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

Optimal. Leaf size=33 \[ \frac{1}{2} a x \sqrt{c x^2}+\frac{1}{3} b x^2 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0078348, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {15, 43} \[ \frac{1}{2} a x \sqrt{c x^2}+\frac{1}{3} b x^2 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x^2]*(a + b*x),x]

[Out]

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

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \sqrt{c x^2} (a+b x) \, dx &=\frac{\sqrt{c x^2} \int x (a+b x) \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (a x+b x^2\right ) \, dx}{x}\\ &=\frac{1}{2} a x \sqrt{c x^2}+\frac{1}{3} b x^2 \sqrt{c x^2}\\ \end{align*}

Mathematica [A]  time = 0.0029974, size = 22, normalized size = 0.67 \[ \frac{1}{6} x \sqrt{c x^2} (3 a+2 b x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c*x^2]*(a + b*x),x]

[Out]

(x*Sqrt[c*x^2]*(3*a + 2*b*x))/6

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Maple [A]  time = 0.004, size = 19, normalized size = 0.6 \begin{align*}{\frac{x \left ( 2\,bx+3\,a \right ) }{6}\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)

[Out]

1/6*x*(2*b*x+3*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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.57154, size = 47, normalized size = 1.42 \begin{align*} \frac{1}{6} \,{\left (2 \, b x^{2} + 3 \, a x\right )} \sqrt{c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b*x^2 + 3*a*x)*sqrt(c*x^2)

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Sympy [A]  time = 0.193054, size = 34, normalized size = 1.03 \begin{align*} \frac{a \sqrt{c} x \sqrt{x^{2}}}{2} + \frac{b \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)*(c*x**2)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.04885, size = 30, normalized size = 0.91 \begin{align*} \frac{1}{6} \,{\left (2 \, b x^{3} \mathrm{sgn}\left (x\right ) + 3 \, a x^{2} \mathrm{sgn}\left (x\right )\right )} \sqrt{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b*x^3*sgn(x) + 3*a*x^2*sgn(x))*sqrt(c)