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

Optimal. Leaf size=66 \[ -\frac{a^2}{4 c x^3 \sqrt{c x^2}}-\frac{2 a b}{3 c x^2 \sqrt{c x^2}}-\frac{b^2}{2 c x \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0358915, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{a^2}{4 c x^3 \sqrt{c x^2}}-\frac{2 a b}{3 c x^2 \sqrt{c x^2}}-\frac{b^2}{2 c x \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.3988, size = 63, normalized size = 0.95 \[ - \frac{a^{2} \sqrt{c x^{2}}}{4 c^{2} x^{5}} - \frac{2 a b \sqrt{c x^{2}}}{3 c^{2} x^{4}} - \frac{b^{2} \sqrt{c x^{2}}}{2 c^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*sqrt(c*x**2)/(4*c**2*x**5) - 2*a*b*sqrt(c*x**2)/(3*c**2*x**4) - b**2*sqrt(
c*x**2)/(2*c**2*x**3)

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Mathematica [A]  time = 0.0165892, size = 38, normalized size = 0.58 \[ -\frac{\sqrt{c x^2} \left (3 a^2+8 a b x+6 b^2 x^2\right )}{12 c^2 x^5} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c*x^2]*(3*a^2 + 8*a*b*x + 6*b^2*x^2))/(12*c^2*x^5)

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Maple [A]  time = 0.006, size = 32, normalized size = 0.5 \[ -{\frac{6\,{b}^{2}{x}^{2}+8\,abx+3\,{a}^{2}}{12\,x} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(6*b^2*x^2+8*a*b*x+3*a^2)/x/(c*x^2)^(3/2)

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Maxima [A]  time = 1.39862, size = 45, normalized size = 0.68 \[ -\frac{b^{2}}{2 \, c^{\frac{3}{2}} x^{2}} - \frac{2 \, a b}{3 \, c^{\frac{3}{2}} x^{3}} - \frac{a^{2}}{4 \, c^{\frac{3}{2}} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.201984, size = 46, normalized size = 0.7 \[ -\frac{{\left (6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}\right )} \sqrt{c x^{2}}}{12 \, c^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(6*b^2*x^2 + 8*a*b*x + 3*a^2)*sqrt(c*x^2)/(c^2*x^5)

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Sympy [A]  time = 2.71032, size = 56, normalized size = 0.85 \[ - \frac{a^{2}}{4 c^{\frac{3}{2}} x \left (x^{2}\right )^{\frac{3}{2}}} - \frac{2 a b}{3 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} - \frac{b^{2} x}{2 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.523941, size = 4, normalized size = 0.06 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x