3.18.90 \(\int \sqrt {1-2 x} (2+3 x)^6 (3+5 x) \, dx\) [1790]

Optimal. Leaf size=105 \[ -\frac {1294139}{384} (1-2 x)^{3/2}+\frac {3916031}{640} (1-2 x)^{5/2}-\frac {725445}{128} (1-2 x)^{7/2}+\frac {406455}{128} (1-2 x)^{9/2}-\frac {1580985 (1-2 x)^{11/2}}{1408}+\frac {409941 (1-2 x)^{13/2}}{1664}-\frac {19683}{640} (1-2 x)^{15/2}+\frac {3645 (1-2 x)^{17/2}}{2176} \]

[Out]

-1294139/384*(1-2*x)^(3/2)+3916031/640*(1-2*x)^(5/2)-725445/128*(1-2*x)^(7/2)+406455/128*(1-2*x)^(9/2)-1580985
/1408*(1-2*x)^(11/2)+409941/1664*(1-2*x)^(13/2)-19683/640*(1-2*x)^(15/2)+3645/2176*(1-2*x)^(17/2)

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Rubi [A]
time = 0.01, antiderivative size = 105, 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 {3645 (1-2 x)^{17/2}}{2176}-\frac {19683}{640} (1-2 x)^{15/2}+\frac {409941 (1-2 x)^{13/2}}{1664}-\frac {1580985 (1-2 x)^{11/2}}{1408}+\frac {406455}{128} (1-2 x)^{9/2}-\frac {725445}{128} (1-2 x)^{7/2}+\frac {3916031}{640} (1-2 x)^{5/2}-\frac {1294139}{384} (1-2 x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]*(2 + 3*x)^6*(3 + 5*x),x]

[Out]

(-1294139*(1 - 2*x)^(3/2))/384 + (3916031*(1 - 2*x)^(5/2))/640 - (725445*(1 - 2*x)^(7/2))/128 + (406455*(1 - 2
*x)^(9/2))/128 - (1580985*(1 - 2*x)^(11/2))/1408 + (409941*(1 - 2*x)^(13/2))/1664 - (19683*(1 - 2*x)^(15/2))/6
40 + (3645*(1 - 2*x)^(17/2))/2176

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)^6 (3+5 x) \, dx &=\int \left (\frac {1294139}{128} \sqrt {1-2 x}-\frac {3916031}{128} (1-2 x)^{3/2}+\frac {5078115}{128} (1-2 x)^{5/2}-\frac {3658095}{128} (1-2 x)^{7/2}+\frac {1580985}{128} (1-2 x)^{9/2}-\frac {409941}{128} (1-2 x)^{11/2}+\frac {59049}{128} (1-2 x)^{13/2}-\frac {3645}{128} (1-2 x)^{15/2}\right ) \, dx\\ &=-\frac {1294139}{384} (1-2 x)^{3/2}+\frac {3916031}{640} (1-2 x)^{5/2}-\frac {725445}{128} (1-2 x)^{7/2}+\frac {406455}{128} (1-2 x)^{9/2}-\frac {1580985 (1-2 x)^{11/2}}{1408}+\frac {409941 (1-2 x)^{13/2}}{1664}-\frac {19683}{640} (1-2 x)^{15/2}+\frac {3645 (1-2 x)^{17/2}}{2176}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 48, normalized size = 0.46 \begin {gather*} -\frac {(1-2 x)^{3/2} \left (23667392+64000896 x+122662080 x^2+172440720 x^3+171389520 x^4+113196204 x^5+44409222 x^6+7818525 x^7\right )}{36465} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - 2*x]*(2 + 3*x)^6*(3 + 5*x),x]

[Out]

-1/36465*((1 - 2*x)^(3/2)*(23667392 + 64000896*x + 122662080*x^2 + 172440720*x^3 + 171389520*x^4 + 113196204*x
^5 + 44409222*x^6 + 7818525*x^7))

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Maple [A]
time = 0.14, size = 74, normalized size = 0.70

method	result	size
gosper	\(-\frac {\left (7818525 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \left (1-2 x \right )^{\frac {3}{2}}}{36465}\)	\(45\)
trager	\(\left (\frac {7290}{17} x^{8}+\frac {188811}{85} x^{7}+\frac {5514642}{1105} x^{6}+\frac {76527612}{12155} x^{5}+\frac {11566128}{2431} x^{4}+\frac {4858896}{2431} x^{3}+\frac {1779904}{12155} x^{2}-\frac {16666112}{36465} x -\frac {23667392}{36465}\right ) \sqrt {1-2 x}\)	\(49\)
risch	\(-\frac {\left (15637050 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}+5339712 x^{2}-16666112 x -23667392\right ) \left (-1+2 x \right )}{36465 \sqrt {1-2 x}}\)	\(55\)
derivativedivides	\(-\frac {1294139 \left (1-2 x \right )^{\frac {3}{2}}}{384}+\frac {3916031 \left (1-2 x \right )^{\frac {5}{2}}}{640}-\frac {725445 \left (1-2 x \right )^{\frac {7}{2}}}{128}+\frac {406455 \left (1-2 x \right )^{\frac {9}{2}}}{128}-\frac {1580985 \left (1-2 x \right )^{\frac {11}{2}}}{1408}+\frac {409941 \left (1-2 x \right )^{\frac {13}{2}}}{1664}-\frac {19683 \left (1-2 x \right )^{\frac {15}{2}}}{640}+\frac {3645 \left (1-2 x \right )^{\frac {17}{2}}}{2176}\)	\(74\)
default	\(-\frac {1294139 \left (1-2 x \right )^{\frac {3}{2}}}{384}+\frac {3916031 \left (1-2 x \right )^{\frac {5}{2}}}{640}-\frac {725445 \left (1-2 x \right )^{\frac {7}{2}}}{128}+\frac {406455 \left (1-2 x \right )^{\frac {9}{2}}}{128}-\frac {1580985 \left (1-2 x \right )^{\frac {11}{2}}}{1408}+\frac {409941 \left (1-2 x \right )^{\frac {13}{2}}}{1664}-\frac {19683 \left (1-2 x \right )^{\frac {15}{2}}}{640}+\frac {3645 \left (1-2 x \right )^{\frac {17}{2}}}{2176}\)	\(74\)
meijerg	\(\frac {64 \sqrt {\pi }-32 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {256 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{\sqrt {\pi }}+\frac {\frac {1248 \sqrt {\pi }}{7}-\frac {156 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{7}}{\sqrt {\pi }}-\frac {1485 \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (280 x^{3}+120 x^{2}+48 x +16\right )}{315}\right )}{2 \sqrt {\pi }}+\frac {\frac {6432 \sqrt {\pi }}{77}-\frac {201 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (5040 x^{4}+2240 x^{3}+960 x^{2}+384 x +128\right )}{308}}{\sqrt {\pi }}-\frac {4131 \left (-\frac {1024 \sqrt {\pi }}{9009}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (22176 x^{5}+10080 x^{4}+4480 x^{3}+1920 x^{2}+768 x +256\right )}{9009}\right )}{16 \sqrt {\pi }}+\frac {\frac {29808 \sqrt {\pi }}{5005}-\frac {1863 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (192192 x^{6}+88704 x^{5}+40320 x^{4}+17920 x^{3}+7680 x^{2}+3072 x +1024\right )}{320320}}{\sqrt {\pi }}-\frac {3645 \left (-\frac {8192 \sqrt {\pi }}{109395}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (823680 x^{7}+384384 x^{6}+177408 x^{5}+80640 x^{4}+35840 x^{3}+15360 x^{2}+6144 x +2048\right )}{109395}\right )}{512 \sqrt {\pi }}\)	\(331\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^6*(3+5*x)*(1-2*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1294139/384*(1-2*x)^(3/2)+3916031/640*(1-2*x)^(5/2)-725445/128*(1-2*x)^(7/2)+406455/128*(1-2*x)^(9/2)-1580985
/1408*(1-2*x)^(11/2)+409941/1664*(1-2*x)^(13/2)-19683/640*(1-2*x)^(15/2)+3645/2176*(1-2*x)^(17/2)

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Maxima [A]
time = 0.27, size = 73, normalized size = 0.70 \begin {gather*} \frac {3645}{2176} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} - \frac {19683}{640} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {409941}{1664} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {1580985}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {406455}{128} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {725445}{128} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {3916031}{640} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {1294139}{384} \, {\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)^6*(3+5*x)*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

3645/2176*(-2*x + 1)^(17/2) - 19683/640*(-2*x + 1)^(15/2) + 409941/1664*(-2*x + 1)^(13/2) - 1580985/1408*(-2*x
+ 1)^(11/2) + 406455/128*(-2*x + 1)^(9/2) - 725445/128*(-2*x + 1)^(7/2) + 3916031/640*(-2*x + 1)^(5/2) - 1294
139/384*(-2*x + 1)^(3/2)

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Fricas [A]
time = 0.86, size = 49, normalized size = 0.47 \begin {gather*} \frac {1}{36465} \, {\left (15637050 \, x^{8} + 80999919 \, x^{7} + 181983186 \, x^{6} + 229582836 \, x^{5} + 173491920 \, x^{4} + 72883440 \, x^{3} + 5339712 \, x^{2} - 16666112 \, x - 23667392\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6*(3+5*x)*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/36465*(15637050*x^8 + 80999919*x^7 + 181983186*x^6 + 229582836*x^5 + 173491920*x^4 + 72883440*x^3 + 5339712*
x^2 - 16666112*x - 23667392)*sqrt(-2*x + 1)

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Sympy [A]
time = 2.74, size = 94, normalized size = 0.90 \begin {gather*} \frac {3645 \left (1 - 2 x\right )^{\frac {17}{2}}}{2176} - \frac {19683 \left (1 - 2 x\right )^{\frac {15}{2}}}{640} + \frac {409941 \left (1 - 2 x\right )^{\frac {13}{2}}}{1664} - \frac {1580985 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} + \frac {406455 \left (1 - 2 x\right )^{\frac {9}{2}}}{128} - \frac {725445 \left (1 - 2 x\right )^{\frac {7}{2}}}{128} + \frac {3916031 \left (1 - 2 x\right )^{\frac {5}{2}}}{640} - \frac {1294139 \left (1 - 2 x\right )^{\frac {3}{2}}}{384} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**6*(3+5*x)*(1-2*x)**(1/2),x)

[Out]

3645*(1 - 2*x)**(17/2)/2176 - 19683*(1 - 2*x)**(15/2)/640 + 409941*(1 - 2*x)**(13/2)/1664 - 1580985*(1 - 2*x)*
*(11/2)/1408 + 406455*(1 - 2*x)**(9/2)/128 - 725445*(1 - 2*x)**(7/2)/128 + 3916031*(1 - 2*x)**(5/2)/640 - 1294
139*(1 - 2*x)**(3/2)/384

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Giac [A]
time = 0.62, size = 122, normalized size = 1.16 \begin {gather*} \frac {3645}{2176} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} + \frac {19683}{640} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {409941}{1664} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {1580985}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {406455}{128} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {725445}{128} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {3916031}{640} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {1294139}{384} \, {\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)^6*(3+5*x)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

3645/2176*(2*x - 1)^8*sqrt(-2*x + 1) + 19683/640*(2*x - 1)^7*sqrt(-2*x + 1) + 409941/1664*(2*x - 1)^6*sqrt(-2*
x + 1) + 1580985/1408*(2*x - 1)^5*sqrt(-2*x + 1) + 406455/128*(2*x - 1)^4*sqrt(-2*x + 1) + 725445/128*(2*x - 1
)^3*sqrt(-2*x + 1) + 3916031/640*(2*x - 1)^2*sqrt(-2*x + 1) - 1294139/384*(-2*x + 1)^(3/2)

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Mupad [B]
time = 1.21, size = 73, normalized size = 0.70 \begin {gather*} \frac {3916031\,{\left (1-2\,x\right )}^{5/2}}{640}-\frac {1294139\,{\left (1-2\,x\right )}^{3/2}}{384}-\frac {725445\,{\left (1-2\,x\right )}^{7/2}}{128}+\frac {406455\,{\left (1-2\,x\right )}^{9/2}}{128}-\frac {1580985\,{\left (1-2\,x\right )}^{11/2}}{1408}+\frac {409941\,{\left (1-2\,x\right )}^{13/2}}{1664}-\frac {19683\,{\left (1-2\,x\right )}^{15/2}}{640}+\frac {3645\,{\left (1-2\,x\right )}^{17/2}}{2176} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(1/2)*(3*x + 2)^6*(5*x + 3),x)

[Out]

(3916031*(1 - 2*x)^(5/2))/640 - (1294139*(1 - 2*x)^(3/2))/384 - (725445*(1 - 2*x)^(7/2))/128 + (406455*(1 - 2*
x)^(9/2))/128 - (1580985*(1 - 2*x)^(11/2))/1408 + (409941*(1 - 2*x)^(13/2))/1664 - (19683*(1 - 2*x)^(15/2))/64
0 + (3645*(1 - 2*x)^(17/2))/2176

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