3.1.70 \(\int \frac {1}{(a+\frac {c}{x^2}+\frac {b}{x}) x^5 (d+e x)^2} \, dx\)

Optimal. Leaf size=372 \[ \frac {\left (a^3 c d^2-a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )+a b^2 e (2 b d+3 c e)+b^4 \left (-e^2\right )\right ) \log \left (a x^2+b x+c\right )}{2 c^3 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (-a^3 c d (3 b d+4 c e)+a^2 b \left (b^2 d^2+8 b c d e+5 c^2 e^2\right )-a b^3 e (2 b d+5 c e)+b^5 e^2\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {\log (x) \left (-c \left (a d^2-3 c e^2\right )+b^2 d^2+2 b c d e\right )}{c^3 d^4}-\frac {e^4 \log (d+e x) \left (5 a d^2-e (4 b d-3 c e)\right )}{d^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {e^4}{d^3 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac {b d+2 c e}{c^2 d^3 x}-\frac {1}{2 c d^2 x^2} \]

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Rubi [A]  time = 0.85, antiderivative size = 372, 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+4 b c d e+c^2 e^2\right )+a^3 c d^2+a b^2 e (2 b d+3 c e)+b^4 \left (-e^2\right )\right ) \log \left (a x^2+b x+c\right )}{2 c^3 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (a^2 b \left (b^2 d^2+8 b c d e+5 c^2 e^2\right )-a^3 c d (3 b d+4 c e)-a b^3 e (2 b d+5 c e)+b^5 e^2\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {\log (x) \left (-c \left (a d^2-3 c e^2\right )+b^2 d^2+2 b c d e\right )}{c^3 d^4}+\frac {e^4}{d^3 (d+e x) \left (a d^2-e (b d-c e)\right )}-\frac {e^4 \log (d+e x) \left (5 a d^2-e (4 b d-3 c e)\right )}{d^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {b d+2 c e}{c^2 d^3 x}-\frac {1}{2 c d^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + c/x^2 + b/x)*x^5*(d + e*x)^2),x]

[Out]

-1/(2*c*d^2*x^2) + (b*d + 2*c*e)/(c^2*d^3*x) + e^4/(d^3*(a*d^2 - e*(b*d - c*e))*(d + e*x)) + ((b^5*e^2 - a^3*c
*d*(3*b*d + 4*c*e) - a*b^3*e*(2*b*d + 5*c*e) + a^2*b*(b^2*d^2 + 8*b*c*d*e + 5*c^2*e^2))*ArcTanh[(b + 2*a*x)/Sq
rt[b^2 - 4*a*c]])/(c^3*Sqrt[b^2 - 4*a*c]*(a*d^2 - e*(b*d - c*e))^2) + ((b^2*d^2 + 2*b*c*d*e - c*(a*d^2 - 3*c*e
^2))*Log[x])/(c^3*d^4) - (e^4*(5*a*d^2 - e*(4*b*d - 3*c*e))*Log[d + e*x])/(d^4*(a*d^2 - e*(b*d - c*e))^2) + ((
a^3*c*d^2 - b^4*e^2 + a*b^2*e*(2*b*d + 3*c*e) - a^2*(b^2*d^2 + 4*b*c*d*e + c^2*e^2))*Log[c + b*x + a*x^2])/(2*
c^3*(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^5 (d+e x)^2} \, dx &=\int \frac {1}{x^3 (d+e x)^2 \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac {1}{c d^2 x^3}+\frac {-b d-2 c e}{c^2 d^3 x^2}+\frac {b^2 d^2+2 b c d e-c \left (a d^2-3 c e^2\right )}{c^3 d^4 x}+\frac {e^5}{d^3 \left (-a d^2+e (b d-c e)\right ) (d+e x)^2}+\frac {e^5 \left (-5 a d^2+e (4 b d-3 c e)\right )}{d^4 \left (a d^2-e (b d-c e)\right )^2 (d+e x)}+\frac {-\left (\left (a b d-b^2 e+a c e\right ) \left (a b^2 d-2 a^2 c d-b^3 e+3 a b c e\right )\right )+a \left (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 )\right ) x}{c^3 \left (a d^2-e (b d-c e)\right )^2 \left (c+b x+a x^2\right )}\right ) \, dx\\ &=-\frac {1}{2 c d^2 x^2}+\frac {b d+2 c e}{c^2 d^3 x}+\frac {e^4}{d^3 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {\left (b^2 d^2+2 b c d e-c \left (a d^2-3 c e^2\right )\right ) \log (x)}{c^3 d^4}-\frac {e^4 \left (5 a d^2-e (4 b d-3 c e)\right ) \log (d+e x)}{d^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\int \frac {-\left (\left (a b d-b^2 e+a c e\right ) \left (a b^2 d-2 a^2 c d-b^3 e+3 a b c e\right )\right )+a \left (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 )\right ) x}{c+b x+a x^2} \, dx}{c^3 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac {1}{2 c d^2 x^2}+\frac {b d+2 c e}{c^2 d^3 x}+\frac {e^4}{d^3 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {\left (b^2 d^2+2 b c d e-c \left (a d^2-3 c e^2\right )\right ) \log (x)}{c^3 d^4}-\frac {e^4 \left (5 a d^2-e (4 b d-3 c e)\right ) \log (d+e x)}{d^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (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 )\right ) \int \frac {b+2 a x}{c+b x+a x^2} \, dx}{2 c^3 \left (a d^2-e (b d-c e)\right )^2}-\frac {\left (b^5 e^2-a^3 c d (3 b d+4 c e)-a b^3 e (2 b d+5 c e)+a^2 b \left (b^2 d^2+8 b c d e+5 c^2 e^2\right )\right ) \int \frac {1}{c+b x+a x^2} \, dx}{2 c^3 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac {1}{2 c d^2 x^2}+\frac {b d+2 c e}{c^2 d^3 x}+\frac {e^4}{d^3 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {\left (b^2 d^2+2 b c d e-c \left (a d^2-3 c e^2\right )\right ) \log (x)}{c^3 d^4}-\frac {e^4 \left (5 a d^2-e (4 b d-3 c e)\right ) \log (d+e x)}{d^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (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 )\right ) \log \left (c+b x+a x^2\right )}{2 c^3 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (b^5 e^2-a^3 c d (3 b d+4 c e)-a b^3 e (2 b d+5 c e)+a^2 b \left (b^2 d^2+8 b c d e+5 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^3 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac {1}{2 c d^2 x^2}+\frac {b d+2 c e}{c^2 d^3 x}+\frac {e^4}{d^3 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {\left (b^5 e^2-a^3 c d (3 b d+4 c e)-a b^3 e (2 b d+5 c e)+a^2 b \left (b^2 d^2+8 b c d e+5 c^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (b^2 d^2+2 b c d e-c \left (a d^2-3 c e^2\right )\right ) \log (x)}{c^3 d^4}-\frac {e^4 \left (5 a d^2-e (4 b d-3 c e)\right ) \log (d+e x)}{d^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (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 )\right ) \log \left (c+b x+a x^2\right )}{2 c^3 \left (a d^2-e (b d-c e)\right )^2}\\ \end {align*}

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Mathematica [A]  time = 0.43, size = 370, normalized size = 0.99 \begin {gather*} -\frac {\left (-a^3 c d^2+a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )-a b^2 e (2 b d+3 c e)+b^4 e^2\right ) \log (x (a x+b)+c)}{2 c^3 \left (a d^2+e (c e-b d)\right )^2}+\frac {\left (a^3 c d (3 b d+4 c e)-a^2 b \left (b^2 d^2+8 b c d e+5 c^2 e^2\right )+a b^3 e (2 b d+5 c e)+b^5 \left (-e^2\right )\right ) \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{c^3 \sqrt {4 a c-b^2} \left (a d^2+e (c e-b d)\right )^2}+\frac {\log (x) \left (c \left (3 c e^2-a d^2\right )+b^2 d^2+2 b c d e\right )}{c^3 d^4}-\frac {e^4 \log (d+e x) \left (5 a d^2+e (3 c e-4 b d)\right )}{d^4 \left (a d^2+e (c e-b d)\right )^2}+\frac {e^4}{d^3 (d+e x) \left (a d^2+e (c e-b d)\right )}+\frac {b d+2 c e}{c^2 d^3 x}-\frac {1}{2 c d^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + c/x^2 + b/x)*x^5*(d + e*x)^2),x]

[Out]

-1/2*1/(c*d^2*x^2) + (b*d + 2*c*e)/(c^2*d^3*x) + e^4/(d^3*(a*d^2 + e*(-(b*d) + c*e))*(d + e*x)) + ((-(b^5*e^2)
+ a^3*c*d*(3*b*d + 4*c*e) + a*b^3*e*(2*b*d + 5*c*e) - a^2*b*(b^2*d^2 + 8*b*c*d*e + 5*c^2*e^2))*ArcTan[(b + 2*
a*x)/Sqrt[-b^2 + 4*a*c]])/(c^3*Sqrt[-b^2 + 4*a*c]*(a*d^2 + e*(-(b*d) + c*e))^2) + ((b^2*d^2 + 2*b*c*d*e + c*(-
(a*d^2) + 3*c*e^2))*Log[x])/(c^3*d^4) - (e^4*(5*a*d^2 + e*(-4*b*d + 3*c*e))*Log[d + e*x])/(d^4*(a*d^2 + e*(-(b
*d) + c*e))^2) - ((-(a^3*c*d^2) + b^4*e^2 - a*b^2*e*(2*b*d + 3*c*e) + a^2*(b^2*d^2 + 4*b*c*d*e + c^2*e^2))*Log
[c + x*(b + a*x)])/(2*c^3*(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^5 (d+e x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/((a + c/x^2 + b/x)*x^5*(d + e*x)^2),x]

[Out]

IntegrateAlgebraic[1/((a + c/x^2 + b/x)*x^5*(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^5/(e*x+d)^2,x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.49, size = 587, normalized size = 1.58 \begin {gather*} \frac {{\left (a^{2} b^{3} d^{2} e^{2} - 3 \, a^{3} b c d^{2} e^{2} - 2 \, a b^{4} d e^{3} + 8 \, a^{2} b^{2} c d e^{3} - 4 \, a^{3} c^{2} d e^{3} + b^{5} e^{4} - 5 \, a b^{3} c e^{4} + 5 \, a^{2} b 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^{3} d^{4} - 2 \, a b c^{3} d^{3} e + b^{2} c^{3} d^{2} e^{2} + 2 \, a c^{4} d^{2} e^{2} - 2 \, b c^{4} d e^{3} + c^{5} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {{\left (a^{2} b^{2} d^{2} - a^{3} c d^{2} - 2 \, a b^{3} d e + 4 \, a^{2} b c d e + b^{4} e^{2} - 3 \, a b^{2} c e^{2} + a^{2} c^{2} 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^{3} d^{4} - 2 \, a b c^{3} d^{3} e + b^{2} c^{3} d^{2} e^{2} + 2 \, a c^{4} d^{2} e^{2} - 2 \, b c^{4} d e^{3} + c^{5} e^{4}\right )}} + \frac {e^{9}}{{\left (a d^{5} e^{5} - b d^{4} e^{6} + c d^{3} e^{7}\right )} {\left (x e + d\right )}} + \frac {{\left (b^{2} d^{2} e - a c d^{2} e + 2 \, b c d e^{2} + 3 \, c^{2} e^{3}\right )} e^{\left (-1\right )} \log \left ({\left | -\frac {d}{x e + d} + 1 \right |}\right )}{c^{3} d^{4}} + \frac {2 \, b c d e + 5 \, c^{2} e^{2} - \frac {2 \, {\left (b c d^{2} e^{2} + 3 \, c^{2} d e^{3}\right )} e^{\left (-1\right )}}{x e + d}}{2 \, c^{3} d^{4} {\left (\frac {d}{x e + d} - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+c/x^2+b/x)/x^5/(e*x+d)^2,x, algorithm="giac")

[Out]

(a^2*b^3*d^2*e^2 - 3*a^3*b*c*d^2*e^2 - 2*a*b^4*d*e^3 + 8*a^2*b^2*c*d*e^3 - 4*a^3*c^2*d*e^3 + b^5*e^4 - 5*a*b^3
*c*e^4 + 5*a^2*b*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^3*d^4 - 2*a*b*c^3*d^3*e + b^2*c^3*d^2*e^2 + 2*a*c^4*d^2*e^2 - 2*b*c^4*
d*e^3 + c^5*e^4)*sqrt(-b^2 + 4*a*c)) - 1/2*(a^2*b^2*d^2 - a^3*c*d^2 - 2*a*b^3*d*e + 4*a^2*b*c*d*e + b^4*e^2 -
3*a*b^2*c*e^2 + a^2*c^2*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^3*d^4 - 2*a*b*c^3*d^3*e + b^2*c^3*d^2*e^2 + 2*a*c^4*d^2*e^2 - 2*b*c^4*d*e^3 + c^5*
e^4) + e^9/((a*d^5*e^5 - b*d^4*e^6 + c*d^3*e^7)*(x*e + d)) + (b^2*d^2*e - a*c*d^2*e + 2*b*c*d*e^2 + 3*c^2*e^3)
*e^(-1)*log(abs(-d/(x*e + d) + 1))/(c^3*d^4) + 1/2*(2*b*c*d*e + 5*c^2*e^2 - 2*(b*c*d^2*e^2 + 3*c^2*d*e^3)*e^(-
1)/(x*e + d))/(c^3*d^4*(d/(x*e + d) - 1)^2)

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maple [B]  time = 0.02, size = 993, normalized size = 2.67 \begin {gather*} \frac {3 a^{3} 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^{2}}+\frac {4 a^{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}-\frac {a^{2} b^{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^{3}}-\frac {8 a^{2} 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^{2}}-\frac {5 a^{2} 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}}\, c}+\frac {2 a \,b^{4} 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^{3}}+\frac {5 a \,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^{2}}-\frac {b^{5} 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^{3}}+\frac {a^{3} 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} b^{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^{3}}-\frac {2 a^{2} 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^{2}}-\frac {a^{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 {a \,b^{3} d e \ln \left (a \,x^{2}+b x +c \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} c^{3}}+\frac {3 a \,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^{2}}-\frac {5 a \,e^{4} \ln \left (e x +d \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} d^{2}}-\frac {b^{4} 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^{3}}+\frac {4 b \,e^{5} \ln \left (e x +d \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} d^{3}}-\frac {3 c \,e^{6} \ln \left (e x +d \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} d^{4}}+\frac {e^{4}}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) \left (e x +d \right ) d^{3}}-\frac {a \ln \relax (x )}{c^{2} d^{2}}+\frac {b^{2} \ln \relax (x )}{c^{3} d^{2}}+\frac {2 b e \ln \relax (x )}{c^{2} d^{3}}+\frac {3 e^{2} \ln \relax (x )}{c \,d^{4}}+\frac {b}{c^{2} d^{2} x}+\frac {2 e}{c \,d^{3} x}-\frac {1}{2 c \,d^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+c/x^2+b/x)/x^5/(e*x+d)^2,x)

[Out]

-8/(a*d^2-b*d*e+c*e^2)^2/c^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*e-1/(a*d^2-b*d*e+
c*e^2)^2/c^3/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*a^2*b^3*d^2-5/(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*e^2+5/(a*d^2-b*d*e+c*e^2)^2/c^2/(4*a*c-b^2)^(1/2)*arcta
n((2*a*x+b)/(4*a*c-b^2)^(1/2))*a*b^3*e^2-2/(a*d^2-b*d*e+c*e^2)^2/c^2*a^2*ln(a*x^2+b*x+c)*b*d*e+1/(a*d^2-b*d*e+
c*e^2)^2/c^3*a*ln(a*x^2+b*x+c)*b^3*d*e+3/(a*d^2-b*d*e+c*e^2)^2/c^2/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b
^2)^(1/2))*a^3*b*d^2+4/(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^3*d*e-1
/c^2/d^2*ln(x)*a+1/c^3/d^2*ln(x)*b^2+3/c/d^4*ln(x)*e^2+1/c^2/d^2/x*b+2/c/d^3/x*e+e^4/(a*d^2-b*d*e+c*e^2)/d^3/(
e*x+d)+2/(a*d^2-b*d*e+c*e^2)^2/c^3/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*a*b^4*d*e+1/2/(a*d^2-
b*d*e+c*e^2)^2/c^2*a^3*ln(a*x^2+b*x+c)*d^2-1/2/c/d^2/x^2-1/2/(a*d^2-b*d*e+c*e^2)^2/c*a^2*ln(a*x^2+b*x+c)*e^2-1
/2/(a*d^2-b*d*e+c*e^2)^2/c^3*ln(a*x^2+b*x+c)*b^4*e^2+2/c^2/d^3*ln(x)*b*e-5*e^4/(a*d^2-b*d*e+c*e^2)^2/d^2*ln(e*
x+d)*a+4*e^5/(a*d^2-b*d*e+c*e^2)^2/d^3*ln(e*x+d)*b-3*e^6/(a*d^2-b*d*e+c*e^2)^2/d^4*ln(e*x+d)*c-1/2/(a*d^2-b*d*
e+c*e^2)^2/c^3*a^2*ln(a*x^2+b*x+c)*b^2*d^2+3/2/(a*d^2-b*d*e+c*e^2)^2/c^2*a*ln(a*x^2+b*x+c)*b^2*e^2-1/(a*d^2-b*
d*e+c*e^2)^2/c^3/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*b^5*e^2

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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^5/(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 = 45.61, size = 7144, normalized size = 19.20

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(d + e*x)^2*(a + b/x + c/x^2)),x)

[Out]

((x*(2*b*d + 3*c*e))/(2*c^2*d^2) - 1/(2*c*d) + (x^2*(3*c^2*e^4 - b^2*d^2*e^2 + a*b*d^3*e - b*c*d*e^3 + 2*a*c*d
^2*e^2))/(c^2*d^3*(a*d^2 + c*e^2 - b*d*e)))/(d*x^2 + e*x^3) - (log(d + e*x)*(3*c*e^6 + 5*a*d^2*e^4 - 4*b*d*e^5
))/(a^2*d^8 + b^2*d^6*e^2 + c^2*d^4*e^4 - 2*a*b*d^7*e + 2*a*c*d^6*e^2 - 2*b*c*d^5*e^3) + (log((((27*a^2*b*c^6*
e^11 - 9*a*b^3*c^5*e^11 - a*b^8*d^5*e^6 - a^6*b^3*d^10*e - 36*a^3*c^6*d*e^10 + 2*a^2*b^7*d^6*e^5 - a^3*b^6*d^7
*e^4 - a^4*b^5*d^8*e^3 + 2*a^5*b^4*d^9*e^2 - 36*a^4*c^5*d^3*e^8 + 4*a^5*c^4*d^5*e^6 + 3*a^6*c^3*d^7*e^4 + a^7*
b*c*d^10*e - 39*a^2*b^3*c^4*d^2*e^9 - 15*a^2*b^4*c^3*d^3*e^8 + 7*a^2*b^5*c^2*d^4*e^7 + 53*a^3*b^2*c^4*d^3*e^8
+ 7*a^3*b^3*c^3*d^4*e^7 - 33*a^3*b^4*c^2*d^5*e^6 + 20*a^4*b^2*c^3*d^5*e^6 + 33*a^4*b^3*c^2*d^6*e^5 - 9*a^5*b^2
*c^2*d^7*e^4 + 6*a*b^4*c^4*d*e^10 - 2*a*b^7*c*d^4*e^7 + 5*a*b^5*c^3*d^2*e^9 + a*b^6*c^2*d^3*e^8 + 12*a^2*b^6*c
*d^5*e^6 + 51*a^3*b*c^5*d^2*e^9 - 16*a^3*b^5*c*d^6*e^5 - 27*a^4*b*c^4*d^4*e^7 + 6*a^4*b^4*c*d^7*e^4 - 19*a^5*b
*c^3*d^6*e^5 + 3*a^5*b^3*c*d^8*e^3 - a^6*b*c^2*d^8*e^3 - 4*a^6*b^2*c*d^9*e^2)/(c^4*d^6*(a*d^2 + c*e^2 - b*d*e)
^2) + (((a*e*(12*a*c^5*e^7 - a^3*b^3*d^7 - 3*b^2*c^4*e^7 + b^6*d^4*e^3 - 3*a*b^5*d^5*e^2 + 3*a^2*b^4*d^6*e + 4
*a^4*c^2*d^6*e + b^3*c^3*d*e^6 + b^5*c*d^3*e^4 + 8*a^2*c^4*d^2*e^5 - 8*a^3*c^3*d^4*e^3 + b^4*c^2*d^2*e^5 + 2*a
^4*b*c*d^7 - 4*a*b*c^4*d*e^6 + 18*a^2*b^2*c^2*d^4*e^3 - 8*a*b^4*c*d^4*e^3 - 10*a^3*b^2*c*d^6*e - 6*a*b^2*c^3*d
^2*e^5 - 7*a*b^3*c^2*d^3*e^4 + 12*a^2*b*c^3*d^3*e^4 + 15*a^2*b^3*c*d^5*e^2 - 16*a^3*b*c^2*d^5*e^2))/(c^2*d^3*(
a*d^2 + c*e^2 - b*d*e)) + (a*e*(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)*(b^6*e^2 + b^5*e^2*(b^2 - 4*a*c)^(1/2) + a^2*b^4*d^2 + 4*a^4*c^2*d^2 - 4*a^3*c^3*e^2
- 5*a^3*b^2*c*d^2 + a^2*b^3*d^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^5*d*e + 13*a^2*b^2*c^2*e^2 - 7*a*b^4*c*e^2 + 12*a
^2*b^3*c*d*e - 16*a^3*b*c^2*d*e - 3*a^3*b*c*d^2*(b^2 - 4*a*c)^(1/2) - 5*a*b^3*c*e^2*(b^2 - 4*a*c)^(1/2) - 4*a^
3*c^2*d*e*(b^2 - 4*a*c)^(1/2) + 5*a^2*b*c^2*e^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^4*d*e*(b^2 - 4*a*c)^(1/2) + 8*a^2*
b^2*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*c^3*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e)^2) - (a*e*x*(2*a^4*b^2*d^7 - 3*a^
5*c*d^7 + 6*b^3*c^3*e^7 - 2*b^6*d^3*e^4 + 4*a*b^5*d^4*e^3 - 4*a^3*b^3*d^6*e + 24*a^2*c^4*d*e^6 - 5*b^4*c^2*d*e
^6 - b^5*c*d^2*e^5 + 32*a^3*c^3*d^3*e^4 - 7*a^4*c^2*d^5*e^2 - 24*a*b*c^4*e^7 + 9*a^4*b*c*d^6*e - 36*a^2*b^2*c^
2*d^3*e^4 + 14*a*b^2*c^3*d*e^6 + 15*a*b^4*c*d^3*e^4 + 16*a*b^3*c^2*d^2*e^5 - 48*a^2*b*c^3*d^2*e^5 - 24*a^2*b^3
*c*d^4*e^3 + 32*a^3*b*c^2*d^4*e^3 + 4*a^3*b^2*c*d^5*e^2))/(c^2*d^3*(a*d^2 + c*e^2 - b*d*e)))*(b^6*e^2 + b^5*e^
2*(b^2 - 4*a*c)^(1/2) + a^2*b^4*d^2 + 4*a^4*c^2*d^2 - 4*a^3*c^3*e^2 - 5*a^3*b^2*c*d^2 + a^2*b^3*d^2*(b^2 - 4*a
*c)^(1/2) - 2*a*b^5*d*e + 13*a^2*b^2*c^2*e^2 - 7*a*b^4*c*e^2 + 12*a^2*b^3*c*d*e - 16*a^3*b*c^2*d*e - 3*a^3*b*c
*d^2*(b^2 - 4*a*c)^(1/2) - 5*a*b^3*c*e^2*(b^2 - 4*a*c)^(1/2) - 4*a^3*c^2*d*e*(b^2 - 4*a*c)^(1/2) + 5*a^2*b*c^2
*e^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^4*d*e*(b^2 - 4*a*c)^(1/2) + 8*a^2*b^2*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*c^3*(4*a
*c - b^2)*(a*d^2 + c*e^2 - b*d*e)^2) - (x*(18*a^3*c^6*e^11 + 9*a*b^4*c^4*e^11 + a*b^8*d^4*e^7 + a^7*b^2*d^10*e
- 36*a^2*b^2*c^5*e^11 - 2*a^2*b^7*d^5*e^6 + a^3*b^6*d^6*e^5 + a^5*b^4*d^8*e^3 - 2*a^6*b^3*d^9*e^2 + 6*a^4*c^5
*d^2*e^9 - 10*a^5*c^4*d^4*e^7 - 12*a^6*c^3*d^6*e^5 + 3*a^7*c^2*d^8*e^3 + 44*a^2*b^4*c^3*d^2*e^9 - 2*a^2*b^5*c^
2*d^3*e^8 - 85*a^3*b^2*c^4*d^2*e^9 - 46*a^3*b^3*c^3*d^3*e^8 + 45*a^3*b^4*c^2*d^4*e^7 - 42*a^4*b^2*c^3*d^4*e^7
- 56*a^4*b^3*c^2*d^5*e^6 + 19*a^5*b^2*c^2*d^6*e^5 - 6*a*b^5*c^3*d*e^10 + 2*a*b^7*c*d^3*e^8 + 42*a^3*b*c^5*d*e^
10 + 2*a^7*b*c*d^9*e^2 - 5*a*b^6*c^2*d^2*e^9 + 6*a^2*b^3*c^4*d*e^10 - 12*a^2*b^6*c*d^4*e^7 + 16*a^3*b^5*c*d^5*
e^6 + 88*a^4*b*c^4*d^3*e^8 - 6*a^4*b^4*c*d^6*e^5 + 62*a^5*b*c^3*d^5*e^6 - 2*a^6*b*c^2*d^7*e^4 - 2*a^6*b^2*c*d^
8*e^3))/(c^4*d^6*(a*d^2 + c*e^2 - b*d*e)^2))*(b^6*e^2 + b^5*e^2*(b^2 - 4*a*c)^(1/2) + a^2*b^4*d^2 + 4*a^4*c^2*
d^2 - 4*a^3*c^3*e^2 - 5*a^3*b^2*c*d^2 + a^2*b^3*d^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^5*d*e + 13*a^2*b^2*c^2*e^2 - 7
*a*b^4*c*e^2 + 12*a^2*b^3*c*d*e - 16*a^3*b*c^2*d*e - 3*a^3*b*c*d^2*(b^2 - 4*a*c)^(1/2) - 5*a*b^3*c*e^2*(b^2 -
4*a*c)^(1/2) - 4*a^3*c^2*d*e*(b^2 - 4*a*c)^(1/2) + 5*a^2*b*c^2*e^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^4*d*e*(b^2 - 4*
a*c)^(1/2) + 8*a^2*b^2*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*c^3*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e)^2) + (a^4*e^4*
(a^2*b^2*d^5 - 9*b*c^3*e^5 - a^3*c*d^5 + 4*b^4*d^3*e^2 + 6*b^2*c^2*d*e^4 + 5*b^3*c*d^2*e^3 + 3*a^2*c^2*d^3*e^2
- 5*a*b^3*d^4*e + 7*a^2*b*c*d^4*e - 12*a*b*c^2*d^2*e^3 - 14*a*b^2*c*d^3*e^2))/(c^4*d^6*(a*d^2 + c*e^2 - b*d*e
)^2) - (a^5*e^5*x*(9*c^3*e^4 + 4*a*b^2*d^4 + a^2*c*d^4 - 4*b^3*d^3*e + 12*a*c^2*d^2*e^2 - 5*b^2*c*d^2*e^2 - 6*
b*c^2*d*e^3 + 8*a*b*c*d^3*e))/(c^4*d^6*(a*d^2 + c*e^2 - b*d*e)^2))*(b^6*e^2 + b^5*e^2*(b^2 - 4*a*c)^(1/2) + a^
2*b^4*d^2 + 4*a^4*c^2*d^2 - 4*a^3*c^3*e^2 - 5*a^3*b^2*c*d^2 + a^2*b^3*d^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^5*d*e +
13*a^2*b^2*c^2*e^2 - 7*a*b^4*c*e^2 + 12*a^2*b^3*c*d*e - 16*a^3*b*c^2*d*e - 3*a^3*b*c*d^2*(b^2 - 4*a*c)^(1/2) -
5*a*b^3*c*e^2*(b^2 - 4*a*c)^(1/2) - 4*a^3*c^2*d*e*(b^2 - 4*a*c)^(1/2) + 5*a^2*b*c^2*e^2*(b^2 - 4*a*c)^(1/2) -
2*a*b^4*d*e*(b^2 - 4*a*c)^(1/2) + 8*a^2*b^2*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*(4*a*c^6*e^4 + 4*a^3*c^4*d^4 - b^2
*c^5*e^4 + 2*b^3*c^4*d*e^3 - a^2*b^2*c^3*d^4 + 8*a^2*c^5*d^2*e^2 - b^4*c^3*d^2*e^2 - 8*a*b*c^5*d*e^3 + 2*a*b^3
*c^3*d^3*e - 8*a^2*b*c^4*d^3*e + 2*a*b^2*c^4*d^2*e^2)) + (log((((27*a^2*b*c^6*e^11 - 9*a*b^3*c^5*e^11 - a*b^8*
d^5*e^6 - a^6*b^3*d^10*e - 36*a^3*c^6*d*e^10 + 2*a^2*b^7*d^6*e^5 - a^3*b^6*d^7*e^4 - a^4*b^5*d^8*e^3 + 2*a^5*b
^4*d^9*e^2 - 36*a^4*c^5*d^3*e^8 + 4*a^5*c^4*d^5*e^6 + 3*a^6*c^3*d^7*e^4 + a^7*b*c*d^10*e - 39*a^2*b^3*c^4*d^2*
e^9 - 15*a^2*b^4*c^3*d^3*e^8 + 7*a^2*b^5*c^2*d^4*e^7 + 53*a^3*b^2*c^4*d^3*e^8 + 7*a^3*b^3*c^3*d^4*e^7 - 33*a^3
*b^4*c^2*d^5*e^6 + 20*a^4*b^2*c^3*d^5*e^6 + 33*a^4*b^3*c^2*d^6*e^5 - 9*a^5*b^2*c^2*d^7*e^4 + 6*a*b^4*c^4*d*e^1
0 - 2*a*b^7*c*d^4*e^7 + 5*a*b^5*c^3*d^2*e^9 + a*b^6*c^2*d^3*e^8 + 12*a^2*b^6*c*d^5*e^6 + 51*a^3*b*c^5*d^2*e^9
- 16*a^3*b^5*c*d^6*e^5 - 27*a^4*b*c^4*d^4*e^7 + 6*a^4*b^4*c*d^7*e^4 - 19*a^5*b*c^3*d^6*e^5 + 3*a^5*b^3*c*d^8*e
^3 - a^6*b*c^2*d^8*e^3 - 4*a^6*b^2*c*d^9*e^2)/(c^4*d^6*(a*d^2 + c*e^2 - b*d*e)^2) + (((a*e*(12*a*c^5*e^7 - a^3
*b^3*d^7 - 3*b^2*c^4*e^7 + b^6*d^4*e^3 - 3*a*b^5*d^5*e^2 + 3*a^2*b^4*d^6*e + 4*a^4*c^2*d^6*e + b^3*c^3*d*e^6 +
b^5*c*d^3*e^4 + 8*a^2*c^4*d^2*e^5 - 8*a^3*c^3*d^4*e^3 + b^4*c^2*d^2*e^5 + 2*a^4*b*c*d^7 - 4*a*b*c^4*d*e^6 + 1
8*a^2*b^2*c^2*d^4*e^3 - 8*a*b^4*c*d^4*e^3 - 10*a^3*b^2*c*d^6*e - 6*a*b^2*c^3*d^2*e^5 - 7*a*b^3*c^2*d^3*e^4 + 1
2*a^2*b*c^3*d^3*e^4 + 15*a^2*b^3*c*d^5*e^2 - 16*a^3*b*c^2*d^5*e^2))/(c^2*d^3*(a*d^2 + c*e^2 - b*d*e)) + (a*e*(
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)*(b^6*e
^2 - b^5*e^2*(b^2 - 4*a*c)^(1/2) + a^2*b^4*d^2 + 4*a^4*c^2*d^2 - 4*a^3*c^3*e^2 - 5*a^3*b^2*c*d^2 - a^2*b^3*d^2
*(b^2 - 4*a*c)^(1/2) - 2*a*b^5*d*e + 13*a^2*b^2*c^2*e^2 - 7*a*b^4*c*e^2 + 12*a^2*b^3*c*d*e - 16*a^3*b*c^2*d*e
+ 3*a^3*b*c*d^2*(b^2 - 4*a*c)^(1/2) + 5*a*b^3*c*e^2*(b^2 - 4*a*c)^(1/2) + 4*a^3*c^2*d*e*(b^2 - 4*a*c)^(1/2) -
5*a^2*b*c^2*e^2*(b^2 - 4*a*c)^(1/2) + 2*a*b^4*d*e*(b^2 - 4*a*c)^(1/2) - 8*a^2*b^2*c*d*e*(b^2 - 4*a*c)^(1/2)))/
(2*c^3*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e)^2) - (a*e*x*(2*a^4*b^2*d^7 - 3*a^5*c*d^7 + 6*b^3*c^3*e^7 - 2*b^6*
d^3*e^4 + 4*a*b^5*d^4*e^3 - 4*a^3*b^3*d^6*e + 24*a^2*c^4*d*e^6 - 5*b^4*c^2*d*e^6 - b^5*c*d^2*e^5 + 32*a^3*c^3*
d^3*e^4 - 7*a^4*c^2*d^5*e^2 - 24*a*b*c^4*e^7 + 9*a^4*b*c*d^6*e - 36*a^2*b^2*c^2*d^3*e^4 + 14*a*b^2*c^3*d*e^6 +
15*a*b^4*c*d^3*e^4 + 16*a*b^3*c^2*d^2*e^5 - 48*a^2*b*c^3*d^2*e^5 - 24*a^2*b^3*c*d^4*e^3 + 32*a^3*b*c^2*d^4*e^
3 + 4*a^3*b^2*c*d^5*e^2))/(c^2*d^3*(a*d^2 + c*e^2 - b*d*e)))*(b^6*e^2 - b^5*e^2*(b^2 - 4*a*c)^(1/2) + a^2*b^4*
d^2 + 4*a^4*c^2*d^2 - 4*a^3*c^3*e^2 - 5*a^3*b^2*c*d^2 - a^2*b^3*d^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^5*d*e + 13*a^2
*b^2*c^2*e^2 - 7*a*b^4*c*e^2 + 12*a^2*b^3*c*d*e - 16*a^3*b*c^2*d*e + 3*a^3*b*c*d^2*(b^2 - 4*a*c)^(1/2) + 5*a*b
^3*c*e^2*(b^2 - 4*a*c)^(1/2) + 4*a^3*c^2*d*e*(b^2 - 4*a*c)^(1/2) - 5*a^2*b*c^2*e^2*(b^2 - 4*a*c)^(1/2) + 2*a*b
^4*d*e*(b^2 - 4*a*c)^(1/2) - 8*a^2*b^2*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*c^3*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e
)^2) - (x*(18*a^3*c^6*e^11 + 9*a*b^4*c^4*e^11 + a*b^8*d^4*e^7 + a^7*b^2*d^10*e - 36*a^2*b^2*c^5*e^11 - 2*a^2*b
^7*d^5*e^6 + a^3*b^6*d^6*e^5 + a^5*b^4*d^8*e^3 - 2*a^6*b^3*d^9*e^2 + 6*a^4*c^5*d^2*e^9 - 10*a^5*c^4*d^4*e^7 -
12*a^6*c^3*d^6*e^5 + 3*a^7*c^2*d^8*e^3 + 44*a^2*b^4*c^3*d^2*e^9 - 2*a^2*b^5*c^2*d^3*e^8 - 85*a^3*b^2*c^4*d^2*e
^9 - 46*a^3*b^3*c^3*d^3*e^8 + 45*a^3*b^4*c^2*d^4*e^7 - 42*a^4*b^2*c^3*d^4*e^7 - 56*a^4*b^3*c^2*d^5*e^6 + 19*a^
5*b^2*c^2*d^6*e^5 - 6*a*b^5*c^3*d*e^10 + 2*a*b^7*c*d^3*e^8 + 42*a^3*b*c^5*d*e^10 + 2*a^7*b*c*d^9*e^2 - 5*a*b^6
*c^2*d^2*e^9 + 6*a^2*b^3*c^4*d*e^10 - 12*a^2*b^6*c*d^4*e^7 + 16*a^3*b^5*c*d^5*e^6 + 88*a^4*b*c^4*d^3*e^8 - 6*a
^4*b^4*c*d^6*e^5 + 62*a^5*b*c^3*d^5*e^6 - 2*a^6*b*c^2*d^7*e^4 - 2*a^6*b^2*c*d^8*e^3))/(c^4*d^6*(a*d^2 + c*e^2
- b*d*e)^2))*(b^6*e^2 - b^5*e^2*(b^2 - 4*a*c)^(1/2) + a^2*b^4*d^2 + 4*a^4*c^2*d^2 - 4*a^3*c^3*e^2 - 5*a^3*b^2*
c*d^2 - a^2*b^3*d^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^5*d*e + 13*a^2*b^2*c^2*e^2 - 7*a*b^4*c*e^2 + 12*a^2*b^3*c*d*e
- 16*a^3*b*c^2*d*e + 3*a^3*b*c*d^2*(b^2 - 4*a*c)^(1/2) + 5*a*b^3*c*e^2*(b^2 - 4*a*c)^(1/2) + 4*a^3*c^2*d*e*(b^
2 - 4*a*c)^(1/2) - 5*a^2*b*c^2*e^2*(b^2 - 4*a*c)^(1/2) + 2*a*b^4*d*e*(b^2 - 4*a*c)^(1/2) - 8*a^2*b^2*c*d*e*(b^
2 - 4*a*c)^(1/2)))/(2*c^3*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e)^2) + (a^4*e^4*(a^2*b^2*d^5 - 9*b*c^3*e^5 - a^3
*c*d^5 + 4*b^4*d^3*e^2 + 6*b^2*c^2*d*e^4 + 5*b^3*c*d^2*e^3 + 3*a^2*c^2*d^3*e^2 - 5*a*b^3*d^4*e + 7*a^2*b*c*d^4
*e - 12*a*b*c^2*d^2*e^3 - 14*a*b^2*c*d^3*e^2))/(c^4*d^6*(a*d^2 + c*e^2 - b*d*e)^2) - (a^5*e^5*x*(9*c^3*e^4 + 4
*a*b^2*d^4 + a^2*c*d^4 - 4*b^3*d^3*e + 12*a*c^2*d^2*e^2 - 5*b^2*c*d^2*e^2 - 6*b*c^2*d*e^3 + 8*a*b*c*d^3*e))/(c
^4*d^6*(a*d^2 + c*e^2 - b*d*e)^2))*(b^6*e^2 - b^5*e^2*(b^2 - 4*a*c)^(1/2) + a^2*b^4*d^2 + 4*a^4*c^2*d^2 - 4*a^
3*c^3*e^2 - 5*a^3*b^2*c*d^2 - a^2*b^3*d^2*(b^2 - 4*a*c)^(1/2) - 2*a*b^5*d*e + 13*a^2*b^2*c^2*e^2 - 7*a*b^4*c*e
^2 + 12*a^2*b^3*c*d*e - 16*a^3*b*c^2*d*e + 3*a^3*b*c*d^2*(b^2 - 4*a*c)^(1/2) + 5*a*b^3*c*e^2*(b^2 - 4*a*c)^(1/
2) + 4*a^3*c^2*d*e*(b^2 - 4*a*c)^(1/2) - 5*a^2*b*c^2*e^2*(b^2 - 4*a*c)^(1/2) + 2*a*b^4*d*e*(b^2 - 4*a*c)^(1/2)
- 8*a^2*b^2*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*(4*a*c^6*e^4 + 4*a^3*c^4*d^4 - b^2*c^5*e^4 + 2*b^3*c^4*d*e^3 - a^2
*b^2*c^3*d^4 + 8*a^2*c^5*d^2*e^2 - b^4*c^3*d^2*e^2 - 8*a*b*c^5*d*e^3 + 2*a*b^3*c^3*d^3*e - 8*a^2*b*c^4*d^3*e +
2*a*b^2*c^4*d^2*e^2)) + (log(x)*(3*c^2*e^2 - d^2*(a*c - b^2) + 2*b*c*d*e))/(c^3*d^4)

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

[Out]

Timed out

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