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

Optimal. Leaf size=27 \[ a \sqrt{c x^2}+\frac{1}{2} b x \sqrt{c x^2} \]

[Out]

a*Sqrt[c*x^2] + (b*x*Sqrt[c*x^2])/2

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Rubi [A]  time = 0.0129056, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ a \sqrt{c x^2}+\frac{1}{2} b x \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

a*Sqrt[c*x^2] + (b*x*Sqrt[c*x^2])/2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00705819, size = 24, normalized size = 0.89 \[ \frac{c x^2 (2 a+b x)}{2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(c*x^2*(2*a + b*x))/(2*Sqrt[c*x^2])

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Maple [A]  time = 0.004, size = 17, normalized size = 0.6 \[{\frac{bx+2\,a}{2}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(c*x^2)^(1/2)/x,x)

[Out]

1/2*(b*x+2*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 = 0.204319, size = 22, normalized size = 0.81 \[ \frac{1}{2} \, \sqrt{c x^{2}}{\left (b x + 2 \, a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(c*x^2)*(b*x + 2*a)

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Sympy [A]  time = 0.542182, size = 29, normalized size = 1.07 \[ a \sqrt{c} \sqrt{x^{2}} + \frac{b \sqrt{c} x \sqrt{x^{2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.212554, size = 23, normalized size = 0.85 \[ \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} \sqrt{c}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^2 + 2*a*x)*sqrt(c)*sign(x)