3.24.6 \(\int \frac {(5-x) (2+5 x+3 x^2)^3}{(3+2 x)^{3/2}} \, dx\)

Optimal. Leaf size=105 \[ -\frac {27 (2 x+3)^{13/2}}{1664}+\frac {567 (2 x+3)^{11/2}}{1408}-\frac {391}{128} (2 x+3)^{9/2}+\frac {10475}{896} (2 x+3)^{7/2}-\frac {17201}{640} (2 x+3)^{5/2}+\frac {5335}{128} (2 x+3)^{3/2}-\frac {7925}{128} \sqrt {2 x+3}-\frac {1625}{128 \sqrt {2 x+3}} \]

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Rubi [A]  time = 0.03, 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 = {771} \begin {gather*} -\frac {27 (2 x+3)^{13/2}}{1664}+\frac {567 (2 x+3)^{11/2}}{1408}-\frac {391}{128} (2 x+3)^{9/2}+\frac {10475}{896} (2 x+3)^{7/2}-\frac {17201}{640} (2 x+3)^{5/2}+\frac {5335}{128} (2 x+3)^{3/2}-\frac {7925}{128} \sqrt {2 x+3}-\frac {1625}{128 \sqrt {2 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1625/(128*Sqrt[3 + 2*x]) - (7925*Sqrt[3 + 2*x])/128 + (5335*(3 + 2*x)^(3/2))/128 - (17201*(3 + 2*x)^(5/2))/64
0 + (10475*(3 + 2*x)^(7/2))/896 - (391*(3 + 2*x)^(9/2))/128 + (567*(3 + 2*x)^(11/2))/1408 - (27*(3 + 2*x)^(13/
2))/1664

Rule 771

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 \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{(3+2 x)^{3/2}} \, dx &=\int \left (\frac {1625}{128 (3+2 x)^{3/2}}-\frac {7925}{128 \sqrt {3+2 x}}+\frac {16005}{128} \sqrt {3+2 x}-\frac {17201}{128} (3+2 x)^{3/2}+\frac {10475}{128} (3+2 x)^{5/2}-\frac {3519}{128} (3+2 x)^{7/2}+\frac {567}{128} (3+2 x)^{9/2}-\frac {27}{128} (3+2 x)^{11/2}\right ) \, dx\\ &=-\frac {1625}{128 \sqrt {3+2 x}}-\frac {7925}{128} \sqrt {3+2 x}+\frac {5335}{128} (3+2 x)^{3/2}-\frac {17201}{640} (3+2 x)^{5/2}+\frac {10475}{896} (3+2 x)^{7/2}-\frac {391}{128} (3+2 x)^{9/2}+\frac {567 (3+2 x)^{11/2}}{1408}-\frac {27 (3+2 x)^{13/2}}{1664}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 48, normalized size = 0.46 \begin {gather*} -\frac {10395 x^7-19845 x^6-180530 x^5-392500 x^4-398339 x^3-256433 x^2+77138 x+431614}{5005 \sqrt {2 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5005*(431614 + 77138*x - 256433*x^2 - 398339*x^3 - 392500*x^4 - 180530*x^5 - 19845*x^6 + 10395*x^7)/Sqrt[3
+ 2*x]

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IntegrateAlgebraic [A]  time = 0.06, size = 76, normalized size = 0.72 \begin {gather*} \frac {-10395 (2 x+3)^7+257985 (2 x+3)^6-1956955 (2 x+3)^5+7489625 (2 x+3)^4-17218201 (2 x+3)^3+26701675 (2 x+3)^2-39664625 (2 x+3)-8133125}{640640 \sqrt {2 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-8133125 - 39664625*(3 + 2*x) + 26701675*(3 + 2*x)^2 - 17218201*(3 + 2*x)^3 + 7489625*(3 + 2*x)^4 - 1956955*(
3 + 2*x)^5 + 257985*(3 + 2*x)^6 - 10395*(3 + 2*x)^7)/(640640*Sqrt[3 + 2*x])

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fricas [A]  time = 0.38, size = 44, normalized size = 0.42 \begin {gather*} -\frac {10395 \, x^{7} - 19845 \, x^{6} - 180530 \, x^{5} - 392500 \, x^{4} - 398339 \, x^{3} - 256433 \, x^{2} + 77138 \, x + 431614}{5005 \, \sqrt {2 \, x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5005*(10395*x^7 - 19845*x^6 - 180530*x^5 - 392500*x^4 - 398339*x^3 - 256433*x^2 + 77138*x + 431614)/sqrt(2*
x + 3)

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giac [A]  time = 0.17, size = 73, normalized size = 0.70 \begin {gather*} -\frac {27}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {567}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {391}{128} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {10475}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {17201}{640} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {5335}{128} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {7925}{128} \, \sqrt {2 \, x + 3} - \frac {1625}{128 \, \sqrt {2 \, x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/1664*(2*x + 3)^(13/2) + 567/1408*(2*x + 3)^(11/2) - 391/128*(2*x + 3)^(9/2) + 10475/896*(2*x + 3)^(7/2) -
17201/640*(2*x + 3)^(5/2) + 5335/128*(2*x + 3)^(3/2) - 7925/128*sqrt(2*x + 3) - 1625/128/sqrt(2*x + 3)

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maple [A]  time = 0.01, size = 45, normalized size = 0.43 \begin {gather*} -\frac {10395 x^{7}-19845 x^{6}-180530 x^{5}-392500 x^{4}-398339 x^{3}-256433 x^{2}+77138 x +431614}{5005 \sqrt {2 x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5005*(10395*x^7-19845*x^6-180530*x^5-392500*x^4-398339*x^3-256433*x^2+77138*x+431614)/(2*x+3)^(1/2)

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maxima [A]  time = 0.51, size = 73, normalized size = 0.70 \begin {gather*} -\frac {27}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {567}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {391}{128} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {10475}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {17201}{640} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {5335}{128} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {7925}{128} \, \sqrt {2 \, x + 3} - \frac {1625}{128 \, \sqrt {2 \, x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/1664*(2*x + 3)^(13/2) + 567/1408*(2*x + 3)^(11/2) - 391/128*(2*x + 3)^(9/2) + 10475/896*(2*x + 3)^(7/2) -
17201/640*(2*x + 3)^(5/2) + 5335/128*(2*x + 3)^(3/2) - 7925/128*sqrt(2*x + 3) - 1625/128/sqrt(2*x + 3)

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mupad [B]  time = 0.04, size = 73, normalized size = 0.70 \begin {gather*} \frac {5335\,{\left (2\,x+3\right )}^{3/2}}{128}-\frac {7925\,\sqrt {2\,x+3}}{128}-\frac {1625}{128\,\sqrt {2\,x+3}}-\frac {17201\,{\left (2\,x+3\right )}^{5/2}}{640}+\frac {10475\,{\left (2\,x+3\right )}^{7/2}}{896}-\frac {391\,{\left (2\,x+3\right )}^{9/2}}{128}+\frac {567\,{\left (2\,x+3\right )}^{11/2}}{1408}-\frac {27\,{\left (2\,x+3\right )}^{13/2}}{1664} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5335*(2*x + 3)^(3/2))/128 - (7925*(2*x + 3)^(1/2))/128 - 1625/(128*(2*x + 3)^(1/2)) - (17201*(2*x + 3)^(5/2))
/640 + (10475*(2*x + 3)^(7/2))/896 - (391*(2*x + 3)^(9/2))/128 + (567*(2*x + 3)^(11/2))/1408 - (27*(2*x + 3)^(
13/2))/1664

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sympy [A]  time = 46.04, size = 94, normalized size = 0.90 \begin {gather*} - \frac {27 \left (2 x + 3\right )^{\frac {13}{2}}}{1664} + \frac {567 \left (2 x + 3\right )^{\frac {11}{2}}}{1408} - \frac {391 \left (2 x + 3\right )^{\frac {9}{2}}}{128} + \frac {10475 \left (2 x + 3\right )^{\frac {7}{2}}}{896} - \frac {17201 \left (2 x + 3\right )^{\frac {5}{2}}}{640} + \frac {5335 \left (2 x + 3\right )^{\frac {3}{2}}}{128} - \frac {7925 \sqrt {2 x + 3}}{128} - \frac {1625}{128 \sqrt {2 x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27*(2*x + 3)**(13/2)/1664 + 567*(2*x + 3)**(11/2)/1408 - 391*(2*x + 3)**(9/2)/128 + 10475*(2*x + 3)**(7/2)/89
6 - 17201*(2*x + 3)**(5/2)/640 + 5335*(2*x + 3)**(3/2)/128 - 7925*sqrt(2*x + 3)/128 - 1625/(128*sqrt(2*x + 3))

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