\(\int \frac {a+b x}{(c+d x)^{3/2}} \, dx\) [1428]

Optimal result

Integrand size = 15, antiderivative size = 38 \[ \int \frac {a+b x}{(c+d x)^{3/2}} \, dx=\frac {2 (b c-a d)}{d^2 \sqrt {c+d x}}+\frac {2 b \sqrt {c+d x}}{d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 38, 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} \[ \int \frac {a+b x}{(c+d x)^{3/2}} \, dx=\frac {2 (b c-a d)}{d^2 \sqrt {c+d x}}+\frac {2 b \sqrt {c+d x}}{d^2} \]

[In]

Int[(a + b*x)/(c + d*x)^(3/2),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.71 \[ \int \frac {a+b x}{(c+d x)^{3/2}} \, dx=\frac {2 (2 b c-a d+b d x)}{d^2 \sqrt {c+d x}} \]

[In]

Integrate[(a + b*x)/(c + d*x)^(3/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.68

method	result	size
gosper	\(-\frac {2 \left (-b d x +a d -2 b c \right )}{\sqrt {d x +c}\, d^{2}}\)	\(26\)
trager	\(-\frac {2 \left (-b d x +a d -2 b c \right )}{\sqrt {d x +c}\, d^{2}}\)	\(26\)
pseudoelliptic	\(\frac {\left (2 b x -2 a \right ) d +4 b c}{\sqrt {d x +c}\, d^{2}}\)	\(27\)
derivativedivides	\(\frac {2 b \sqrt {d x +c}-\frac {2 \left (a d -b c \right )}{\sqrt {d x +c}}}{d^{2}}\)	\(33\)
default	\(\frac {2 b \sqrt {d x +c}-\frac {2 \left (a d -b c \right )}{\sqrt {d x +c}}}{d^{2}}\)	\(33\)
risch	\(\frac {2 b \sqrt {d x +c}}{d^{2}}-\frac {2 \left (a d -b c \right )}{d^{2} \sqrt {d x +c}}\)	\(35\)

[In]

int((b*x+a)/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \frac {a+b x}{(c+d x)^{3/2}} \, dx=\frac {2 \, {\left (b d x + 2 \, b c - a d\right )} \sqrt {d x + c}}{d^{3} x + c d^{2}} \]

[In]

integrate((b*x+a)/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.58 \[ \int \frac {a+b x}{(c+d x)^{3/2}} \, dx=\begin {cases} - \frac {2 a}{d \sqrt {c + d x}} + \frac {4 b c}{d^{2} \sqrt {c + d x}} + \frac {2 b x}{d \sqrt {c + d x}} & \text {for}\: d \neq 0 \\\frac {a x + \frac {b x^{2}}{2}}{c^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-2*a/(d*sqrt(c + d*x)) + 4*b*c/(d**2*sqrt(c + d*x)) + 2*b*x/(d*sqrt(c + d*x)), Ne(d, 0)), ((a*x + b
*x**2/2)/c**(3/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {a+b x}{(c+d x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {\sqrt {d x + c} b}{d} + \frac {b c - a d}{\sqrt {d x + c} d}\right )}}{d} \]

[In]

integrate((b*x+a)/(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {a+b x}{(c+d x)^{3/2}} \, dx=\frac {2 \, \sqrt {d x + c} b}{d^{2}} + \frac {2 \, {\left (b c - a d\right )}}{\sqrt {d x + c} d^{2}} \]

[In]

integrate((b*x+a)/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66 \[ \int \frac {a+b x}{(c+d x)^{3/2}} \, dx=\frac {4\,b\,c-2\,a\,d+2\,b\,d\,x}{d^2\,\sqrt {c+d\,x}} \]

[In]

int((a + b*x)/(c + d*x)^(3/2),x)

[Out]

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