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

Optimal. Leaf size=58 \[ \frac{1}{2} a^2 c x \sqrt{c x^2}+\frac{2}{3} a b c x^2 \sqrt{c x^2}+\frac{1}{4} b^2 c x^3 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0342071, antiderivative size = 58, 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 \[ \frac{1}{2} a^2 c x \sqrt{c x^2}+\frac{2}{3} a b c x^2 \sqrt{c x^2}+\frac{1}{4} b^2 c x^3 \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} c \sqrt{c x^{2}} \int x\, dx}{x} + \frac{2 a b c x^{2} \sqrt{c x^{2}}}{3} + \frac{b^{2} c x^{3} \sqrt{c x^{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00370924, size = 34, normalized size = 0.59 \[ \frac{1}{12} c x \sqrt{c x^2} \left (6 a^2+8 a b x+3 b^2 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 32, normalized size = 0.6 \[{\frac{3\,{b}^{2}{x}^{2}+8\,abx+6\,{a}^{2}}{12\,x} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12/x*(3*b^2*x^2+8*a*b*x+6*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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.205095, size = 46, normalized size = 0.79 \[ \frac{1}{12} \,{\left (3 \, b^{2} c x^{3} + 8 \, a b c x^{2} + 6 \, a^{2} c x\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.3194, size = 54, normalized size = 0.93 \[ \frac{a^{2} c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{2 x} + \frac{2 a b c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{3} + \frac{b^{2} c^{\frac{3}{2}} x \left (x^{2}\right )^{\frac{3}{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.204901, size = 47, normalized size = 0.81 \[ \frac{1}{12} \,{\left (3 \, b^{2} x^{4}{\rm sign}\left (x\right ) + 8 \, a b x^{3}{\rm sign}\left (x\right ) + 6 \, a^{2} x^{2}{\rm sign}\left (x\right )\right )} c^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(3*b^2*x^4*sign(x) + 8*a*b*x^3*sign(x) + 6*a^2*x^2*sign(x))*c^(3/2)