3.264 \(\int \frac{(A+B x) \left (a+c x^2\right )^2}{x^3} \, dx\)

Optimal. Leaf size=56 \[ -\frac{a^2 A}{2 x^2}-\frac{a^2 B}{x}+2 a A c \log (x)+2 a B c x+\frac{1}{2} A c^2 x^2+\frac{1}{3} B c^2 x^3 \]

[Out]

-(a^2*A)/(2*x^2) - (a^2*B)/x + 2*a*B*c*x + (A*c^2*x^2)/2 + (B*c^2*x^3)/3 + 2*a*A
*c*Log[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0760609, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{a^2 A}{2 x^2}-\frac{a^2 B}{x}+2 a A c \log (x)+2 a B c x+\frac{1}{2} A c^2 x^2+\frac{1}{3} B c^2 x^3 \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^2)/x^3,x]

[Out]

-(a^2*A)/(2*x^2) - (a^2*B)/x + 2*a*B*c*x + (A*c^2*x^2)/2 + (B*c^2*x^3)/3 + 2*a*A
*c*Log[x]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{2}}{2 x^{2}} + 2 A a c \log{\left (x \right )} + A c^{2} \int x\, dx - \frac{B a^{2}}{x} + 2 B a c x + \frac{B c^{2} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**2/(2*x**2) + 2*A*a*c*log(x) + A*c**2*Integral(x, x) - B*a**2/x + 2*B*a*c*x
+ B*c**2*x**3/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.0117511, size = 56, normalized size = 1. \[ -\frac{a^2 A}{2 x^2}-\frac{a^2 B}{x}+2 a A c \log (x)+2 a B c x+\frac{1}{2} A c^2 x^2+\frac{1}{3} B c^2 x^3 \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^2)/x^3,x]

[Out]

-(a^2*A)/(2*x^2) - (a^2*B)/x + 2*a*B*c*x + (A*c^2*x^2)/2 + (B*c^2*x^3)/3 + 2*a*A
*c*Log[x]

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 51, normalized size = 0.9 \[ -{\frac{A{a}^{2}}{2\,{x}^{2}}}-{\frac{{a}^{2}B}{x}}+2\,aBcx+{\frac{A{c}^{2}{x}^{2}}{2}}+{\frac{B{c}^{2}{x}^{3}}{3}}+2\,aAc\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^2/x^3,x)

[Out]

-1/2*a^2*A/x^2-a^2*B/x+2*a*B*c*x+1/2*A*c^2*x^2+1/3*B*c^2*x^3+2*a*A*c*ln(x)

_______________________________________________________________________________________

Maxima [A]  time = 0.686973, size = 68, normalized size = 1.21 \[ \frac{1}{3} \, B c^{2} x^{3} + \frac{1}{2} \, A c^{2} x^{2} + 2 \, B a c x + 2 \, A a c \log \left (x\right ) - \frac{2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*B*c^2*x^3 + 1/2*A*c^2*x^2 + 2*B*a*c*x + 2*A*a*c*log(x) - 1/2*(2*B*a^2*x + A*
a^2)/x^2

_______________________________________________________________________________________

Fricas [A]  time = 0.27465, size = 74, normalized size = 1.32 \[ \frac{2 \, B c^{2} x^{5} + 3 \, A c^{2} x^{4} + 12 \, B a c x^{3} + 12 \, A a c x^{2} \log \left (x\right ) - 6 \, B a^{2} x - 3 \, A a^{2}}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*B*c^2*x^5 + 3*A*c^2*x^4 + 12*B*a*c*x^3 + 12*A*a*c*x^2*log(x) - 6*B*a^2*x
- 3*A*a^2)/x^2

_______________________________________________________________________________________

Sympy [A]  time = 1.522, size = 56, normalized size = 1. \[ 2 A a c \log{\left (x \right )} + \frac{A c^{2} x^{2}}{2} + 2 B a c x + \frac{B c^{2} x^{3}}{3} - \frac{A a^{2} + 2 B a^{2} x}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**2/x**3,x)

[Out]

2*A*a*c*log(x) + A*c**2*x**2/2 + 2*B*a*c*x + B*c**2*x**3/3 - (A*a**2 + 2*B*a**2*
x)/(2*x**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.269166, size = 69, normalized size = 1.23 \[ \frac{1}{3} \, B c^{2} x^{3} + \frac{1}{2} \, A c^{2} x^{2} + 2 \, B a c x + 2 \, A a c{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2*(B*x + A)/x^3,x, algorithm="giac")

[Out]

1/3*B*c^2*x^3 + 1/2*A*c^2*x^2 + 2*B*a*c*x + 2*A*a*c*ln(abs(x)) - 1/2*(2*B*a^2*x
+ A*a^2)/x^2