3.19.17 \(\int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3 \, dx\) [1817]

Optimal. Leaf size=79 \[ -\frac {65219}{96} (1-2 x)^{3/2}+\frac {144837}{160} (1-2 x)^{5/2}-\frac {64317}{112} (1-2 x)^{7/2}+\frac {28555}{144} (1-2 x)^{9/2}-\frac {12675}{352} (1-2 x)^{11/2}+\frac {1125}{416} (1-2 x)^{13/2} \]

[Out]

-65219/96*(1-2*x)^(3/2)+144837/160*(1-2*x)^(5/2)-64317/112*(1-2*x)^(7/2)+28555/144*(1-2*x)^(9/2)-12675/352*(1-
2*x)^(11/2)+1125/416*(1-2*x)^(13/2)

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Rubi [A]
time = 0.01, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {90} \begin {gather*} \frac {1125}{416} (1-2 x)^{13/2}-\frac {12675}{352} (1-2 x)^{11/2}+\frac {28555}{144} (1-2 x)^{9/2}-\frac {64317}{112} (1-2 x)^{7/2}+\frac {144837}{160} (1-2 x)^{5/2}-\frac {65219}{96} (1-2 x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-65219*(1 - 2*x)^(3/2))/96 + (144837*(1 - 2*x)^(5/2))/160 - (64317*(1 - 2*x)^(7/2))/112 + (28555*(1 - 2*x)^(9
/2))/144 - (12675*(1 - 2*x)^(11/2))/352 + (1125*(1 - 2*x)^(13/2))/416

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*} \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3 \, dx &=\int \left (\frac {65219}{32} \sqrt {1-2 x}-\frac {144837}{32} (1-2 x)^{3/2}+\frac {64317}{16} (1-2 x)^{5/2}-\frac {28555}{16} (1-2 x)^{7/2}+\frac {12675}{32} (1-2 x)^{9/2}-\frac {1125}{32} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac {65219}{96} (1-2 x)^{3/2}+\frac {144837}{160} (1-2 x)^{5/2}-\frac {64317}{112} (1-2 x)^{7/2}+\frac {28555}{144} (1-2 x)^{9/2}-\frac {12675}{352} (1-2 x)^{11/2}+\frac {1125}{416} (1-2 x)^{13/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {(1-2 x)^{3/2} \left (8261156+19918608 x+30337080 x^2+29300075 x^3+16206750 x^4+3898125 x^5\right )}{45045} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/45045*((1 - 2*x)^(3/2)*(8261156 + 19918608*x + 30337080*x^2 + 29300075*x^3 + 16206750*x^4 + 3898125*x^5))

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Maple [A]
time = 0.10, size = 56, normalized size = 0.71

method	result	size
gosper	\(-\frac {\left (3898125 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x +8261156\right ) \left (1-2 x \right )^{\frac {3}{2}}}{45045}\)	\(35\)
trager	\(\left (\frac {2250}{13} x^{6}+\frac {90525}{143} x^{5}+\frac {1211240}{1287} x^{4}+\frac {6274817}{9009} x^{3}+\frac {3166712}{15015} x^{2}-\frac {3396296}{45045} x -\frac {8261156}{45045}\right ) \sqrt {1-2 x}\)	\(39\)
risch	\(-\frac {\left (7796250 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}+9500136 x^{2}-3396296 x -8261156\right ) \left (-1+2 x \right )}{45045 \sqrt {1-2 x}}\)	\(45\)
derivativedivides	\(-\frac {65219 \left (1-2 x \right )^{\frac {3}{2}}}{96}+\frac {144837 \left (1-2 x \right )^{\frac {5}{2}}}{160}-\frac {64317 \left (1-2 x \right )^{\frac {7}{2}}}{112}+\frac {28555 \left (1-2 x \right )^{\frac {9}{2}}}{144}-\frac {12675 \left (1-2 x \right )^{\frac {11}{2}}}{352}+\frac {1125 \left (1-2 x \right )^{\frac {13}{2}}}{416}\)	\(56\)
default	\(-\frac {65219 \left (1-2 x \right )^{\frac {3}{2}}}{96}+\frac {144837 \left (1-2 x \right )^{\frac {5}{2}}}{160}-\frac {64317 \left (1-2 x \right )^{\frac {7}{2}}}{112}+\frac {28555 \left (1-2 x \right )^{\frac {9}{2}}}{144}-\frac {12675 \left (1-2 x \right )^{\frac {11}{2}}}{352}+\frac {1125 \left (1-2 x \right )^{\frac {13}{2}}}{416}\)	\(56\)
meijerg	\(\frac {36 \sqrt {\pi }-18 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {108 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{\sqrt {\pi }}+\frac {\frac {1842 \sqrt {\pi }}{35}-\frac {921 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{140}}{\sqrt {\pi }}-\frac {4415 \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (280 x^{3}+120 x^{2}+48 x +16\right )}{315}\right )}{32 \sqrt {\pi }}+\frac {\frac {1880 \sqrt {\pi }}{231}-\frac {235 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (5040 x^{4}+2240 x^{3}+960 x^{2}+384 x +128\right )}{3696}}{\sqrt {\pi }}-\frac {1125 \left (-\frac {1024 \sqrt {\pi }}{9009}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (22176 x^{5}+10080 x^{4}+4480 x^{3}+1920 x^{2}+768 x +256\right )}{9009}\right )}{128 \sqrt {\pi }}\)	\(220\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-65219/96*(1-2*x)^(3/2)+144837/160*(1-2*x)^(5/2)-64317/112*(1-2*x)^(7/2)+28555/144*(1-2*x)^(9/2)-12675/352*(1-
2*x)^(11/2)+1125/416*(1-2*x)^(13/2)

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Maxima [A]
time = 0.27, size = 55, normalized size = 0.70 \begin {gather*} \frac {1125}{416} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {12675}{352} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {28555}{144} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {64317}{112} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {144837}{160} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {65219}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1125/416*(-2*x + 1)^(13/2) - 12675/352*(-2*x + 1)^(11/2) + 28555/144*(-2*x + 1)^(9/2) - 64317/112*(-2*x + 1)^(
7/2) + 144837/160*(-2*x + 1)^(5/2) - 65219/96*(-2*x + 1)^(3/2)

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Fricas [A]
time = 1.06, size = 39, normalized size = 0.49 \begin {gather*} \frac {1}{45045} \, {\left (7796250 \, x^{6} + 28515375 \, x^{5} + 42393400 \, x^{4} + 31374085 \, x^{3} + 9500136 \, x^{2} - 3396296 \, x - 8261156\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45045*(7796250*x^6 + 28515375*x^5 + 42393400*x^4 + 31374085*x^3 + 9500136*x^2 - 3396296*x - 8261156)*sqrt(-2
*x + 1)

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Sympy [A]
time = 1.40, size = 70, normalized size = 0.89 \begin {gather*} \frac {1125 \left (1 - 2 x\right )^{\frac {13}{2}}}{416} - \frac {12675 \left (1 - 2 x\right )^{\frac {11}{2}}}{352} + \frac {28555 \left (1 - 2 x\right )^{\frac {9}{2}}}{144} - \frac {64317 \left (1 - 2 x\right )^{\frac {7}{2}}}{112} + \frac {144837 \left (1 - 2 x\right )^{\frac {5}{2}}}{160} - \frac {65219 \left (1 - 2 x\right )^{\frac {3}{2}}}{96} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1125*(1 - 2*x)**(13/2)/416 - 12675*(1 - 2*x)**(11/2)/352 + 28555*(1 - 2*x)**(9/2)/144 - 64317*(1 - 2*x)**(7/2)
/112 + 144837*(1 - 2*x)**(5/2)/160 - 65219*(1 - 2*x)**(3/2)/96

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Giac [A]
time = 1.72, size = 90, normalized size = 1.14 \begin {gather*} \frac {1125}{416} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {12675}{352} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {28555}{144} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {64317}{112} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {144837}{160} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {65219}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1125/416*(2*x - 1)^6*sqrt(-2*x + 1) + 12675/352*(2*x - 1)^5*sqrt(-2*x + 1) + 28555/144*(2*x - 1)^4*sqrt(-2*x +
1) + 64317/112*(2*x - 1)^3*sqrt(-2*x + 1) + 144837/160*(2*x - 1)^2*sqrt(-2*x + 1) - 65219/96*(-2*x + 1)^(3/2)

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Mupad [B]
time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {144837\,{\left (1-2\,x\right )}^{5/2}}{160}-\frac {65219\,{\left (1-2\,x\right )}^{3/2}}{96}-\frac {64317\,{\left (1-2\,x\right )}^{7/2}}{112}+\frac {28555\,{\left (1-2\,x\right )}^{9/2}}{144}-\frac {12675\,{\left (1-2\,x\right )}^{11/2}}{352}+\frac {1125\,{\left (1-2\,x\right )}^{13/2}}{416} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(144837*(1 - 2*x)^(5/2))/160 - (65219*(1 - 2*x)^(3/2))/96 - (64317*(1 - 2*x)^(7/2))/112 + (28555*(1 - 2*x)^(9/
2))/144 - (12675*(1 - 2*x)^(11/2))/352 + (1125*(1 - 2*x)^(13/2))/416

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