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

Optimal. Leaf size=57 \[ \frac{1}{3} a^2 x^2 \sqrt{c x^2}+\frac{1}{2} a b x^3 \sqrt{c x^2}+\frac{1}{5} b^2 x^4 \sqrt{c x^2} \]

[Out]

(a^2*x^2*Sqrt[c*x^2])/3 + (a*b*x^3*Sqrt[c*x^2])/2 + (b^2*x^4*Sqrt[c*x^2])/5

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Rubi [A]  time = 0.0365248, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{3} a^2 x^2 \sqrt{c x^2}+\frac{1}{2} a b x^3 \sqrt{c x^2}+\frac{1}{5} b^2 x^4 \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*x^2*Sqrt[c*x^2])/3 + (a*b*x^3*Sqrt[c*x^2])/2 + (b^2*x^4*Sqrt[c*x^2])/5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{c x^{2}} \left (a + b x\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00772535, size = 35, normalized size = 0.61 \[ \frac{1}{30} x^2 \sqrt{c x^2} \left (10 a^2+15 a b x+6 b^2 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*Sqrt[c*x^2]*(10*a^2 + 15*a*b*x + 6*b^2*x^2))/30

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Maple [A]  time = 0.005, size = 32, normalized size = 0.6 \[{\frac{{x}^{2} \left ( 6\,{b}^{2}{x}^{2}+15\,abx+10\,{a}^{2} \right ) }{30}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*x^2*(6*b^2*x^2+15*a*b*x+10*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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.197611, size = 45, normalized size = 0.79 \[ \frac{1}{30} \,{\left (6 \, b^{2} x^{4} + 15 \, a b x^{3} + 10 \, a^{2} x^{2}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(6*b^2*x^4 + 15*a*b*x^3 + 10*a^2*x^2)*sqrt(c*x^2)

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Sympy [A]  time = 1.01472, size = 60, normalized size = 1.05 \[ \frac{a^{2} \sqrt{c} x^{2} \sqrt{x^{2}}}{3} + \frac{a b \sqrt{c} x^{3} \sqrt{x^{2}}}{2} + \frac{b^{2} \sqrt{c} x^{4} \sqrt{x^{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.207698, size = 47, normalized size = 0.82 \[ \frac{1}{30} \,{\left (6 \, b^{2} x^{5}{\rm sign}\left (x\right ) + 15 \, a b x^{4}{\rm sign}\left (x\right ) + 10 \, a^{2} x^{3}{\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(6*b^2*x^5*sign(x) + 15*a*b*x^4*sign(x) + 10*a^2*x^3*sign(x))*sqrt(c)