3.1.39 \(\int \sqrt {c+d (a+b x)} \, dx\) [39]

Optimal. Leaf size=23 \[ \frac {2 (c+d (a+b x))^{3/2}}{3 b d} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {33, 32} \begin {gather*} \frac {2 (d (a+b x)+c)^{3/2}}{3 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*(a + b*x)],x]

[Out]

(2*(c + d*(a + b*x))^(3/2))/(3*b*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]

Rule 33

Int[((a_.) + (b_.)*(u_))^(m_), x_Symbol] :> Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x)^m, x], x, u], x]
/; FreeQ[{a, b, m}, x] && LinearQ[u, x] && NeQ[u, x]

Rubi steps

\begin {align*} \int \sqrt {c+d (a+b x)} \, dx &=\frac {\text {Subst}\left (\int \sqrt {c+d x} \, dx,x,a+b x\right )}{b}\\ &=\frac {2 (c+d (a+b x))^{3/2}}{3 b d}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 (c+a d+b d x)^{3/2}}{3 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*(a + b*x)],x]

[Out]

(2*(c + a*d + b*d*x)^(3/2))/(3*b*d)

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Maple [A]
time = 0.09, size = 20, normalized size = 0.87

method	result	size
gosper	\(\frac {2 \left (b d x +a d +c \right )^{\frac {3}{2}}}{3 b d}\)	\(20\)
derivativedivides	\(\frac {2 \left (b d x +a d +c \right )^{\frac {3}{2}}}{3 b d}\)	\(20\)
default	\(\frac {2 \left (b d x +a d +c \right )^{\frac {3}{2}}}{3 b d}\)	\(20\)
trager	\(\frac {2 \left (b d x +a d +c \right )^{\frac {3}{2}}}{3 b d}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 19, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left ({\left (b x + a\right )} d + c\right )}^{\frac {3}{2}}}{3 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*((b*x + a)*d + c)^(3/2)/(b*d)

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Fricas [A]
time = 1.06, size = 19, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left (b d x + a d + c\right )}^{\frac {3}{2}}}{3 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b*d*x + a*d + c)^(3/2)/(b*d)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (17) = 34\).
time = 0.09, size = 78, normalized size = 3.39 \begin {gather*} \begin {cases} \sqrt {c} x & \text {for}\: d = 0 \wedge \left (b = 0 \vee d = 0\right ) \\x \sqrt {a d + c} & \text {for}\: b = 0 \\\frac {2 a \sqrt {a d + b d x + c}}{3 b} + \frac {2 x \sqrt {a d + b d x + c}}{3} + \frac {2 c \sqrt {a d + b d x + c}}{3 b d} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sqrt(c)*x, Eq(d, 0) & (Eq(b, 0) | Eq(d, 0))), (x*sqrt(a*d + c), Eq(b, 0)), (2*a*sqrt(a*d + b*d*x +
c)/(3*b) + 2*x*sqrt(a*d + b*d*x + c)/3 + 2*c*sqrt(a*d + b*d*x + c)/(3*b*d), True))

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Giac [A]
time = 1.83, size = 19, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left (b d x + a d + c\right )}^{\frac {3}{2}}}{3 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b*d*x + a*d + c)^(3/2)/(b*d)

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Mupad [B]
time = 0.08, size = 19, normalized size = 0.83 \begin {gather*} \frac {2\,{\left (c+d\,\left (a+b\,x\right )\right )}^{3/2}}{3\,b\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*(a + b*x))^(1/2),x)

[Out]

(2*(c + d*(a + b*x))^(3/2))/(3*b*d)

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