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

Optimal. Leaf size=52 \[ -\frac{a^2 A}{x}+a^2 B \log (x)+2 a A c x+a B c x^2+\frac{1}{3} A c^2 x^3+\frac{1}{4} B c^2 x^4 \]

[Out]

-((a^2*A)/x) + 2*a*A*c*x + a*B*c*x^2 + (A*c^2*x^3)/3 + (B*c^2*x^4)/4 + a^2*B*Log
[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0724793, antiderivative size = 52, 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 \[ -\frac{a^2 A}{x}+a^2 B \log (x)+2 a A c x+a B c x^2+\frac{1}{3} A c^2 x^3+\frac{1}{4} B c^2 x^4 \]

Antiderivative was successfully verified.

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

[Out]

-((a^2*A)/x) + 2*a*A*c*x + a*B*c*x^2 + (A*c^2*x^3)/3 + (B*c^2*x^4)/4 + a^2*B*Log
[x]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{2}}{x} + 2 A a c x + \frac{A c^{2} x^{3}}{3} + B a^{2} \log{\left (x \right )} + 2 B a c \int x\, dx + \frac{B c^{2} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**2/x + 2*A*a*c*x + A*c**2*x**3/3 + B*a**2*log(x) + 2*B*a*c*Integral(x, x) +
B*c**2*x**4/4

_______________________________________________________________________________________

Mathematica [A]  time = 0.011442, size = 52, normalized size = 1. \[ -\frac{a^2 A}{x}+a^2 B \log (x)+2 a A c x+a B c x^2+\frac{1}{3} A c^2 x^3+\frac{1}{4} B c^2 x^4 \]

Antiderivative was successfully verified.

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

[Out]

-((a^2*A)/x) + 2*a*A*c*x + a*B*c*x^2 + (A*c^2*x^3)/3 + (B*c^2*x^4)/4 + a^2*B*Log
[x]

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 49, normalized size = 0.9 \[ -{\frac{A{a}^{2}}{x}}+2\,aAcx+aBc{x}^{2}+{\frac{A{c}^{2}{x}^{3}}{3}}+{\frac{B{c}^{2}{x}^{4}}{4}}+{a}^{2}B\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^2/x^2,x)

[Out]

-a^2*A/x+2*a*A*c*x+a*B*c*x^2+1/3*A*c^2*x^3+1/4*B*c^2*x^4+a^2*B*ln(x)

_______________________________________________________________________________________

Maxima [A]  time = 0.689293, size = 65, normalized size = 1.25 \[ \frac{1}{4} \, B c^{2} x^{4} + \frac{1}{3} \, A c^{2} x^{3} + B a c x^{2} + 2 \, A a c x + B a^{2} \log \left (x\right ) - \frac{A a^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.296472, size = 74, normalized size = 1.42 \[ \frac{3 \, B c^{2} x^{5} + 4 \, A c^{2} x^{4} + 12 \, B a c x^{3} + 24 \, A a c x^{2} + 12 \, B a^{2} x \log \left (x\right ) - 12 \, A a^{2}}{12 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(3*B*c^2*x^5 + 4*A*c^2*x^4 + 12*B*a*c*x^3 + 24*A*a*c*x^2 + 12*B*a^2*x*log(x
) - 12*A*a^2)/x

_______________________________________________________________________________________

Sympy [A]  time = 1.21955, size = 51, normalized size = 0.98 \[ - \frac{A a^{2}}{x} + 2 A a c x + \frac{A c^{2} x^{3}}{3} + B a^{2} \log{\left (x \right )} + B a c x^{2} + \frac{B c^{2} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**2/x + 2*A*a*c*x + A*c**2*x**3/3 + B*a**2*log(x) + B*a*c*x**2 + B*c**2*x**4
/4

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270536, size = 66, normalized size = 1.27 \[ \frac{1}{4} \, B c^{2} x^{4} + \frac{1}{3} \, A c^{2} x^{3} + B a c x^{2} + 2 \, A a c x + B a^{2}{\rm ln}\left ({\left | x \right |}\right ) - \frac{A a^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*B*c^2*x^4 + 1/3*A*c^2*x^3 + B*a*c*x^2 + 2*A*a*c*x + B*a^2*ln(abs(x)) - A*a^2
/x