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

Optimal. Leaf size=27 \[ \frac{x (a+b x)^3}{3 b c \sqrt{c x^2}} \]

[Out]

(x*(a + b*x)^3)/(3*b*c*Sqrt[c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0042818, antiderivative size = 27, 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} \[ \frac{x (a+b x)^3}{3 b c \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0041706, size = 26, normalized size = 0.96 \[ \frac{x^3 (a+b x)^3}{3 b \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(a + b*x)^3)/(3*b*(c*x^2)^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 31, normalized size = 1.2 \begin{align*}{\frac{{x}^{4} \left ({b}^{2}{x}^{2}+3\,abx+3\,{a}^{2} \right ) }{3} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^4*(b^2*x^2+3*a*b*x+3*a^2)/(c*x^2)^(3/2)

________________________________________________________________________________________

Maxima [B]  time = 1.06627, size = 70, normalized size = 2.59 \begin{align*} \frac{b^{2} x^{4}}{3 \, \sqrt{c x^{2}} c} + \frac{a b x^{3}}{\sqrt{c x^{2}} c} + \frac{a^{2} x^{2}}{\sqrt{c x^{2}} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2*x^4/(sqrt(c*x^2)*c) + a*b*x^3/(sqrt(c*x^2)*c) + a^2*x^2/(sqrt(c*x^2)*c)

________________________________________________________________________________________

Fricas [A]  time = 1.49582, size = 66, normalized size = 2.44 \begin{align*} \frac{{\left (b^{2} x^{2} + 3 \, a b x + 3 \, a^{2}\right )} \sqrt{c x^{2}}}{3 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 0.758949, size = 56, normalized size = 2.07 \begin{align*} \frac{a^{2} x^{4}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} + \frac{a b x^{5}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} + \frac{b^{2} x^{6}}{3 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**4/(c**(3/2)*(x**2)**(3/2)) + a*b*x**5/(c**(3/2)*(x**2)**(3/2)) + b**2*x**6/(3*c**(3/2)*(x**2)**(3/2))

________________________________________________________________________________________

Giac [A]  time = 1.07006, size = 53, normalized size = 1.96 \begin{align*} \frac{\sqrt{c x^{2}}{\left ({\left (\frac{b^{2} x}{c} + \frac{3 \, a b}{c}\right )} x + \frac{3 \, a^{2}}{c}\right )}}{3 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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