3.839 \(\int \frac{(a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=58 \[ -\frac{a^2}{2 c x \sqrt{c x^2}}-\frac{2 a b}{c \sqrt{c x^2}}+\frac{b^2 x \log (x)}{c \sqrt{c x^2}} \]

[Out]

(-2*a*b)/(c*Sqrt[c*x^2]) - a^2/(2*c*x*Sqrt[c*x^2]) + (b^2*x*Log[x])/(c*Sqrt[c*x^
2])

_______________________________________________________________________________________

Rubi [A]  time = 0.0348897, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{a^2}{2 c x \sqrt{c x^2}}-\frac{2 a b}{c \sqrt{c x^2}}+\frac{b^2 x \log (x)}{c \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a*b)/(c*Sqrt[c*x^2]) - a^2/(2*c*x*Sqrt[c*x^2]) + (b^2*x*Log[x])/(c*Sqrt[c*x^
2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.1233, size = 60, normalized size = 1.03 \[ - \frac{a^{2} \sqrt{c x^{2}}}{2 c^{2} x^{3}} - \frac{2 a b \sqrt{c x^{2}}}{c^{2} x^{2}} + \frac{b^{2} \sqrt{c x^{2}} \log{\left (x \right )}}{c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00745528, size = 34, normalized size = 0.59 \[ \frac{x \left (2 b^2 x^2 \log (x)-a (a+4 b x)\right )}{2 \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 32, normalized size = 0.6 \[{\frac{x \left ( 2\,{b}^{2}\ln \left ( x \right ){x}^{2}-4\,abx-{a}^{2} \right ) }{2} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/(c*x^2)^(3/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34639, size = 47, normalized size = 0.81 \[ \frac{b^{2} \log \left (x\right )}{c^{\frac{3}{2}}} - \frac{2 \, a b}{\sqrt{c x^{2}} c} - \frac{a^{2}}{2 \, c^{\frac{3}{2}} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.212627, size = 49, normalized size = 0.84 \[ \frac{{\left (2 \, b^{2} x^{2} \log \left (x\right ) - 4 \, a b x - a^{2}\right )} \sqrt{c x^{2}}}{2 \, c^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)**2/(c*x**2)**(3/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.501621, size = 4, normalized size = 0.07 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x