\(\int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx\) [2076]

Optimal result

Integrand size = 15, antiderivative size = 27 \[ \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx=\frac {11}{2 \sqrt {1-2 x}}+\frac {5}{2} \sqrt {1-2 x} \]

[Out]

11/2/(1-2*x)^(1/2)+5/2*(1-2*x)^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 27, 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} \[ \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx=\frac {5}{2} \sqrt {1-2 x}+\frac {11}{2 \sqrt {1-2 x}} \]

[In]

Int[(3 + 5*x)/(1 - 2*x)^(3/2),x]

[Out]

11/(2*Sqrt[1 - 2*x]) + (5*Sqrt[1 - 2*x])/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 {11}{2 (1-2 x)^{3/2}}-\frac {5}{2 \sqrt {1-2 x}}\right ) \, dx \\ & = \frac {11}{2 \sqrt {1-2 x}}+\frac {5}{2} \sqrt {1-2 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.56 \[ \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx=\frac {8-5 x}{\sqrt {1-2 x}} \]

[In]

Integrate[(3 + 5*x)/(1 - 2*x)^(3/2),x]

[Out]

(8 - 5*x)/Sqrt[1 - 2*x]

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52

method	result	size
pseudoelliptic	\(\frac {8-5 x}{\sqrt {1-2 x}}\)	\(14\)
gosper	\(-\frac {-8+5 x}{\sqrt {1-2 x}}\)	\(15\)
risch	\(-\frac {-8+5 x}{\sqrt {1-2 x}}\)	\(15\)
derivativedivides	\(\frac {11}{2 \sqrt {1-2 x}}+\frac {5 \sqrt {1-2 x}}{2}\)	\(20\)
default	\(\frac {11}{2 \sqrt {1-2 x}}+\frac {5 \sqrt {1-2 x}}{2}\)	\(20\)
trager	\(\frac {\left (-8+5 x \right ) \sqrt {1-2 x}}{-1+2 x}\)	\(21\)
meijerg	\(-\frac {3 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-5 \sqrt {\pi }+\frac {5 \sqrt {\pi }\, \left (-8 x +8\right )}{8 \sqrt {1-2 x}}}{\sqrt {\pi }}\)	\(51\)

[In]

int((3+5*x)/(1-2*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

(8-5*x)/(1-2*x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx=\frac {{\left (5 \, x - 8\right )} \sqrt {-2 \, x + 1}}{2 \, x - 1} \]

[In]

integrate((3+5*x)/(1-2*x)^(3/2),x, algorithm="fricas")

[Out]

(5*x - 8)*sqrt(-2*x + 1)/(2*x - 1)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx=\frac {5 x \sqrt {1 - 2 x}}{2 x - 1} - \frac {8 \sqrt {1 - 2 x}}{2 x - 1} \]

[In]

integrate((3+5*x)/(1-2*x)**(3/2),x)

[Out]

5*x*sqrt(1 - 2*x)/(2*x - 1) - 8*sqrt(1 - 2*x)/(2*x - 1)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx=\frac {5}{2} \, \sqrt {-2 \, x + 1} + \frac {11}{2 \, \sqrt {-2 \, x + 1}} \]

[In]

integrate((3+5*x)/(1-2*x)^(3/2),x, algorithm="maxima")

[Out]

5/2*sqrt(-2*x + 1) + 11/2/sqrt(-2*x + 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx=\frac {5}{2} \, \sqrt {-2 \, x + 1} + \frac {11}{2 \, \sqrt {-2 \, x + 1}} \]

[In]

integrate((3+5*x)/(1-2*x)^(3/2),x, algorithm="giac")

[Out]

5/2*sqrt(-2*x + 1) + 11/2/sqrt(-2*x + 1)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx=-\frac {5\,x-8}{\sqrt {1-2\,x}} \]

[In]

int((5*x + 3)/(1 - 2*x)^(3/2),x)

[Out]

-(5*x - 8)/(1 - 2*x)^(1/2)