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

Optimal. Leaf size=33 \[ \frac{1}{2} a x \sqrt{c x^2}+\frac{1}{3} b x^2 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.021011, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{2} a x \sqrt{c x^2}+\frac{1}{3} b x^2 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c*x^2]*(a + b*x),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00440649, size = 22, normalized size = 0.67 \[ \frac{1}{6} x \sqrt{c x^2} (3 a+2 b x) \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[c*x^2]*(3*a + 2*b*x))/6

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Maple [A]  time = 0.03, size = 19, normalized size = 0.6 \[{\frac{x \left ( 2\,bx+3\,a \right ) }{6}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(c*x^2)^(1/2),x)

[Out]

1/6*x*(2*b*x+3*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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.209206, size = 27, normalized size = 0.82 \[ \frac{1}{6} \,{\left (2 \, b x^{2} + 3 \, a x\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*b*x^2 + 3*a*x)*sqrt(c*x^2)

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Sympy [A]  time = 0.536257, size = 34, normalized size = 1.03 \[ \frac{a \sqrt{c} x \sqrt{x^{2}}}{2} + \frac{b \sqrt{c} x^{2} \sqrt{x^{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.206292, size = 30, normalized size = 0.91 \[ \frac{1}{6} \,{\left (2 \, b x^{3}{\rm sign}\left (x\right ) + 3 \, a x^{2}{\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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