3.106 \(\int \frac {A+B x^2}{x^3 (a+b x^2+c x^4)} \, dx\)

Optimal. Leaf size=112 \[ -\frac {\left (-2 a A c-a b B+A b^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \sqrt {b^2-4 a c}}+\frac {(A b-a B) \log \left (a+b x^2+c x^4\right )}{4 a^2}-\frac {\log (x) (A b-a B)}{a^2}-\frac {A}{2 a x^2} \]

[Out]

-1/2*A/a/x^2-(A*b-B*a)*ln(x)/a^2+1/4*(A*b-B*a)*ln(c*x^4+b*x^2+a)/a^2-1/2*(-2*A*a*c+A*b^2-B*a*b)*arctanh((2*c*x
^2+b)/(-4*a*c+b^2)^(1/2))/a^2/(-4*a*c+b^2)^(1/2)

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Rubi [A]  time = 0.25, antiderivative size = 112, 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 = {1251, 800, 634, 618, 206, 628} \[ -\frac {\left (-2 a A c-a b B+A b^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \sqrt {b^2-4 a c}}+\frac {(A b-a B) \log \left (a+b x^2+c x^4\right )}{4 a^2}-\frac {\log (x) (A b-a B)}{a^2}-\frac {A}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^3*(a + b*x^2 + c*x^4)),x]

[Out]

-A/(2*a*x^2) - ((A*b^2 - a*b*B - 2*a*A*c)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^2*Sqrt[b^2 - 4*a*c])
- ((A*b - a*B)*Log[x])/a^2 + ((A*b - a*B)*Log[a + b*x^2 + c*x^4])/(4*a^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 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 1251

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

Rubi steps

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

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Mathematica [A]  time = 0.22, size = 186, normalized size = 1.66 \[ \frac {\frac {\left (A \left (b \sqrt {b^2-4 a c}-2 a c+b^2\right )-a B \left (\sqrt {b^2-4 a c}+b\right )\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c x^2\right )}{\sqrt {b^2-4 a c}}+\frac {\left (A \left (b \sqrt {b^2-4 a c}+2 a c-b^2\right )+a B \left (b-\sqrt {b^2-4 a c}\right )\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c x^2\right )}{\sqrt {b^2-4 a c}}+4 \log (x) (a B-A b)-\frac {2 a A}{x^2}}{4 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^3*(a + b*x^2 + c*x^4)),x]

[Out]

((-2*a*A)/x^2 + 4*(-(A*b) + a*B)*Log[x] + ((-(a*B*(b + Sqrt[b^2 - 4*a*c])) + A*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a
*c]))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c] + ((a*B*(b - Sqrt[b^2 - 4*a*c]) + A*(-b^2 + 2*a*
c + b*Sqrt[b^2 - 4*a*c]))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/(4*a^2)

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fricas [A]  time = 1.01, size = 385, normalized size = 3.44 \[ \left [\frac {{\left (B a b - A b^{2} + 2 \, A a c\right )} \sqrt {b^{2} - 4 \, a c} x^{2} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - 2 \, A a b^{2} + 8 \, A a^{2} c - {\left (B a b^{2} - A b^{3} - 4 \, {\left (B a^{2} - A a b\right )} c\right )} x^{2} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \, {\left (B a b^{2} - A b^{3} - 4 \, {\left (B a^{2} - A a b\right )} c\right )} x^{2} \log \relax (x)}{4 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2}}, \frac {2 \, {\left (B a b - A b^{2} + 2 \, A a c\right )} \sqrt {-b^{2} + 4 \, a c} x^{2} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - 2 \, A a b^{2} + 8 \, A a^{2} c - {\left (B a b^{2} - A b^{3} - 4 \, {\left (B a^{2} - A a b\right )} c\right )} x^{2} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \, {\left (B a b^{2} - A b^{3} - 4 \, {\left (B a^{2} - A a b\right )} c\right )} x^{2} \log \relax (x)}{4 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^3/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

[1/4*((B*a*b - A*b^2 + 2*A*a*c)*sqrt(b^2 - 4*a*c)*x^2*log((2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^2 + b)
*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)) - 2*A*a*b^2 + 8*A*a^2*c - (B*a*b^2 - A*b^3 - 4*(B*a^2 - A*a*b)*c)*x^2
*log(c*x^4 + b*x^2 + a) + 4*(B*a*b^2 - A*b^3 - 4*(B*a^2 - A*a*b)*c)*x^2*log(x))/((a^2*b^2 - 4*a^3*c)*x^2), 1/4
*(2*(B*a*b - A*b^2 + 2*A*a*c)*sqrt(-b^2 + 4*a*c)*x^2*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) -
2*A*a*b^2 + 8*A*a^2*c - (B*a*b^2 - A*b^3 - 4*(B*a^2 - A*a*b)*c)*x^2*log(c*x^4 + b*x^2 + a) + 4*(B*a*b^2 - A*b
^3 - 4*(B*a^2 - A*a*b)*c)*x^2*log(x))/((a^2*b^2 - 4*a^3*c)*x^2)]

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giac [A]  time = 1.87, size = 124, normalized size = 1.11 \[ -\frac {{\left (B a - A b\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{2}} + \frac {{\left (B a - A b\right )} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac {{\left (B a b - A b^{2} + 2 \, A a c\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} a^{2}} - \frac {B a x^{2} - A b x^{2} + A a}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^3/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

-1/4*(B*a - A*b)*log(c*x^4 + b*x^2 + a)/a^2 + 1/2*(B*a - A*b)*log(x^2)/a^2 - 1/2*(B*a*b - A*b^2 + 2*A*a*c)*arc
tan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^2) - 1/2*(B*a*x^2 - A*b*x^2 + A*a)/(a^2*x^2)

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maple [A]  time = 0.01, size = 191, normalized size = 1.71 \[ -\frac {A c \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a}+\frac {A \,b^{2} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, a^{2}}-\frac {B b \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, a}-\frac {A b \ln \relax (x )}{a^{2}}+\frac {A b \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 a^{2}}+\frac {B \ln \relax (x )}{a}-\frac {B \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 a}-\frac {A}{2 a \,x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/x^3/(c*x^4+b*x^2+a),x)

[Out]

-1/2*A/a/x^2-1/a^2*ln(x)*A*b+1/a*ln(x)*B+1/4/a^2*ln(c*x^4+b*x^2+a)*A*b-1/4/a*ln(c*x^4+b*x^2+a)*B-1/a/(4*a*c-b^
2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*A*c+1/2/a^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1
/2))*A*b^2-1/2/a/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b*B

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^3/(c*x^4+b*x^2+a),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 = 4.85, size = 3729, normalized size = 33.29 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2)/(x^3*(a + b*x^2 + c*x^4)),x)

[Out]

- A/(2*a*x^2) - (log(x)*(A*b - B*a))/a^2 - (log(((A^3*c^5*x^2)/a^3 - (((((4*b*c^2*(A*a*c - A*b^2 + B*a*b))/a -
(2*c^3*x^2*(A*b^2 + 10*A*a*c - 5*B*a*b))/a + (b*c^2*(a*b + 3*b^2*x^2 - 10*a*c*x^2)*(B*a - A*b + a^2*(-(2*A*a*
c - A*b^2 + B*a*b)^2/(a^4*(4*a*c - b^2)))^(1/2)))/a^2)*(B*a - A*b + a^2*(-(2*A*a*c - A*b^2 + B*a*b)^2/(a^4*(4*
a*c - b^2)))^(1/2)))/(4*a^2) + (A*c^3*(A*a*c - 4*A*b^2 + 4*B*a*b))/a^2 - (A*c^4*x^2*(6*A*b - 5*B*a))/a^2)*(B*a
- A*b + a^2*(-(2*A*a*c - A*b^2 + B*a*b)^2/(a^4*(4*a*c - b^2)))^(1/2)))/(4*a^2) + (A^2*c^4*(A*b - B*a))/a^3)*(
(((((2*c^3*x^2*(A*b^2 + 10*A*a*c - 5*B*a*b))/a - (4*b*c^2*(A*a*c - A*b^2 + B*a*b))/a + (b*c^2*(a*b + 3*b^2*x^2
- 10*a*c*x^2)*(A*b - B*a + a^2*(-(2*A*a*c - A*b^2 + B*a*b)^2/(a^4*(4*a*c - b^2)))^(1/2)))/a^2)*(A*b - B*a + a
^2*(-(2*A*a*c - A*b^2 + B*a*b)^2/(a^4*(4*a*c - b^2)))^(1/2)))/(4*a^2) + (A*c^3*(A*a*c - 4*A*b^2 + 4*B*a*b))/a^
2 - (A*c^4*x^2*(6*A*b - 5*B*a))/a^2)*(A*b - B*a + a^2*(-(2*A*a*c - A*b^2 + B*a*b)^2/(a^4*(4*a*c - b^2)))^(1/2)
))/(4*a^2) + (A^3*c^5*x^2)/a^3 + (A^2*c^4*(A*b - B*a))/a^3))*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c))/(2
*(16*a^3*c - 4*a^2*b^2)) - (atan((16*a^6*x^2*((((((5*A*B*a^2*c^4 - 6*A^2*a*b*c^4)/a^3 - (((20*A*a^3*c^4 - 10*B
*a^3*b*c^3 + 2*A*a^2*b^2*c^3)/a^3 + ((40*a^4*b*c^3 - 12*a^3*b^3*c^2)*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*
b*c))/(2*a^3*(16*a^3*c - 4*a^2*b^2)))*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c))/(2*(16*a^3*c - 4*a^2*b^2)
))*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c))/(2*(16*a^3*c - 4*a^2*b^2)) - (A^3*c^5)/a^3 + (((((20*A*a^3*c
^4 - 10*B*a^3*b*c^3 + 2*A*a^2*b^2*c^3)/a^3 + ((40*a^4*b*c^3 - 12*a^3*b^3*c^2)*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c
- 8*A*a*b*c))/(2*a^3*(16*a^3*c - 4*a^2*b^2)))*(2*A*a*c - A*b^2 + B*a*b))/(4*a^2*(4*a*c - b^2)^(1/2)) + ((40*a
^4*b*c^3 - 12*a^3*b^3*c^2)*(2*A*a*c - A*b^2 + B*a*b)*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c))/(8*a^5*(4*
a*c - b^2)^(1/2)*(16*a^3*c - 4*a^2*b^2)))*(2*A*a*c - A*b^2 + B*a*b))/(4*a^2*(4*a*c - b^2)^(1/2)) + ((40*a^4*b*
c^3 - 12*a^3*b^3*c^2)*(2*A*a*c - A*b^2 + B*a*b)^2*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c))/(32*a^7*(4*a*
c - b^2)*(16*a^3*c - 4*a^2*b^2)))*(3*A*b^4 + A*a^2*c^2 - 3*B*a*b^3 - 9*A*a*b^2*c + 8*B*a^2*b*c))/(8*a^3*c^2*(2
5*B^2*a^3*c - 6*A^2*b^4 + A^2*a^2*c^2 - 6*B^2*a^2*b^2 + 12*A*B*a*b^3 + 24*A^2*a*b^2*c - 49*A*B*a^2*b*c)) + (((
((5*A*B*a^2*c^4 - 6*A^2*a*b*c^4)/a^3 - (((20*A*a^3*c^4 - 10*B*a^3*b*c^3 + 2*A*a^2*b^2*c^3)/a^3 + ((40*a^4*b*c^
3 - 12*a^3*b^3*c^2)*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c))/(2*a^3*(16*a^3*c - 4*a^2*b^2)))*(2*A*b^3 -
2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c))/(2*(16*a^3*c - 4*a^2*b^2)))*(2*A*a*c - A*b^2 + B*a*b))/(4*a^2*(4*a*c - b^2
)^(1/2)) - (((((20*A*a^3*c^4 - 10*B*a^3*b*c^3 + 2*A*a^2*b^2*c^3)/a^3 + ((40*a^4*b*c^3 - 12*a^3*b^3*c^2)*(2*A*b
^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c))/(2*a^3*(16*a^3*c - 4*a^2*b^2)))*(2*A*a*c - A*b^2 + B*a*b))/(4*a^2*(4*
a*c - b^2)^(1/2)) + ((40*a^4*b*c^3 - 12*a^3*b^3*c^2)*(2*A*a*c - A*b^2 + B*a*b)*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*
c - 8*A*a*b*c))/(8*a^5*(4*a*c - b^2)^(1/2)*(16*a^3*c - 4*a^2*b^2)))*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b
*c))/(2*(16*a^3*c - 4*a^2*b^2)) + ((40*a^4*b*c^3 - 12*a^3*b^3*c^2)*(2*A*a*c - A*b^2 + B*a*b)^3)/(64*a^9*(4*a*c
- b^2)^(3/2)))*(6*A*b^5 - 20*B*a^3*c^2 - 6*B*a*b^4 - 30*A*a*b^3*c + 26*A*a^2*b*c^2 + 28*B*a^2*b^2*c))/(16*a^3
*c^2*(4*a*c - b^2)^(1/2)*(25*B^2*a^3*c - 6*A^2*b^4 + A^2*a^2*c^2 - 6*B^2*a^2*b^2 + 12*A*B*a*b^3 + 24*A^2*a*b^2
*c - 49*A*B*a^2*b*c)))*(4*a*c - b^2)^(3/2))/(4*A^2*a^2*c^4 + A^2*b^4*c^2 + B^2*a^2*b^2*c^2 - 4*A^2*a*b^2*c^3 -
2*A*B*a*b^3*c^2 + 4*A*B*a^2*b*c^3) + (a^3*(4*a*c - b^2)*((((((4*A*a^3*b*c^3 - 4*A*a^2*b^3*c^2 + 4*B*a^3*b^2*c
^2)/a^3 + (2*a*b^2*c^2*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c))/(16*a^3*c - 4*a^2*b^2))*(2*A*a*c - A*b^2
+ B*a*b))/(4*a^2*(4*a*c - b^2)^(1/2)) + (b^2*c^2*(2*A*a*c - A*b^2 + B*a*b)*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c -
8*A*a*b*c))/(2*a*(4*a*c - b^2)^(1/2)*(16*a^3*c - 4*a^2*b^2)))*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c))/
(2*(16*a^3*c - 4*a^2*b^2)) + (((A^2*a^2*c^4 - 4*A^2*a*b^2*c^3 + 4*A*B*a^2*b*c^3)/a^3 + (((4*A*a^3*b*c^3 - 4*A*
a^2*b^3*c^2 + 4*B*a^3*b^2*c^2)/a^3 + (2*a*b^2*c^2*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c))/(16*a^3*c - 4
*a^2*b^2))*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c))/(2*(16*a^3*c - 4*a^2*b^2)))*(2*A*a*c - A*b^2 + B*a*b
))/(4*a^2*(4*a*c - b^2)^(1/2)) - (b^2*c^2*(2*A*a*c - A*b^2 + B*a*b)^3)/(16*a^5*(4*a*c - b^2)^(3/2)))*(6*A*b^5
- 20*B*a^3*c^2 - 6*B*a*b^4 - 30*A*a*b^3*c + 26*A*a^2*b*c^2 + 28*B*a^2*b^2*c))/(c^2*(4*A^2*a^2*c^4 + A^2*b^4*c^
2 + B^2*a^2*b^2*c^2 - 4*A^2*a*b^2*c^3 - 2*A*B*a*b^3*c^2 + 4*A*B*a^2*b*c^3)*(25*B^2*a^3*c - 6*A^2*b^4 + A^2*a^2
*c^2 - 6*B^2*a^2*b^2 + 12*A*B*a*b^3 + 24*A^2*a*b^2*c - 49*A*B*a^2*b*c)) - (2*a^3*(4*a*c - b^2)^(3/2)*((A^3*b*c
^4 - A^2*B*a*c^4)/a^3 - (((A^2*a^2*c^4 - 4*A^2*a*b^2*c^3 + 4*A*B*a^2*b*c^3)/a^3 + (((4*A*a^3*b*c^3 - 4*A*a^2*b
^3*c^2 + 4*B*a^3*b^2*c^2)/a^3 + (2*a*b^2*c^2*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c))/(16*a^3*c - 4*a^2*
b^2))*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c))/(2*(16*a^3*c - 4*a^2*b^2)))*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^
2*c - 8*A*a*b*c))/(2*(16*a^3*c - 4*a^2*b^2)) + (((((4*A*a^3*b*c^3 - 4*A*a^2*b^3*c^2 + 4*B*a^3*b^2*c^2)/a^3 + (
2*a*b^2*c^2*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c))/(16*a^3*c - 4*a^2*b^2))*(2*A*a*c - A*b^2 + B*a*b))/
(4*a^2*(4*a*c - b^2)^(1/2)) + (b^2*c^2*(2*A*a*c - A*b^2 + B*a*b)*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c)
)/(2*a*(4*a*c - b^2)^(1/2)*(16*a^3*c - 4*a^2*b^2)))*(2*A*a*c - A*b^2 + B*a*b))/(4*a^2*(4*a*c - b^2)^(1/2)) + (
b^2*c^2*(2*A*a*c - A*b^2 + B*a*b)^2*(2*A*b^3 - 2*B*a*b^2 + 8*B*a^2*c - 8*A*a*b*c))/(8*a^3*(4*a*c - b^2)*(16*a^
3*c - 4*a^2*b^2)))*(3*A*b^4 + A*a^2*c^2 - 3*B*a*b^3 - 9*A*a*b^2*c + 8*B*a^2*b*c))/(c^2*(4*A^2*a^2*c^4 + A^2*b^
4*c^2 + B^2*a^2*b^2*c^2 - 4*A^2*a*b^2*c^3 - 2*A*B*a*b^3*c^2 + 4*A*B*a^2*b*c^3)*(25*B^2*a^3*c - 6*A^2*b^4 + A^2
*a^2*c^2 - 6*B^2*a^2*b^2 + 12*A*B*a*b^3 + 24*A^2*a*b^2*c - 49*A*B*a^2*b*c)))*(2*A*a*c - A*b^2 + B*a*b))/(2*a^2
*(4*a*c - b^2)^(1/2))

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sympy [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((B*x**2+A)/x**3/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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