3.769 \(\int \frac{(c x^2)^{3/2} (a+b x)}{x^2} \, dx\)

Optimal. Leaf size=35 \[ \frac{1}{2} a c x \sqrt{c x^2}+\frac{1}{3} b c x^2 \sqrt{c x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0084287, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 43} \[ \frac{1}{2} a c x \sqrt{c x^2}+\frac{1}{3} b c x^2 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]

Int[((c*x^2)^(3/2)*(a + b*x))/x^2,x]

[Out]

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

Mathematica [A]  time = 0.0020262, size = 23, normalized size = 0.66 \[ \frac{1}{6} c x \sqrt{c x^2} (3 a+2 b x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*Sqrt[c*x^2]*(3*a + 2*b*x))/6

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 21, normalized size = 0.6 \begin{align*}{\frac{2\,bx+3\,a}{6\,x} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.52435, size = 53, normalized size = 1.51 \begin{align*} \frac{1}{6} \,{\left (2 \, b c x^{2} + 3 \, a c x\right )} \sqrt{c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b*c*x^2 + 3*a*c*x)*sqrt(c*x^2)

________________________________________________________________________________________

Sympy [A]  time = 0.51171, size = 31, normalized size = 0.89 \begin{align*} \frac{a c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{2 x} + \frac{b c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2)**(3/2)*(b*x+a)/x**2,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.07138, size = 30, normalized size = 0.86 \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 )} c^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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