(x)/(c*d^2)
+ e^2/(d*(d + e*x)*(a*d^2 + c*e^2 - b*d*e))

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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**3/(e*x+d)**2,x)

[Out]

Timed out

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