\(\int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx\) [2005]

Optimal result

Integrand size = 20, antiderivative size = 40 \[ \int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {77}{4} \sqrt {1-2 x}+\frac {17}{3} (1-2 x)^{3/2}-\frac {3}{4} (1-2 x)^{5/2} \]

[Out]

17/3*(1-2*x)^(3/2)-3/4*(1-2*x)^(5/2)-77/4*(1-2*x)^(1/2)

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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {3}{4} (1-2 x)^{5/2}+\frac {17}{3} (1-2 x)^{3/2}-\frac {77}{4} \sqrt {1-2 x} \]

[In]

Int[((2 + 3*x)*(3 + 5*x))/Sqrt[1 - 2*x],x]

[Out]

(-77*Sqrt[1 - 2*x])/4 + (17*(1 - 2*x)^(3/2))/3 - (3*(1 - 2*x)^(5/2))/4

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {77}{4 \sqrt {1-2 x}}-17 \sqrt {1-2 x}+\frac {15}{4} (1-2 x)^{3/2}\right ) \, dx \\ & = -\frac {77}{4} \sqrt {1-2 x}+\frac {17}{3} (1-2 x)^{3/2}-\frac {3}{4} (1-2 x)^{5/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.58 \[ \int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {1}{3} \sqrt {1-2 x} \left (43+25 x+9 x^2\right ) \]

[In]

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

[Out]

-1/3*(Sqrt[1 - 2*x]*(43 + 25*x + 9*x^2))

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.48

method	result	size
trager	\(\left (-3 x^{2}-\frac {25}{3} x -\frac {43}{3}\right ) \sqrt {1-2 x}\)	\(19\)
gosper	\(-\frac {\sqrt {1-2 x}\, \left (9 x^{2}+25 x +43\right )}{3}\)	\(20\)
pseudoelliptic	\(-\frac {\sqrt {1-2 x}\, \left (9 x^{2}+25 x +43\right )}{3}\)	\(20\)
risch	\(\frac {\left (-1+2 x \right ) \left (9 x^{2}+25 x +43\right )}{3 \sqrt {1-2 x}}\)	\(25\)
derivativedivides	\(\frac {17 \left (1-2 x \right )^{\frac {3}{2}}}{3}-\frac {3 \left (1-2 x \right )^{\frac {5}{2}}}{4}-\frac {77 \sqrt {1-2 x}}{4}\)	\(29\)
default	\(\frac {17 \left (1-2 x \right )^{\frac {3}{2}}}{3}-\frac {3 \left (1-2 x \right )^{\frac {5}{2}}}{4}-\frac {77 \sqrt {1-2 x}}{4}\)	\(29\)
meijerg	\(-\frac {3 \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-2 x}\right )}{\sqrt {\pi }}+\frac {\frac {19 \sqrt {\pi }}{3}-\frac {19 \sqrt {\pi }\, \left (8 x +8\right ) \sqrt {1-2 x}}{24}}{\sqrt {\pi }}-\frac {15 \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{15}\right )}{8 \sqrt {\pi }}\)	\(86\)

[In]

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

[Out]

(-3*x^2-25/3*x-43/3)*(1-2*x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.48 \[ \int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {1}{3} \, {\left (9 \, x^{2} + 25 \, x + 43\right )} \sqrt {-2 \, x + 1} \]

[In]

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

[Out]

-1/3*(9*x^2 + 25*x + 43)*sqrt(-2*x + 1)

Sympy [A] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx=- \frac {3 \left (1 - 2 x\right )^{\frac {5}{2}}}{4} + \frac {17 \left (1 - 2 x\right )^{\frac {3}{2}}}{3} - \frac {77 \sqrt {1 - 2 x}}{4} \]

[In]

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

[Out]

-3*(1 - 2*x)**(5/2)/4 + 17*(1 - 2*x)**(3/2)/3 - 77*sqrt(1 - 2*x)/4

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {3}{4} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {17}{3} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {77}{4} \, \sqrt {-2 \, x + 1} \]

[In]

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

[Out]

-3/4*(-2*x + 1)^(5/2) + 17/3*(-2*x + 1)^(3/2) - 77/4*sqrt(-2*x + 1)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {3}{4} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {17}{3} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {77}{4} \, \sqrt {-2 \, x + 1} \]

[In]

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

[Out]

-3/4*(2*x - 1)^2*sqrt(-2*x + 1) + 17/3*(-2*x + 1)^(3/2) - 77/4*sqrt(-2*x + 1)

Mupad [B] (verification not implemented)

Time = 1.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.58 \[ \int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {\sqrt {1-2\,x}\,\left (136\,x+9\,{\left (2\,x-1\right )}^2+163\right )}{12} \]

[In]

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

[Out]

-((1 - 2*x)^(1/2)*(136*x + 9*(2*x - 1)^2 + 163))/12