3.834 \(\int \frac{(a+b x)^2}{x^3 \sqrt{c x^2}} \, dx\)

Optimal. Leaf size=26 \[ -\frac{(a+b x)^3}{3 a x^2 \sqrt{c x^2}} \]

[Out]

-(a + b*x)^3/(3*a*x^2*Sqrt[c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.015093, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{(a+b x)^3}{3 a x^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-(a + b*x)^3/(3*a*x^2*Sqrt[c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.3658, size = 24, normalized size = 0.92 \[ - \frac{\sqrt{c x^{2}} \left (a + b x\right )^{3}}{3 a c x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2/x**3/(c*x**2)**(1/2),x)

[Out]

-sqrt(c*x**2)*(a + b*x)**3/(3*a*c*x**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0161121, size = 33, normalized size = 1.27 \[ \frac{c \left (-a^2-3 a b x-3 b^2 x^2\right )}{3 \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(c*(-a^2 - 3*a*b*x - 3*b^2*x^2))/(3*(c*x^2)^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 30, normalized size = 1.2 \[ -{\frac{3\,{b}^{2}{x}^{2}+3\,abx+{a}^{2}}{3\,{x}^{2}}{\frac{1}{\sqrt{c{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/x^3/(c*x^2)^(1/2),x)

[Out]

-1/3*(3*b^2*x^2+3*a*b*x+a^2)/x^2/(c*x^2)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.342, size = 45, normalized size = 1.73 \[ -\frac{b^{2}}{\sqrt{c} x} - \frac{a b}{\sqrt{c} x^{2}} - \frac{a^{2}}{3 \, \sqrt{c} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b^2/(sqrt(c)*x) - a*b/(sqrt(c)*x^2) - 1/3*a^2/(sqrt(c)*x^3)

_______________________________________________________________________________________

Fricas [A]  time = 0.2129, size = 43, normalized size = 1.65 \[ -\frac{{\left (3 \, b^{2} x^{2} + 3 \, a b x + a^{2}\right )} \sqrt{c x^{2}}}{3 \, c x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(3*b^2*x^2 + 3*a*b*x + a^2)*sqrt(c*x^2)/(c*x^4)

_______________________________________________________________________________________

Sympy [A]  time = 2.44276, size = 53, normalized size = 2.04 \[ - \frac{a^{2}}{3 \sqrt{c} x^{2} \sqrt{x^{2}}} - \frac{a b}{\sqrt{c} x \sqrt{x^{2}}} - \frac{b^{2}}{\sqrt{c} \sqrt{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2/x**3/(c*x**2)**(1/2),x)

[Out]

-a**2/(3*sqrt(c)*x**2*sqrt(x**2)) - a*b/(sqrt(c)*x*sqrt(x**2)) - b**2/(sqrt(c)*s
qrt(x**2))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{2}}{\sqrt{c x^{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^2/(sqrt(c*x^2)*x^3), x)