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

Optimal. Leaf size=29 \[ -\frac{(a+b x)^3}{3 a c x^2 \sqrt{c x^2}} \]

[Out]

-(a + b*x)^3/(3*a*c*x^2*Sqrt[c*x^2])

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Rubi [A]  time = 0.0176509, antiderivative size = 29, 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 \[ -\frac{(a+b x)^3}{3 a c x^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-(a + b*x)^3/(3*a*c*x^2*Sqrt[c*x^2])

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Rubi in Sympy [A]  time = 13.1476, size = 26, normalized size = 0.9 \[ - \frac{\sqrt{c x^{2}} \left (a + b x\right )^{3}}{3 a c^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0193321, size = 36, normalized size = 1.24 \[ \frac{c x^2 \left (-a^2-3 a b x-3 b^2 x^2\right )}{3 \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(c*x^2*(-a^2 - 3*a*b*x - 3*b^2*x^2))/(3*(c*x^2)^(5/2))

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Maple [A]  time = 0.007, size = 27, normalized size = 0.9 \[ -{\frac{3\,{b}^{2}{x}^{2}+3\,abx+{a}^{2}}{3} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35064, size = 50, normalized size = 1.72 \[ -\frac{b^{2}}{\sqrt{c x^{2}} c} - \frac{a b}{c^{\frac{3}{2}} x^{2}} - \frac{a^{2}}{3 \, c^{\frac{3}{2}} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.207315, size = 43, normalized size = 1.48 \[ -\frac{{\left (3 \, b^{2} x^{2} + 3 \, a b x + a^{2}\right )} \sqrt{c x^{2}}}{3 \, c^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.29853, size = 53, normalized size = 1.83 \[ - \frac{a^{2}}{3 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} - \frac{a b x}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} - \frac{b^{2} x^{2}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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