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

Optimal. Leaf size=78 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{b^2-4 a c}}-\frac{A \log \left (a+b x^2+c x^4\right )}{4 a}+\frac{A \log (x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.138659, antiderivative size = 78, 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} \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{b^2-4 a c}}-\frac{A \log \left (a+b x^2+c x^4\right )}{4 a}+\frac{A \log (x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.108057, size = 128, normalized size = 1.64 \[ \frac{-\left (A \left (\sqrt{b^2-4 a c}+b\right )-2 a B\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )+\left (A \left (b-\sqrt{b^2-4 a c}\right )-2 a B\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )+4 A \log (x) \sqrt{b^2-4 a c}}{4 a \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 105, normalized size = 1.4 \begin{align*} -{\frac{A\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{4\,a}}-{\frac{Ab}{2\,a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{B\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{A\ln \left ( x \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.73968, size = 567, normalized size = 7.27 \begin{align*} \left [-\frac{{\left (2 \, B a - A b\right )} \sqrt{b^{2} - 4 \, a c} \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 ) +{\left (A b^{2} - 4 \, A a c\right )} \log \left (c x^{4} + b x^{2} + a\right ) - 4 \,{\left (A b^{2} - 4 \, A a c\right )} \log \left (x\right )}{4 \,{\left (a b^{2} - 4 \, a^{2} c\right )}}, -\frac{2 \,{\left (2 \, B a - A b\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left (A b^{2} - 4 \, A a c\right )} \log \left (c x^{4} + b x^{2} + a\right ) - 4 \,{\left (A b^{2} - 4 \, A a c\right )} \log \left (x\right )}{4 \,{\left (a b^{2} - 4 \, a^{2} c\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 53.6247, size = 330, normalized size = 4.23 \begin{align*} \frac{A \log{\left (x \right )}}{a} + \left (- \frac{A}{4 a} - \frac{\left (- A b + 2 B a\right ) \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{2 A a c - A b^{2} + B a b + 8 a^{2} c \left (- \frac{A}{4 a} - \frac{\left (- A b + 2 B a\right ) \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )}\right ) - 2 a b^{2} \left (- \frac{A}{4 a} - \frac{\left (- A b + 2 B a\right ) \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )}\right )}{- A b c + 2 B a c} \right )} + \left (- \frac{A}{4 a} + \frac{\left (- A b + 2 B a\right ) \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{2 A a c - A b^{2} + B a b + 8 a^{2} c \left (- \frac{A}{4 a} + \frac{\left (- A b + 2 B a\right ) \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )}\right ) - 2 a b^{2} \left (- \frac{A}{4 a} + \frac{\left (- A b + 2 B a\right ) \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )}\right )}{- A b c + 2 B a c} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x/(c*x**4+b*x**2+a),x)

[Out]

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

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Giac [A]  time = 1.19475, size = 105, normalized size = 1.35 \begin{align*} -\frac{A \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a} + \frac{A \log \left (x^{2}\right )}{2 \, a} + \frac{{\left (2 \, B a - A b\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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