3.99 \(\int \frac{A+B x+C x^2+D x^3}{x (a+b x^2)^2} \, dx\)

Optimal. Leaf size=95 \[ -\frac{A \log \left (a+b x^2\right )}{2 a^2}+\frac{A \log (x)}{a^2}+\frac{(a D+b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{3/2}}+\frac{x (b B-a D)-a C+A b}{2 a b \left (a+b x^2\right )} \]

[Out]

(A*b - a*C + (b*B - a*D)*x)/(2*a*b*(a + b*x^2)) + ((b*B + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(3/2)
) + (A*Log[x])/a^2 - (A*Log[a + b*x^2])/(2*a^2)

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Rubi [A]  time = 0.121648, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {1805, 801, 635, 205, 260} \[ -\frac{A \log \left (a+b x^2\right )}{2 a^2}+\frac{A \log (x)}{a^2}+\frac{(a D+b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{3/2}}+\frac{x (b B-a D)-a C+A b}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b - a*C + (b*B - a*D)*x)/(2*a*b*(a + b*x^2)) + ((b*B + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(3/2)
) + (A*Log[x])/a^2 - (A*Log[a + b*x^2])/(2*a^2)

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{A+B x+C x^2+D x^3}{x \left (a+b x^2\right )^2} \, dx &=\frac{A b-a C+(b B-a D) x}{2 a b \left (a+b x^2\right )}-\frac{\int \frac{-2 A-\frac{(b B+a D) x}{b}}{x \left (a+b x^2\right )} \, dx}{2 a}\\ &=\frac{A b-a C+(b B-a D) x}{2 a b \left (a+b x^2\right )}-\frac{\int \left (-\frac{2 A}{a x}+\frac{-a b B-a^2 D+2 A b^2 x}{a b \left (a+b x^2\right )}\right ) \, dx}{2 a}\\ &=\frac{A b-a C+(b B-a D) x}{2 a b \left (a+b x^2\right )}+\frac{A \log (x)}{a^2}-\frac{\int \frac{-a b B-a^2 D+2 A b^2 x}{a+b x^2} \, dx}{2 a^2 b}\\ &=\frac{A b-a C+(b B-a D) x}{2 a b \left (a+b x^2\right )}+\frac{A \log (x)}{a^2}-\frac{(A b) \int \frac{x}{a+b x^2} \, dx}{a^2}+\frac{(b B+a D) \int \frac{1}{a+b x^2} \, dx}{2 a b}\\ &=\frac{A b-a C+(b B-a D) x}{2 a b \left (a+b x^2\right )}+\frac{(b B+a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{3/2}}+\frac{A \log (x)}{a^2}-\frac{A \log \left (a+b x^2\right )}{2 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0706707, size = 85, normalized size = 0.89 \[ \frac{\frac{a (-a (C+D x)+A b+b B x)}{b \left (a+b x^2\right )}-A \log \left (a+b x^2\right )+\frac{\sqrt{a} (a D+b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}}+2 A \log (x)}{2 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(A*b + b*B*x - a*(C + D*x)))/(b*(a + b*x^2)) + (Sqrt[a]*(b*B + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2) +
2*A*Log[x] - A*Log[a + b*x^2])/(2*a^2)

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Maple [A]  time = 0.01, size = 125, normalized size = 1.3 \begin{align*}{\frac{A\ln \left ( x \right ) }{{a}^{2}}}+{\frac{Bx}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{xD}{ \left ( 2\,b{x}^{2}+2\,a \right ) b}}+{\frac{A}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{C}{ \left ( 2\,b{x}^{2}+2\,a \right ) b}}-{\frac{A\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2}}}+{\frac{B}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{D}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((D*x^3+C*x^2+B*x+A)/x/(b*x^2+a)^2,x)

[Out]

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

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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((D*x^3+C*x^2+B*x+A)/x/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [B]  time = 8.1076, size = 797, normalized size = 8.39 \begin{align*} \frac{A \log{\left (x \right )}}{a^{2}} + \left (- \frac{A}{2 a^{2}} - \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right ) \log{\left (x + \frac{48 A^{3} b^{4} + 48 A^{2} a^{2} b^{4} \left (- \frac{A}{2 a^{2}} - \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right ) - 4 A B^{2} a b^{3} - 8 A B D a^{2} b^{2} - 4 A D^{2} a^{3} b - 96 A a^{4} b^{4} \left (- \frac{A}{2 a^{2}} - \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right )^{2} + 4 B^{2} a^{3} b^{3} \left (- \frac{A}{2 a^{2}} - \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right ) + 8 B D a^{4} b^{2} \left (- \frac{A}{2 a^{2}} - \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right ) + 4 D^{2} a^{5} b \left (- \frac{A}{2 a^{2}} - \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right )}{36 A^{2} B b^{4} + 36 A^{2} D a b^{3} + B^{3} a b^{3} + 3 B^{2} D a^{2} b^{2} + 3 B D^{2} a^{3} b + D^{3} a^{4}} \right )} + \left (- \frac{A}{2 a^{2}} + \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right ) \log{\left (x + \frac{48 A^{3} b^{4} + 48 A^{2} a^{2} b^{4} \left (- \frac{A}{2 a^{2}} + \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right ) - 4 A B^{2} a b^{3} - 8 A B D a^{2} b^{2} - 4 A D^{2} a^{3} b - 96 A a^{4} b^{4} \left (- \frac{A}{2 a^{2}} + \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right )^{2} + 4 B^{2} a^{3} b^{3} \left (- \frac{A}{2 a^{2}} + \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right ) + 8 B D a^{4} b^{2} \left (- \frac{A}{2 a^{2}} + \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right ) + 4 D^{2} a^{5} b \left (- \frac{A}{2 a^{2}} + \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right )}{36 A^{2} B b^{4} + 36 A^{2} D a b^{3} + B^{3} a b^{3} + 3 B^{2} D a^{2} b^{2} + 3 B D^{2} a^{3} b + D^{3} a^{4}} \right )} - \frac{- A b + C a + x \left (- B b + D a\right )}{2 a^{2} b + 2 a b^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x**3+C*x**2+B*x+A)/x/(b*x**2+a)**2,x)

[Out]

A*log(x)/a**2 + (-A/(2*a**2) - sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3))*log(x + (48*A**3*b**4 + 48*A**2*a**
2*b**4*(-A/(2*a**2) - sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3)) - 4*A*B**2*a*b**3 - 8*A*B*D*a**2*b**2 - 4*A*
D**2*a**3*b - 96*A*a**4*b**4*(-A/(2*a**2) - sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3))**2 + 4*B**2*a**3*b**3*
(-A/(2*a**2) - sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3)) + 8*B*D*a**4*b**2*(-A/(2*a**2) - sqrt(-a**5*b**3)*(
B*b + D*a)/(4*a**4*b**3)) + 4*D**2*a**5*b*(-A/(2*a**2) - sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3)))/(36*A**2
*B*b**4 + 36*A**2*D*a*b**3 + B**3*a*b**3 + 3*B**2*D*a**2*b**2 + 3*B*D**2*a**3*b + D**3*a**4)) + (-A/(2*a**2) +
sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3))*log(x + (48*A**3*b**4 + 48*A**2*a**2*b**4*(-A/(2*a**2) + sqrt(-a*
*5*b**3)*(B*b + D*a)/(4*a**4*b**3)) - 4*A*B**2*a*b**3 - 8*A*B*D*a**2*b**2 - 4*A*D**2*a**3*b - 96*A*a**4*b**4*(
-A/(2*a**2) + sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3))**2 + 4*B**2*a**3*b**3*(-A/(2*a**2) + sqrt(-a**5*b**3
)*(B*b + D*a)/(4*a**4*b**3)) + 8*B*D*a**4*b**2*(-A/(2*a**2) + sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3)) + 4*
D**2*a**5*b*(-A/(2*a**2) + sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3)))/(36*A**2*B*b**4 + 36*A**2*D*a*b**3 + B
**3*a*b**3 + 3*B**2*D*a**2*b**2 + 3*B*D**2*a**3*b + D**3*a**4)) - (-A*b + C*a + x*(-B*b + D*a))/(2*a**2*b + 2*
a*b**2*x**2)

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Giac [A]  time = 1.20465, size = 126, normalized size = 1.33 \begin{align*} -\frac{A \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac{A \log \left ({\left | x \right |}\right )}{a^{2}} + \frac{{\left (D a + B b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a b} - \frac{C a^{2} - A a b +{\left (D a^{2} - B a b\right )} x}{2 \,{\left (b x^{2} + a\right )} a^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*A*log(b*x^2 + a)/a^2 + A*log(abs(x))/a^2 + 1/2*(D*a + B*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b) - 1/2*(C
*a^2 - A*a*b + (D*a^2 - B*a*b)*x)/((b*x^2 + a)*a^2*b)