3.26.43 \(\int (5-x) (3+2 x)^{5/2} (2+5 x+3 x^2)^3 \, dx\) [2543]

Optimal. Leaf size=105 \[ \frac {1625}{896} (3+2 x)^{7/2}-\frac {7925 (3+2 x)^{9/2}}{1152}+\frac {1455}{128} (3+2 x)^{11/2}-\frac {17201 (3+2 x)^{13/2}}{1664}+\frac {2095}{384} (3+2 x)^{15/2}-\frac {207}{128} (3+2 x)^{17/2}+\frac {567 (3+2 x)^{19/2}}{2432}-\frac {9}{896} (3+2 x)^{21/2} \]

[Out]

1625/896*(3+2*x)^(7/2)-7925/1152*(3+2*x)^(9/2)+1455/128*(3+2*x)^(11/2)-17201/1664*(3+2*x)^(13/2)+2095/384*(3+2
*x)^(15/2)-207/128*(3+2*x)^(17/2)+567/2432*(3+2*x)^(19/2)-9/896*(3+2*x)^(21/2)

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Rubi [A]
time = 0.02, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {785} \begin {gather*} -\frac {9}{896} (2 x+3)^{21/2}+\frac {567 (2 x+3)^{19/2}}{2432}-\frac {207}{128} (2 x+3)^{17/2}+\frac {2095}{384} (2 x+3)^{15/2}-\frac {17201 (2 x+3)^{13/2}}{1664}+\frac {1455}{128} (2 x+3)^{11/2}-\frac {7925 (2 x+3)^{9/2}}{1152}+\frac {1625}{896} (2 x+3)^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1625*(3 + 2*x)^(7/2))/896 - (7925*(3 + 2*x)^(9/2))/1152 + (1455*(3 + 2*x)^(11/2))/128 - (17201*(3 + 2*x)^(13/
2))/1664 + (2095*(3 + 2*x)^(15/2))/384 - (207*(3 + 2*x)^(17/2))/128 + (567*(3 + 2*x)^(19/2))/2432 - (9*(3 + 2*
x)^(21/2))/896

Rule 785

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^3 \, dx &=\int \left (\frac {1625}{128} (3+2 x)^{5/2}-\frac {7925}{128} (3+2 x)^{7/2}+\frac {16005}{128} (3+2 x)^{9/2}-\frac {17201}{128} (3+2 x)^{11/2}+\frac {10475}{128} (3+2 x)^{13/2}-\frac {3519}{128} (3+2 x)^{15/2}+\frac {567}{128} (3+2 x)^{17/2}-\frac {27}{128} (3+2 x)^{19/2}\right ) \, dx\\ &=\frac {1625}{896} (3+2 x)^{7/2}-\frac {7925 (3+2 x)^{9/2}}{1152}+\frac {1455}{128} (3+2 x)^{11/2}-\frac {17201 (3+2 x)^{13/2}}{1664}+\frac {2095}{384} (3+2 x)^{15/2}-\frac {207}{128} (3+2 x)^{17/2}+\frac {567 (3+2 x)^{19/2}}{2432}-\frac {9}{896} (3+2 x)^{21/2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 48, normalized size = 0.46 \begin {gather*} -\frac {(3+2 x)^{7/2} \left (-20346-160006 x-517293 x^2-871983 x^3-791700 x^4-339066 x^5-22113 x^6+20007 x^7\right )}{15561} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/15561*((3 + 2*x)^(7/2)*(-20346 - 160006*x - 517293*x^2 - 871983*x^3 - 791700*x^4 - 339066*x^5 - 22113*x^6 +
20007*x^7))

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Maple [A]
time = 0.09, size = 74, normalized size = 0.70

method	result	size
gosper	\(-\frac {\left (20007 x^{7}-22113 x^{6}-339066 x^{5}-791700 x^{4}-871983 x^{3}-517293 x^{2}-160006 x -20346\right ) \left (3+2 x \right )^{\frac {7}{2}}}{15561}\)	\(45\)
trager	\(\left (-\frac {72}{7} x^{10}-\frac {4644}{133} x^{9}+\frac {20754}{133} x^{8}+\frac {492151}{399} x^{7}+\frac {863233}{247} x^{6}+\frac {1387878}{247} x^{5}+\frac {12623654}{2223} x^{4}+\frac {19133449}{5187} x^{3}+\frac {2593299}{1729} x^{2}+\frac {602094}{1729} x +\frac {61038}{1729}\right ) \sqrt {3+2 x}\)	\(59\)
risch	\(-\frac {\left (160056 x^{10}+543348 x^{9}-2428218 x^{8}-19193889 x^{7}-54383679 x^{6}-87436314 x^{5}-88365578 x^{4}-57400347 x^{3}-23339691 x^{2}-5418846 x -549342\right ) \sqrt {3+2 x}}{15561}\)	\(60\)
derivativedivides	\(\frac {1625 \left (3+2 x \right )^{\frac {7}{2}}}{896}-\frac {7925 \left (3+2 x \right )^{\frac {9}{2}}}{1152}+\frac {1455 \left (3+2 x \right )^{\frac {11}{2}}}{128}-\frac {17201 \left (3+2 x \right )^{\frac {13}{2}}}{1664}+\frac {2095 \left (3+2 x \right )^{\frac {15}{2}}}{384}-\frac {207 \left (3+2 x \right )^{\frac {17}{2}}}{128}+\frac {567 \left (3+2 x \right )^{\frac {19}{2}}}{2432}-\frac {9 \left (3+2 x \right )^{\frac {21}{2}}}{896}\)	\(74\)
default	\(\frac {1625 \left (3+2 x \right )^{\frac {7}{2}}}{896}-\frac {7925 \left (3+2 x \right )^{\frac {9}{2}}}{1152}+\frac {1455 \left (3+2 x \right )^{\frac {11}{2}}}{128}-\frac {17201 \left (3+2 x \right )^{\frac {13}{2}}}{1664}+\frac {2095 \left (3+2 x \right )^{\frac {15}{2}}}{384}-\frac {207 \left (3+2 x \right )^{\frac {17}{2}}}{128}+\frac {567 \left (3+2 x \right )^{\frac {19}{2}}}{2432}-\frac {9 \left (3+2 x \right )^{\frac {21}{2}}}{896}\)	\(74\)
meijerg	\(-\frac {14641965 \sqrt {3}\, \left (-\frac {256 \sqrt {\pi }}{45045}+\frac {2 \sqrt {\pi }\, \left (-\frac {39424}{243} x^{6}-\frac {1792}{3} x^{5}-\frac {47488}{81} x^{4}-\frac {320}{27} x^{3}+\frac {64}{3} x^{2}-\frac {128}{3} x +128\right ) \sqrt {1+\frac {2 x}{3}}}{45045}\right )}{128 \sqrt {\pi }}-\frac {17878725 \sqrt {3}\, \left (\frac {2048 \sqrt {\pi }}{675675}-\frac {8 \sqrt {\pi }\, \left (\frac {256256}{729} x^{7}+\frac {305536}{243} x^{6}+\frac {31808}{27} x^{5}+\frac {1120}{81} x^{4}-\frac {640}{27} x^{3}+\frac {128}{3} x^{2}-\frac {256}{3} x +256\right ) \sqrt {1+\frac {2 x}{3}}}{675675}\right )}{128 \sqrt {\pi }}-\frac {9743085 \sqrt {3}\, \left (-\frac {4096 \sqrt {\pi }}{2297295}+\frac {4 \sqrt {\pi }\, \left (-\frac {1025024}{729} x^{8}-\frac {3587584}{729} x^{7}-\frac {1084160}{243} x^{6}-\frac {896}{27} x^{5}+\frac {4480}{81} x^{4}-\frac {2560}{27} x^{3}+\frac {512}{3} x^{2}-\frac {1024}{3} x +1024\right ) \sqrt {1+\frac {2 x}{3}}}{2297295}\right )}{128 \sqrt {\pi }}-\frac {1585575 \sqrt {3}\, \left (\frac {128 \sqrt {\pi }}{10395}-\frac {8 \sqrt {\pi }\, \left (\frac {448}{27} x^{5}+\frac {5152}{81} x^{4}+\frac {1808}{27} x^{3}+\frac {8}{3} x^{2}-\frac {16}{3} x +16\right ) \sqrt {1+\frac {2 x}{3}}}{10395}\right )}{32 \sqrt {\pi }}-\frac {88695 \sqrt {3}\, \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-\frac {448}{81} x^{4}-\frac {608}{27} x^{3}-\frac {80}{3} x^{2}-\frac {8}{3} x +8\right ) \sqrt {1+\frac {2 x}{3}}}{945}\right )}{8 \sqrt {\pi }}-\frac {2025 \sqrt {3}\, \left (\frac {16 \sqrt {\pi }}{105}-\frac {8 \sqrt {\pi }\, \left (\frac {16}{27} x^{3}+\frac {8}{3} x^{2}+4 x +2\right ) \sqrt {1+\frac {2 x}{3}}}{105}\right )}{2 \sqrt {\pi }}+\frac {23914845 \sqrt {3}\, \left (-\frac {32768 \sqrt {\pi }}{43648605}+\frac {\sqrt {\pi }\, \left (-\frac {756760576}{19683} x^{10}-\frac {856334336}{6561} x^{9}-\frac {27529216}{243} x^{8}-\frac {292864}{729} x^{7}+\frac {157696}{243} x^{6}-\frac {28672}{27} x^{5}+\frac {143360}{81} x^{4}-\frac {81920}{27} x^{3}+\frac {16384}{3} x^{2}-\frac {32768}{3} x +32768\right ) \sqrt {1+\frac {2 x}{3}}}{43648605}\right )}{2048 \sqrt {\pi }}\)	\(399\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1625/896*(3+2*x)^(7/2)-7925/1152*(3+2*x)^(9/2)+1455/128*(3+2*x)^(11/2)-17201/1664*(3+2*x)^(13/2)+2095/384*(3+2
*x)^(15/2)-207/128*(3+2*x)^(17/2)+567/2432*(3+2*x)^(19/2)-9/896*(3+2*x)^(21/2)

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Maxima [A]
time = 0.27, size = 73, normalized size = 0.70 \begin {gather*} -\frac {9}{896} \, {\left (2 \, x + 3\right )}^{\frac {21}{2}} + \frac {567}{2432} \, {\left (2 \, x + 3\right )}^{\frac {19}{2}} - \frac {207}{128} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} + \frac {2095}{384} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} - \frac {17201}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {1455}{128} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {7925}{1152} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {1625}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/896*(2*x + 3)^(21/2) + 567/2432*(2*x + 3)^(19/2) - 207/128*(2*x + 3)^(17/2) + 2095/384*(2*x + 3)^(15/2) - 1
7201/1664*(2*x + 3)^(13/2) + 1455/128*(2*x + 3)^(11/2) - 7925/1152*(2*x + 3)^(9/2) + 1625/896*(2*x + 3)^(7/2)

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Fricas [A]
time = 3.44, size = 59, normalized size = 0.56 \begin {gather*} -\frac {1}{15561} \, {\left (160056 \, x^{10} + 543348 \, x^{9} - 2428218 \, x^{8} - 19193889 \, x^{7} - 54383679 \, x^{6} - 87436314 \, x^{5} - 88365578 \, x^{4} - 57400347 \, x^{3} - 23339691 \, x^{2} - 5418846 \, x - 549342\right )} \sqrt {2 \, x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15561*(160056*x^10 + 543348*x^9 - 2428218*x^8 - 19193889*x^7 - 54383679*x^6 - 87436314*x^5 - 88365578*x^4 -
57400347*x^3 - 23339691*x^2 - 5418846*x - 549342)*sqrt(2*x + 3)

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Sympy [A]
time = 23.16, size = 94, normalized size = 0.90 \begin {gather*} - \frac {9 \left (2 x + 3\right )^{\frac {21}{2}}}{896} + \frac {567 \left (2 x + 3\right )^{\frac {19}{2}}}{2432} - \frac {207 \left (2 x + 3\right )^{\frac {17}{2}}}{128} + \frac {2095 \left (2 x + 3\right )^{\frac {15}{2}}}{384} - \frac {17201 \left (2 x + 3\right )^{\frac {13}{2}}}{1664} + \frac {1455 \left (2 x + 3\right )^{\frac {11}{2}}}{128} - \frac {7925 \left (2 x + 3\right )^{\frac {9}{2}}}{1152} + \frac {1625 \left (2 x + 3\right )^{\frac {7}{2}}}{896} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*(2*x + 3)**(21/2)/896 + 567*(2*x + 3)**(19/2)/2432 - 207*(2*x + 3)**(17/2)/128 + 2095*(2*x + 3)**(15/2)/384
- 17201*(2*x + 3)**(13/2)/1664 + 1455*(2*x + 3)**(11/2)/128 - 7925*(2*x + 3)**(9/2)/1152 + 1625*(2*x + 3)**(7
/2)/896

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Giac [A]
time = 1.14, size = 73, normalized size = 0.70 \begin {gather*} -\frac {9}{896} \, {\left (2 \, x + 3\right )}^{\frac {21}{2}} + \frac {567}{2432} \, {\left (2 \, x + 3\right )}^{\frac {19}{2}} - \frac {207}{128} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} + \frac {2095}{384} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} - \frac {17201}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {1455}{128} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {7925}{1152} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {1625}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/896*(2*x + 3)^(21/2) + 567/2432*(2*x + 3)^(19/2) - 207/128*(2*x + 3)^(17/2) + 2095/384*(2*x + 3)^(15/2) - 1
7201/1664*(2*x + 3)^(13/2) + 1455/128*(2*x + 3)^(11/2) - 7925/1152*(2*x + 3)^(9/2) + 1625/896*(2*x + 3)^(7/2)

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Mupad [B]
time = 0.03, size = 73, normalized size = 0.70 \begin {gather*} \frac {1625\,{\left (2\,x+3\right )}^{7/2}}{896}-\frac {7925\,{\left (2\,x+3\right )}^{9/2}}{1152}+\frac {1455\,{\left (2\,x+3\right )}^{11/2}}{128}-\frac {17201\,{\left (2\,x+3\right )}^{13/2}}{1664}+\frac {2095\,{\left (2\,x+3\right )}^{15/2}}{384}-\frac {207\,{\left (2\,x+3\right )}^{17/2}}{128}+\frac {567\,{\left (2\,x+3\right )}^{19/2}}{2432}-\frac {9\,{\left (2\,x+3\right )}^{21/2}}{896} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1625*(2*x + 3)^(7/2))/896 - (7925*(2*x + 3)^(9/2))/1152 + (1455*(2*x + 3)^(11/2))/128 - (17201*(2*x + 3)^(13/
2))/1664 + (2095*(2*x + 3)^(15/2))/384 - (207*(2*x + 3)^(17/2))/128 + (567*(2*x + 3)^(19/2))/2432 - (9*(2*x +
3)^(21/2))/896

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