\(\int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx\) [2004]

Optimal result

Integrand size = 22, antiderivative size = 53 \[ \int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {539}{8} \sqrt {1-2 x}+\frac {707}{24} (1-2 x)^{3/2}-\frac {309}{40} (1-2 x)^{5/2}+\frac {45}{56} (1-2 x)^{7/2} \]

[Out]

707/24*(1-2*x)^(3/2)-309/40*(1-2*x)^(5/2)+45/56*(1-2*x)^(7/2)-539/8*(1-2*x)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78} \[ \int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx=\frac {45}{56} (1-2 x)^{7/2}-\frac {309}{40} (1-2 x)^{5/2}+\frac {707}{24} (1-2 x)^{3/2}-\frac {539}{8} \sqrt {1-2 x} \]

[In]

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

[Out]

(-539*Sqrt[1 - 2*x])/8 + (707*(1 - 2*x)^(3/2))/24 - (309*(1 - 2*x)^(5/2))/40 + (45*(1 - 2*x)^(7/2))/56

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 {539}{8 \sqrt {1-2 x}}-\frac {707}{8} \sqrt {1-2 x}+\frac {309}{8} (1-2 x)^{3/2}-\frac {45}{8} (1-2 x)^{5/2}\right ) \, dx \\ & = -\frac {539}{8} \sqrt {1-2 x}+\frac {707}{24} (1-2 x)^{3/2}-\frac {309}{40} (1-2 x)^{5/2}+\frac {45}{56} (1-2 x)^{7/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.53 \[ \int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {1}{105} \sqrt {1-2 x} \left (4708+3448 x+2232 x^2+675 x^3\right ) \]

[In]

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

[Out]

-1/105*(Sqrt[1 - 2*x]*(4708 + 3448*x + 2232*x^2 + 675*x^3))

Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.45

method	result	size
trager	\(\left (-\frac {45}{7} x^{3}-\frac {744}{35} x^{2}-\frac {3448}{105} x -\frac {4708}{105}\right ) \sqrt {1-2 x}\)	\(24\)
gosper	\(-\frac {\sqrt {1-2 x}\, \left (675 x^{3}+2232 x^{2}+3448 x +4708\right )}{105}\)	\(25\)
pseudoelliptic	\(-\frac {\sqrt {1-2 x}\, \left (675 x^{3}+2232 x^{2}+3448 x +4708\right )}{105}\)	\(25\)
risch	\(\frac {\left (-1+2 x \right ) \left (675 x^{3}+2232 x^{2}+3448 x +4708\right )}{105 \sqrt {1-2 x}}\)	\(30\)
derivativedivides	\(\frac {707 \left (1-2 x \right )^{\frac {3}{2}}}{24}-\frac {309 \left (1-2 x \right )^{\frac {5}{2}}}{40}+\frac {45 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {539 \sqrt {1-2 x}}{8}\)	\(38\)
default	\(\frac {707 \left (1-2 x \right )^{\frac {3}{2}}}{24}-\frac {309 \left (1-2 x \right )^{\frac {5}{2}}}{40}+\frac {45 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {539 \sqrt {1-2 x}}{8}\)	\(38\)
meijerg	\(-\frac {6 \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-2 x}\right )}{\sqrt {\pi }}+\frac {\frac {56 \sqrt {\pi }}{3}-\frac {7 \sqrt {\pi }\, \left (8 x +8\right ) \sqrt {1-2 x}}{3}}{\sqrt {\pi }}-\frac {87 \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{15}\right )}{8 \sqrt {\pi }}+\frac {\frac {18 \sqrt {\pi }}{7}-\frac {9 \sqrt {\pi }\, \left (320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{448}}{\sqrt {\pi }}\)	\(124\)

[In]

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

[Out]

(-45/7*x^3-744/35*x^2-3448/105*x-4708/105)*(1-2*x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.45 \[ \int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {1}{105} \, {\left (675 \, x^{3} + 2232 \, x^{2} + 3448 \, x + 4708\right )} \sqrt {-2 \, x + 1} \]

[In]

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

[Out]

-1/105*(675*x^3 + 2232*x^2 + 3448*x + 4708)*sqrt(-2*x + 1)

Sympy [A] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.87 \[ \int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx=\frac {45 \left (1 - 2 x\right )^{\frac {7}{2}}}{56} - \frac {309 \left (1 - 2 x\right )^{\frac {5}{2}}}{40} + \frac {707 \left (1 - 2 x\right )^{\frac {3}{2}}}{24} - \frac {539 \sqrt {1 - 2 x}}{8} \]

[In]

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

[Out]

45*(1 - 2*x)**(7/2)/56 - 309*(1 - 2*x)**(5/2)/40 + 707*(1 - 2*x)**(3/2)/24 - 539*sqrt(1 - 2*x)/8

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx=\frac {45}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {309}{40} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {707}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {539}{8} \, \sqrt {-2 \, x + 1} \]

[In]

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

[Out]

45/56*(-2*x + 1)^(7/2) - 309/40*(-2*x + 1)^(5/2) + 707/24*(-2*x + 1)^(3/2) - 539/8*sqrt(-2*x + 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.96 \[ \int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {45}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {309}{40} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {707}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {539}{8} \, \sqrt {-2 \, x + 1} \]

[In]

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

[Out]

-45/56*(2*x - 1)^3*sqrt(-2*x + 1) - 309/40*(2*x - 1)^2*sqrt(-2*x + 1) + 707/24*(-2*x + 1)^(3/2) - 539/8*sqrt(-
2*x + 1)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx=\frac {707\,{\left (1-2\,x\right )}^{3/2}}{24}-\frac {539\,\sqrt {1-2\,x}}{8}-\frac {309\,{\left (1-2\,x\right )}^{5/2}}{40}+\frac {45\,{\left (1-2\,x\right )}^{7/2}}{56} \]

[In]

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

[Out]

(707*(1 - 2*x)^(3/2))/24 - (539*(1 - 2*x)^(1/2))/8 - (309*(1 - 2*x)^(5/2))/40 + (45*(1 - 2*x)^(7/2))/56