3.19.4 \(\int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2 \, dx\) [1804]

Optimal. Leaf size=79 \[ -\frac {41503}{96} (1-2 x)^{3/2}+\frac {91091}{160} (1-2 x)^{5/2}-\frac {5711}{16} (1-2 x)^{7/2}+\frac {1949}{16} (1-2 x)^{9/2}-\frac {7695}{352} (1-2 x)^{11/2}+\frac {675}{416} (1-2 x)^{13/2} \]

[Out]

-41503/96*(1-2*x)^(3/2)+91091/160*(1-2*x)^(5/2)-5711/16*(1-2*x)^(7/2)+1949/16*(1-2*x)^(9/2)-7695/352*(1-2*x)^(
11/2)+675/416*(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 {675}{416} (1-2 x)^{13/2}-\frac {7695}{352} (1-2 x)^{11/2}+\frac {1949}{16} (1-2 x)^{9/2}-\frac {5711}{16} (1-2 x)^{7/2}+\frac {91091}{160} (1-2 x)^{5/2}-\frac {41503}{96} (1-2 x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-41503*(1 - 2*x)^(3/2))/96 + (91091*(1 - 2*x)^(5/2))/160 - (5711*(1 - 2*x)^(7/2))/16 + (1949*(1 - 2*x)^(9/2))
/16 - (7695*(1 - 2*x)^(11/2))/352 + (675*(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)^3 (3+5 x)^2 \, dx &=\int \left (\frac {41503}{32} \sqrt {1-2 x}-\frac {91091}{32} (1-2 x)^{3/2}+\frac {39977}{16} (1-2 x)^{5/2}-\frac {17541}{16} (1-2 x)^{7/2}+\frac {7695}{32} (1-2 x)^{9/2}-\frac {675}{32} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac {41503}{96} (1-2 x)^{3/2}+\frac {91091}{160} (1-2 x)^{5/2}-\frac {5711}{16} (1-2 x)^{7/2}+\frac {1949}{16} (1-2 x)^{9/2}-\frac {7695}{352} (1-2 x)^{11/2}+\frac {675}{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 (253898+607254 x+913245 x^2+868215 x^3+471825 x^4+111375 x^5\right )}{2145} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2145*((1 - 2*x)^(3/2)*(253898 + 607254*x + 913245*x^2 + 868215*x^3 + 471825*x^4 + 111375*x^5))

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

method	result	size
gosper	\(-\frac {\left (111375 x^{5}+471825 x^{4}+868215 x^{3}+913245 x^{2}+607254 x +253898\right ) \left (1-2 x \right )^{\frac {3}{2}}}{2145}\)	\(35\)
trager	\(\left (\frac {1350}{13} x^{6}+\frac {55485}{143} x^{5}+\frac {84307}{143} x^{4}+\frac {63885}{143} x^{3}+\frac {100421}{715} x^{2}-\frac {99458}{2145} x -\frac {253898}{2145}\right ) \sqrt {1-2 x}\)	\(39\)
risch	\(-\frac {\left (222750 x^{6}+832275 x^{5}+1264605 x^{4}+958275 x^{3}+301263 x^{2}-99458 x -253898\right ) \left (-1+2 x \right )}{2145 \sqrt {1-2 x}}\)	\(45\)
derivativedivides	\(-\frac {41503 \left (1-2 x \right )^{\frac {3}{2}}}{96}+\frac {91091 \left (1-2 x \right )^{\frac {5}{2}}}{160}-\frac {5711 \left (1-2 x \right )^{\frac {7}{2}}}{16}+\frac {1949 \left (1-2 x \right )^{\frac {9}{2}}}{16}-\frac {7695 \left (1-2 x \right )^{\frac {11}{2}}}{352}+\frac {675 \left (1-2 x \right )^{\frac {13}{2}}}{416}\)	\(56\)
default	\(-\frac {41503 \left (1-2 x \right )^{\frac {3}{2}}}{96}+\frac {91091 \left (1-2 x \right )^{\frac {5}{2}}}{160}-\frac {5711 \left (1-2 x \right )^{\frac {7}{2}}}{16}+\frac {1949 \left (1-2 x \right )^{\frac {9}{2}}}{16}-\frac {7695 \left (1-2 x \right )^{\frac {11}{2}}}{352}+\frac {675 \left (1-2 x \right )^{\frac {13}{2}}}{416}\)	\(56\)
meijerg	\(\frac {24 \sqrt {\pi }-12 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {141 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{2 \sqrt {\pi }}+\frac {\frac {3532 \sqrt {\pi }}{105}-\frac {883 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{210}}{\sqrt {\pi }}-\frac {2763 \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (280 x^{3}+120 x^{2}+48 x +16\right )}{315}\right )}{32 \sqrt {\pi }}+\frac {\frac {384 \sqrt {\pi }}{77}-\frac {3 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (5040 x^{4}+2240 x^{3}+960 x^{2}+384 x +128\right )}{77}}{\sqrt {\pi }}-\frac {675 \left (-\frac {1024 \sqrt {\pi }}{9009}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (22176 x^{5}+10080 x^{4}+4480 x^{3}+1920 x^{2}+768 x +256\right )}{9009}\right )}{128 \sqrt {\pi }}\)	\(220\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-41503/96*(1-2*x)^(3/2)+91091/160*(1-2*x)^(5/2)-5711/16*(1-2*x)^(7/2)+1949/16*(1-2*x)^(9/2)-7695/352*(1-2*x)^(
11/2)+675/416*(1-2*x)^(13/2)

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Maxima [A]
time = 0.35, size = 55, normalized size = 0.70 \begin {gather*} \frac {675}{416} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {7695}{352} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {1949}{16} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {5711}{16} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {91091}{160} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {41503}{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)^3*(3+5*x)^2*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

675/416*(-2*x + 1)^(13/2) - 7695/352*(-2*x + 1)^(11/2) + 1949/16*(-2*x + 1)^(9/2) - 5711/16*(-2*x + 1)^(7/2) +
91091/160*(-2*x + 1)^(5/2) - 41503/96*(-2*x + 1)^(3/2)

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Fricas [A]
time = 0.83, size = 39, normalized size = 0.49 \begin {gather*} \frac {1}{2145} \, {\left (222750 \, x^{6} + 832275 \, x^{5} + 1264605 \, x^{4} + 958275 \, x^{3} + 301263 \, x^{2} - 99458 \, x - 253898\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2145*(222750*x^6 + 832275*x^5 + 1264605*x^4 + 958275*x^3 + 301263*x^2 - 99458*x - 253898)*sqrt(-2*x + 1)

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Sympy [A]
time = 2.47, size = 70, normalized size = 0.89 \begin {gather*} \frac {675 \left (1 - 2 x\right )^{\frac {13}{2}}}{416} - \frac {7695 \left (1 - 2 x\right )^{\frac {11}{2}}}{352} + \frac {1949 \left (1 - 2 x\right )^{\frac {9}{2}}}{16} - \frac {5711 \left (1 - 2 x\right )^{\frac {7}{2}}}{16} + \frac {91091 \left (1 - 2 x\right )^{\frac {5}{2}}}{160} - \frac {41503 \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)**3*(3+5*x)**2*(1-2*x)**(1/2),x)

[Out]

675*(1 - 2*x)**(13/2)/416 - 7695*(1 - 2*x)**(11/2)/352 + 1949*(1 - 2*x)**(9/2)/16 - 5711*(1 - 2*x)**(7/2)/16 +
91091*(1 - 2*x)**(5/2)/160 - 41503*(1 - 2*x)**(3/2)/96

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Giac [A]
time = 0.78, size = 90, normalized size = 1.14 \begin {gather*} \frac {675}{416} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {7695}{352} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {1949}{16} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {5711}{16} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {91091}{160} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {41503}{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)^3*(3+5*x)^2*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

675/416*(2*x - 1)^6*sqrt(-2*x + 1) + 7695/352*(2*x - 1)^5*sqrt(-2*x + 1) + 1949/16*(2*x - 1)^4*sqrt(-2*x + 1)
+ 5711/16*(2*x - 1)^3*sqrt(-2*x + 1) + 91091/160*(2*x - 1)^2*sqrt(-2*x + 1) - 41503/96*(-2*x + 1)^(3/2)

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Mupad [B]
time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {91091\,{\left (1-2\,x\right )}^{5/2}}{160}-\frac {41503\,{\left (1-2\,x\right )}^{3/2}}{96}-\frac {5711\,{\left (1-2\,x\right )}^{7/2}}{16}+\frac {1949\,{\left (1-2\,x\right )}^{9/2}}{16}-\frac {7695\,{\left (1-2\,x\right )}^{11/2}}{352}+\frac {675\,{\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)^3*(5*x + 3)^2,x)

[Out]

(91091*(1 - 2*x)^(5/2))/160 - (41503*(1 - 2*x)^(3/2))/96 - (5711*(1 - 2*x)^(7/2))/16 + (1949*(1 - 2*x)^(9/2))/
16 - (7695*(1 - 2*x)^(11/2))/352 + (675*(1 - 2*x)^(13/2))/416

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