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

Optimal. Leaf size=32 \[ \frac{a x^2}{\sqrt{c x^2}}+\frac{b x^3}{2 \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0042455, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {15} \[ \frac{a x^2}{\sqrt{c x^2}}+\frac{b x^3}{2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*x))/Sqrt[c*x^2],x]

[Out]

(a*x^2)/Sqrt[c*x^2] + (b*x^3)/(2*Sqrt[c*x^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{x (a+b x)}{\sqrt{c x^2}} \, dx &=\frac{x \int (a+b x) \, dx}{\sqrt{c x^2}}\\ &=\frac{a x^2}{\sqrt{c x^2}}+\frac{b x^3}{2 \sqrt{c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0014175, size = 23, normalized size = 0.72 \[ \frac{x^2 (2 a+b x)}{2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(2*a + b*x))/(2*Sqrt[c*x^2])

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Maple [A]  time = 0.002, size = 20, normalized size = 0.6 \begin{align*}{\frac{{x}^{2} \left ( bx+2\,a \right ) }{2}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00629, size = 30, normalized size = 0.94 \begin{align*} \frac{b x^{2}}{2 \, \sqrt{c}} + \frac{\sqrt{c x^{2}} a}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2/sqrt(c) + sqrt(c*x^2)*a/c

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Fricas [A]  time = 1.66548, size = 42, normalized size = 1.31 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x + 2 \, a\right )}}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(c*x^2)*(b*x + 2*a)/c

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Sympy [A]  time = 0.437677, size = 34, normalized size = 1.06 \begin{align*} \frac{a x^{2}}{\sqrt{c} \sqrt{x^{2}}} + \frac{b x^{3}}{2 \sqrt{c} \sqrt{x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07768, size = 30, normalized size = 0.94 \begin{align*} \frac{1}{2} \, \sqrt{c x^{2}}{\left (\frac{b x}{c} + \frac{2 \, a}{c}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(c*x^2)*(b*x/c + 2*a/c)