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

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

[Out]

(2*(a + b*x)^(5/2))/(5*b)

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Rubi [A]  time = 0.0013847, 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} \[ \frac{2 (a+b x)^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(3/2),x]

[Out]

(2*(a + b*x)^(5/2))/(5*b)

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

Mathematica [A]  time = 0.0101099, size = 16, normalized size = 1. \[ \frac{2 (a+b x)^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(3/2),x]

[Out]

(2*(a + b*x)^(5/2))/(5*b)

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Maple [A]  time = 0.002, size = 13, normalized size = 0.8 \begin{align*}{\frac{2}{5\,b} \left ( bx+a \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(b*x+a)^(5/2)/b

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Maxima [A]  time = 1.06051, size = 16, normalized size = 1. \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(b*x + a)^(5/2)/b

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Fricas [B]  time = 1.45074, size = 63, normalized size = 3.94 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a)/b

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Sympy [A]  time = 0.069356, size = 12, normalized size = 0.75 \begin{align*} \frac{2 \left (a + b x\right )^{\frac{5}{2}}}{5 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(a + b*x)**(5/2)/(5*b)

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Giac [A]  time = 1.20759, size = 16, normalized size = 1. \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(b*x + a)^(5/2)/b