3.19.83 \(\int (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^3 \, dx\) [1883]

Optimal. Leaf size=92 \[ -\frac {456533}{320} (1-2 x)^{5/2}+\frac {43197}{16} (1-2 x)^{7/2}-\frac {444983}{192} (1-2 x)^{9/2}+\frac {98209}{88} (1-2 x)^{11/2}-\frac {260055}{832} (1-2 x)^{13/2}+\frac {765}{16} (1-2 x)^{15/2}-\frac {3375 (1-2 x)^{17/2}}{1088} \]

[Out]

-456533/320*(1-2*x)^(5/2)+43197/16*(1-2*x)^(7/2)-444983/192*(1-2*x)^(9/2)+98209/88*(1-2*x)^(11/2)-260055/832*(
1-2*x)^(13/2)+765/16*(1-2*x)^(15/2)-3375/1088*(1-2*x)^(17/2)

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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 {3375 (1-2 x)^{17/2}}{1088}+\frac {765}{16} (1-2 x)^{15/2}-\frac {260055}{832} (1-2 x)^{13/2}+\frac {98209}{88} (1-2 x)^{11/2}-\frac {444983}{192} (1-2 x)^{9/2}+\frac {43197}{16} (1-2 x)^{7/2}-\frac {456533}{320} (1-2 x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-456533*(1 - 2*x)^(5/2))/320 + (43197*(1 - 2*x)^(7/2))/16 - (444983*(1 - 2*x)^(9/2))/192 + (98209*(1 - 2*x)^(
11/2))/88 - (260055*(1 - 2*x)^(13/2))/832 + (765*(1 - 2*x)^(15/2))/16 - (3375*(1 - 2*x)^(17/2))/1088

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 (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^3 \, dx &=\int \left (\frac {456533}{64} (1-2 x)^{3/2}-\frac {302379}{16} (1-2 x)^{5/2}+\frac {1334949}{64} (1-2 x)^{7/2}-\frac {98209}{8} (1-2 x)^{9/2}+\frac {260055}{64} (1-2 x)^{11/2}-\frac {11475}{16} (1-2 x)^{13/2}+\frac {3375}{64} (1-2 x)^{15/2}\right ) \, dx\\ &=-\frac {456533}{320} (1-2 x)^{5/2}+\frac {43197}{16} (1-2 x)^{7/2}-\frac {444983}{192} (1-2 x)^{9/2}+\frac {98209}{88} (1-2 x)^{11/2}-\frac {260055}{832} (1-2 x)^{13/2}+\frac {765}{16} (1-2 x)^{15/2}-\frac {3375 (1-2 x)^{17/2}}{1088}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 43, normalized size = 0.47 \begin {gather*} -\frac {(1-2 x)^{5/2} \left (7158706+27917090 x+60296725 x^2+82215885 x^3+70032600 x^4+34073325 x^5+7239375 x^6\right )}{36465} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/36465*((1 - 2*x)^(5/2)*(7158706 + 27917090*x + 60296725*x^2 + 82215885*x^3 + 70032600*x^4 + 34073325*x^5 +
7239375*x^6))

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Maple [A]
time = 0.10, size = 65, normalized size = 0.71

method	result	size
gosper	\(-\frac {\left (7239375 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x +7158706\right ) \left (1-2 x \right )^{\frac {5}{2}}}{36465}\)	\(40\)
trager	\(\left (-\frac {13500}{17} x^{8}-\frac {50040}{17} x^{7}-\frac {915615}{221} x^{6}-\frac {5520431}{2431} x^{5}+\frac {3528808}{7293} x^{4}+\frac {9460531}{7293} x^{3}+\frac {7578937}{12155} x^{2}+\frac {717734}{36465} x -\frac {7158706}{36465}\right ) \sqrt {1-2 x}\)	\(49\)
risch	\(\frac {\left (28957500 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}-17644040 x^{4}-47302655 x^{3}-22736811 x^{2}-717734 x +7158706\right ) \left (-1+2 x \right )}{36465 \sqrt {1-2 x}}\)	\(55\)
derivativedivides	\(-\frac {456533 \left (1-2 x \right )^{\frac {5}{2}}}{320}+\frac {43197 \left (1-2 x \right )^{\frac {7}{2}}}{16}-\frac {444983 \left (1-2 x \right )^{\frac {9}{2}}}{192}+\frac {98209 \left (1-2 x \right )^{\frac {11}{2}}}{88}-\frac {260055 \left (1-2 x \right )^{\frac {13}{2}}}{832}+\frac {765 \left (1-2 x \right )^{\frac {15}{2}}}{16}-\frac {3375 \left (1-2 x \right )^{\frac {17}{2}}}{1088}\)	\(65\)
default	\(-\frac {456533 \left (1-2 x \right )^{\frac {5}{2}}}{320}+\frac {43197 \left (1-2 x \right )^{\frac {7}{2}}}{16}-\frac {444983 \left (1-2 x \right )^{\frac {9}{2}}}{192}+\frac {98209 \left (1-2 x \right )^{\frac {11}{2}}}{88}-\frac {260055 \left (1-2 x \right )^{\frac {13}{2}}}{832}+\frac {765 \left (1-2 x \right )^{\frac {15}{2}}}{16}-\frac {3375 \left (1-2 x \right )^{\frac {17}{2}}}{1088}\)	\(65\)
meijerg	\(-\frac {81 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (8 x^{2}-8 x +2\right ) \sqrt {1-2 x}}{15}\right )}{\sqrt {\pi }}+\frac {\frac {2052 \sqrt {\pi }}{35}-\frac {513 \sqrt {\pi }\, \left (160 x^{3}-128 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{70}}{\sqrt {\pi }}-\frac {12177 \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (1120 x^{4}-800 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{945}\right )}{16 \sqrt {\pi }}+\frac {\frac {34238 \sqrt {\pi }}{1155}-\frac {17119 \sqrt {\pi }\, \left (26880 x^{5}-17920 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{73920}}{\sqrt {\pi }}-\frac {60885 \left (-\frac {1024 \sqrt {\pi }}{45045}+\frac {4 \sqrt {\pi }\, \left (147840 x^{6}-94080 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{45045}\right )}{128 \sqrt {\pi }}+\frac {\frac {2280 \sqrt {\pi }}{1001}-\frac {285 \sqrt {\pi }\, \left (1537536 x^{7}-946176 x^{6}+8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{128128}}{\sqrt {\pi }}-\frac {10125 \left (-\frac {8192 \sqrt {\pi }}{765765}+\frac {4 \sqrt {\pi }\, \left (7687680 x^{8}-4612608 x^{7}+29568 x^{6}+16128 x^{5}+8960 x^{4}+5120 x^{3}+3072 x^{2}+2048 x +2048\right ) \sqrt {1-2 x}}{765765}\right )}{512 \sqrt {\pi }}\)	\(338\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-456533/320*(1-2*x)^(5/2)+43197/16*(1-2*x)^(7/2)-444983/192*(1-2*x)^(9/2)+98209/88*(1-2*x)^(11/2)-260055/832*(
1-2*x)^(13/2)+765/16*(1-2*x)^(15/2)-3375/1088*(1-2*x)^(17/2)

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Maxima [A]
time = 0.31, size = 64, normalized size = 0.70 \begin {gather*} -\frac {3375}{1088} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} + \frac {765}{16} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {260055}{832} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {98209}{88} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {444983}{192} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {43197}{16} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {456533}{320} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375/1088*(-2*x + 1)^(17/2) + 765/16*(-2*x + 1)^(15/2) - 260055/832*(-2*x + 1)^(13/2) + 98209/88*(-2*x + 1)^(
11/2) - 444983/192*(-2*x + 1)^(9/2) + 43197/16*(-2*x + 1)^(7/2) - 456533/320*(-2*x + 1)^(5/2)

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Fricas [A]
time = 1.54, size = 49, normalized size = 0.53 \begin {gather*} -\frac {1}{36465} \, {\left (28957500 \, x^{8} + 107335800 \, x^{7} + 151076475 \, x^{6} + 82806465 \, x^{5} - 17644040 \, x^{4} - 47302655 \, x^{3} - 22736811 \, x^{2} - 717734 \, x + 7158706\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36465*(28957500*x^8 + 107335800*x^7 + 151076475*x^6 + 82806465*x^5 - 17644040*x^4 - 47302655*x^3 - 22736811
*x^2 - 717734*x + 7158706)*sqrt(-2*x + 1)

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Sympy [A]
time = 17.46, size = 82, normalized size = 0.89 \begin {gather*} - \frac {3375 \left (1 - 2 x\right )^{\frac {17}{2}}}{1088} + \frac {765 \left (1 - 2 x\right )^{\frac {15}{2}}}{16} - \frac {260055 \left (1 - 2 x\right )^{\frac {13}{2}}}{832} + \frac {98209 \left (1 - 2 x\right )^{\frac {11}{2}}}{88} - \frac {444983 \left (1 - 2 x\right )^{\frac {9}{2}}}{192} + \frac {43197 \left (1 - 2 x\right )^{\frac {7}{2}}}{16} - \frac {456533 \left (1 - 2 x\right )^{\frac {5}{2}}}{320} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375*(1 - 2*x)**(17/2)/1088 + 765*(1 - 2*x)**(15/2)/16 - 260055*(1 - 2*x)**(13/2)/832 + 98209*(1 - 2*x)**(11/
2)/88 - 444983*(1 - 2*x)**(9/2)/192 + 43197*(1 - 2*x)**(7/2)/16 - 456533*(1 - 2*x)**(5/2)/320

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Giac [A]
time = 1.24, size = 113, normalized size = 1.23 \begin {gather*} -\frac {3375}{1088} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} - \frac {765}{16} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {260055}{832} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {98209}{88} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {444983}{192} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {43197}{16} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {456533}{320} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375/1088*(2*x - 1)^8*sqrt(-2*x + 1) - 765/16*(2*x - 1)^7*sqrt(-2*x + 1) - 260055/832*(2*x - 1)^6*sqrt(-2*x +
1) - 98209/88*(2*x - 1)^5*sqrt(-2*x + 1) - 444983/192*(2*x - 1)^4*sqrt(-2*x + 1) - 43197/16*(2*x - 1)^3*sqrt(
-2*x + 1) - 456533/320*(2*x - 1)^2*sqrt(-2*x + 1)

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Mupad [B]
time = 0.03, size = 64, normalized size = 0.70 \begin {gather*} \frac {43197\,{\left (1-2\,x\right )}^{7/2}}{16}-\frac {456533\,{\left (1-2\,x\right )}^{5/2}}{320}-\frac {444983\,{\left (1-2\,x\right )}^{9/2}}{192}+\frac {98209\,{\left (1-2\,x\right )}^{11/2}}{88}-\frac {260055\,{\left (1-2\,x\right )}^{13/2}}{832}+\frac {765\,{\left (1-2\,x\right )}^{15/2}}{16}-\frac {3375\,{\left (1-2\,x\right )}^{17/2}}{1088} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(43197*(1 - 2*x)^(7/2))/16 - (456533*(1 - 2*x)^(5/2))/320 - (444983*(1 - 2*x)^(9/2))/192 + (98209*(1 - 2*x)^(1
1/2))/88 - (260055*(1 - 2*x)^(13/2))/832 + (765*(1 - 2*x)^(15/2))/16 - (3375*(1 - 2*x)^(17/2))/1088

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