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

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

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Rubi [A]  time = 0.00, antiderivative size = 32, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {a x^2}{\sqrt {c x^2}}+\frac {b x^3}{2 \sqrt {c x^2}} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 23, normalized size = 0.72 \begin {gather*} \frac {x^2 (2 a+b x)}{2 \sqrt {c x^2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 23, normalized size = 0.72 \begin {gather*} \frac {\sqrt {c x^2} (2 a+b x)}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c*x^2]*(2*a + b*x))/(2*c)

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fricas [A]  time = 1.18, size = 19, normalized size = 0.59 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x + 2 \, a\right )}}{2 \, c} \end {gather*}

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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giac [A]  time = 1.11, size = 22, normalized size = 0.69 \begin {gather*} \frac {1}{2} \, \sqrt {c x^{2}} {\left (\frac {b x}{c} + \frac {2 \, a}{c}\right )} \end {gather*}

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)

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maple [A]  time = 0.00, size = 20, normalized size = 0.62 \begin {gather*} \frac {\left (b x +2 a \right ) x^{2}}{2 \sqrt {c \,x^{2}}} \end {gather*}

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 = 1.30, size = 22, normalized size = 0.69 \begin {gather*} \frac {b x^{2}}{2 \, \sqrt {c}} + \frac {\sqrt {c x^{2}} a}{c} \end {gather*}

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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mupad [B]  time = 0.22, size = 19, normalized size = 0.59 \begin {gather*} \frac {2\,a\,\relax |x|+b\,x\,\sqrt {x^2}}{2\,\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a + b*x))/(c*x^2)^(1/2),x)

[Out]

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

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sympy [A]  time = 0.46, size = 34, normalized size = 1.06 \begin {gather*} \frac {a x^{2}}{\sqrt {c} \sqrt {x^{2}}} + \frac {b x^{3}}{2 \sqrt {c} \sqrt {x^{2}}} \end {gather*}

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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