3.2.87 \(\int (a+b x)^{p/2} \, dx\) [187]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2 (a+b x)^{\frac {p+2}{2}}}{b (p+2)} \end {gather*}

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*}

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Mathematica [A]
time = 0.02, size = 24, normalized size = 1.04 \begin {gather*} \frac {2 (a+b x)^{1+\frac {p}{2}}}{2 b+b p} \end {gather*}

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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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: } \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[Sqrt[a+b*x]^p,x]')

[Out]

cought exception:

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Maple [A]
time = 0.03, size = 25, normalized size = 1.09

method	result	size
gosper	\(\frac {2 \left (b x +a \right ) \left (b x +a \right )^{\frac {p}{2}}}{b \left (2+p \right )}\)	\(25\)
risch	\(\frac {2 \left (b x +a \right ) \left (b x +a \right )^{\frac {p}{2}}}{b \left (2+p \right )}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 21, normalized size = 0.91 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {1}{2} \, p + 1}}{b {\left (p + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b*x + a)^(1/2*p + 1)/(b*(p + 2))

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Fricas [A]
time = 0.32, size = 25, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (b x + a\right )} \sqrt {b x + a}^{p}}{b p + 2 \, b} \end {gather*}

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 = 0.03, size = 24, normalized size = 1.04 \begin {gather*} \frac {\begin {cases} \frac {\left (a + b x\right )^{\frac {p}{2} + 1}}{\frac {p}{2} + 1} & \text {for}\: p \neq -2 \\\log {\left (a + b x \right )} & \text {otherwise} \end {cases}}{b} \end {gather*}

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)), (log(a + b*x), True))/b

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Giac [A]
time = 0.00, size = 22, normalized size = 0.96 \begin {gather*} \frac {\left (b x+a\right )^{\frac {p}{2}+1}}{b \left (\frac {p}{2}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b*x + a)^(1/2*p + 1)/(b*(p + 2))

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Mupad [B]
time = 0.25, size = 21, normalized size = 0.91 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{\frac {p}{2}+1}}{b\,\left (p+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(p/2),x)

[Out]

(2*(a + b*x)^(p/2 + 1))/(b*(p + 2))

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