3.2.63 \(\int \frac {\alpha +x \beta }{\sqrt {a+b x}} \, dx\) [163]

Optimal. Leaf size=44 \[ \frac {2 \sqrt {a+b x} \alpha }{b}+\frac {2 \sqrt {a+b x} \left (-a+\frac {1}{3} (a+b x)\right ) \beta }{b^2} \]

[Out]

2*alpha*(b*x+a)^(1/2)/b+2*beta*(1/3*b*x-2/3*a)*(b*x+a)^(1/2)/b^2

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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 0.91, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \begin {gather*} \frac {2 \sqrt {a+b x} (\alpha b-a \beta )}{b^2}+\frac {2 \beta (a+b x)^{3/2}}{3 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(\[Alpha] + x*\[Beta])/Sqrt[a + b*x],x]

[Out]

(2*(a + b*x)^(3/2)*\[Beta])/(3*b^2) + (2*Sqrt[a + b*x]*(b*\[Alpha] - a*\[Beta]))/b^2

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\int \left (\frac {\sqrt {a+b x} \beta }{b}+\frac {b \alpha -a \beta }{b \sqrt {a+b x}}\right ) \, dx\\ &=\frac {2 (a+b x)^{3/2} \beta }{3 b^2}+\frac {2 \sqrt {a+b x} (b \alpha -a \beta )}{b^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 29, normalized size = 0.66 \begin {gather*} \frac {2 \sqrt {a+b x} (3 b \alpha -2 a \beta +b x \beta )}{3 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(\[Alpha] + x*\[Beta])/Sqrt[a + b*x],x]

[Out]

(2*Sqrt[a + b*x]*(3*b*\[Alpha] - 2*a*\[Beta] + b*x*\[Beta]))/(3*b^2)

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Maple [A]
time = 0.03, size = 38, normalized size = 0.86

method	result	size
gosper	\(-\frac {2 \sqrt {b x +a}\, \left (-b \beta x +2 a \beta -3 \alpha b \right )}{3 b^{2}}\)	\(27\)
trager	\(-\frac {2 \sqrt {b x +a}\, \left (-b \beta x +2 a \beta -3 \alpha b \right )}{3 b^{2}}\)	\(27\)
risch	\(-\frac {2 \sqrt {b x +a}\, \left (-b \beta x +2 a \beta -3 \alpha b \right )}{3 b^{2}}\)	\(27\)
pseudoelliptic	\(-\frac {2 \sqrt {b x +a}\, \left (-b \beta x +2 a \beta -3 \alpha b \right )}{3 b^{2}}\)	\(27\)
derivativedivides	\(\frac {\frac {2 \beta \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a \beta \sqrt {b x +a}+2 \alpha b \sqrt {b x +a}}{b^{2}}\)	\(38\)
default	\(\frac {\frac {2 \beta \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a \beta \sqrt {b x +a}+2 \alpha b \sqrt {b x +a}}{b^{2}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/b^2*(1/3*beta*(b*x+a)^(3/2)+alpha*b*(b*x+a)^(1/2)-a*beta*(b*x+a)^(1/2))

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Maxima [A]
time = 0.34, size = 39, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (3 \, \sqrt {b x + a} \alpha + \frac {{\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} \beta }{b}\right )}}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(3*sqrt(b*x + a)*alpha + ((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*beta/b)/b

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Fricas [A]
time = 0.57, size = 25, normalized size = 0.57 \begin {gather*} \frac {2 \, {\left (b \beta x + 3 \, \alpha b - 2 \, a \beta \right )} \sqrt {b x + a}}{3 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b*beta*x + 3*alpha*b - 2*a*beta)*sqrt(b*x + a)/b^2

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Sympy [A]
time = 0.41, size = 53, normalized size = 1.20 \begin {gather*} \begin {cases} \frac {2 \alpha \sqrt {a + b x} + \frac {2 \beta \left (- a \sqrt {a + b x} + \frac {\left (a + b x\right )^{\frac {3}{2}}}{3}\right )}{b}}{b} & \text {for}\: b \neq 0 \\\frac {\alpha x + \frac {\beta x^{2}}{2}}{\sqrt {a}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((2*ALPHA*sqrt(a + b*x) + 2*BETA*(-a*sqrt(a + b*x) + (a + b*x)**(3/2)/3)/b)/b, Ne(b, 0)), ((ALPHA*x
+ BETA*x**2/2)/sqrt(a), True))

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Giac [A]
time = 0.49, size = 39, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (3 \, \sqrt {b x + a} \alpha + \frac {{\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} \beta }{b}\right )}}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(3*sqrt(b*x + a)*alpha + ((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*beta/b)/b

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Mupad [B]
time = 0.19, size = 28, normalized size = 0.64 \begin {gather*} \frac {2\,\sqrt {a+b\,x}\,\left (3\,\alpha \,b+\left (a+b\,x\right )\,\beta -3\,a\,\beta \right )}{3\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(a + b*x)^(1/2)*(3*alpha*b + (a + b*x)*beta - 3*a*beta))/(3*b^2)

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