3.807 \(\int \sqrt{c x^2} (a+b x)^2 \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{2} a^2 x \sqrt{c x^2}+\frac{2}{3} a b x^2 \sqrt{c x^2}+\frac{1}{4} b^2 x^3 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.03524, antiderivative size = 55, 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 \[ \frac{1}{2} a^2 x \sqrt{c x^2}+\frac{2}{3} a b x^2 \sqrt{c x^2}+\frac{1}{4} b^2 x^3 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c*x^2]*(a + b*x)^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2*(c*x**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0102158, size = 33, normalized size = 0.6 \[ \frac{1}{12} x \sqrt{c x^2} \left (6 a^2+8 a b x+3 b^2 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(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 = 0.004, size = 30, normalized size = 0.6 \[{\frac{x \left ( 3\,{b}^{2}{x}^{2}+8\,abx+6\,{a}^{2} \right ) }{12}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(c*x^2)^(1/2),x)

[Out]

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

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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(sqrt(c*x^2)*(b*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.195615, size = 42, normalized size = 0.76 \[ \frac{1}{12} \,{\left (3 \, b^{2} x^{3} + 8 \, a b x^{2} + 6 \, a^{2} x\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.87286, size = 60, normalized size = 1.09 \[ \frac{a^{2} \sqrt{c} x \sqrt{x^{2}}}{2} + \frac{2 a b \sqrt{c} x^{2} \sqrt{x^{2}}}{3} + \frac{b^{2} \sqrt{c} x^{3} \sqrt{x^{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2*(c*x**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.208969, size = 47, normalized size = 0.85 \[ \frac{1}{12} \,{\left (3 \, b^{2} x^{4}{\rm sign}\left (x\right ) + 8 \, a b x^{3}{\rm sign}\left (x\right ) + 6 \, a^{2} x^{2}{\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2)*(b*x + a)^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))*sqrt(c)