3.1.40 \(\int \frac {1}{\sqrt {c+d (a+b x)}} \, dx\) [40]

Optimal. Leaf size=21 \[ \frac {2 \sqrt {c+d (a+b x)}}{b d} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 21, 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 \sqrt {d (a+b x)+c}}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[c + d*(a + b*x)],x]

[Out]

(2*Sqrt[c + d*(a + b*x)])/(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 \frac {1}{\sqrt {c+d (a+b x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {c+d x}} \, dx,x,a+b x\right )}{b}\\ &=\frac {2 \sqrt {c+d (a+b x)}}{b d}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {c+a d+b d x}}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[c + d*(a + b*x)],x]

[Out]

(2*Sqrt[c + a*d + b*d*x])/(b*d)

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Maple [A]
time = 0.09, size = 20, normalized size = 0.95

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 19, normalized size = 0.90 \begin {gather*} \frac {2 \, \sqrt {{\left (b x + a\right )} d + c}}{b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt((b*x + a)*d + c)/(b*d)

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Fricas [A]
time = 0.58, size = 19, normalized size = 0.90 \begin {gather*} \frac {2 \, \sqrt {b d x + a d + c}}{b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*d*x + a*d + c)/(b*d)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).
time = 0.61, size = 31, normalized size = 1.48 \begin {gather*} \begin {cases} \frac {x}{\sqrt {a d + c}} & \text {for}\: b = 0 \\\frac {x}{\sqrt {c}} & \text {for}\: d = 0 \\\frac {2 \sqrt {c + d \left (a + b x\right )}}{b d} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.21, size = 19, normalized size = 0.90 \begin {gather*} \frac {2 \, \sqrt {b d x + a d + c}}{b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*d*x + a*d + c)/(b*d)

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Mupad [B]
time = 0.11, size = 19, normalized size = 0.90 \begin {gather*} \frac {2\,\sqrt {c+d\,\left (a+b\,x\right )}}{b\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c + d*(a + b*x))^(1/2),x)

[Out]

(2*(c + d*(a + b*x))^(1/2))/(b*d)

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