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]

-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)

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Rubi [A]  time = 0.245491, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, 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 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]

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 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 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 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 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]

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*}

Mathematica [A]  time = 0.226514, 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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Maple [A]  time = 0.01, size = 191, normalized size = 1.7 \begin{align*}{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) Ab}{4\,{a}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) B}{4\,a}}-{\frac{Ac}{a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{A{b}^{2}}{2\,{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{bB}{2\,a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{A}{2\,a{x}^{2}}}-{\frac{\ln \left ( x \right ) Ab}{{a}^{2}}}+{\frac{\ln \left ( x \right ) B}{a}} \end{align*}

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]

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

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

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

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Fricas [A]  time = 2.70211, size = 834, normalized size = 7.45 \begin{align*} \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 \left (x\right )}{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 \left (x\right )}{4 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2}}\right ] \end{align*}

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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Sympy [B]  time = 170.625, size = 495, normalized size = 4.42 \begin{align*} - \frac{A}{2 a x^{2}} + \left (- \frac{- A b + B a}{4 a^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 A a c - A b^{2} + B a b\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{3 A a b c - A b^{3} - 2 B a^{2} c + B a b^{2} - 8 a^{3} c \left (- \frac{- A b + B a}{4 a^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 A a c - A b^{2} + B a b\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 2 a^{2} b^{2} \left (- \frac{- A b + B a}{4 a^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 A a c - A b^{2} + B a b\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right )}{2 A a c^{2} - A b^{2} c + B a b c} \right )} + \left (- \frac{- A b + B a}{4 a^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 A a c - A b^{2} + B a b\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{3 A a b c - A b^{3} - 2 B a^{2} c + B a b^{2} - 8 a^{3} c \left (- \frac{- A b + B a}{4 a^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 A a c - A b^{2} + B a b\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 2 a^{2} b^{2} \left (- \frac{- A b + B a}{4 a^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 A a c - A b^{2} + B a b\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right )}{2 A a c^{2} - A b^{2} c + B a b c} \right )} + \frac{\left (- A b + B a\right ) \log{\left (x \right )}}{a^{2}} \end{align*}

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]

-A/(2*a*x**2) + (-(-A*b + B*a)/(4*a**2) - sqrt(-4*a*c + b**2)*(2*A*a*c - A*b**2 + B*a*b)/(4*a**2*(4*a*c - b**2
)))*log(x**2 + (3*A*a*b*c - A*b**3 - 2*B*a**2*c + B*a*b**2 - 8*a**3*c*(-(-A*b + B*a)/(4*a**2) - sqrt(-4*a*c +
b**2)*(2*A*a*c - A*b**2 + B*a*b)/(4*a**2*(4*a*c - b**2))) + 2*a**2*b**2*(-(-A*b + B*a)/(4*a**2) - sqrt(-4*a*c
+ b**2)*(2*A*a*c - A*b**2 + B*a*b)/(4*a**2*(4*a*c - b**2))))/(2*A*a*c**2 - A*b**2*c + B*a*b*c)) + (-(-A*b + B*
a)/(4*a**2) + sqrt(-4*a*c + b**2)*(2*A*a*c - A*b**2 + B*a*b)/(4*a**2*(4*a*c - b**2)))*log(x**2 + (3*A*a*b*c -
A*b**3 - 2*B*a**2*c + B*a*b**2 - 8*a**3*c*(-(-A*b + B*a)/(4*a**2) + sqrt(-4*a*c + b**2)*(2*A*a*c - A*b**2 + B*
a*b)/(4*a**2*(4*a*c - b**2))) + 2*a**2*b**2*(-(-A*b + B*a)/(4*a**2) + sqrt(-4*a*c + b**2)*(2*A*a*c - A*b**2 +
B*a*b)/(4*a**2*(4*a*c - b**2))))/(2*A*a*c**2 - A*b**2*c + B*a*b*c)) + (-A*b + B*a)*log(x)/a**2

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Giac [A]  time = 1.16742, size = 167, normalized size = 1.49 \begin{align*} -\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}} \end{align*}

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)