\(\int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2 \, dx\) [1805]

Optimal result

Integrand size = 24, antiderivative size = 66 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2 \, dx=-\frac {5929}{48} (1-2 x)^{3/2}+\frac {1309}{10} (1-2 x)^{5/2}-\frac {3467}{56} (1-2 x)^{7/2}+\frac {85}{6} (1-2 x)^{9/2}-\frac {225}{176} (1-2 x)^{11/2} \]

[Out]

-5929/48*(1-2*x)^(3/2)+1309/10*(1-2*x)^(5/2)-3467/56*(1-2*x)^(7/2)+85/6*(1-2*x)^(9/2)-225/176*(1-2*x)^(11/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {90} \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2 \, dx=-\frac {225}{176} (1-2 x)^{11/2}+\frac {85}{6} (1-2 x)^{9/2}-\frac {3467}{56} (1-2 x)^{7/2}+\frac {1309}{10} (1-2 x)^{5/2}-\frac {5929}{48} (1-2 x)^{3/2} \]

[In]

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

[Out]

(-5929*(1 - 2*x)^(3/2))/48 + (1309*(1 - 2*x)^(5/2))/10 - (3467*(1 - 2*x)^(7/2))/56 + (85*(1 - 2*x)^(9/2))/6 -
(225*(1 - 2*x)^(11/2))/176

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {5929}{16} \sqrt {1-2 x}-\frac {1309}{2} (1-2 x)^{3/2}+\frac {3467}{8} (1-2 x)^{5/2}-\frac {255}{2} (1-2 x)^{7/2}+\frac {225}{16} (1-2 x)^{9/2}\right ) \, dx \\ & = -\frac {5929}{48} (1-2 x)^{3/2}+\frac {1309}{10} (1-2 x)^{5/2}-\frac {3467}{56} (1-2 x)^{7/2}+\frac {85}{6} (1-2 x)^{9/2}-\frac {225}{176} (1-2 x)^{11/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.50 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2 \, dx=-\frac {(1-2 x)^{3/2} \left (48098+102714 x+125115 x^2+83650 x^3+23625 x^4\right )}{1155} \]

[In]

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

[Out]

-1/1155*((1 - 2*x)^(3/2)*(48098 + 102714*x + 125115*x^2 + 83650*x^3 + 23625*x^4))

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.45

method	result	size
gosper	\(-\frac {\left (1-2 x \right )^{\frac {3}{2}} \left (23625 x^{4}+83650 x^{3}+125115 x^{2}+102714 x +48098\right )}{1155}\)	\(30\)
trager	\(\left (\frac {450}{11} x^{5}+\frac {4105}{33} x^{4}+\frac {33316}{231} x^{3}+\frac {26771}{385} x^{2}-\frac {6518}{1155} x -\frac {48098}{1155}\right ) \sqrt {1-2 x}\)	\(34\)
pseudoelliptic	\(\frac {\sqrt {1-2 x}\, \left (47250 x^{5}+143675 x^{4}+166580 x^{3}+80313 x^{2}-6518 x -48098\right )}{1155}\)	\(35\)
risch	\(-\frac {\left (47250 x^{5}+143675 x^{4}+166580 x^{3}+80313 x^{2}-6518 x -48098\right ) \left (-1+2 x \right )}{1155 \sqrt {1-2 x}}\)	\(40\)
derivativedivides	\(-\frac {5929 \left (1-2 x \right )^{\frac {3}{2}}}{48}+\frac {1309 \left (1-2 x \right )^{\frac {5}{2}}}{10}-\frac {3467 \left (1-2 x \right )^{\frac {7}{2}}}{56}+\frac {85 \left (1-2 x \right )^{\frac {9}{2}}}{6}-\frac {225 \left (1-2 x \right )^{\frac {11}{2}}}{176}\)	\(47\)
default	\(-\frac {5929 \left (1-2 x \right )^{\frac {3}{2}}}{48}+\frac {1309 \left (1-2 x \right )^{\frac {5}{2}}}{10}-\frac {3467 \left (1-2 x \right )^{\frac {7}{2}}}{56}+\frac {85 \left (1-2 x \right )^{\frac {9}{2}}}{6}-\frac {225 \left (1-2 x \right )^{\frac {11}{2}}}{176}\)	\(47\)
meijerg	\(\frac {12 \sqrt {\pi }-6 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {57 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{2 \sqrt {\pi }}+\frac {\frac {1082 \sqrt {\pi }}{105}-\frac {541 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{420}}{\sqrt {\pi }}-\frac {285 \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (280 x^{3}+120 x^{2}+48 x +16\right )}{315}\right )}{16 \sqrt {\pi }}+\frac {\frac {40 \sqrt {\pi }}{77}-\frac {5 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (5040 x^{4}+2240 x^{3}+960 x^{2}+384 x +128\right )}{1232}}{\sqrt {\pi }}\)	\(172\)

[In]

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

[Out]

-1/1155*(1-2*x)^(3/2)*(23625*x^4+83650*x^3+125115*x^2+102714*x+48098)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.52 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2 \, dx=\frac {1}{1155} \, {\left (47250 \, x^{5} + 143675 \, x^{4} + 166580 \, x^{3} + 80313 \, x^{2} - 6518 \, x - 48098\right )} \sqrt {-2 \, x + 1} \]

[In]

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

[Out]

1/1155*(47250*x^5 + 143675*x^4 + 166580*x^3 + 80313*x^2 - 6518*x - 48098)*sqrt(-2*x + 1)

Sympy [A] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2 \, dx=- \frac {225 \left (1 - 2 x\right )^{\frac {11}{2}}}{176} + \frac {85 \left (1 - 2 x\right )^{\frac {9}{2}}}{6} - \frac {3467 \left (1 - 2 x\right )^{\frac {7}{2}}}{56} + \frac {1309 \left (1 - 2 x\right )^{\frac {5}{2}}}{10} - \frac {5929 \left (1 - 2 x\right )^{\frac {3}{2}}}{48} \]

[In]

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

[Out]

-225*(1 - 2*x)**(11/2)/176 + 85*(1 - 2*x)**(9/2)/6 - 3467*(1 - 2*x)**(7/2)/56 + 1309*(1 - 2*x)**(5/2)/10 - 592
9*(1 - 2*x)**(3/2)/48

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2 \, dx=-\frac {225}{176} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {85}{6} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {3467}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {1309}{10} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {5929}{48} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

-225/176*(-2*x + 1)^(11/2) + 85/6*(-2*x + 1)^(9/2) - 3467/56*(-2*x + 1)^(7/2) + 1309/10*(-2*x + 1)^(5/2) - 592
9/48*(-2*x + 1)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.12 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2 \, dx=\frac {225}{176} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {85}{6} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {3467}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {1309}{10} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {5929}{48} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

225/176*(2*x - 1)^5*sqrt(-2*x + 1) + 85/6*(2*x - 1)^4*sqrt(-2*x + 1) + 3467/56*(2*x - 1)^3*sqrt(-2*x + 1) + 13
09/10*(2*x - 1)^2*sqrt(-2*x + 1) - 5929/48*(-2*x + 1)^(3/2)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2 \, dx=\frac {1309\,{\left (1-2\,x\right )}^{5/2}}{10}-\frac {5929\,{\left (1-2\,x\right )}^{3/2}}{48}-\frac {3467\,{\left (1-2\,x\right )}^{7/2}}{56}+\frac {85\,{\left (1-2\,x\right )}^{9/2}}{6}-\frac {225\,{\left (1-2\,x\right )}^{11/2}}{176} \]

[In]

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

[Out]

(1309*(1 - 2*x)^(5/2))/10 - (5929*(1 - 2*x)^(3/2))/48 - (3467*(1 - 2*x)^(7/2))/56 + (85*(1 - 2*x)^(9/2))/6 - (
225*(1 - 2*x)^(11/2))/176