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

Optimal. Leaf size=26 \[ \frac{\sqrt{c x^2} (a+b x)^3}{3 b x} \]

[Out]

(Sqrt[c*x^2]*(a + b*x)^3)/(3*b*x)

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Rubi [A]  time = 0.0036536, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 32} \[ \frac{\sqrt{c x^2} (a+b x)^3}{3 b x} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c*x^2]*(a + b*x)^2)/x,x]

[Out]

(Sqrt[c*x^2]*(a + b*x)^3)/(3*b*x)

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

Mathematica [A]  time = 0.005424, size = 25, normalized size = 0.96 \[ \frac{c x (a+b x)^3}{3 b \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(a + b*x)^3)/(3*b*Sqrt[c*x^2])

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Maple [A]  time = 0.003, size = 28, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{x}^{2}+3\,abx+3\,{a}^{2}}{3}\sqrt{c{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^2*x^2+3*a*b*x+3*a^2)*(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)^2*(c*x^2)^(1/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.55154, size = 61, normalized size = 2.35 \begin{align*} \frac{1}{3} \,{\left (b^{2} x^{2} + 3 \, a b x + 3 \, a^{2}\right )} \sqrt{c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.258034, size = 51, normalized size = 1.96 \begin{align*} a^{2} \sqrt{c} \sqrt{x^{2}} + a b \sqrt{c} x \sqrt{x^{2}} + \frac{b^{2} \sqrt{c} x^{2} \sqrt{x^{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.04565, size = 39, normalized size = 1.5 \begin{align*} \frac{1}{3} \,{\left (\frac{{\left (b x + a\right )}^{3} \mathrm{sgn}\left (x\right )}{b} - \frac{a^{3} \mathrm{sgn}\left (x\right )}{b}\right )} \sqrt{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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