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

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

[Out]

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

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Rubi [A]  time = 0.0206731, antiderivative size = 35, 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 \[ \frac{a x^3}{2 \sqrt{c x^2}}+\frac{b x^4}{3 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0063251, size = 24, normalized size = 0.69 \[ \frac{x^3 (3 a+2 b x)}{6 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(3*a + 2*b*x))/(6*Sqrt[c*x^2])

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Maple [A]  time = 0.004, size = 21, normalized size = 0.6 \[{\frac{{x}^{3} \left ( 2\,bx+3\,a \right ) }{6}{\frac{1}{\sqrt{c{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.33397, size = 35, normalized size = 1. \[ \frac{\sqrt{c x^{2}} b x^{2}}{3 \, c} + \frac{a x^{2}}{2 \, \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.195584, size = 31, normalized size = 0.89 \[ \frac{{\left (2 \, b x^{2} + 3 \, a x\right )} \sqrt{c x^{2}}}{6 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*b*x^2 + 3*a*x)*sqrt(c*x^2)/c

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Sympy [A]  time = 1.85788, size = 36, normalized size = 1.03 \[ \frac{a x^{3}}{2 \sqrt{c} \sqrt{x^{2}}} + \frac{b x^{4}}{3 \sqrt{c} \sqrt{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.210782, size = 32, normalized size = 0.91 \[ \frac{1}{6} \, \sqrt{c x^{2}}{\left (\frac{2 \, b x}{c} + \frac{3 \, a}{c}\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(c*x^2)*(2*b*x/c + 3*a/c)*x