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

Optimal. Leaf size=110 \[ -\frac{(3 A b-a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{A}{a^2 x}-\frac{B \log \left (a+b x^2\right )}{2 a^2}+\frac{B \log (x)}{a^2}+\frac{-b x \left (\frac{A b}{a}-C\right )-a D+b B}{2 a b \left (a+b x^2\right )} \]

[Out]

-(A/(a^2*x)) + (b*B - a*D - b*((A*b)/a - C)*x)/(2*a*b*(a + b*x^2)) - ((3*A*b - a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]
])/(2*a^(5/2)*Sqrt[b]) + (B*Log[x])/a^2 - (B*Log[a + b*x^2])/(2*a^2)

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Rubi [A]  time = 0.141581, antiderivative size = 110, 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, 1802, 635, 205, 260} \[ -\frac{(3 A b-a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{A}{a^2 x}-\frac{B \log \left (a+b x^2\right )}{2 a^2}+\frac{B \log (x)}{a^2}+\frac{-b x \left (\frac{A b}{a}-C\right )-a D+b B}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A/(a^2*x)) + (b*B - a*D - b*((A*b)/a - C)*x)/(2*a*b*(a + b*x^2)) - ((3*A*b - a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]
])/(2*a^(5/2)*Sqrt[b]) + (B*Log[x])/a^2 - (B*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 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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^2 \left (a+b x^2\right )^2} \, dx &=\frac{b B-a D-b \left (\frac{A b}{a}-C\right ) x}{2 a b \left (a+b x^2\right )}-\frac{\int \frac{-2 A-2 B x+\left (\frac{A b}{a}-C\right ) x^2}{x^2 \left (a+b x^2\right )} \, dx}{2 a}\\ &=\frac{b B-a D-b \left (\frac{A b}{a}-C\right ) x}{2 a b \left (a+b x^2\right )}-\frac{\int \left (-\frac{2 A}{a x^2}-\frac{2 B}{a x}+\frac{3 A b-a C+2 b B x}{a \left (a+b x^2\right )}\right ) \, dx}{2 a}\\ &=-\frac{A}{a^2 x}+\frac{b B-a D-b \left (\frac{A b}{a}-C\right ) x}{2 a b \left (a+b x^2\right )}+\frac{B \log (x)}{a^2}-\frac{\int \frac{3 A b-a C+2 b B x}{a+b x^2} \, dx}{2 a^2}\\ &=-\frac{A}{a^2 x}+\frac{b B-a D-b \left (\frac{A b}{a}-C\right ) x}{2 a b \left (a+b x^2\right )}+\frac{B \log (x)}{a^2}-\frac{(b B) \int \frac{x}{a+b x^2} \, dx}{a^2}-\frac{(3 A b-a C) \int \frac{1}{a+b x^2} \, dx}{2 a^2}\\ &=-\frac{A}{a^2 x}+\frac{b B-a D-b \left (\frac{A b}{a}-C\right ) x}{2 a b \left (a+b x^2\right )}-\frac{(3 A b-a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}+\frac{B \log (x)}{a^2}-\frac{B \log \left (a+b x^2\right )}{2 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0704589, size = 110, normalized size = 1. \[ \frac{a^2 (-D)+a b B+a b C x-A b^2 x}{2 a^2 b \left (a+b x^2\right )}+\frac{(a C-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{A}{a^2 x}-\frac{B \log \left (a+b x^2\right )}{2 a^2}+\frac{B \log (x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A/(a^2*x)) + (a*b*B - a^2*D - A*b^2*x + a*b*C*x)/(2*a^2*b*(a + b*x^2)) + ((-3*A*b + a*C)*ArcTan[(Sqrt[b]*x)/
Sqrt[a]])/(2*a^(5/2)*Sqrt[b]) + (B*Log[x])/a^2 - (B*Log[a + b*x^2])/(2*a^2)

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Maple [A]  time = 0.012, size = 136, normalized size = 1.2 \begin{align*} -{\frac{A}{{a}^{2}x}}+{\frac{B\ln \left ( x \right ) }{{a}^{2}}}-{\frac{Abx}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{Cx}{2\,a \left ( b{x}^{2}+a \right ) }}+{\frac{B}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{D}{ \left ( 2\,b{x}^{2}+2\,a \right ) b}}-{\frac{B\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2}}}-{\frac{3\,Ab}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{C}{2\,a}\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^2/(b*x^2+a)^2,x)

[Out]

-A/a^2/x+B*ln(x)/a^2-1/2/a^2*x/(b*x^2+a)*A*b+1/2/a/(b*x^2+a)*C*x+1/2/a/(b*x^2+a)*B-1/2/(b*x^2+a)/b*D-1/2*B*ln(
b*x^2+a)/a^2-3/2/a^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*A*b+1/2/a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*C

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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^2/(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^2/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [B]  time = 8.45161, size = 782, normalized size = 7.11 \begin{align*} \frac{B \log{\left (x \right )}}{a^{2}} + \left (- \frac{B}{2 a^{2}} - \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right ) \log{\left (x + \frac{- 36 A^{2} B a b^{2} + 36 A^{2} a^{3} b^{2} \left (- \frac{B}{2 a^{2}} - \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right ) + 24 A B C a^{2} b - 24 A C a^{4} b \left (- \frac{B}{2 a^{2}} - \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right ) + 48 B^{3} a^{2} b + 48 B^{2} a^{4} b \left (- \frac{B}{2 a^{2}} - \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right ) - 4 B C^{2} a^{3} - 96 B a^{6} b \left (- \frac{B}{2 a^{2}} - \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right )^{2} + 4 C^{2} a^{5} \left (- \frac{B}{2 a^{2}} - \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right )}{- 27 A^{3} b^{3} + 27 A^{2} C a b^{2} - 108 A B^{2} a b^{2} - 9 A C^{2} a^{2} b + 36 B^{2} C a^{2} b + C^{3} a^{3}} \right )} + \left (- \frac{B}{2 a^{2}} + \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right ) \log{\left (x + \frac{- 36 A^{2} B a b^{2} + 36 A^{2} a^{3} b^{2} \left (- \frac{B}{2 a^{2}} + \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right ) + 24 A B C a^{2} b - 24 A C a^{4} b \left (- \frac{B}{2 a^{2}} + \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right ) + 48 B^{3} a^{2} b + 48 B^{2} a^{4} b \left (- \frac{B}{2 a^{2}} + \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right ) - 4 B C^{2} a^{3} - 96 B a^{6} b \left (- \frac{B}{2 a^{2}} + \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right )^{2} + 4 C^{2} a^{5} \left (- \frac{B}{2 a^{2}} + \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right )}{- 27 A^{3} b^{3} + 27 A^{2} C a b^{2} - 108 A B^{2} a b^{2} - 9 A C^{2} a^{2} b + 36 B^{2} C a^{2} b + C^{3} a^{3}} \right )} + \frac{- 2 A a b + x^{2} \left (- 3 A b^{2} + C a b\right ) + x \left (B a b - D a^{2}\right )}{2 a^{3} b x + 2 a^{2} b^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*log(x)/a**2 + (-B/(2*a**2) - sqrt(-a**5*b)*(-3*A*b + C*a)/(4*a**5*b))*log(x + (-36*A**2*B*a*b**2 + 36*A**2*a
**3*b**2*(-B/(2*a**2) - sqrt(-a**5*b)*(-3*A*b + C*a)/(4*a**5*b)) + 24*A*B*C*a**2*b - 24*A*C*a**4*b*(-B/(2*a**2
) - sqrt(-a**5*b)*(-3*A*b + C*a)/(4*a**5*b)) + 48*B**3*a**2*b + 48*B**2*a**4*b*(-B/(2*a**2) - sqrt(-a**5*b)*(-
3*A*b + C*a)/(4*a**5*b)) - 4*B*C**2*a**3 - 96*B*a**6*b*(-B/(2*a**2) - sqrt(-a**5*b)*(-3*A*b + C*a)/(4*a**5*b))
**2 + 4*C**2*a**5*(-B/(2*a**2) - sqrt(-a**5*b)*(-3*A*b + C*a)/(4*a**5*b)))/(-27*A**3*b**3 + 27*A**2*C*a*b**2 -
108*A*B**2*a*b**2 - 9*A*C**2*a**2*b + 36*B**2*C*a**2*b + C**3*a**3)) + (-B/(2*a**2) + sqrt(-a**5*b)*(-3*A*b +
C*a)/(4*a**5*b))*log(x + (-36*A**2*B*a*b**2 + 36*A**2*a**3*b**2*(-B/(2*a**2) + sqrt(-a**5*b)*(-3*A*b + C*a)/(
4*a**5*b)) + 24*A*B*C*a**2*b - 24*A*C*a**4*b*(-B/(2*a**2) + sqrt(-a**5*b)*(-3*A*b + C*a)/(4*a**5*b)) + 48*B**3
*a**2*b + 48*B**2*a**4*b*(-B/(2*a**2) + sqrt(-a**5*b)*(-3*A*b + C*a)/(4*a**5*b)) - 4*B*C**2*a**3 - 96*B*a**6*b
*(-B/(2*a**2) + sqrt(-a**5*b)*(-3*A*b + C*a)/(4*a**5*b))**2 + 4*C**2*a**5*(-B/(2*a**2) + sqrt(-a**5*b)*(-3*A*b
+ C*a)/(4*a**5*b)))/(-27*A**3*b**3 + 27*A**2*C*a*b**2 - 108*A*B**2*a*b**2 - 9*A*C**2*a**2*b + 36*B**2*C*a**2*
b + C**3*a**3)) + (-2*A*a*b + x**2*(-3*A*b**2 + C*a*b) + x*(B*a*b - D*a**2))/(2*a**3*b*x + 2*a**2*b**2*x**3)

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Giac [A]  time = 1.18563, size = 139, normalized size = 1.26 \begin{align*} -\frac{B \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac{B \log \left ({\left | x \right |}\right )}{a^{2}} + \frac{{\left (C a - 3 \, A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2}} + \frac{C a b x^{2} - 3 \, A b^{2} x^{2} - D a^{2} x + B a b x - 2 \, A a b}{2 \,{\left (b x^{3} + a x\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^2/(b*x^2+a)^2,x, algorithm="giac")

[Out]

-1/2*B*log(b*x^2 + a)/a^2 + B*log(abs(x))/a^2 + 1/2*(C*a - 3*A*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2) + 1/2*
(C*a*b*x^2 - 3*A*b^2*x^2 - D*a^2*x + B*a*b*x - 2*A*a*b)/((b*x^3 + a*x)*a^2*b)