\(\int \frac {3+5 x}{(1-2 x)^{5/2}} \, dx\) [2140]

Optimal result

Integrand size = 15, antiderivative size = 27 \[ \int \frac {3+5 x}{(1-2 x)^{5/2}} \, dx=\frac {11}{6 (1-2 x)^{3/2}}-\frac {5}{2 \sqrt {1-2 x}} \]

[Out]

11/6/(1-2*x)^(3/2)-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} \[ \int \frac {3+5 x}{(1-2 x)^{5/2}} \, dx=\frac {11}{6 (1-2 x)^{3/2}}-\frac {5}{2 \sqrt {1-2 x}} \]

[In]

Int[(3 + 5*x)/(1 - 2*x)^(5/2),x]

[Out]

11/(6*(1 - 2*x)^(3/2)) - 5/(2*Sqrt[1 - 2*x])

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)^{5/2}}-\frac {5}{2 (1-2 x)^{3/2}}\right ) \, dx \\ & = \frac {11}{6 (1-2 x)^{3/2}}-\frac {5}{2 \sqrt {1-2 x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \frac {3+5 x}{(1-2 x)^{5/2}} \, dx=\frac {-2+15 x}{3 (1-2 x)^{3/2}} \]

[In]

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

[Out]

(-2 + 15*x)/(3*(1 - 2*x)^(3/2))

Maple [A] (verified)

Time = 3.38 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.56

method	result	size
gosper	\(\frac {-2+15 x}{3 \left (1-2 x \right )^{\frac {3}{2}}}\)	\(15\)
pseudoelliptic	\(\frac {-2+15 x}{3 \left (1-2 x \right )^{\frac {3}{2}}}\)	\(15\)
derivativedivides	\(\frac {11}{6 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {5}{2 \sqrt {1-2 x}}\)	\(20\)
default	\(\frac {11}{6 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {5}{2 \sqrt {1-2 x}}\)	\(20\)
trager	\(\frac {\left (-2+15 x \right ) \sqrt {1-2 x}}{3 \left (-1+2 x \right )^{2}}\)	\(22\)
risch	\(-\frac {-2+15 x}{3 \left (-1+2 x \right ) \sqrt {1-2 x}}\)	\(22\)
meijerg	\(-\frac {2 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {\frac {5 \sqrt {\pi }}{3}-\frac {5 \sqrt {\pi }\, \left (-24 x +8\right )}{24 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}\)	\(51\)

[In]

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

[Out]

1/3*(-2+15*x)/(1-2*x)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {3+5 x}{(1-2 x)^{5/2}} \, dx=\frac {{\left (15 \, x - 2\right )} \sqrt {-2 \, x + 1}}{3 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

[In]

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

[Out]

1/3*(15*x - 2)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {3+5 x}{(1-2 x)^{5/2}} \, dx=\frac {15 x \sqrt {1 - 2 x}}{12 x^{2} - 12 x + 3} - \frac {2 \sqrt {1 - 2 x}}{12 x^{2} - 12 x + 3} \]

[In]

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

[Out]

15*x*sqrt(1 - 2*x)/(12*x**2 - 12*x + 3) - 2*sqrt(1 - 2*x)/(12*x**2 - 12*x + 3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {3+5 x}{(1-2 x)^{5/2}} \, dx=\frac {15 \, x - 2}{3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

1/3*(15*x - 2)/(-2*x + 1)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {3+5 x}{(1-2 x)^{5/2}} \, dx=-\frac {15 \, x - 2}{3 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \]

[In]

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

[Out]

-1/3*(15*x - 2)/((2*x - 1)*sqrt(-2*x + 1))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.48 \[ \int \frac {3+5 x}{(1-2 x)^{5/2}} \, dx=\frac {5\,x-\frac {2}{3}}{{\left (1-2\,x\right )}^{3/2}} \]

[In]

int((5*x + 3)/(1 - 2*x)^(5/2),x)

[Out]

(5*x - 2/3)/(1 - 2*x)^(3/2)