3.2.5 \(\int \frac {A+B x^2}{x (a+b x^2+c x^4)} \, dx\) [105]

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

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 78, 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 = {1265, 814, 648, 632, 212, 642} \begin {gather*} \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} \end {gather*}

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

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 642

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 648

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 814

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 1265

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 \left (a+b x^2+c x^4\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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 {\text {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 \text {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) \text {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) \text {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*}

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Mathematica [A]
time = 0.07, size = 128, normalized size = 1.64 \begin {gather*} \frac {4 A \sqrt {b^2-4 a c} \log (x)-\left (-2 a B+A \left (b+\sqrt {b^2-4 a c}\right )\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )+\left (-2 a B+A \left (b-\sqrt {b^2-4 a c}\right )\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{4 a \sqrt {b^2-4 a c}} \end {gather*}

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 = 0.04, size = 76, normalized size = 0.97

method	result	size
default	\(-\frac {\frac {A \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2}+\frac {2 \left (\frac {A b}{2}-a B \right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 a}+\frac {A \ln \left (x \right )}{a}\)	\(76\)
risch	\(\frac {A \ln \left (x \right )}{a}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (4 a^{2} c -a \,b^{2}\right ) \textit {\_Z}^{2}+\left (4 a c A -A \,b^{2}\right ) \textit {\_Z} +A^{2} c -b B A +B^{2} a \right )}{\sum }\textit {\_R} \ln \left (\left (\left (10 a c -3 b^{2}\right ) \textit {\_R}^{2}+\left (5 A c -b B \right ) \textit {\_R} +2 B^{2}\right ) x^{2}-b \,\textit {\_R}^{2} a +\left (2 A b -a B \right ) \textit {\_R} +2 A B \right )\right )}{2}\)	\(123\)

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,method=_RETURNVERBOSE)

[Out]

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

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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((B*x^2+A)/x/(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 deta

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Fricas [A]
time = 0.37, size = 249, normalized size = 3.19 \begin {gather*} \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 {gather*}

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 [F(-1)] Timed out
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((B*x**2+A)/x/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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Giac [A]
time = 5.88, size = 78, normalized size = 1.00 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 4.48, size = 2424, normalized size = 31.08 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(A*log(x))/a - (log((A*B^2*c^2 + ((A + a*(-(A*b - 2*B*a)^2/(a^2*(4*a*c - b^2)))^(1/2))*(B^2*a*c^2 + ((A + a*(-
(A*b - 2*B*a)^2/(a^2*(4*a*c - b^2)))^(1/2))*(4*A*b^2*c^2 + 2*c^2*x^2*(5*A*b*c - 4*B*b^2 + 10*B*a*c) - 4*B*a*b*
c^2 + (b*c^2*(A + a*(-(A*b - 2*B*a)^2/(a^2*(4*a*c - b^2)))^(1/2))*(a*b + 3*b^2*x^2 - 10*a*c*x^2))/a))/(4*a) -
4*A*B*b*c^2 - B*c^2*x^2*(5*A*c + B*b)))/(4*a) + B^3*c^2*x^2)*(A*B^2*c^2 + ((A - a*(-(A*b - 2*B*a)^2/(a^2*(4*a*
c - b^2)))^(1/2))*(B^2*a*c^2 + ((A - a*(-(A*b - 2*B*a)^2/(a^2*(4*a*c - b^2)))^(1/2))*(4*A*b^2*c^2 + 2*c^2*x^2*
(5*A*b*c - 4*B*b^2 + 10*B*a*c) - 4*B*a*b*c^2 + (b*c^2*(A - a*(-(A*b - 2*B*a)^2/(a^2*(4*a*c - b^2)))^(1/2))*(a*
b + 3*b^2*x^2 - 10*a*c*x^2))/a))/(4*a) - 4*A*B*b*c^2 - B*c^2*x^2*(5*A*c + B*b)))/(4*a) + B^3*c^2*x^2))*(2*A*b^
2 - 8*A*a*c))/(2*(4*a*b^2 - 16*a^2*c)) - (atan((2*(4*a*c - b^2)^(3/2)*(3*A*b^3 - B*a*b^2 + B*a^2*c - 8*A*a*b*c
)*(A*B^2*c^2 + ((2*A*b^2 - 8*A*a*c)*(((2*A*b^2 - 8*A*a*c)*(4*A*b^2*c^2 - 4*B*a*b*c^2 + (2*a*b^2*c^2*(2*A*b^2 -
8*A*a*c))/(4*a*b^2 - 16*a^2*c)))/(2*(4*a*b^2 - 16*a^2*c)) + B^2*a*c^2 - 4*A*B*b*c^2))/(2*(4*a*b^2 - 16*a^2*c)
) - ((A*b - 2*B*a)*(((A*b - 2*B*a)*(4*A*b^2*c^2 - 4*B*a*b*c^2 + (2*a*b^2*c^2*(2*A*b^2 - 8*A*a*c))/(4*a*b^2 - 1
6*a^2*c)))/(4*a*(4*a*c - b^2)^(1/2)) + (b^2*c^2*(2*A*b^2 - 8*A*a*c)*(A*b - 2*B*a))/(2*(4*a*b^2 - 16*a^2*c)*(4*
a*c - b^2)^(1/2))))/(4*a*(4*a*c - b^2)^(1/2)) - (b^2*c^2*(2*A*b^2 - 8*A*a*c)*(A*b - 2*B*a)^2)/(8*a*(4*a*b^2 -
16*a^2*c)*(4*a*c - b^2))))/(c^2*(A^2*b^2*c^2 + 4*B^2*a^2*c^2 - 4*A*B*a*b*c^2)*(6*A^2*b^2 - B^2*a^2 - 25*A^2*a*
c + A*B*a*b)) - (16*a^3*x^2*(((3*A*b^3 - B*a*b^2 + B*a^2*c - 8*A*a*b*c)*(((2*A*b^2 - 8*A*a*c)*(B^2*b*c^2 + 5*A
*B*c^3 - ((2*A*b^2 - 8*A*a*c)*(((2*A*b^2 - 8*A*a*c)*(12*b^3*c^2 - 40*a*b*c^3))/(2*(4*a*b^2 - 16*a^2*c)) - 8*B*
b^2*c^2 + 10*A*b*c^3 + 20*B*a*c^3))/(2*(4*a*b^2 - 16*a^2*c))))/(2*(4*a*b^2 - 16*a^2*c)) - B^3*c^2 + ((A*b - 2*
B*a)*(((A*b - 2*B*a)*(((2*A*b^2 - 8*A*a*c)*(12*b^3*c^2 - 40*a*b*c^3))/(2*(4*a*b^2 - 16*a^2*c)) - 8*B*b^2*c^2 +
10*A*b*c^3 + 20*B*a*c^3))/(4*a*(4*a*c - b^2)^(1/2)) + ((2*A*b^2 - 8*A*a*c)*(12*b^3*c^2 - 40*a*b*c^3)*(A*b - 2
*B*a))/(8*a*(4*a*b^2 - 16*a^2*c)*(4*a*c - b^2)^(1/2))))/(4*a*(4*a*c - b^2)^(1/2)) + ((2*A*b^2 - 8*A*a*c)*(12*b
^3*c^2 - 40*a*b*c^3)*(A*b - 2*B*a)^2)/(32*a^2*(4*a*b^2 - 16*a^2*c)*(4*a*c - b^2))))/(8*a^3*c^2*(6*A^2*b^2 - B^
2*a^2 - 25*A^2*a*c + A*B*a*b)) + ((((12*b^3*c^2 - 40*a*b*c^3)*(A*b - 2*B*a)^3)/(64*a^3*(4*a*c - b^2)^(3/2)) -
((2*A*b^2 - 8*A*a*c)*(((A*b - 2*B*a)*(((2*A*b^2 - 8*A*a*c)*(12*b^3*c^2 - 40*a*b*c^3))/(2*(4*a*b^2 - 16*a^2*c))
- 8*B*b^2*c^2 + 10*A*b*c^3 + 20*B*a*c^3))/(4*a*(4*a*c - b^2)^(1/2)) + ((2*A*b^2 - 8*A*a*c)*(12*b^3*c^2 - 40*a
*b*c^3)*(A*b - 2*B*a))/(8*a*(4*a*b^2 - 16*a^2*c)*(4*a*c - b^2)^(1/2))))/(2*(4*a*b^2 - 16*a^2*c)) + ((A*b - 2*B
*a)*(B^2*b*c^2 + 5*A*B*c^3 - ((2*A*b^2 - 8*A*a*c)*(((2*A*b^2 - 8*A*a*c)*(12*b^3*c^2 - 40*a*b*c^3))/(2*(4*a*b^2
- 16*a^2*c)) - 8*B*b^2*c^2 + 10*A*b*c^3 + 20*B*a*c^3))/(2*(4*a*b^2 - 16*a^2*c))))/(4*a*(4*a*c - b^2)^(1/2)))*
(3*A*b^4 + 10*A*a^2*c^2 - B*a*b^3 - 14*A*a*b^2*c + 3*B*a^2*b*c))/(8*a^3*c^2*(4*a*c - b^2)^(1/2)*(6*A^2*b^2 - B
^2*a^2 - 25*A^2*a*c + A*B*a*b)))*(4*a*c - b^2)^(3/2))/(A^2*b^2*c^2 + 4*B^2*a^2*c^2 - 4*A*B*a*b*c^2) + (2*(4*a*
c - b^2)*(((2*A*b^2 - 8*A*a*c)*(((A*b - 2*B*a)*(4*A*b^2*c^2 - 4*B*a*b*c^2 + (2*a*b^2*c^2*(2*A*b^2 - 8*A*a*c))/
(4*a*b^2 - 16*a^2*c)))/(4*a*(4*a*c - b^2)^(1/2)) + (b^2*c^2*(2*A*b^2 - 8*A*a*c)*(A*b - 2*B*a))/(2*(4*a*b^2 - 1
6*a^2*c)*(4*a*c - b^2)^(1/2))))/(2*(4*a*b^2 - 16*a^2*c)) + ((A*b - 2*B*a)*(((2*A*b^2 - 8*A*a*c)*(4*A*b^2*c^2 -
4*B*a*b*c^2 + (2*a*b^2*c^2*(2*A*b^2 - 8*A*a*c))/(4*a*b^2 - 16*a^2*c)))/(2*(4*a*b^2 - 16*a^2*c)) + B^2*a*c^2 -
4*A*B*b*c^2))/(4*a*(4*a*c - b^2)^(1/2)) - (b^2*c^2*(A*b - 2*B*a)^3)/(16*a^2*(4*a*c - b^2)^(3/2)))*(3*A*b^4 +
10*A*a^2*c^2 - B*a*b^3 - 14*A*a*b^2*c + 3*B*a^2*b*c))/(c^2*(A^2*b^2*c^2 + 4*B^2*a^2*c^2 - 4*A*B*a*b*c^2)*(6*A^
2*b^2 - B^2*a^2 - 25*A^2*a*c + A*B*a*b)))*(A*b - 2*B*a))/(2*a*(4*a*c - b^2)^(1/2))

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