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

Optimal. Leaf size=27 \[ \frac{c \sqrt{c x^2} (a+b x)^3}{3 b x} \]

[Out]

(c*Sqrt[c*x^2]*(a + b*x)^3)/(3*b*x)

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Rubi [A]  time = 0.0126876, antiderivative size = 27, 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 \[ \frac{c \sqrt{c x^2} (a+b x)^3}{3 b x} \]

Antiderivative was successfully verified.

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

[Out]

(c*Sqrt[c*x^2]*(a + b*x)^3)/(3*b*x)

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Rubi in Sympy [A]  time = 12.2651, size = 20, normalized size = 0.74 \[ \frac{c \sqrt{c x^{2}} \left (a + b x\right )^{3}}{3 b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*sqrt(c*x**2)*(a + b*x)**3/(3*b*x)

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Mathematica [A]  time = 0.00818805, size = 26, normalized size = 0.96 \[ \frac{\left (c x^2\right )^{3/2} (a+b x)^3}{3 b x^3} \]

Antiderivative was successfully verified.

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

[Out]

((c*x^2)^(3/2)*(a + b*x)^3)/(3*b*x^3)

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Maple [A]  time = 0.003, size = 31, normalized size = 1.2 \[{\frac{{b}^{2}{x}^{2}+3\,abx+3\,{a}^{2}}{3\,{x}^{2}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.206881, size = 41, normalized size = 1.52 \[ \frac{1}{3} \,{\left (b^{2} c x^{2} + 3 \, a b c x + 3 \, a^{2} c\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.21202, size = 51, normalized size = 1.89 \[ \frac{a^{2} c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{x^{2}} + \frac{a b c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{x} + \frac{b^{2} c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.206975, size = 39, normalized size = 1.44 \[ \frac{1}{3} \,{\left (\frac{{\left (b x + a\right )}^{3}{\rm sign}\left (x\right )}{b} - \frac{a^{3}{\rm sign}\left (x\right )}{b}\right )} c^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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