3.1392 \(\int (c+d x)^{3/2} \, dx\)

Optimal. Leaf size=16 \[ \frac{2 (c+d x)^{5/2}}{5 d} \]

[Out]

(2*(c + d*x)^(5/2))/(5*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0014835, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {32} \[ \frac{2 (c+d x)^{5/2}}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(3/2),x]

[Out]

(2*(c + d*x)^(5/2))/(5*d)

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 (c+d x)^{3/2} \, dx &=\frac{2 (c+d x)^{5/2}}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.0046322, size = 16, normalized size = 1. \[ \frac{2 (c+d x)^{5/2}}{5 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(3/2),x]

[Out]

(2*(c + d*x)^(5/2))/(5*d)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \begin{align*}{\frac{2}{5\,d} \left ( dx+c \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(d*x+c)^(5/2)/d

________________________________________________________________________________________

Maxima [A]  time = 0.964246, size = 16, normalized size = 1. \begin{align*} \frac{2 \,{\left (d x + c\right )}^{\frac{5}{2}}}{5 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(d*x + c)^(5/2)/d

________________________________________________________________________________________

Fricas [B]  time = 1.81845, size = 63, normalized size = 3.94 \begin{align*} \frac{2 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \sqrt{d x + c}}{5 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(d^2*x^2 + 2*c*d*x + c^2)*sqrt(d*x + c)/d

________________________________________________________________________________________

Sympy [A]  time = 0.057442, size = 12, normalized size = 0.75 \begin{align*} \frac{2 \left (c + d x\right )^{\frac{5}{2}}}{5 d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(3/2),x)

[Out]

2*(c + d*x)**(5/2)/(5*d)

________________________________________________________________________________________

Giac [A]  time = 1.06122, size = 16, normalized size = 1. \begin{align*} \frac{2 \,{\left (d x + c\right )}^{\frac{5}{2}}}{5 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(d*x + c)^(5/2)/d