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]

1/2*c*(b*x+a)^(5/3)*(c*(b*x+a)^(2/3))^(1/2)/b

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Rubi [A]  time = 0.01, antiderivative size = 32, 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} \[ \frac {c (a+b x)^{5/3} \sqrt {c (a+b x)^{2/3}}}{2 b} \]

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 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)^{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*}

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Mathematica [A]  time = 0.02, 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))

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fricas [A]  time = 0.80, size = 17, normalized size = 0.53 \[ \frac {1}{2} \, {\left (b c x^{2} + 2 \, a c x\right )} \sqrt {c} \]

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)

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giac [A]  time = 0.15, size = 15, normalized size = 0.47 \[ \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c^{\frac {3}{2}} \]

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)

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maple [A]  time = 0.00, size = 29, normalized size = 0.91 \[ \frac {\left (b x +2 a \right ) \left (\left (b x +a \right )^{\frac {2}{3}} c \right )^{\frac {3}{2}} x}{2 b x +2 a} \]

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)

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maxima [A]  time = 0.52, size = 15, normalized size = 0.47 \[ \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c^{\frac {3}{2}} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (c\,{\left (a+b\,x\right )}^{2/3}\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*(a + b*x)^(2/3))^(3/2),x)

[Out]

int((c*(a + b*x)^(2/3))^(3/2), x)

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sympy [A]  time = 4.60, size = 65, normalized size = 2.03 \[ \begin {cases} \frac {2 a^{2} c^{\frac {3}{2}} x}{2 a + 2 b x} + \frac {3 a b c^{\frac {3}{2}} x^{2}}{2 a + 2 b x} + \frac {b^{2} c^{\frac {3}{2}} x^{3}}{2 a + 2 b x} & \text {for}\: a \neq 0 \vee b \neq 0 \\0 & \text {otherwise} \end {cases} \]

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

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