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

Optimal. Leaf size=174 \[ -\frac{4 (2 A b-a C)+x (7 b B-3 a D)}{8 a^3 \left (a+b x^2\right )}+\frac{(3 A b-a C) \log \left (a+b x^2\right )}{2 a^4}-\frac{\log (x) (3 A b-a C)}{a^4}-\frac{A}{2 a^3 x^2}-\frac{3 (5 b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} \sqrt{b}}-\frac{B}{a^3 x}-\frac{\frac{A b}{a}+x \left (\frac{b B}{a}-D\right )-C}{4 a \left (a+b x^2\right )^2} \]

[Out]

-A/(2*a^3*x^2) - B/(a^3*x) - ((A*b)/a - C + ((b*B)/a - D)*x)/(4*a*(a + b*x^2)^2) - (4*(2*A*b - a*C) + (7*b*B -
3*a*D)*x)/(8*a^3*(a + b*x^2)) - (3*(5*b*B - a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*Sqrt[b]) - ((3*A*b -
a*C)*Log[x])/a^4 + ((3*A*b - a*C)*Log[a + b*x^2])/(2*a^4)

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Rubi [A]  time = 0.311011, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{4 (2 A b-a C)+x (7 b B-3 a D)}{8 a^3 \left (a+b x^2\right )}+\frac{(3 A b-a C) \log \left (a+b x^2\right )}{2 a^4}-\frac{\log (x) (3 A b-a C)}{a^4}-\frac{A}{2 a^3 x^2}-\frac{3 (5 b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} \sqrt{b}}-\frac{B}{a^3 x}-\frac{\frac{A b}{a}+x \left (\frac{b B}{a}-D\right )-C}{4 a \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-A/(2*a^3*x^2) - B/(a^3*x) - ((A*b)/a - C + ((b*B)/a - D)*x)/(4*a*(a + b*x^2)^2) - (4*(2*A*b - a*C) + (7*b*B -
3*a*D)*x)/(8*a^3*(a + b*x^2)) - (3*(5*b*B - a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*Sqrt[b]) - ((3*A*b -
a*C)*Log[x])/a^4 + ((3*A*b - a*C)*Log[a + b*x^2])/(2*a^4)

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 )^3} \, dx &=-\frac{\frac{A b}{a}-C+\left (\frac{b B}{a}-D\right ) x}{4 a \left (a+b x^2\right )^2}-\frac{\int \frac{-4 A-4 B x+4 \left (\frac{A b}{a}-C\right ) x^2+3 \left (\frac{b B}{a}-D\right ) x^3}{x^3 \left (a+b x^2\right )^2} \, dx}{4 a}\\ &=-\frac{\frac{A b}{a}-C+\left (\frac{b B}{a}-D\right ) x}{4 a \left (a+b x^2\right )^2}-\frac{4 (2 A b-a C)+(7 b B-3 a D) x}{8 a^3 \left (a+b x^2\right )}+\frac{\int \frac{8 A+8 B x-8 \left (\frac{2 A b}{a}-C\right ) x^2-\left (\frac{7 b B}{a}-3 D\right ) x^3}{x^3 \left (a+b x^2\right )} \, dx}{8 a^2}\\ &=-\frac{\frac{A b}{a}-C+\left (\frac{b B}{a}-D\right ) x}{4 a \left (a+b x^2\right )^2}-\frac{4 (2 A b-a C)+(7 b B-3 a D) x}{8 a^3 \left (a+b x^2\right )}+\frac{\int \left (\frac{8 A}{a x^3}+\frac{8 B}{a x^2}+\frac{8 (-3 A b+a C)}{a^2 x}+\frac{-3 a (5 b B-a D)+8 b (3 A b-a C) x}{a^2 \left (a+b x^2\right )}\right ) \, dx}{8 a^2}\\ &=-\frac{A}{2 a^3 x^2}-\frac{B}{a^3 x}-\frac{\frac{A b}{a}-C+\left (\frac{b B}{a}-D\right ) x}{4 a \left (a+b x^2\right )^2}-\frac{4 (2 A b-a C)+(7 b B-3 a D) x}{8 a^3 \left (a+b x^2\right )}-\frac{(3 A b-a C) \log (x)}{a^4}+\frac{\int \frac{-3 a (5 b B-a D)+8 b (3 A b-a C) x}{a+b x^2} \, dx}{8 a^4}\\ &=-\frac{A}{2 a^3 x^2}-\frac{B}{a^3 x}-\frac{\frac{A b}{a}-C+\left (\frac{b B}{a}-D\right ) x}{4 a \left (a+b x^2\right )^2}-\frac{4 (2 A b-a C)+(7 b B-3 a D) x}{8 a^3 \left (a+b x^2\right )}-\frac{(3 A b-a C) \log (x)}{a^4}+\frac{(b (3 A b-a C)) \int \frac{x}{a+b x^2} \, dx}{a^4}-\frac{(3 (5 b B-a D)) \int \frac{1}{a+b x^2} \, dx}{8 a^3}\\ &=-\frac{A}{2 a^3 x^2}-\frac{B}{a^3 x}-\frac{\frac{A b}{a}-C+\left (\frac{b B}{a}-D\right ) x}{4 a \left (a+b x^2\right )^2}-\frac{4 (2 A b-a C)+(7 b B-3 a D) x}{8 a^3 \left (a+b x^2\right )}-\frac{3 (5 b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} \sqrt{b}}-\frac{(3 A b-a C) \log (x)}{a^4}+\frac{(3 A b-a C) \log \left (a+b x^2\right )}{2 a^4}\\ \end{align*}

Mathematica [A]  time = 0.150128, size = 147, normalized size = 0.84 \[ \frac{\frac{2 a^2 (a (C+D x)-A b-b B x)}{\left (a+b x^2\right )^2}+\frac{a (4 a C+3 a D x-8 A b-7 b B x)}{a+b x^2}+4 (3 A b-a C) \log \left (a+b x^2\right )+8 \log (x) (a C-3 A b)-\frac{4 a A}{x^2}+\frac{3 \sqrt{a} (a D-5 b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{8 a B}{x}}{8 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*a*A)/x^2 - (8*a*B)/x + (a*(-8*A*b + 4*a*C - 7*b*B*x + 3*a*D*x))/(a + b*x^2) + (2*a^2*(-(A*b) - b*B*x + a*
(C + D*x)))/(a + b*x^2)^2 + (3*Sqrt[a]*(-5*b*B + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] + 8*(-3*A*b + a*C)*
Log[x] + 4*(3*A*b - a*C)*Log[a + b*x^2])/(8*a^4)

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Maple [A]  time = 0.017, size = 250, normalized size = 1.4 \begin{align*} -{\frac{A}{2\,{a}^{3}{x}^{2}}}-{\frac{B}{{a}^{3}x}}-3\,{\frac{A\ln \left ( x \right ) b}{{a}^{4}}}+{\frac{\ln \left ( x \right ) C}{{a}^{3}}}-{\frac{7\,B{x}^{3}{b}^{2}}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,bD{x}^{3}}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{A{x}^{2}{b}^{2}}{{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{bC{x}^{2}}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{9\,bBx}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,Dx}{8\,a \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,Ab}{4\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,C}{4\,a \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,b\ln \left ( b{x}^{2}+a \right ) A}{2\,{a}^{4}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) C}{2\,{a}^{3}}}-{\frac{15\,Bb}{8\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,D}{8\,{a}^{2}}\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)^3,x)

[Out]

-1/2*A/a^3/x^2-B/a^3/x-3/a^4*ln(x)*A*b+1/a^3*ln(x)*C-7/8/a^3/(b*x^2+a)^2*B*x^3*b^2+3/8/a^2/(b*x^2+a)^2*D*x^3*b
-1/a^3/(b*x^2+a)^2*A*x^2*b^2+1/2/a^2/(b*x^2+a)^2*C*x^2*b-9/8/a^2/(b*x^2+a)^2*B*x*b+5/8/a/(b*x^2+a)^2*D*x-5/4/a
^2/(b*x^2+a)^2*A*b+3/4/a/(b*x^2+a)^2*C+3/2/a^4*b*ln(b*x^2+a)*A-1/2/a^3*ln(b*x^2+a)*C-15/8/a^3/(a*b)^(1/2)*arct
an(b*x/(a*b)^(1/2))*B*b+3/8/a^2/(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)^3,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)^3,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [B]  time = 60.1891, size = 1904, normalized size = 10.94 \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)**3,x)

[Out]

(-(-3*A*b + C*a)/(2*a**4) - 3*sqrt(-a**9*b)*(-5*B*b + D*a)/(16*a**8*b))*log(x + (-27648*A**3*b**4 + 27648*A**2
*C*a*b**3 + 9216*A**2*a**4*b**3*(-(-3*A*b + C*a)/(2*a**4) - 3*sqrt(-a**9*b)*(-5*B*b + D*a)/(16*a**8*b)) + 3600
*A*B**2*a*b**3 - 1440*A*B*D*a**2*b**2 - 9216*A*C**2*a**2*b**2 - 6144*A*C*a**5*b**2*(-(-3*A*b + C*a)/(2*a**4) -
3*sqrt(-a**9*b)*(-5*B*b + D*a)/(16*a**8*b)) + 144*A*D**2*a**3*b + 6144*A*a**8*b**2*(-(-3*A*b + C*a)/(2*a**4)
- 3*sqrt(-a**9*b)*(-5*B*b + D*a)/(16*a**8*b))**2 - 1200*B**2*C*a**2*b**2 + 1200*B**2*a**5*b**2*(-(-3*A*b + C*a
)/(2*a**4) - 3*sqrt(-a**9*b)*(-5*B*b + D*a)/(16*a**8*b)) + 480*B*C*D*a**3*b - 480*B*D*a**6*b*(-(-3*A*b + C*a)/
(2*a**4) - 3*sqrt(-a**9*b)*(-5*B*b + D*a)/(16*a**8*b)) + 1024*C**3*a**3*b + 1024*C**2*a**6*b*(-(-3*A*b + C*a)/
(2*a**4) - 3*sqrt(-a**9*b)*(-5*B*b + D*a)/(16*a**8*b)) - 48*C*D**2*a**4 - 2048*C*a**9*b*(-(-3*A*b + C*a)/(2*a*
*4) - 3*sqrt(-a**9*b)*(-5*B*b + D*a)/(16*a**8*b))**2 + 48*D**2*a**7*(-(-3*A*b + C*a)/(2*a**4) - 3*sqrt(-a**9*b
)*(-5*B*b + D*a)/(16*a**8*b)))/(-25920*A**2*B*b**4 + 5184*A**2*D*a*b**3 + 17280*A*B*C*a*b**3 - 3456*A*C*D*a**2
*b**2 - 1125*B**3*a*b**3 + 675*B**2*D*a**2*b**2 - 2880*B*C**2*a**2*b**2 - 135*B*D**2*a**3*b + 576*C**2*D*a**3*
b + 9*D**3*a**4)) + (-(-3*A*b + C*a)/(2*a**4) + 3*sqrt(-a**9*b)*(-5*B*b + D*a)/(16*a**8*b))*log(x + (-27648*A*
*3*b**4 + 27648*A**2*C*a*b**3 + 9216*A**2*a**4*b**3*(-(-3*A*b + C*a)/(2*a**4) + 3*sqrt(-a**9*b)*(-5*B*b + D*a)
/(16*a**8*b)) + 3600*A*B**2*a*b**3 - 1440*A*B*D*a**2*b**2 - 9216*A*C**2*a**2*b**2 - 6144*A*C*a**5*b**2*(-(-3*A
*b + C*a)/(2*a**4) + 3*sqrt(-a**9*b)*(-5*B*b + D*a)/(16*a**8*b)) + 144*A*D**2*a**3*b + 6144*A*a**8*b**2*(-(-3*
A*b + C*a)/(2*a**4) + 3*sqrt(-a**9*b)*(-5*B*b + D*a)/(16*a**8*b))**2 - 1200*B**2*C*a**2*b**2 + 1200*B**2*a**5*
b**2*(-(-3*A*b + C*a)/(2*a**4) + 3*sqrt(-a**9*b)*(-5*B*b + D*a)/(16*a**8*b)) + 480*B*C*D*a**3*b - 480*B*D*a**6
*b*(-(-3*A*b + C*a)/(2*a**4) + 3*sqrt(-a**9*b)*(-5*B*b + D*a)/(16*a**8*b)) + 1024*C**3*a**3*b + 1024*C**2*a**6
*b*(-(-3*A*b + C*a)/(2*a**4) + 3*sqrt(-a**9*b)*(-5*B*b + D*a)/(16*a**8*b)) - 48*C*D**2*a**4 - 2048*C*a**9*b*(-
(-3*A*b + C*a)/(2*a**4) + 3*sqrt(-a**9*b)*(-5*B*b + D*a)/(16*a**8*b))**2 + 48*D**2*a**7*(-(-3*A*b + C*a)/(2*a*
*4) + 3*sqrt(-a**9*b)*(-5*B*b + D*a)/(16*a**8*b)))/(-25920*A**2*B*b**4 + 5184*A**2*D*a*b**3 + 17280*A*B*C*a*b*
*3 - 3456*A*C*D*a**2*b**2 - 1125*B**3*a*b**3 + 675*B**2*D*a**2*b**2 - 2880*B*C**2*a**2*b**2 - 135*B*D**2*a**3*
b + 576*C**2*D*a**3*b + 9*D**3*a**4)) + (-4*A*a**2 - 8*B*a**2*x + x**5*(-15*B*b**2 + 3*D*a*b) + x**4*(-12*A*b*
*2 + 4*C*a*b) + x**3*(-25*B*a*b + 5*D*a**2) + x**2*(-18*A*a*b + 6*C*a**2))/(8*a**5*x**2 + 16*a**4*b*x**4 + 8*a
**3*b**2*x**6) + (-3*A*b + C*a)*log(x + (-27648*A**3*b**4 + 27648*A**2*C*a*b**3 + 9216*A**2*b**3*(-3*A*b + C*a
) + 3600*A*B**2*a*b**3 - 1440*A*B*D*a**2*b**2 - 9216*A*C**2*a**2*b**2 - 6144*A*C*a*b**2*(-3*A*b + C*a) + 144*A
*D**2*a**3*b + 6144*A*b**2*(-3*A*b + C*a)**2 - 1200*B**2*C*a**2*b**2 + 1200*B**2*a*b**2*(-3*A*b + C*a) + 480*B
*C*D*a**3*b - 480*B*D*a**2*b*(-3*A*b + C*a) + 1024*C**3*a**3*b + 1024*C**2*a**2*b*(-3*A*b + C*a) - 48*C*D**2*a
**4 - 2048*C*a*b*(-3*A*b + C*a)**2 + 48*D**2*a**3*(-3*A*b + C*a))/(-25920*A**2*B*b**4 + 5184*A**2*D*a*b**3 + 1
7280*A*B*C*a*b**3 - 3456*A*C*D*a**2*b**2 - 1125*B**3*a*b**3 + 675*B**2*D*a**2*b**2 - 2880*B*C**2*a**2*b**2 - 1
35*B*D**2*a**3*b + 576*C**2*D*a**3*b + 9*D**3*a**4))/a**4

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Giac [A]  time = 1.19892, size = 219, normalized size = 1.26 \begin{align*} \frac{3 \,{\left (D a - 5 \, B b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{3}} - \frac{{\left (C a - 3 \, A b\right )} \log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac{{\left (C a - 3 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac{3 \, D a b x^{5} - 15 \, B b^{2} x^{5} + 4 \, C a b x^{4} - 12 \, A b^{2} x^{4} + 5 \, D a^{2} x^{3} - 25 \, B a b x^{3} + 6 \, C a^{2} x^{2} - 18 \, A a b x^{2} - 8 \, B a^{2} x - 4 \, A a^{2}}{8 \,{\left (b x^{3} + a x\right )}^{2} a^{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^3/(b*x^2+a)^3,x, algorithm="giac")

[Out]

3/8*(D*a - 5*B*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^3) - 1/2*(C*a - 3*A*b)*log(b*x^2 + a)/a^4 + (C*a - 3*A*b)
*log(abs(x))/a^4 + 1/8*(3*D*a*b*x^5 - 15*B*b^2*x^5 + 4*C*a*b*x^4 - 12*A*b^2*x^4 + 5*D*a^2*x^3 - 25*B*a*b*x^3 +
6*C*a^2*x^2 - 18*A*a*b*x^2 - 8*B*a^2*x - 4*A*a^2)/((b*x^3 + a*x)^2*a^3)