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

Optimal. Leaf size=35 \[ \frac{1}{3} a x^2 \sqrt{c x^2}+\frac{1}{4} b x^3 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0243507, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{1}{3} a x^2 \sqrt{c x^2}+\frac{1}{4} b x^3 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[c*x^2]*(a + b*x),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*sqrt(c*x**2)*(a + b*x), x)

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Mathematica [A]  time = 0.00500997, size = 24, normalized size = 0.69 \[ \frac{1}{12} x^2 \sqrt{c x^2} (4 a+3 b x) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*Sqrt[c*x^2]*(4*a + 3*b*x))/12

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Maple [A]  time = 0.004, size = 21, normalized size = 0.6 \[{\frac{{x}^{2} \left ( 3\,bx+4\,a \right ) }{12}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)*(c*x^2)^(1/2),x)

[Out]

1/12*x^2*(3*b*x+4*a)*(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)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.199645, size = 30, normalized size = 0.86 \[ \frac{1}{12} \,{\left (3 \, b x^{3} + 4 \, a x^{2}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(3*b*x^3 + 4*a*x^2)*sqrt(c*x^2)

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Sympy [A]  time = 0.685655, size = 36, normalized size = 1.03 \[ \frac{a \sqrt{c} x^{2} \sqrt{x^{2}}}{3} + \frac{b \sqrt{c} x^{3} \sqrt{x^{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.206327, size = 30, normalized size = 0.86 \[ \frac{1}{12} \,{\left (3 \, b x^{4}{\rm sign}\left (x\right ) + 4 \, a x^{3}{\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(3*b*x^4*sign(x) + 4*a*x^3*sign(x))*sqrt(c)