3.101 \(\int \frac{A+B x+C x^2+D x^3}{x^3 (a+b x^2)^2} \, dx\)
Optimal. Leaf size=135 \[ \frac{(2 A b-a C) \log \left (a+b x^2\right )}{2 a^3}-\frac{\log (x) (2 A b-a C)}{a^3}-\frac{A}{2 a^2 x^2}-\frac{(3 b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{B}{a^2 x}-\frac{\frac{A b}{a}+x \left (\frac{b B}{a}-D\right )-C}{2 a \left (a+b x^2\right )} \]
[Out]
-A/(2*a^2*x^2) - B/(a^2*x) - ((A*b)/a - C + ((b*B)/a - D)*x)/(2*a*(a + b*x^2)) - ((3*b*B - a*D)*ArcTan[(Sqrt[b
]*x)/Sqrt[a]])/(2*a^(5/2)*Sqrt[b]) - ((2*A*b - a*C)*Log[x])/a^3 + ((2*A*b - a*C)*Log[a + b*x^2])/(2*a^3)
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Rubi [A] time = 0.204027, antiderivative size = 135, 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} \[ \frac{(2 A b-a C) \log \left (a+b x^2\right )}{2 a^3}-\frac{\log (x) (2 A b-a C)}{a^3}-\frac{A}{2 a^2 x^2}-\frac{(3 b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{B}{a^2 x}-\frac{\frac{A b}{a}+x \left (\frac{b B}{a}-D\right )-C}{2 a \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x + C*x^2 + D*x^3)/(x^3*(a + b*x^2)^2),x]
[Out]
-A/(2*a^2*x^2) - B/(a^2*x) - ((A*b)/a - C + ((b*B)/a - D)*x)/(2*a*(a + b*x^2)) - ((3*b*B - a*D)*ArcTan[(Sqrt[b
]*x)/Sqrt[a]])/(2*a^(5/2)*Sqrt[b]) - ((2*A*b - a*C)*Log[x])/a^3 + ((2*A*b - a*C)*Log[a + b*x^2])/(2*a^3)
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^3 \left (a+b x^2\right )^2} \, dx &=-\frac{\frac{A b}{a}-C+\left (\frac{b B}{a}-D\right ) x}{2 a \left (a+b x^2\right )}-\frac{\int \frac{-2 A-2 B x+2 \left (\frac{A b}{a}-C\right ) x^2+\left (\frac{b B}{a}-D\right ) x^3}{x^3 \left (a+b x^2\right )} \, dx}{2 a}\\ &=-\frac{\frac{A b}{a}-C+\left (\frac{b B}{a}-D\right ) x}{2 a \left (a+b x^2\right )}-\frac{\int \left (-\frac{2 A}{a x^3}-\frac{2 B}{a x^2}-\frac{2 (-2 A b+a C)}{a^2 x}+\frac{a (3 b B-a D)-2 b (2 A b-a C) x}{a^2 \left (a+b x^2\right )}\right ) \, dx}{2 a}\\ &=-\frac{A}{2 a^2 x^2}-\frac{B}{a^2 x}-\frac{\frac{A b}{a}-C+\left (\frac{b B}{a}-D\right ) x}{2 a \left (a+b x^2\right )}-\frac{(2 A b-a C) \log (x)}{a^3}-\frac{\int \frac{a (3 b B-a D)-2 b (2 A b-a C) x}{a+b x^2} \, dx}{2 a^3}\\ &=-\frac{A}{2 a^2 x^2}-\frac{B}{a^2 x}-\frac{\frac{A b}{a}-C+\left (\frac{b B}{a}-D\right ) x}{2 a \left (a+b x^2\right )}-\frac{(2 A b-a C) \log (x)}{a^3}+\frac{(b (2 A b-a C)) \int \frac{x}{a+b x^2} \, dx}{a^3}-\frac{(3 b B-a D) \int \frac{1}{a+b x^2} \, dx}{2 a^2}\\ &=-\frac{A}{2 a^2 x^2}-\frac{B}{a^2 x}-\frac{\frac{A b}{a}-C+\left (\frac{b B}{a}-D\right ) x}{2 a \left (a+b x^2\right )}-\frac{(3 b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{(2 A b-a C) \log (x)}{a^3}+\frac{(2 A b-a C) \log \left (a+b x^2\right )}{2 a^3}\\ \end{align*}
Mathematica [A] time = 0.101593, size = 112, normalized size = 0.83 \[ \frac{\frac{a (a (C+D x)-A b-b B x)}{a+b x^2}+(2 A b-a C) \log \left (a+b x^2\right )+2 \log (x) (a C-2 A b)-\frac{a A}{x^2}+\frac{\sqrt{a} (a D-3 b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{2 a B}{x}}{2 a^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x + C*x^2 + D*x^3)/(x^3*(a + b*x^2)^2),x]
[Out]
(-((a*A)/x^2) - (2*a*B)/x + (a*(-(A*b) - b*B*x + a*(C + D*x)))/(a + b*x^2) + (Sqrt[a]*(-3*b*B + a*D)*ArcTan[(S
qrt[b]*x)/Sqrt[a]])/Sqrt[b] + 2*(-2*A*b + a*C)*Log[x] + (2*A*b - a*C)*Log[a + b*x^2])/(2*a^3)
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Maple [A] time = 0.015, size = 169, normalized size = 1.3 \begin{align*} -{\frac{A}{2\,{a}^{2}{x}^{2}}}-{\frac{B}{{a}^{2}x}}-2\,{\frac{A\ln \left ( x \right ) b}{{a}^{3}}}+{\frac{\ln \left ( x \right ) C}{{a}^{2}}}-{\frac{bBx}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{Dx}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{Ab}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{C}{2\,a \left ( b{x}^{2}+a \right ) }}+{\frac{b\ln \left ( b{x}^{2}+a \right ) A}{{a}^{3}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) C}{2\,{a}^{2}}}-{\frac{3\,Bb}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{D}{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^3/(b*x^2+a)^2,x)
[Out]
-1/2*A/a^2/x^2-B/a^2/x-2/a^3*ln(x)*A*b+1/a^2*ln(x)*C-1/2/a^2/(b*x^2+a)*B*x*b+1/2/a/(b*x^2+a)*D*x-1/2/a^2/(b*x^
2+a)*A*b+1/2/a/(b*x^2+a)*C+1/a^3*b*ln(b*x^2+a)*A-1/2/a^2*ln(b*x^2+a)*C-3/2/a^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1
/2))*B*b+1/2/a/(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)^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^3/(b*x^2+a)^2,x, algorithm="fricas")
[Out]
Exception raised: UnboundLocalError
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Sympy [B] time = 39.5762, size = 1807, normalized size = 13.39 \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)**2,x)
[Out]
(-(-2*A*b + C*a)/(2*a**3) - sqrt(-a**7*b)*(-3*B*b + D*a)/(4*a**6*b))*log(x + (-384*A**3*b**4 + 576*A**2*C*a*b*
*3 + 192*A**2*a**3*b**3*(-(-2*A*b + C*a)/(2*a**3) - sqrt(-a**7*b)*(-3*B*b + D*a)/(4*a**6*b)) + 72*A*B**2*a*b**
3 - 48*A*B*D*a**2*b**2 - 288*A*C**2*a**2*b**2 - 192*A*C*a**4*b**2*(-(-2*A*b + C*a)/(2*a**3) - sqrt(-a**7*b)*(-
3*B*b + D*a)/(4*a**6*b)) + 8*A*D**2*a**3*b + 192*A*a**6*b**2*(-(-2*A*b + C*a)/(2*a**3) - sqrt(-a**7*b)*(-3*B*b
+ D*a)/(4*a**6*b))**2 - 36*B**2*C*a**2*b**2 + 36*B**2*a**4*b**2*(-(-2*A*b + C*a)/(2*a**3) - sqrt(-a**7*b)*(-3
*B*b + D*a)/(4*a**6*b)) + 24*B*C*D*a**3*b - 24*B*D*a**5*b*(-(-2*A*b + C*a)/(2*a**3) - sqrt(-a**7*b)*(-3*B*b +
D*a)/(4*a**6*b)) + 48*C**3*a**3*b + 48*C**2*a**5*b*(-(-2*A*b + C*a)/(2*a**3) - sqrt(-a**7*b)*(-3*B*b + D*a)/(4
*a**6*b)) - 4*C*D**2*a**4 - 96*C*a**7*b*(-(-2*A*b + C*a)/(2*a**3) - sqrt(-a**7*b)*(-3*B*b + D*a)/(4*a**6*b))**
2 + 4*D**2*a**6*(-(-2*A*b + C*a)/(2*a**3) - sqrt(-a**7*b)*(-3*B*b + D*a)/(4*a**6*b)))/(-432*A**2*B*b**4 + 144*
A**2*D*a*b**3 + 432*A*B*C*a*b**3 - 144*A*C*D*a**2*b**2 - 27*B**3*a*b**3 + 27*B**2*D*a**2*b**2 - 108*B*C**2*a**
2*b**2 - 9*B*D**2*a**3*b + 36*C**2*D*a**3*b + D**3*a**4)) + (-(-2*A*b + C*a)/(2*a**3) + sqrt(-a**7*b)*(-3*B*b
+ D*a)/(4*a**6*b))*log(x + (-384*A**3*b**4 + 576*A**2*C*a*b**3 + 192*A**2*a**3*b**3*(-(-2*A*b + C*a)/(2*a**3)
+ sqrt(-a**7*b)*(-3*B*b + D*a)/(4*a**6*b)) + 72*A*B**2*a*b**3 - 48*A*B*D*a**2*b**2 - 288*A*C**2*a**2*b**2 - 19
2*A*C*a**4*b**2*(-(-2*A*b + C*a)/(2*a**3) + sqrt(-a**7*b)*(-3*B*b + D*a)/(4*a**6*b)) + 8*A*D**2*a**3*b + 192*A
*a**6*b**2*(-(-2*A*b + C*a)/(2*a**3) + sqrt(-a**7*b)*(-3*B*b + D*a)/(4*a**6*b))**2 - 36*B**2*C*a**2*b**2 + 36*
B**2*a**4*b**2*(-(-2*A*b + C*a)/(2*a**3) + sqrt(-a**7*b)*(-3*B*b + D*a)/(4*a**6*b)) + 24*B*C*D*a**3*b - 24*B*D
*a**5*b*(-(-2*A*b + C*a)/(2*a**3) + sqrt(-a**7*b)*(-3*B*b + D*a)/(4*a**6*b)) + 48*C**3*a**3*b + 48*C**2*a**5*b
*(-(-2*A*b + C*a)/(2*a**3) + sqrt(-a**7*b)*(-3*B*b + D*a)/(4*a**6*b)) - 4*C*D**2*a**4 - 96*C*a**7*b*(-(-2*A*b
+ C*a)/(2*a**3) + sqrt(-a**7*b)*(-3*B*b + D*a)/(4*a**6*b))**2 + 4*D**2*a**6*(-(-2*A*b + C*a)/(2*a**3) + sqrt(-
a**7*b)*(-3*B*b + D*a)/(4*a**6*b)))/(-432*A**2*B*b**4 + 144*A**2*D*a*b**3 + 432*A*B*C*a*b**3 - 144*A*C*D*a**2*
b**2 - 27*B**3*a*b**3 + 27*B**2*D*a**2*b**2 - 108*B*C**2*a**2*b**2 - 9*B*D**2*a**3*b + 36*C**2*D*a**3*b + D**3
*a**4)) + (-A*a - 2*B*a*x + x**3*(-3*B*b + D*a) + x**2*(-2*A*b + C*a))/(2*a**3*x**2 + 2*a**2*b*x**4) + (-2*A*b
+ C*a)*log(x + (-384*A**3*b**4 + 576*A**2*C*a*b**3 + 192*A**2*b**3*(-2*A*b + C*a) + 72*A*B**2*a*b**3 - 48*A*B
*D*a**2*b**2 - 288*A*C**2*a**2*b**2 - 192*A*C*a*b**2*(-2*A*b + C*a) + 8*A*D**2*a**3*b + 192*A*b**2*(-2*A*b + C
*a)**2 - 36*B**2*C*a**2*b**2 + 36*B**2*a*b**2*(-2*A*b + C*a) + 24*B*C*D*a**3*b - 24*B*D*a**2*b*(-2*A*b + C*a)
+ 48*C**3*a**3*b + 48*C**2*a**2*b*(-2*A*b + C*a) - 4*C*D**2*a**4 - 96*C*a*b*(-2*A*b + C*a)**2 + 4*D**2*a**3*(-
2*A*b + C*a))/(-432*A**2*B*b**4 + 144*A**2*D*a*b**3 + 432*A*B*C*a*b**3 - 144*A*C*D*a**2*b**2 - 27*B**3*a*b**3
+ 27*B**2*D*a**2*b**2 - 108*B*C**2*a**2*b**2 - 9*B*D**2*a**3*b + 36*C**2*D*a**3*b + D**3*a**4))/a**3
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Giac [A] time = 1.16835, size = 170, normalized size = 1.26 \begin{align*} \frac{{\left (D a - 3 \, B b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2}} - \frac{{\left (C a - 2 \, A b\right )} \log \left (b x^{2} + a\right )}{2 \, a^{3}} + \frac{{\left (C a - 2 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac{2 \, B a^{2} x -{\left (D a^{2} - 3 \, B a b\right )} x^{3} + A a^{2} -{\left (C a^{2} - 2 \, A a b\right )} x^{2}}{2 \,{\left (b x^{2} + a\right )} a^{3} 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)^2,x, algorithm="giac")
[Out]
1/2*(D*a - 3*B*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2) - 1/2*(C*a - 2*A*b)*log(b*x^2 + a)/a^3 + (C*a - 2*A*b)
*log(abs(x))/a^3 - 1/2*(2*B*a^2*x - (D*a^2 - 3*B*a*b)*x^3 + A*a^2 - (C*a^2 - 2*A*a*b)*x^2)/((b*x^2 + a)*a^3*x^
2)