3.2838 \(\int (c (a+b x)^{2/3})^{3/2} \, dx\)

Optimal. Leaf size=32 \[ \frac{c (a+b x)^{5/3} \sqrt{c (a+b x)^{2/3}}}{2 b} \]

[Out]

(c*Sqrt[c*(a + b*x)^(2/3)]*(a + b*x)^(5/3))/(2*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0097607, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {247, 15, 30} \[ \frac{c (a+b x)^{5/3} \sqrt{c (a+b x)^{2/3}}}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[(c*(a + b*x)^(2/3))^(3/2),x]

[Out]

(c*Sqrt[c*(a + b*x)^(2/3)]*(a + b*x)^(5/3))/(2*b)

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, 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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \left (c (a+b x)^{2/3}\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (c x^{2/3}\right )^{3/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c \sqrt{c (a+b x)^{2/3}}\right ) \operatorname{Subst}(\int x \, dx,x,a+b x)}{b \sqrt [3]{a+b x}}\\ &=\frac{c \sqrt{c (a+b x)^{2/3}} (a+b x)^{5/3}}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0166755, size = 34, normalized size = 1.06 \[ \frac{x (2 a+b x) \left (c (a+b x)^{2/3}\right )^{3/2}}{2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 29, normalized size = 0.9 \begin{align*}{\frac{x \left ( bx+2\,a \right ) }{2\,bx+2\,a} \left ( c \left ( bx+a \right ) ^{{\frac{2}{3}}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.06698, size = 20, normalized size = 0.62 \begin{align*} \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x^2 + 2*a*x)*c^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 1.27893, size = 45, normalized size = 1.41 \begin{align*} \frac{1}{2} \,{\left (b c x^{2} + 2 \, a c x\right )} \sqrt{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*c*x^2 + 2*a*c*x)*sqrt(c)

________________________________________________________________________________________

Sympy [A]  time = 6.06757, size = 109, normalized size = 3.41 \begin{align*} \begin{cases} \frac{2 a^{3} c^{\frac{3}{2}}}{2 a b + 2 b^{2} x} + \frac{4 a^{2} b c^{\frac{3}{2}} x}{2 a b + 2 b^{2} x} + \frac{3 a b^{2} c^{\frac{3}{2}} x^{2}}{2 a b + 2 b^{2} x} + \frac{b^{3} c^{\frac{3}{2}} x^{3}}{2 a b + 2 b^{2} x} & \text{for}\: b \neq 0 \\x \left (a^{\frac{2}{3}} c\right )^{\frac{3}{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*a**3*c**(3/2)/(2*a*b + 2*b**2*x) + 4*a**2*b*c**(3/2)*x/(2*a*b + 2*b**2*x) + 3*a*b**2*c**(3/2)*x**
2/(2*a*b + 2*b**2*x) + b**3*c**(3/2)*x**3/(2*a*b + 2*b**2*x), Ne(b, 0)), (x*(a**(2/3)*c)**(3/2), True))

________________________________________________________________________________________

Giac [A]  time = 1.09666, size = 20, normalized size = 0.62 \begin{align*} \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x^2 + 2*a*x)*c^(3/2)