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

Optimal. Leaf size=24 \[ \frac{x (a+b x)^3}{3 b \sqrt{c x^2}} \]

[Out]

(x*(a + b*x)^3)/(3*b*Sqrt[c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0033883, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 32} \[ \frac{x (a+b x)^3}{3 b \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*x)^2)/Sqrt[c*x^2],x]

[Out]

(x*(a + b*x)^3)/(3*b*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]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.001783, size = 24, normalized size = 1. \[ \frac{x (a+b x)^3}{3 b \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a + b*x)^3)/(3*b*Sqrt[c*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 31, normalized size = 1.3 \begin{align*}{\frac{{x}^{2} \left ({b}^{2}{x}^{2}+3\,abx+3\,{a}^{2} \right ) }{3}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.04516, size = 57, normalized size = 2.38 \begin{align*} \frac{\sqrt{c x^{2}} b^{2} x^{2}}{3 \, c} + \frac{a b x^{2}}{\sqrt{c}} + \frac{\sqrt{c x^{2}} a^{2}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.45534, size = 63, normalized size = 2.62 \begin{align*} \frac{{\left (b^{2} x^{2} + 3 \, a b x + 3 \, a^{2}\right )} \sqrt{c x^{2}}}{3 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 0.507174, size = 56, normalized size = 2.33 \begin{align*} \frac{a^{2} x^{2}}{\sqrt{c} \sqrt{x^{2}}} + \frac{a b x^{3}}{\sqrt{c} \sqrt{x^{2}}} + \frac{b^{2} x^{4}}{3 \sqrt{c} \sqrt{x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.06458, size = 49, normalized size = 2.04 \begin{align*} \frac{1}{3} \, \sqrt{c x^{2}}{\left ({\left (\frac{b^{2} x}{c} + \frac{3 \, a b}{c}\right )} x + \frac{3 \, a^{2}}{c}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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