\(\int \frac {A+B x}{\sqrt {a+b x}} \, dx\) [427]

Optimal result

Integrand size = 15, antiderivative size = 40 \[ \int \frac {A+B x}{\sqrt {a+b x}} \, dx=\frac {2 (A b-a B) \sqrt {a+b x}}{b^2}+\frac {2 B (a+b x)^{3/2}}{3 b^2} \]

[Out]

2/3*B*(b*x+a)^(3/2)/b^2+2*(A*b-B*a)*(b*x+a)^(1/2)/b^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \frac {A+B x}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} (A b-a B)}{b^2}+\frac {2 B (a+b x)^{3/2}}{3 b^2} \]

[In]

Int[(A + B*x)/Sqrt[a + b*x],x]

[Out]

(2*(A*b - a*B)*Sqrt[a + b*x])/b^2 + (2*B*(a + b*x)^(3/2))/(3*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{align*} \text {integral}& = \int \left (\frac {A b-a B}{b \sqrt {a+b x}}+\frac {B \sqrt {a+b x}}{b}\right ) \, dx \\ & = \frac {2 (A b-a B) \sqrt {a+b x}}{b^2}+\frac {2 B (a+b x)^{3/2}}{3 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72 \[ \int \frac {A+B x}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} (3 A b-2 a B+b B x)}{3 b^2} \]

[In]

Integrate[(A + B*x)/Sqrt[a + b*x],x]

[Out]

(2*Sqrt[a + b*x]*(3*A*b - 2*a*B + b*B*x))/(3*b^2)

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.65

method	result	size
gosper	\(\frac {2 \sqrt {b x +a}\, \left (b B x +3 A b -2 B a \right )}{3 b^{2}}\)	\(26\)
trager	\(\frac {2 \sqrt {b x +a}\, \left (b B x +3 A b -2 B a \right )}{3 b^{2}}\)	\(26\)
risch	\(\frac {2 \sqrt {b x +a}\, \left (b B x +3 A b -2 B a \right )}{3 b^{2}}\)	\(26\)
pseudoelliptic	\(\frac {2 \left (\left (\frac {B x}{3}+A \right ) b -\frac {2 B a}{3}\right ) \sqrt {b x +a}}{b^{2}}\)	\(26\)
derivativedivides	\(\frac {\frac {2 B \left (b x +a \right )^{\frac {3}{2}}}{3}+2 A b \sqrt {b x +a}-2 B a \sqrt {b x +a}}{b^{2}}\)	\(38\)
default	\(\frac {\frac {2 B \left (b x +a \right )^{\frac {3}{2}}}{3}+2 A b \sqrt {b x +a}-2 B a \sqrt {b x +a}}{b^{2}}\)	\(38\)

[In]

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

[Out]

2/3*(b*x+a)^(1/2)*(B*b*x+3*A*b-2*B*a)/b^2

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.62 \[ \int \frac {A+B x}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (B b x - 2 \, B a + 3 \, A b\right )} \sqrt {b x + a}}{3 \, b^{2}} \]

[In]

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

[Out]

2/3*(B*b*x - 2*B*a + 3*A*b)*sqrt(b*x + a)/b^2

Sympy [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.32 \[ \int \frac {A+B x}{\sqrt {a+b x}} \, dx=\begin {cases} \frac {2 A \sqrt {a + b x} + \frac {2 B \left (- a \sqrt {a + b x} + \frac {\left (a + b x\right )^{\frac {3}{2}}}{3}\right )}{b}}{b} & \text {for}\: b \neq 0 \\\frac {A x + \frac {B x^{2}}{2}}{\sqrt {a}} & \text {otherwise} \end {cases} \]

[In]

integrate((B*x+A)/(b*x+a)**(1/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {b x + a} A + \frac {{\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} B}{b}\right )}}{3 \, b} \]

[In]

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

[Out]

2/3*(3*sqrt(b*x + a)*A + ((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*B/b)/b

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {b x + a} A + \frac {{\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} B}{b}\right )}}{3 \, b} \]

[In]

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

[Out]

2/3*(3*sqrt(b*x + a)*A + ((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*B/b)/b

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.70 \[ \int \frac {A+B x}{\sqrt {a+b x}} \, dx=\frac {2\,\sqrt {a+b\,x}\,\left (3\,A\,b-3\,B\,a+B\,\left (a+b\,x\right )\right )}{3\,b^2} \]

[In]

int((A + B*x)/(a + b*x)^(1/2),x)

[Out]

(2*(a + b*x)^(1/2)*(3*A*b - 3*B*a + B*(a + b*x)))/(3*b^2)