3.93 \(\int \frac{A+B x+C x^2+D x^3}{x^3 (a+b x^2)} \, dx\)
Optimal. Leaf size=92 \[ \frac{(A b-a C) \log \left (a+b x^2\right )}{2 a^2}-\frac{\log (x) (A b-a C)}{a^2}-\frac{(b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{A}{2 a x^2}-\frac{B}{a x} \]
[Out]
-A/(2*a*x^2) - B/(a*x) - ((b*B - a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]) - ((A*b - a*C)*Log[x])/a^
2 + ((A*b - a*C)*Log[a + b*x^2])/(2*a^2)
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Rubi [A] time = 0.108623, antiderivative size = 92, 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} \[ \frac{(A b-a C) \log \left (a+b x^2\right )}{2 a^2}-\frac{\log (x) (A b-a C)}{a^2}-\frac{(b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{A}{2 a x^2}-\frac{B}{a x} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x + C*x^2 + D*x^3)/(x^3*(a + b*x^2)),x]
[Out]
-A/(2*a*x^2) - B/(a*x) - ((b*B - a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]) - ((A*b - a*C)*Log[x])/a^
2 + ((A*b - a*C)*Log[a + b*x^2])/(2*a^2)
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^3 \left (a+b x^2\right )} \, dx &=\int \left (\frac{A}{a x^3}+\frac{B}{a x^2}+\frac{-A b+a C}{a^2 x}+\frac{-a (b B-a D)+b (A b-a C) x}{a^2 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{A}{2 a x^2}-\frac{B}{a x}-\frac{(A b-a C) \log (x)}{a^2}+\frac{\int \frac{-a (b B-a D)+b (A b-a C) x}{a+b x^2} \, dx}{a^2}\\ &=-\frac{A}{2 a x^2}-\frac{B}{a x}-\frac{(A b-a C) \log (x)}{a^2}+\frac{(b (A b-a C)) \int \frac{x}{a+b x^2} \, dx}{a^2}-\frac{(b B-a D) \int \frac{1}{a+b x^2} \, dx}{a}\\ &=-\frac{A}{2 a x^2}-\frac{B}{a x}-\frac{(b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{(A b-a C) \log (x)}{a^2}+\frac{(A b-a C) \log \left (a+b x^2\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0799407, size = 84, normalized size = 0.91 \[ \frac{(A b-a C) \log \left (a+b x^2\right )+2 \log (x) (a C-A b)-\frac{a A}{x^2}+\frac{2 \sqrt{a} (a D-b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{2 a B}{x}}{2 a^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x + C*x^2 + D*x^3)/(x^3*(a + b*x^2)),x]
[Out]
(-((a*A)/x^2) - (2*a*B)/x + (2*Sqrt[a]*(-(b*B) + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] + 2*(-(A*b) + a*C)*
Log[x] + (A*b - a*C)*Log[a + b*x^2])/(2*a^2)
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Maple [A] time = 0.009, size = 102, normalized size = 1.1 \begin{align*} -{\frac{A}{2\,a{x}^{2}}}-{\frac{B}{ax}}-{\frac{A\ln \left ( x \right ) b}{{a}^{2}}}+{\frac{\ln \left ( x \right ) C}{a}}+{\frac{b\ln \left ( b{x}^{2}+a \right ) A}{2\,{a}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) C}{2\,a}}-{\frac{Bb}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{D\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^3/(b*x^2+a),x)
[Out]
-1/2*A/a/x^2-B/a/x-1/a^2*ln(x)*A*b+1/a*ln(x)*C+1/2/a^2*b*ln(b*x^2+a)*A-1/2/a*ln(b*x^2+a)*C-1/a/(a*b)^(1/2)*arc
tan(b*x/(a*b)^(1/2))*B*b+1/(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^3/(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^3/(b*x^2+a),x, algorithm="fricas")
[Out]
Exception raised: UnboundLocalError
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Sympy [B] time = 24.6077, size = 1686, normalized size = 18.33 \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**3/(b*x**2+a),x)
[Out]
(-(-A*b + C*a)/(2*a**2) - sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b))*log(x + (-6*A**3*b**4 + 18*A**2*C*a*b**3 + 6*
A**2*a**2*b**3*(-(-A*b + C*a)/(2*a**2) - sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b)) + 2*A*B**2*a*b**3 - 4*A*B*D*a*
*2*b**2 - 18*A*C**2*a**2*b**2 - 12*A*C*a**3*b**2*(-(-A*b + C*a)/(2*a**2) - sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*
b)) + 2*A*D**2*a**3*b + 12*A*a**4*b**2*(-(-A*b + C*a)/(2*a**2) - sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b))**2 - 2
*B**2*C*a**2*b**2 + 2*B**2*a**3*b**2*(-(-A*b + C*a)/(2*a**2) - sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b)) + 4*B*C*
D*a**3*b - 4*B*D*a**4*b*(-(-A*b + C*a)/(2*a**2) - sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b)) + 6*C**3*a**3*b + 6*C
**2*a**4*b*(-(-A*b + C*a)/(2*a**2) - sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b)) - 2*C*D**2*a**4 - 12*C*a**5*b*(-(-
A*b + C*a)/(2*a**2) - sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b))**2 + 2*D**2*a**5*(-(-A*b + C*a)/(2*a**2) - sqrt(-
a**5*b)*(-B*b + D*a)/(2*a**4*b)))/(-9*A**2*B*b**4 + 9*A**2*D*a*b**3 + 18*A*B*C*a*b**3 - 18*A*C*D*a**2*b**2 - B
**3*a*b**3 + 3*B**2*D*a**2*b**2 - 9*B*C**2*a**2*b**2 - 3*B*D**2*a**3*b + 9*C**2*D*a**3*b + D**3*a**4)) + (-(-A
*b + C*a)/(2*a**2) + sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b))*log(x + (-6*A**3*b**4 + 18*A**2*C*a*b**3 + 6*A**2*
a**2*b**3*(-(-A*b + C*a)/(2*a**2) + sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b)) + 2*A*B**2*a*b**3 - 4*A*B*D*a**2*b*
*2 - 18*A*C**2*a**2*b**2 - 12*A*C*a**3*b**2*(-(-A*b + C*a)/(2*a**2) + sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b)) +
2*A*D**2*a**3*b + 12*A*a**4*b**2*(-(-A*b + C*a)/(2*a**2) + sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b))**2 - 2*B**2
*C*a**2*b**2 + 2*B**2*a**3*b**2*(-(-A*b + C*a)/(2*a**2) + sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b)) + 4*B*C*D*a**
3*b - 4*B*D*a**4*b*(-(-A*b + C*a)/(2*a**2) + sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b)) + 6*C**3*a**3*b + 6*C**2*a
**4*b*(-(-A*b + C*a)/(2*a**2) + sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b)) - 2*C*D**2*a**4 - 12*C*a**5*b*(-(-A*b +
C*a)/(2*a**2) + sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b))**2 + 2*D**2*a**5*(-(-A*b + C*a)/(2*a**2) + sqrt(-a**5*
b)*(-B*b + D*a)/(2*a**4*b)))/(-9*A**2*B*b**4 + 9*A**2*D*a*b**3 + 18*A*B*C*a*b**3 - 18*A*C*D*a**2*b**2 - B**3*a
*b**3 + 3*B**2*D*a**2*b**2 - 9*B*C**2*a**2*b**2 - 3*B*D**2*a**3*b + 9*C**2*D*a**3*b + D**3*a**4)) - (A + 2*B*x
)/(2*a*x**2) + (-A*b + C*a)*log(x + (-6*A**3*b**4 + 18*A**2*C*a*b**3 + 6*A**2*b**3*(-A*b + C*a) + 2*A*B**2*a*b
**3 - 4*A*B*D*a**2*b**2 - 18*A*C**2*a**2*b**2 - 12*A*C*a*b**2*(-A*b + C*a) + 2*A*D**2*a**3*b + 12*A*b**2*(-A*b
+ C*a)**2 - 2*B**2*C*a**2*b**2 + 2*B**2*a*b**2*(-A*b + C*a) + 4*B*C*D*a**3*b - 4*B*D*a**2*b*(-A*b + C*a) + 6*
C**3*a**3*b + 6*C**2*a**2*b*(-A*b + C*a) - 2*C*D**2*a**4 - 12*C*a*b*(-A*b + C*a)**2 + 2*D**2*a**3*(-A*b + C*a)
)/(-9*A**2*B*b**4 + 9*A**2*D*a*b**3 + 18*A*B*C*a*b**3 - 18*A*C*D*a**2*b**2 - B**3*a*b**3 + 3*B**2*D*a**2*b**2
- 9*B*C**2*a**2*b**2 - 3*B*D**2*a**3*b + 9*C**2*D*a**3*b + D**3*a**4))/a**2
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Giac [A] time = 1.16396, size = 108, normalized size = 1.17 \begin{align*} \frac{{\left (D a - B b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{{\left (C a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac{{\left (C a - A b\right )} \log \left ({\left | x \right |}\right )}{a^{2}} - \frac{2 \, B a x + A a}{2 \, a^{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^3/(b*x^2+a),x, algorithm="giac")
[Out]
(D*a - B*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a) - 1/2*(C*a - A*b)*log(b*x^2 + a)/a^2 + (C*a - A*b)*log(abs(x))
/a^2 - 1/2*(2*B*a*x + A*a)/(a^2*x^2)