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

Optimal. Leaf size=72 \[ -\frac{(A b-a C) \log \left (a+b x^2\right )}{2 a b}+\frac{A \log (x)}{a}+\frac{(b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{D x}{b} \]

[Out]

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

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Rubi [A]  time = 0.0977419, antiderivative size = 72, 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} \[ -\frac{(A b-a C) \log \left (a+b x^2\right )}{2 a b}+\frac{A \log (x)}{a}+\frac{(b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{D x}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0511625, size = 73, normalized size = 1.01 \[ \frac{(a C-A b) \log \left (a+b x^2\right )}{2 a b}+\frac{A \log (x)}{a}-\frac{(a D-b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{D x}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 80, normalized size = 1.1 \begin{align*}{\frac{Dx}{b}}+{\frac{A\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( b{x}^{2}+a \right ) A}{2\,a}}+{\frac{\ln \left ( b{x}^{2}+a \right ) C}{2\,b}}+{B\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{aD}{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),x)

[Out]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [B]  time = 25.4488, size = 1268, normalized size = 17.61 \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/(b*x**2+a),x)

[Out]

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

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Giac [A]  time = 1.12289, size = 89, normalized size = 1.24 \begin{align*} \frac{D x}{b} + \frac{A \log \left ({\left | x \right |}\right )}{a} - \frac{{\left (D a - B b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b} + \frac{{\left (C a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, a 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),x, algorithm="giac")

[Out]

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