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

Optimal. Leaf size=76 \[ -\frac{(A b-a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{A}{a x}-\frac{(b B-a D) \log \left (a+b x^2\right )}{2 a b}+\frac{B \log (x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.0984595, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1802, 635, 205, 260} \[ -\frac{(A b-a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{A}{a x}-\frac{(b B-a D) \log \left (a+b x^2\right )}{2 a b}+\frac{B \log (x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0459417, size = 75, normalized size = 0.99 \[ \frac{(a C-A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{A}{a x}+\frac{(a D-b B) \log \left (a+b x^2\right )}{2 a b}+\frac{B \log (x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 83, normalized size = 1.1 \begin{align*} -{\frac{A}{ax}}+{\frac{B\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( b{x}^{2}+a \right ) B}{2\,a}}+{\frac{\ln \left ( b{x}^{2}+a \right ) D}{2\,b}}-{\frac{Ab}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{C\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),x)

[Out]

-A/a/x+B*ln(x)/a-1/2/a*ln(b*x^2+a)*B+1/2/b*ln(b*x^2+a)*D-1/a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*A*b+1/(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),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),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [B]  time = 23.3426, size = 1258, normalized size = 16.55 \begin{align*} \text{result too large to display} \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),x)

[Out]

-A/(a*x) + B*log(x)/a + ((-B*b + D*a)/(2*a*b) - sqrt(-a**3*b**3)*(-A*b + C*a)/(2*a**3*b**2))*log(x + (-2*A**2*
B*a*b**3 + 2*A**2*a**2*b**3*((-B*b + D*a)/(2*a*b) - sqrt(-a**3*b**3)*(-A*b + C*a)/(2*a**3*b**2)) + 4*A*B*C*a**
2*b**2 - 4*A*C*a**3*b**2*((-B*b + D*a)/(2*a*b) - sqrt(-a**3*b**3)*(-A*b + C*a)/(2*a**3*b**2)) + 6*B**3*a**2*b*
*2 - 8*B**2*D*a**3*b + 6*B**2*a**3*b**2*((-B*b + D*a)/(2*a*b) - sqrt(-a**3*b**3)*(-A*b + C*a)/(2*a**3*b**2)) -
 2*B*C**2*a**3*b + 2*B*D**2*a**4 + 4*B*D*a**4*b*((-B*b + D*a)/(2*a*b) - sqrt(-a**3*b**3)*(-A*b + C*a)/(2*a**3*
b**2)) - 12*B*a**4*b**2*((-B*b + D*a)/(2*a*b) - sqrt(-a**3*b**3)*(-A*b + C*a)/(2*a**3*b**2))**2 + 2*C**2*a**4*
b*((-B*b + D*a)/(2*a*b) - sqrt(-a**3*b**3)*(-A*b + C*a)/(2*a**3*b**2)) - 2*D**2*a**5*((-B*b + D*a)/(2*a*b) - s
qrt(-a**3*b**3)*(-A*b + C*a)/(2*a**3*b**2)) + 4*D*a**5*b*((-B*b + D*a)/(2*a*b) - sqrt(-a**3*b**3)*(-A*b + C*a)
/(2*a**3*b**2))**2)/(-A**3*b**4 + 3*A**2*C*a*b**3 - 9*A*B**2*a*b**3 + 6*A*B*D*a**2*b**2 - 3*A*C**2*a**2*b**2 -
 A*D**2*a**3*b + 9*B**2*C*a**2*b**2 - 6*B*C*D*a**3*b + C**3*a**3*b + C*D**2*a**4)) + ((-B*b + D*a)/(2*a*b) + s
qrt(-a**3*b**3)*(-A*b + C*a)/(2*a**3*b**2))*log(x + (-2*A**2*B*a*b**3 + 2*A**2*a**2*b**3*((-B*b + D*a)/(2*a*b)
 + sqrt(-a**3*b**3)*(-A*b + C*a)/(2*a**3*b**2)) + 4*A*B*C*a**2*b**2 - 4*A*C*a**3*b**2*((-B*b + D*a)/(2*a*b) +
sqrt(-a**3*b**3)*(-A*b + C*a)/(2*a**3*b**2)) + 6*B**3*a**2*b**2 - 8*B**2*D*a**3*b + 6*B**2*a**3*b**2*((-B*b +
D*a)/(2*a*b) + sqrt(-a**3*b**3)*(-A*b + C*a)/(2*a**3*b**2)) - 2*B*C**2*a**3*b + 2*B*D**2*a**4 + 4*B*D*a**4*b*(
(-B*b + D*a)/(2*a*b) + sqrt(-a**3*b**3)*(-A*b + C*a)/(2*a**3*b**2)) - 12*B*a**4*b**2*((-B*b + D*a)/(2*a*b) + s
qrt(-a**3*b**3)*(-A*b + C*a)/(2*a**3*b**2))**2 + 2*C**2*a**4*b*((-B*b + D*a)/(2*a*b) + sqrt(-a**3*b**3)*(-A*b
+ C*a)/(2*a**3*b**2)) - 2*D**2*a**5*((-B*b + D*a)/(2*a*b) + sqrt(-a**3*b**3)*(-A*b + C*a)/(2*a**3*b**2)) + 4*D
*a**5*b*((-B*b + D*a)/(2*a*b) + sqrt(-a**3*b**3)*(-A*b + C*a)/(2*a**3*b**2))**2)/(-A**3*b**4 + 3*A**2*C*a*b**3
 - 9*A*B**2*a*b**3 + 6*A*B*D*a**2*b**2 - 3*A*C**2*a**2*b**2 - A*D**2*a**3*b + 9*B**2*C*a**2*b**2 - 6*B*C*D*a**
3*b + C**3*a**3*b + C*D**2*a**4))

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Giac [A]  time = 1.15784, size = 92, normalized size = 1.21 \begin{align*} \frac{B \log \left ({\left | x \right |}\right )}{a} + \frac{{\left (C a - A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a} + \frac{{\left (D a - B b\right )} \log \left (b x^{2} + a\right )}{2 \, a b} - \frac{A}{a x} \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),x, algorithm="giac")

[Out]

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

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