3.19.18 \(\int \sqrt {1-2 x} (2+3 x) (3+5 x)^3 \, dx\) [1818]

Optimal. Leaf size=66 \[ -\frac {9317}{48} (1-2 x)^{3/2}+\frac {8349}{40} (1-2 x)^{5/2}-\frac {2805}{28} (1-2 x)^{7/2}+\frac {1675}{72} (1-2 x)^{9/2}-\frac {375}{176} (1-2 x)^{11/2} \]

[Out]

-9317/48*(1-2*x)^(3/2)+8349/40*(1-2*x)^(5/2)-2805/28*(1-2*x)^(7/2)+1675/72*(1-2*x)^(9/2)-375/176*(1-2*x)^(11/2
)

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Rubi [A]
time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78} \begin {gather*} -\frac {375}{176} (1-2 x)^{11/2}+\frac {1675}{72} (1-2 x)^{9/2}-\frac {2805}{28} (1-2 x)^{7/2}+\frac {8349}{40} (1-2 x)^{5/2}-\frac {9317}{48} (1-2 x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-9317*(1 - 2*x)^(3/2))/48 + (8349*(1 - 2*x)^(5/2))/40 - (2805*(1 - 2*x)^(7/2))/28 + (1675*(1 - 2*x)^(9/2))/72
- (375*(1 - 2*x)^(11/2))/176

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*} \int \sqrt {1-2 x} (2+3 x) (3+5 x)^3 \, dx &=\int \left (\frac {9317}{16} \sqrt {1-2 x}-\frac {8349}{8} (1-2 x)^{3/2}+\frac {2805}{4} (1-2 x)^{5/2}-\frac {1675}{8} (1-2 x)^{7/2}+\frac {375}{16} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac {9317}{48} (1-2 x)^{3/2}+\frac {8349}{40} (1-2 x)^{5/2}-\frac {2805}{28} (1-2 x)^{7/2}+\frac {1675}{72} (1-2 x)^{9/2}-\frac {375}{176} (1-2 x)^{11/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 33, normalized size = 0.50 \begin {gather*} -\frac {(1-2 x)^{3/2} \left (223231+482583 x+598350 x^2+408625 x^3+118125 x^4\right )}{3465} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3465*((1 - 2*x)^(3/2)*(223231 + 482583*x + 598350*x^2 + 408625*x^3 + 118125*x^4))

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Maple [A]
time = 0.09, size = 47, normalized size = 0.71

method	result	size
gosper	\(-\frac {\left (118125 x^{4}+408625 x^{3}+598350 x^{2}+482583 x +223231\right ) \left (1-2 x \right )^{\frac {3}{2}}}{3465}\)	\(30\)
trager	\(\left (\frac {750}{11} x^{5}+\frac {19975}{99} x^{4}+\frac {157615}{693} x^{3}+\frac {122272}{1155} x^{2}-\frac {36121}{3465} x -\frac {223231}{3465}\right ) \sqrt {1-2 x}\)	\(34\)
risch	\(-\frac {\left (236250 x^{5}+699125 x^{4}+788075 x^{3}+366816 x^{2}-36121 x -223231\right ) \left (-1+2 x \right )}{3465 \sqrt {1-2 x}}\)	\(40\)
derivativedivides	\(-\frac {9317 \left (1-2 x \right )^{\frac {3}{2}}}{48}+\frac {8349 \left (1-2 x \right )^{\frac {5}{2}}}{40}-\frac {2805 \left (1-2 x \right )^{\frac {7}{2}}}{28}+\frac {1675 \left (1-2 x \right )^{\frac {9}{2}}}{72}-\frac {375 \left (1-2 x \right )^{\frac {11}{2}}}{176}\)	\(47\)
default	\(-\frac {9317 \left (1-2 x \right )^{\frac {3}{2}}}{48}+\frac {8349 \left (1-2 x \right )^{\frac {5}{2}}}{40}-\frac {2805 \left (1-2 x \right )^{\frac {7}{2}}}{28}+\frac {1675 \left (1-2 x \right )^{\frac {9}{2}}}{72}-\frac {375 \left (1-2 x \right )^{\frac {11}{2}}}{176}\)	\(47\)
meijerg	\(\frac {18 \sqrt {\pi }-9 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {351 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{8 \sqrt {\pi }}+\frac {\frac {114 \sqrt {\pi }}{7}-\frac {57 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{28}}{\sqrt {\pi }}-\frac {925 \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (280 x^{3}+120 x^{2}+48 x +16\right )}{315}\right )}{32 \sqrt {\pi }}+\frac {\frac {200 \sqrt {\pi }}{231}-\frac {25 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (5040 x^{4}+2240 x^{3}+960 x^{2}+384 x +128\right )}{3696}}{\sqrt {\pi }}\)	\(172\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9317/48*(1-2*x)^(3/2)+8349/40*(1-2*x)^(5/2)-2805/28*(1-2*x)^(7/2)+1675/72*(1-2*x)^(9/2)-375/176*(1-2*x)^(11/2
)

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Maxima [A]
time = 0.38, size = 46, normalized size = 0.70 \begin {gather*} -\frac {375}{176} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {1675}{72} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {2805}{28} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {8349}{40} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {9317}{48} \, {\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)*(3+5*x)^3*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

-375/176*(-2*x + 1)^(11/2) + 1675/72*(-2*x + 1)^(9/2) - 2805/28*(-2*x + 1)^(7/2) + 8349/40*(-2*x + 1)^(5/2) -
9317/48*(-2*x + 1)^(3/2)

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Fricas [A]
time = 0.63, size = 34, normalized size = 0.52 \begin {gather*} \frac {1}{3465} \, {\left (236250 \, x^{5} + 699125 \, x^{4} + 788075 \, x^{3} + 366816 \, x^{2} - 36121 \, x - 223231\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3465*(236250*x^5 + 699125*x^4 + 788075*x^3 + 366816*x^2 - 36121*x - 223231)*sqrt(-2*x + 1)

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Sympy [A]
time = 1.22, size = 58, normalized size = 0.88 \begin {gather*} - \frac {375 \left (1 - 2 x\right )^{\frac {11}{2}}}{176} + \frac {1675 \left (1 - 2 x\right )^{\frac {9}{2}}}{72} - \frac {2805 \left (1 - 2 x\right )^{\frac {7}{2}}}{28} + \frac {8349 \left (1 - 2 x\right )^{\frac {5}{2}}}{40} - \frac {9317 \left (1 - 2 x\right )^{\frac {3}{2}}}{48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-375*(1 - 2*x)**(11/2)/176 + 1675*(1 - 2*x)**(9/2)/72 - 2805*(1 - 2*x)**(7/2)/28 + 8349*(1 - 2*x)**(5/2)/40 -
9317*(1 - 2*x)**(3/2)/48

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Giac [A]
time = 1.37, size = 74, normalized size = 1.12 \begin {gather*} \frac {375}{176} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {1675}{72} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {2805}{28} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {8349}{40} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {9317}{48} \, {\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)*(3+5*x)^3*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

375/176*(2*x - 1)^5*sqrt(-2*x + 1) + 1675/72*(2*x - 1)^4*sqrt(-2*x + 1) + 2805/28*(2*x - 1)^3*sqrt(-2*x + 1) +
8349/40*(2*x - 1)^2*sqrt(-2*x + 1) - 9317/48*(-2*x + 1)^(3/2)

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Mupad [B]
time = 0.02, size = 46, normalized size = 0.70 \begin {gather*} \frac {8349\,{\left (1-2\,x\right )}^{5/2}}{40}-\frac {9317\,{\left (1-2\,x\right )}^{3/2}}{48}-\frac {2805\,{\left (1-2\,x\right )}^{7/2}}{28}+\frac {1675\,{\left (1-2\,x\right )}^{9/2}}{72}-\frac {375\,{\left (1-2\,x\right )}^{11/2}}{176} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8349*(1 - 2*x)^(5/2))/40 - (9317*(1 - 2*x)^(3/2))/48 - (2805*(1 - 2*x)^(7/2))/28 + (1675*(1 - 2*x)^(9/2))/72
- (375*(1 - 2*x)^(11/2))/176

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