3.2837 \(\int (c (a+b x)^{3/2})^{2/3} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {247, 15, 30} \[ \frac {(a+b x) \left (c (a+b x)^{3/2}\right )^{2/3}}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[(c*(a + b*x)^(3/2))^(2/3),x]

[Out]

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

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]

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 34, normalized size = 1.26 \[ \frac {x (2 a+b x) \left (c (a+b x)^{3/2}\right )^{2/3}}{2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.86, size = 37, normalized size = 1.37 \[ \frac {{\left (b x^{2} + 2 \, a x\right )} \left ({\left (b c x + a c\right )} \sqrt {b x + a}\right )^{\frac {2}{3}}}{2 \, {\left (b x + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.24, size = 15, normalized size = 0.56 \[ \frac {{\left (b x + a\right )}^{2} c^{\frac {2}{3}}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x + a)^2*c^(2/3)/b

________________________________________________________________________________________

maple [A]  time = 0.00, size = 29, normalized size = 1.07 \[ \frac {\left (b x +2 a \right ) \left (\left (b x +a \right )^{\frac {3}{2}} c \right )^{\frac {2}{3}} x}{2 b x +2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 21, normalized size = 0.78 \[ \frac {\left ({\left (b x + a\right )}^{\frac {3}{2}} c\right )^{\frac {2}{3}} {\left (b x + a\right )}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\left (c\,{\left (a+b\,x\right )}^{3/2}\right )}^{2/3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*(a + b*x)^(3/2))^(2/3),x)

[Out]

int((c*(a + b*x)^(3/2))^(2/3), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \left (a + b x\right )^{\frac {3}{2}}\right )^{\frac {2}{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________