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 = 0.0040651, 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, Rules used = {15} \[ a \sqrt{c x^2}+\frac{1}{2} b x \sqrt{c x^2} \]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rubi steps

\begin{align*} \int \frac{\sqrt{c x^2} (a+b x)}{x} \, dx &=\frac{\sqrt{c x^2} \int (a+b x) \, dx}{x}\\ &=a \sqrt{c x^2}+\frac{1}{2} b x \sqrt{c x^2}\\ \end{align*}

Mathematica [A]  time = 0.0045681, 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 = 0.002, size = 17, normalized size = 0.6 \begin{align*}{\frac{bx+2\,a}{2}\sqrt{c{x}^{2}}} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(c*x^2)^(1/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.4784, size = 39, normalized size = 1.44 \begin{align*} \frac{1}{2} \, \sqrt{c x^{2}}{\left (b x + 2 \, a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(c*x^2)^(1/2)/x,x, algorithm="fricas")

[Out]

1/2*sqrt(c*x^2)*(b*x + 2*a)

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Sympy [A]  time = 0.194566, size = 29, normalized size = 1.07 \begin{align*} a \sqrt{c} \sqrt{x^{2}} + \frac{b \sqrt{c} x \sqrt{x^{2}}}{2} \end{align*}

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 [A]  time = 1.0545, size = 23, normalized size = 0.85 \begin{align*} \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} \sqrt{c} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(c*x^2)^(1/2)/x,x, algorithm="giac")

[Out]

1/2*(b*x^2 + 2*a*x)*sqrt(c)*sgn(x)