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

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

[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 = 0.628237, 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 \[ -\frac{3 (5 b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} \sqrt{b}}+\frac{(3 A b-a C) \log \left (a+b x^2\right )}{2 a^4}-\frac{\log (x) (3 A b-a C)}{a^4}-\frac{4 (2 A b-a C)+x (7 b B-3 a D)}{8 a^3 \left (a+b x^2\right )}-\frac{A}{2 a^3 x^2}-\frac{B}{a^3 x}-\frac{\frac{A b}{a}+x \left (\frac{b B}{a}-D\right )-C}{4 a \left (a+b x^2\right )^2} \]

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)

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Rubi in Sympy [A]  time = 74.1163, size = 136, normalized size = 0.78 \[ - \frac{C}{a^{2} b x^{2}} - \frac{2 C \log{\left (x \right )}}{a^{3}} + \frac{C \log{\left (a + b x^{2} \right )}}{a^{3}} - \frac{3 D}{2 a^{2} b x} - \frac{3 D \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}} \sqrt{b}} + \frac{x \left (\frac{A b}{x^{3}} + \frac{B b}{x^{2}} - \frac{C a}{x^{3}} - \frac{D a}{x^{2}}\right )}{4 a b \left (a + b x^{2}\right )^{2}} + \frac{C + D x}{2 a b x^{2} \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-C/(a**2*b*x**2) - 2*C*log(x)/a**3 + C*log(a + b*x**2)/a**3 - 3*D/(2*a**2*b*x) -
3*D*atan(sqrt(b)*x/sqrt(a))/(2*a**(5/2)*sqrt(b)) + x*(A*b/x**3 + B*b/x**2 - C*a
/x**3 - D*a/x**2)/(4*a*b*(a + b*x**2)**2) + (C + D*x)/(2*a*b*x**2*(a + b*x**2))

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Mathematica [A]  time = 0.321675, 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 = 0.024, size = 250, normalized size = 1.4 \[ -{\frac{A}{2\,{a}^{3}{x}^{2}}}-{\frac{B}{{a}^{3}x}}-3\,{\frac{A\ln \left ( x \right ) b}{{a}^{4}}}+{\frac{\ln \left ( x \right ) C}{{a}^{3}}}-{\frac{7\,B{b}^{2}{x}^{3}}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,bD{x}^{3}}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{A{x}^{2}{b}^{2}}{{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{bC{x}^{2}}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{9\,bBx}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,Dx}{8\,a \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,Ab}{4\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,C}{4\,a \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,b\ln \left ( b{x}^{2}+a \right ) A}{2\,{a}^{4}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) C}{2\,{a}^{3}}}-{\frac{15\,Bb}{8\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,D}{8\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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/(b*x^2+a)^2*A+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)*arctan(x*b/(a*b)^(1/2))*b*B+3/8/a^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*D

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)/((b*x^2 + a)^3*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.299975, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)/((b*x^2 + a)^3*x^3),x, algorithm="fricas")

[Out]

[-1/16*(3*((D*a^2*b^2 - 5*B*a*b^3)*x^6 + 2*(D*a^3*b - 5*B*a^2*b^2)*x^4 + (D*a^4
- 5*B*a^3*b)*x^2)*log(-(2*a*b*x - (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) - 2*(3*(D
*a^2*b - 5*B*a*b^2)*x^5 - 8*B*a^3*x + 4*(C*a^2*b - 3*A*a*b^2)*x^4 - 4*A*a^3 + 5*
(D*a^3 - 5*B*a^2*b)*x^3 + 6*(C*a^3 - 3*A*a^2*b)*x^2 - 4*((C*a*b^2 - 3*A*b^3)*x^6
+ 2*(C*a^2*b - 3*A*a*b^2)*x^4 + (C*a^3 - 3*A*a^2*b)*x^2)*log(b*x^2 + a) + 8*((C
*a*b^2 - 3*A*b^3)*x^6 + 2*(C*a^2*b - 3*A*a*b^2)*x^4 + (C*a^3 - 3*A*a^2*b)*x^2)*l
og(x))*sqrt(-a*b))/((a^4*b^2*x^6 + 2*a^5*b*x^4 + a^6*x^2)*sqrt(-a*b)), 1/8*(3*((
D*a^2*b^2 - 5*B*a*b^3)*x^6 + 2*(D*a^3*b - 5*B*a^2*b^2)*x^4 + (D*a^4 - 5*B*a^3*b)
*x^2)*arctan(sqrt(a*b)*x/a) + (3*(D*a^2*b - 5*B*a*b^2)*x^5 - 8*B*a^3*x + 4*(C*a^
2*b - 3*A*a*b^2)*x^4 - 4*A*a^3 + 5*(D*a^3 - 5*B*a^2*b)*x^3 + 6*(C*a^3 - 3*A*a^2*
b)*x^2 - 4*((C*a*b^2 - 3*A*b^3)*x^6 + 2*(C*a^2*b - 3*A*a*b^2)*x^4 + (C*a^3 - 3*A
*a^2*b)*x^2)*log(b*x^2 + a) + 8*((C*a*b^2 - 3*A*b^3)*x^6 + 2*(C*a^2*b - 3*A*a*b^
2)*x^4 + (C*a^3 - 3*A*a^2*b)*x^2)*log(x))*sqrt(a*b))/((a^4*b^2*x^6 + 2*a^5*b*x^4
+ a^6*x^2)*sqrt(a*b))]

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Sympy [A]  time = 96.7322, size = 1904, normalized size = 10.94 \[ \text{result too large to display} \]

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 - 144
0*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 + 614
4*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)) + 102
4*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)) + 4
80*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 - 92
16*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 + 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))/a**4

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GIAC/XCAS [A]  time = 0.225452, size = 219, normalized size = 1.26 \[ \frac{3 \,{\left (D a - 5 \, B b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{3}} - \frac{{\left (C a - 3 \, A b\right )}{\rm ln}\left (b x^{2} + a\right )}{2 \, a^{4}} + \frac{{\left (C a - 3 \, A b\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} + \frac{3 \, D a b x^{5} - 15 \, B b^{2} x^{5} + 4 \, C a b x^{4} - 12 \, A b^{2} x^{4} + 5 \, D a^{2} x^{3} - 25 \, B a b x^{3} + 6 \, C a^{2} x^{2} - 18 \, A a b x^{2} - 8 \, B a^{2} x - 4 \, A a^{2}}{8 \,{\left (b x^{3} + a x\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)/((b*x^2 + a)^3*x^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)*ln(b
*x^2 + a)/a^4 + (C*a - 3*A*b)*ln(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)