3.19.16 \(\int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3 \, dx\) [1816]

Optimal. Leaf size=92 \[ -\frac {456533}{192} (1-2 x)^{3/2}+\frac {302379}{80} (1-2 x)^{5/2}-\frac {190707}{64} (1-2 x)^{7/2}+\frac {98209}{72} (1-2 x)^{9/2}-\frac {260055}{704} (1-2 x)^{11/2}+\frac {11475}{208} (1-2 x)^{13/2}-\frac {225}{64} (1-2 x)^{15/2} \]

[Out]

-456533/192*(1-2*x)^(3/2)+302379/80*(1-2*x)^(5/2)-190707/64*(1-2*x)^(7/2)+98209/72*(1-2*x)^(9/2)-260055/704*(1
-2*x)^(11/2)+11475/208*(1-2*x)^(13/2)-225/64*(1-2*x)^(15/2)

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Rubi [A]
time = 0.01, antiderivative size = 92, 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 {225}{64} (1-2 x)^{15/2}+\frac {11475}{208} (1-2 x)^{13/2}-\frac {260055}{704} (1-2 x)^{11/2}+\frac {98209}{72} (1-2 x)^{9/2}-\frac {190707}{64} (1-2 x)^{7/2}+\frac {302379}{80} (1-2 x)^{5/2}-\frac {456533}{192} (1-2 x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-456533*(1 - 2*x)^(3/2))/192 + (302379*(1 - 2*x)^(5/2))/80 - (190707*(1 - 2*x)^(7/2))/64 + (98209*(1 - 2*x)^(
9/2))/72 - (260055*(1 - 2*x)^(11/2))/704 + (11475*(1 - 2*x)^(13/2))/208 - (225*(1 - 2*x)^(15/2))/64

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)^3 \, dx &=\int \left (\frac {456533}{64} \sqrt {1-2 x}-\frac {302379}{16} (1-2 x)^{3/2}+\frac {1334949}{64} (1-2 x)^{5/2}-\frac {98209}{8} (1-2 x)^{7/2}+\frac {260055}{64} (1-2 x)^{9/2}-\frac {11475}{16} (1-2 x)^{11/2}+\frac {3375}{64} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac {456533}{192} (1-2 x)^{3/2}+\frac {302379}{80} (1-2 x)^{5/2}-\frac {190707}{64} (1-2 x)^{7/2}+\frac {98209}{72} (1-2 x)^{9/2}-\frac {260055}{704} (1-2 x)^{11/2}+\frac {11475}{208} (1-2 x)^{13/2}-\frac {225}{64} (1-2 x)^{15/2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 43, normalized size = 0.47 \begin {gather*} -\frac {(1-2 x)^{3/2} \left (3420622+8871906 x+15577455 x^2+18934285 x^3+15061950 x^4+7016625 x^5+1447875 x^6\right )}{6435} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6435*((1 - 2*x)^(3/2)*(3420622 + 8871906*x + 15577455*x^2 + 18934285*x^3 + 15061950*x^4 + 7016625*x^5 + 144
7875*x^6))

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Maple [A]
time = 0.12, size = 65, normalized size = 0.71

method	result	size
gosper	\(-\frac {\left (1447875 x^{6}+7016625 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}+8871906 x +3420622\right ) \left (1-2 x \right )^{\frac {3}{2}}}{6435}\)	\(40\)
trager	\(\left (450 x^{7}+\frac {25425}{13} x^{6}+\frac {513495}{143} x^{5}+\frac {4561324}{1287} x^{4}+\frac {2444125}{1287} x^{3}+\frac {722119}{2145} x^{2}-\frac {2030662}{6435} x -\frac {3420622}{6435}\right ) \sqrt {1-2 x}\)	\(44\)
risch	\(-\frac {\left (2895750 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}+2166357 x^{2}-2030662 x -3420622\right ) \left (-1+2 x \right )}{6435 \sqrt {1-2 x}}\)	\(50\)
derivativedivides	\(-\frac {456533 \left (1-2 x \right )^{\frac {3}{2}}}{192}+\frac {302379 \left (1-2 x \right )^{\frac {5}{2}}}{80}-\frac {190707 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {98209 \left (1-2 x \right )^{\frac {9}{2}}}{72}-\frac {260055 \left (1-2 x \right )^{\frac {11}{2}}}{704}+\frac {11475 \left (1-2 x \right )^{\frac {13}{2}}}{208}-\frac {225 \left (1-2 x \right )^{\frac {15}{2}}}{64}\)	\(65\)
default	\(-\frac {456533 \left (1-2 x \right )^{\frac {3}{2}}}{192}+\frac {302379 \left (1-2 x \right )^{\frac {5}{2}}}{80}-\frac {190707 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {98209 \left (1-2 x \right )^{\frac {9}{2}}}{72}-\frac {260055 \left (1-2 x \right )^{\frac {11}{2}}}{704}+\frac {11475 \left (1-2 x \right )^{\frac {13}{2}}}{208}-\frac {225 \left (1-2 x \right )^{\frac {15}{2}}}{64}\)	\(65\)
meijerg	\(\frac {72 \sqrt {\pi }-36 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {513 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{2 \sqrt {\pi }}+\frac {\frac {5412 \sqrt {\pi }}{35}-\frac {1353 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{70}}{\sqrt {\pi }}-\frac {17119 \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (280 x^{3}+120 x^{2}+48 x +16\right )}{315}\right )}{32 \sqrt {\pi }}+\frac {\frac {328 \sqrt {\pi }}{7}-\frac {41 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (5040 x^{4}+2240 x^{3}+960 x^{2}+384 x +128\right )}{112}}{\sqrt {\pi }}-\frac {12825 \left (-\frac {1024 \sqrt {\pi }}{9009}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (22176 x^{5}+10080 x^{4}+4480 x^{3}+1920 x^{2}+768 x +256\right )}{9009}\right )}{128 \sqrt {\pi }}+\frac {\frac {1200 \sqrt {\pi }}{1001}-\frac {75 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (192192 x^{6}+88704 x^{5}+40320 x^{4}+17920 x^{3}+7680 x^{2}+3072 x +1024\right )}{64064}}{\sqrt {\pi }}\)	\(273\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-456533/192*(1-2*x)^(3/2)+302379/80*(1-2*x)^(5/2)-190707/64*(1-2*x)^(7/2)+98209/72*(1-2*x)^(9/2)-260055/704*(1
-2*x)^(11/2)+11475/208*(1-2*x)^(13/2)-225/64*(1-2*x)^(15/2)

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Maxima [A]
time = 0.30, size = 64, normalized size = 0.70 \begin {gather*} -\frac {225}{64} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {11475}{208} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {260055}{704} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {98209}{72} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {190707}{64} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {302379}{80} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {456533}{192} \, {\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)^3*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

-225/64*(-2*x + 1)^(15/2) + 11475/208*(-2*x + 1)^(13/2) - 260055/704*(-2*x + 1)^(11/2) + 98209/72*(-2*x + 1)^(
9/2) - 190707/64*(-2*x + 1)^(7/2) + 302379/80*(-2*x + 1)^(5/2) - 456533/192*(-2*x + 1)^(3/2)

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Fricas [A]
time = 1.24, size = 44, normalized size = 0.48 \begin {gather*} \frac {1}{6435} \, {\left (2895750 \, x^{7} + 12585375 \, x^{6} + 23107275 \, x^{5} + 22806620 \, x^{4} + 12220625 \, x^{3} + 2166357 \, x^{2} - 2030662 \, x - 3420622\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)^3*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/6435*(2895750*x^7 + 12585375*x^6 + 23107275*x^5 + 22806620*x^4 + 12220625*x^3 + 2166357*x^2 - 2030662*x - 34
20622)*sqrt(-2*x + 1)

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Sympy [A]
time = 2.57, size = 82, normalized size = 0.89 \begin {gather*} - \frac {225 \left (1 - 2 x\right )^{\frac {15}{2}}}{64} + \frac {11475 \left (1 - 2 x\right )^{\frac {13}{2}}}{208} - \frac {260055 \left (1 - 2 x\right )^{\frac {11}{2}}}{704} + \frac {98209 \left (1 - 2 x\right )^{\frac {9}{2}}}{72} - \frac {190707 \left (1 - 2 x\right )^{\frac {7}{2}}}{64} + \frac {302379 \left (1 - 2 x\right )^{\frac {5}{2}}}{80} - \frac {456533 \left (1 - 2 x\right )^{\frac {3}{2}}}{192} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-225*(1 - 2*x)**(15/2)/64 + 11475*(1 - 2*x)**(13/2)/208 - 260055*(1 - 2*x)**(11/2)/704 + 98209*(1 - 2*x)**(9/2
)/72 - 190707*(1 - 2*x)**(7/2)/64 + 302379*(1 - 2*x)**(5/2)/80 - 456533*(1 - 2*x)**(3/2)/192

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Giac [A]
time = 1.11, size = 106, normalized size = 1.15 \begin {gather*} \frac {225}{64} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {11475}{208} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {260055}{704} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {98209}{72} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {190707}{64} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {302379}{80} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {456533}{192} \, {\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)^3*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

225/64*(2*x - 1)^7*sqrt(-2*x + 1) + 11475/208*(2*x - 1)^6*sqrt(-2*x + 1) + 260055/704*(2*x - 1)^5*sqrt(-2*x +
1) + 98209/72*(2*x - 1)^4*sqrt(-2*x + 1) + 190707/64*(2*x - 1)^3*sqrt(-2*x + 1) + 302379/80*(2*x - 1)^2*sqrt(-
2*x + 1) - 456533/192*(-2*x + 1)^(3/2)

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Mupad [B]
time = 0.04, size = 64, normalized size = 0.70 \begin {gather*} \frac {302379\,{\left (1-2\,x\right )}^{5/2}}{80}-\frac {456533\,{\left (1-2\,x\right )}^{3/2}}{192}-\frac {190707\,{\left (1-2\,x\right )}^{7/2}}{64}+\frac {98209\,{\left (1-2\,x\right )}^{9/2}}{72}-\frac {260055\,{\left (1-2\,x\right )}^{11/2}}{704}+\frac {11475\,{\left (1-2\,x\right )}^{13/2}}{208}-\frac {225\,{\left (1-2\,x\right )}^{15/2}}{64} \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)^3,x)

[Out]

(302379*(1 - 2*x)^(5/2))/80 - (456533*(1 - 2*x)^(3/2))/192 - (190707*(1 - 2*x)^(7/2))/64 + (98209*(1 - 2*x)^(9
/2))/72 - (260055*(1 - 2*x)^(11/2))/704 + (11475*(1 - 2*x)^(13/2))/208 - (225*(1 - 2*x)^(15/2))/64

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