3.14.80 \(\int \sqrt {c+d x} \, dx\) [1380]

Optimal. Leaf size=16 \[ \frac {2 (c+d x)^{3/2}}{3 d} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2 (c+d x)^{3/2}}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x],x]

[Out]

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

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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 (c+d x)^{3/2}}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*x],x]

[Out]

(2*(c + d*x)^(3/2))/(3*d)

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Maple [A]
time = 0.12, size = 13, normalized size = 0.81

method	result	size
gosper	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3 d}\)	\(13\)
derivativedivides	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3 d}\)	\(13\)
default	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3 d}\)	\(13\)
trager	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3 d}\)	\(13\)
risch	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3 d}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2),x, algorithm="maxima")

[Out]

2/3*(d*x + c)^(3/2)/d

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Fricas [A]
time = 0.69, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2),x, algorithm="fricas")

[Out]

2/3*(d*x + c)^(3/2)/d

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Sympy [A]
time = 0.01, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \left (c + d x\right )^{\frac {3}{2}}}{3 d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(1/2),x)

[Out]

2*(c + d*x)**(3/2)/(3*d)

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Giac [A]
time = 1.54, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2),x, algorithm="giac")

[Out]

2/3*(d*x + c)^(3/2)/d

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Mupad [B]
time = 0.02, size = 12, normalized size = 0.75 \begin {gather*} \frac {2\,{\left (c+d\,x\right )}^{3/2}}{3\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(1/2),x)

[Out]

(2*(c + d*x)^(3/2))/(3*d)

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