\(\int \frac {A+B x+C x^2+D x^3}{x^3 (a+b x^2)^2} \, dx\) [52]

Optimal result

Integrand size = 28, antiderivative size = 131 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {A}{2 a^2 x^2}-\frac {B}{a^2 x}-\frac {A b-a C+(b B-a D) x}{2 a^2 \left (a+b x^2\right )}-\frac {(3 b B-a D) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b}}-\frac {(2 A b-a C) \log (x)}{a^3}+\frac {(2 A b-a C) \log \left (a+b x^2\right )}{2 a^3} \] Output:

-1/2*A/a^2/x^2-B/a^2/x-1/2*(A*b-C*a+(B*b-D*a)*x)/a^2/(b*x^2+a)-1/2*(3*B*b- 
D*a)*arctan(b^(1/2)*x/a^(1/2))/a^(5/2)/b^(1/2)-(2*A*b-C*a)*ln(x)/a^3+1/2*( 
2*A*b-C*a)*ln(b*x^2+a)/a^3
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {-\frac {a A}{x^2}-\frac {2 a B}{x}+\frac {a (-A b-b B x+a (C+D x))}{a+b x^2}+\frac {\sqrt {a} (-3 b B+a D) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+2 (-2 A b+a C) \log (x)+(2 A b-a C) \log \left (a+b x^2\right )}{2 a^3} \] Input:

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

Output:

(-((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[(Sqrt[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)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2336, 25, 2333, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2336

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2333

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (3 b B-a D)}{a^{3/2} \sqrt {b}}+\frac {(2 A b-a C) \log \left (a+b x^2\right )}{a^2}-\frac {2 \log (x) (2 A b-a C)}{a^2}-\frac {A}{a x^2}-\frac {2 B}{a x}}{2 a}-\frac {\frac {A b}{a}+x \left (\frac {b B}{a}-D\right )-C}{2 a \left (a+b x^2\right )}\)

Input:

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

Output:

-1/2*((A*b)/a - C + ((b*B)/a - D)*x)/(a*(a + b*x^2)) + (-(A/(a*x^2)) - (2* 
B)/(a*x) - ((3*b*B - a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]) - 
 (2*(2*A*b - a*C)*Log[x])/a^2 + ((2*A*b - a*C)*Log[a + b*x^2])/a^2)/(2*a)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2333

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 2336

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[PolynomialRema 
inder[(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] + Simp[1/(2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex 
pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F 
reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
 

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {\frac {\left (-\frac {1}{2} a b B +\frac {1}{2} D a^{2}\right ) x -\frac {a \left (A b -C a \right )}{2}}{b \,x^{2}+a}+\frac {\left (4 b^{2} A -2 C a b \right ) \ln \left (b \,x^{2}+a \right )}{4 b}+\frac {\left (-3 a b B +D a^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}}{a^{3}}-\frac {A}{2 a^{2} x^{2}}-\frac {B}{a^{2} x}+\frac {\left (-2 A b +C a \right ) \ln \left (x \right )}{a^{3}}\)	\(127\)

Input:

int((D*x^3+C*x^2+B*x+A)/x^3/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/a^3*(((-1/2*a*b*B+1/2*D*a^2)*x-1/2*a*(A*b-C*a))/(b*x^2+a)+1/4*(4*A*b^2-2 
*C*a*b)/b*ln(b*x^2+a)+1/2*(-3*B*a*b+D*a^2)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1 
/2)))-1/2*A/a^2/x^2-B/a^2/x+(-2*A*b+C*a)/a^3*ln(x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 442, normalized size of antiderivative = 3.37 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^2} \, dx=\left [-\frac {4 \, B a^{2} b x + 2 \, A a^{2} b - 2 \, {\left (D a^{2} b - 3 \, B a b^{2}\right )} x^{3} - 2 \, {\left (C a^{2} b - 2 \, A a b^{2}\right )} x^{2} - {\left ({\left (D a b - 3 \, B b^{2}\right )} x^{4} + {\left (D a^{2} - 3 \, B a b\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left ({\left (C a b^{2} - 2 \, A b^{3}\right )} x^{4} + {\left (C a^{2} b - 2 \, A a b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 4 \, {\left ({\left (C a b^{2} - 2 \, A b^{3}\right )} x^{4} + {\left (C a^{2} b - 2 \, A a b^{2}\right )} x^{2}\right )} \log \left (x\right )}{4 \, {\left (a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )}}, -\frac {2 \, B a^{2} b x + A a^{2} b - {\left (D a^{2} b - 3 \, B a b^{2}\right )} x^{3} - {\left (C a^{2} b - 2 \, A a b^{2}\right )} x^{2} - {\left ({\left (D a b - 3 \, B b^{2}\right )} x^{4} + {\left (D a^{2} - 3 \, B a b\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left ({\left (C a b^{2} - 2 \, A b^{3}\right )} x^{4} + {\left (C a^{2} b - 2 \, A a b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left ({\left (C a b^{2} - 2 \, A b^{3}\right )} x^{4} + {\left (C a^{2} b - 2 \, A a b^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )}}\right ] \] Input:

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

Output:

[-1/4*(4*B*a^2*b*x + 2*A*a^2*b - 2*(D*a^2*b - 3*B*a*b^2)*x^3 - 2*(C*a^2*b 
- 2*A*a*b^2)*x^2 - ((D*a*b - 3*B*b^2)*x^4 + (D*a^2 - 3*B*a*b)*x^2)*sqrt(-a 
*b)*log((b*x^2 + 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 2*((C*a*b^2 - 2*A*b^3) 
*x^4 + (C*a^2*b - 2*A*a*b^2)*x^2)*log(b*x^2 + a) - 4*((C*a*b^2 - 2*A*b^3)* 
x^4 + (C*a^2*b - 2*A*a*b^2)*x^2)*log(x))/(a^3*b^2*x^4 + a^4*b*x^2), -1/2*( 
2*B*a^2*b*x + A*a^2*b - (D*a^2*b - 3*B*a*b^2)*x^3 - (C*a^2*b - 2*A*a*b^2)* 
x^2 - ((D*a*b - 3*B*b^2)*x^4 + (D*a^2 - 3*B*a*b)*x^2)*sqrt(a*b)*arctan(sqr 
t(a*b)*x/a) + ((C*a*b^2 - 2*A*b^3)*x^4 + (C*a^2*b - 2*A*a*b^2)*x^2)*log(b* 
x^2 + a) - 2*((C*a*b^2 - 2*A*b^3)*x^4 + (C*a^2*b - 2*A*a*b^2)*x^2)*log(x)) 
/(a^3*b^2*x^4 + a^4*b*x^2)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((D*x**3+C*x**2+B*x+A)/x**3/(b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.89 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {{\left (D a - 3 \, B b\right )} x^{3} - 2 \, B a x + {\left (C a - 2 \, A b\right )} x^{2} - A a}{2 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} + \frac {{\left (D a - 3 \, B b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2}} - \frac {{\left (C a - 2 \, A b\right )} \log \left (b x^{2} + a\right )}{2 \, a^{3}} + \frac {{\left (C a - 2 \, A b\right )} \log \left (x\right )}{a^{3}} \] Input:

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

Output:

1/2*((D*a - 3*B*b)*x^3 - 2*B*a*x + (C*a - 2*A*b)*x^2 - A*a)/(a^2*b*x^4 + a 
^3*x^2) + 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(x)/a^3
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {{\left (D a - 3 \, B b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2}} - \frac {{\left (C a - 2 \, A b\right )} \log \left (b x^{2} + a\right )}{2 \, a^{3}} + \frac {{\left (C a - 2 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {2 \, B a^{2} x - {\left (D a^{2} - 3 \, B a b\right )} x^{3} + A a^{2} - {\left (C a^{2} - 2 \, A a b\right )} x^{2}}{2 \, {\left (b x^{2} + a\right )} a^{3} x^{2}} \] Input:

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

Output:

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)
 

Mupad [B] (verification not implemented)

Time = 2.41 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.21 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {C}{2\,a\,\left (b\,x^2+a\right )}-\frac {\frac {A}{2\,a}+\frac {A\,b\,x^2}{a^2}}{b\,x^4+a\,x^2}-\frac {\frac {B}{a}+\frac {3\,B\,b\,x^2}{2\,a^2}}{b\,x^3+a\,x}-\frac {C\,\ln \left (b\,x^2+a\right )}{2\,a^2}+\frac {C\,\ln \left (x\right )}{a^2}+\frac {A\,b\,\ln \left (b\,x^2+a\right )}{a^3}-\frac {2\,A\,b\,\ln \left (x\right )}{a^3}+\frac {x\,D\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},2;\ \frac {3}{2};\ -\frac {b\,x^2}{a}\right )}{a^2}-\frac {3\,B\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,a^{5/2}} \] Input:

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

Output:

C/(2*a*(a + b*x^2)) - (A/(2*a) + (A*b*x^2)/a^2)/(a*x^2 + b*x^4) - (B/a + ( 
3*B*b*x^2)/(2*a^2))/(a*x + b*x^3) - (C*log(a + b*x^2))/(2*a^2) + (C*log(x) 
)/a^2 + (A*b*log(a + b*x^2))/a^3 - (2*A*b*log(x))/a^3 + (x*D*hypergeom([1/ 
2, 2], 3/2, -(b*x^2)/a))/a^2 - (3*B*b^(1/2)*atan((b^(1/2)*x)/a^(1/2)))/(2* 
a^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.21 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} d \,x^{2}-3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} x^{2}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b d \,x^{4}-3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{3} x^{4}+2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{2} x^{2}-\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b c \,x^{2}+2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{3} x^{4}-\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{2} c \,x^{4}-4 \,\mathrm {log}\left (x \right ) a^{2} b^{2} x^{2}+2 \,\mathrm {log}\left (x \right ) a^{2} b c \,x^{2}-4 \,\mathrm {log}\left (x \right ) a \,b^{3} x^{4}+2 \,\mathrm {log}\left (x \right ) a \,b^{2} c \,x^{4}-a^{3} b -2 a^{2} b^{2} x +a^{2} b d \,x^{3}+2 a \,b^{3} x^{4}-3 a \,b^{3} x^{3}-a \,b^{2} c \,x^{4}}{2 a^{3} b \,x^{2} \left (b \,x^{2}+a \right )} \] Input:

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

Output:

(sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*d*x**2 - 3*sqrt(b)*sqr 
t(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**2*x**2 + sqrt(b)*sqrt(a)*atan((b*x 
)/(sqrt(b)*sqrt(a)))*a*b*d*x**4 - 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sq 
rt(a)))*b**3*x**4 + 2*log(a + b*x**2)*a**2*b**2*x**2 - log(a + b*x**2)*a** 
2*b*c*x**2 + 2*log(a + b*x**2)*a*b**3*x**4 - log(a + b*x**2)*a*b**2*c*x**4 
 - 4*log(x)*a**2*b**2*x**2 + 2*log(x)*a**2*b*c*x**2 - 4*log(x)*a*b**3*x**4 
 + 2*log(x)*a*b**2*c*x**4 - a**3*b - 2*a**2*b**2*x + a**2*b*d*x**3 + 2*a*b 
**3*x**4 - 3*a*b**3*x**3 - a*b**2*c*x**4)/(2*a**3*b*x**2*(a + b*x**2))