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

Optimal. Leaf size=105 \[ -\frac {27 (2 x+3)^{11/2}}{1408}+\frac {63}{128} (2 x+3)^{9/2}-\frac {3519}{896} (2 x+3)^{7/2}+\frac {2095}{128} (2 x+3)^{5/2}-\frac {17201}{384} (2 x+3)^{3/2}+\frac {16005}{128} \sqrt {2 x+3}+\frac {7925}{128 \sqrt {2 x+3}}-\frac {1625}{384 (2 x+3)^{3/2}} \]

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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)^{11/2}}{1408}+\frac {63}{128} (2 x+3)^{9/2}-\frac {3519}{896} (2 x+3)^{7/2}+\frac {2095}{128} (2 x+3)^{5/2}-\frac {17201}{384} (2 x+3)^{3/2}+\frac {16005}{128} \sqrt {2 x+3}+\frac {7925}{128 \sqrt {2 x+3}}-\frac {1625}{384 (2 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1625/(384*(3 + 2*x)^(3/2)) + 7925/(128*Sqrt[3 + 2*x]) + (16005*Sqrt[3 + 2*x])/128 - (17201*(3 + 2*x)^(3/2))/3
84 + (2095*(3 + 2*x)^(5/2))/128 - (3519*(3 + 2*x)^(7/2))/896 + (63*(3 + 2*x)^(9/2))/128 - (27*(3 + 2*x)^(11/2)
)/1408

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)^{5/2}} \, dx &=\int \left (\frac {1625}{128 (3+2 x)^{5/2}}-\frac {7925}{128 (3+2 x)^{3/2}}+\frac {16005}{128 \sqrt {3+2 x}}-\frac {17201}{128} \sqrt {3+2 x}+\frac {10475}{128} (3+2 x)^{3/2}-\frac {3519}{128} (3+2 x)^{5/2}+\frac {567}{128} (3+2 x)^{7/2}-\frac {27}{128} (3+2 x)^{9/2}\right ) \, dx\\ &=-\frac {1625}{384 (3+2 x)^{3/2}}+\frac {7925}{128 \sqrt {3+2 x}}+\frac {16005}{128} \sqrt {3+2 x}-\frac {17201}{384} (3+2 x)^{3/2}+\frac {2095}{128} (3+2 x)^{5/2}-\frac {3519}{896} (3+2 x)^{7/2}+\frac {63}{128} (3+2 x)^{9/2}-\frac {27 (3+2 x)^{11/2}}{1408}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 48, normalized size = 0.46 \begin {gather*} -\frac {567 x^7-1323 x^6-9666 x^5-21360 x^4-17663 x^3-42003 x^2-184566 x-181486}{231 (2 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/231*(-181486 - 184566*x - 42003*x^2 - 17663*x^3 - 21360*x^4 - 9666*x^5 - 1323*x^6 + 567*x^7)/(3 + 2*x)^(3/2
)

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IntegrateAlgebraic [A]  time = 0.05, size = 76, normalized size = 0.72 \begin {gather*} \frac {-567 (2 x+3)^7+14553 (2 x+3)^6-116127 (2 x+3)^5+483945 (2 x+3)^4-1324477 (2 x+3)^3+3697155 (2 x+3)^2+1830675 (2 x+3)-125125}{29568 (2 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-125125 + 1830675*(3 + 2*x) + 3697155*(3 + 2*x)^2 - 1324477*(3 + 2*x)^3 + 483945*(3 + 2*x)^4 - 116127*(3 + 2*
x)^5 + 14553*(3 + 2*x)^6 - 567*(3 + 2*x)^7)/(29568*(3 + 2*x)^(3/2))

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fricas [A]  time = 0.39, size = 56, normalized size = 0.53 \begin {gather*} -\frac {{\left (567 \, x^{7} - 1323 \, x^{6} - 9666 \, x^{5} - 21360 \, x^{4} - 17663 \, x^{3} - 42003 \, x^{2} - 184566 \, x - 181486\right )} \sqrt {2 \, x + 3}}{231 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \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)^(5/2),x, algorithm="fricas")

[Out]

-1/231*(567*x^7 - 1323*x^6 - 9666*x^5 - 21360*x^4 - 17663*x^3 - 42003*x^2 - 184566*x - 181486)*sqrt(2*x + 3)/(
4*x^2 + 12*x + 9)

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giac [A]  time = 0.18, size = 69, normalized size = 0.66 \begin {gather*} -\frac {27}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {63}{128} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {3519}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {2095}{128} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {17201}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {16005}{128} \, \sqrt {2 \, x + 3} + \frac {25 \, {\left (951 \, x + 1394\right )}}{192 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}}} \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)^(5/2),x, algorithm="giac")

[Out]

-27/1408*(2*x + 3)^(11/2) + 63/128*(2*x + 3)^(9/2) - 3519/896*(2*x + 3)^(7/2) + 2095/128*(2*x + 3)^(5/2) - 172
01/384*(2*x + 3)^(3/2) + 16005/128*sqrt(2*x + 3) + 25/192*(951*x + 1394)/(2*x + 3)^(3/2)

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maple [A]  time = 0.01, size = 45, normalized size = 0.43 \begin {gather*} -\frac {567 x^{7}-1323 x^{6}-9666 x^{5}-21360 x^{4}-17663 x^{3}-42003 x^{2}-184566 x -181486}{231 \left (2 x +3\right )^{\frac {3}{2}}} \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)^(5/2),x)

[Out]

-1/231*(567*x^7-1323*x^6-9666*x^5-21360*x^4-17663*x^3-42003*x^2-184566*x-181486)/(2*x+3)^(3/2)

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maxima [A]  time = 0.56, size = 69, normalized size = 0.66 \begin {gather*} -\frac {27}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {63}{128} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {3519}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {2095}{128} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {17201}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {16005}{128} \, \sqrt {2 \, x + 3} + \frac {25 \, {\left (951 \, x + 1394\right )}}{192 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}}} \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)^(5/2),x, algorithm="maxima")

[Out]

-27/1408*(2*x + 3)^(11/2) + 63/128*(2*x + 3)^(9/2) - 3519/896*(2*x + 3)^(7/2) + 2095/128*(2*x + 3)^(5/2) - 172
01/384*(2*x + 3)^(3/2) + 16005/128*sqrt(2*x + 3) + 25/192*(951*x + 1394)/(2*x + 3)^(3/2)

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mupad [B]  time = 0.03, size = 68, normalized size = 0.65 \begin {gather*} \frac {\frac {7925\,x}{64}+\frac {17425}{96}}{{\left (2\,x+3\right )}^{3/2}}+\frac {16005\,\sqrt {2\,x+3}}{128}-\frac {17201\,{\left (2\,x+3\right )}^{3/2}}{384}+\frac {2095\,{\left (2\,x+3\right )}^{5/2}}{128}-\frac {3519\,{\left (2\,x+3\right )}^{7/2}}{896}+\frac {63\,{\left (2\,x+3\right )}^{9/2}}{128}-\frac {27\,{\left (2\,x+3\right )}^{11/2}}{1408} \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)^(5/2),x)

[Out]

((7925*x)/64 + 17425/96)/(2*x + 3)^(3/2) + (16005*(2*x + 3)^(1/2))/128 - (17201*(2*x + 3)^(3/2))/384 + (2095*(
2*x + 3)^(5/2))/128 - (3519*(2*x + 3)^(7/2))/896 + (63*(2*x + 3)^(9/2))/128 - (27*(2*x + 3)^(11/2))/1408

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sympy [A]  time = 56.09, size = 94, normalized size = 0.90 \begin {gather*} - \frac {27 \left (2 x + 3\right )^{\frac {11}{2}}}{1408} + \frac {63 \left (2 x + 3\right )^{\frac {9}{2}}}{128} - \frac {3519 \left (2 x + 3\right )^{\frac {7}{2}}}{896} + \frac {2095 \left (2 x + 3\right )^{\frac {5}{2}}}{128} - \frac {17201 \left (2 x + 3\right )^{\frac {3}{2}}}{384} + \frac {16005 \sqrt {2 x + 3}}{128} + \frac {7925}{128 \sqrt {2 x + 3}} - \frac {1625}{384 \left (2 x + 3\right )^{\frac {3}{2}}} \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)**(5/2),x)

[Out]

-27*(2*x + 3)**(11/2)/1408 + 63*(2*x + 3)**(9/2)/128 - 3519*(2*x + 3)**(7/2)/896 + 2095*(2*x + 3)**(5/2)/128 -
17201*(2*x + 3)**(3/2)/384 + 16005*sqrt(2*x + 3)/128 + 7925/(128*sqrt(2*x + 3)) - 1625/(384*(2*x + 3)**(3/2))

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