3.187 \(\int (a+b x)^{p/2} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 (a+b x)^{\frac{p+2}{2}}}{b (p+2)} \]

[Out]

(2*(a + b*x)^((2 + p)/2))/(b*(2 + p))

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Rubi [A]  time = 0.0036895, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {32} \[ \frac{2 (a+b x)^{\frac{p+2}{2}}}{b (p+2)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(p/2),x]

[Out]

(2*(a + b*x)^((2 + p)/2))/(b*(2 + p))

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

Mathematica [A]  time = 0.0118408, size = 24, normalized size = 1.04 \[ \frac{2 (a+b x)^{\frac{p}{2}+1}}{b p+2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(p/2),x]

[Out]

(2*(a + b*x)^(1 + p/2))/(2*b + b*p)

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Maple [A]  time = 0.001, size = 25, normalized size = 1.1 \begin{align*} 2\,{\frac{ \left ( bx+a \right ) \left ( \sqrt{bx+a} \right ) ^{p}}{b \left ( 2+p \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x+a)^(1/2))^p,x)

[Out]

2*(b*x+a)*((b*x+a)^(1/2))^p/b/(2+p)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)^(1/2))^p,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.70099, size = 55, normalized size = 2.39 \begin{align*} \frac{2 \,{\left (b x + a\right )} \sqrt{b x + a}^{p}}{b p + 2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)^(1/2))^p,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.058297, size = 26, normalized size = 1.13 \begin{align*} \frac{\begin{cases} \frac{\left (a + b x\right )^{\frac{p}{2} + 1}}{\frac{p}{2} + 1} & \text{for}\: \frac{p}{2} \neq -1 \\\log{\left (a + b x \right )} & \text{otherwise} \end{cases}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)**(1/2))**p,x)

[Out]

Piecewise(((a + b*x)**(p/2 + 1)/(p/2 + 1), Ne(p/2, -1)), (log(a + b*x), True))/b

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Giac [A]  time = 1.07583, size = 28, normalized size = 1.22 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{1}{2} \, p + 1}}{b{\left (p + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)^(1/2))^p,x, algorithm="giac")

[Out]

2*(b*x + a)^(1/2*p + 1)/(b*(p + 2))