\(\int \frac {a+b x}{\sqrt {c+d x}} \, dx\) [1417]

Optimal result

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

[Out]

2/3*b*(d*x+c)^(3/2)/d^2-2*(-a*d+b*c)*(d*x+c)^(1/2)/d^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} \[ \int \frac {a+b x}{\sqrt {c+d x}} \, dx=\frac {2 b (c+d x)^{3/2}}{3 d^2}-\frac {2 \sqrt {c+d x} (b c-a d)}{d^2} \]

[In]

Int[(a + b*x)/Sqrt[c + d*x],x]

[Out]

(-2*(b*c - a*d)*Sqrt[c + d*x])/d^2 + (2*b*(c + d*x)^(3/2))/(3*d^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 {-b c+a d}{d \sqrt {c+d x}}+\frac {b \sqrt {c+d x}}{d}\right ) \, dx \\ & = -\frac {2 (b c-a d) \sqrt {c+d x}}{d^2}+\frac {2 b (c+d x)^{3/2}}{3 d^2} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[(a + b*x)/Sqrt[c + d*x],x]

[Out]

(2*Sqrt[c + d*x]*(-2*b*c + 3*a*d + b*d*x))/(3*d^2)

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.65

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

[In]

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

[Out]

2/3*(d*x+c)^(1/2)*(b*d*x+3*a*d-2*b*c)/d^2

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.62 \[ \int \frac {a+b x}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (b d x - 2 \, b c + 3 \, a d\right )} \sqrt {d x + c}}{3 \, d^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.32 \[ \int \frac {a+b x}{\sqrt {c+d x}} \, dx=\begin {cases} \frac {2 a \sqrt {c + d x} + \frac {2 b \left (- c \sqrt {c + d x} + \frac {\left (c + d x\right )^{\frac {3}{2}}}{3}\right )}{d}}{d} & \text {for}\: d \neq 0 \\\frac {a x + \frac {b x^{2}}{2}}{\sqrt {c}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {a+b x}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {d x + c} a + \frac {{\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} b}{d}\right )}}{3 \, d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

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

[In]

int((a + b*x)/(c + d*x)^(1/2),x)

[Out]

(2*(c + d*x)^(1/2)*(3*a*d - 3*b*c + b*(c + d*x)))/(3*d^2)