3.24.8 \(\int \frac {(5-x) (2+5 x+3 x^2)^3}{(3+2 x)^{7/2}} \, dx\)

Optimal. Leaf size=105 \[ -\frac {3}{128} (2 x+3)^{9/2}+\frac {81}{128} (2 x+3)^{7/2}-\frac {3519}{640} (2 x+3)^{5/2}+\frac {10475}{384} (2 x+3)^{3/2}-\frac {17201}{128} \sqrt {2 x+3}-\frac {16005}{128 \sqrt {2 x+3}}+\frac {7925}{384 (2 x+3)^{3/2}}-\frac {325}{128 (2 x+3)^{5/2}} \]

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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 {3}{128} (2 x+3)^{9/2}+\frac {81}{128} (2 x+3)^{7/2}-\frac {3519}{640} (2 x+3)^{5/2}+\frac {10475}{384} (2 x+3)^{3/2}-\frac {17201}{128} \sqrt {2 x+3}-\frac {16005}{128 \sqrt {2 x+3}}+\frac {7925}{384 (2 x+3)^{3/2}}-\frac {325}{128 (2 x+3)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^3)/(3 + 2*x)^(7/2),x]

[Out]

-325/(128*(3 + 2*x)^(5/2)) + 7925/(384*(3 + 2*x)^(3/2)) - 16005/(128*Sqrt[3 + 2*x]) - (17201*Sqrt[3 + 2*x])/12
8 + (10475*(3 + 2*x)^(3/2))/384 - (3519*(3 + 2*x)^(5/2))/640 + (81*(3 + 2*x)^(7/2))/128 - (3*(3 + 2*x)^(9/2))/
128

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)^{7/2}} \, dx &=\int \left (\frac {1625}{128 (3+2 x)^{7/2}}-\frac {7925}{128 (3+2 x)^{5/2}}+\frac {16005}{128 (3+2 x)^{3/2}}-\frac {17201}{128 \sqrt {3+2 x}}+\frac {10475}{128} \sqrt {3+2 x}-\frac {3519}{128} (3+2 x)^{3/2}+\frac {567}{128} (3+2 x)^{5/2}-\frac {27}{128} (3+2 x)^{7/2}\right ) \, dx\\ &=-\frac {325}{128 (3+2 x)^{5/2}}+\frac {7925}{384 (3+2 x)^{3/2}}-\frac {16005}{128 \sqrt {3+2 x}}-\frac {17201}{128} \sqrt {3+2 x}+\frac {10475}{384} (3+2 x)^{3/2}-\frac {3519}{640} (3+2 x)^{5/2}+\frac {81}{128} (3+2 x)^{7/2}-\frac {3}{128} (3+2 x)^{9/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 48, normalized size = 0.46 \begin {gather*} -\frac {45 x^7-135 x^6-702 x^5-1940 x^4+3195 x^3+41805 x^2+85070 x+51162}{15 (2 x+3)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^3)/(3 + 2*x)^(7/2),x]

[Out]

-1/15*(51162 + 85070*x + 41805*x^2 + 3195*x^3 - 1940*x^4 - 702*x^5 - 135*x^6 + 45*x^7)/(3 + 2*x)^(5/2)

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IntegrateAlgebraic [A]  time = 0.05, size = 76, normalized size = 0.72 \begin {gather*} \frac {-45 (2 x+3)^7+1215 (2 x+3)^6-10557 (2 x+3)^5+52375 (2 x+3)^4-258015 (2 x+3)^3-240075 (2 x+3)^2+39625 (2 x+3)-4875}{1920 (2 x+3)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 5*x + 3*x^2)^3)/(3 + 2*x)^(7/2),x]

[Out]

(-4875 + 39625*(3 + 2*x) - 240075*(3 + 2*x)^2 - 258015*(3 + 2*x)^3 + 52375*(3 + 2*x)^4 - 10557*(3 + 2*x)^5 + 1
215*(3 + 2*x)^6 - 45*(3 + 2*x)^7)/(1920*(3 + 2*x)^(5/2))

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fricas [A]  time = 0.38, size = 61, normalized size = 0.58 \begin {gather*} -\frac {{\left (45 \, x^{7} - 135 \, x^{6} - 702 \, x^{5} - 1940 \, x^{4} + 3195 \, x^{3} + 41805 \, x^{2} + 85070 \, x + 51162\right )} \sqrt {2 \, x + 3}}{15 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\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)^(7/2),x, algorithm="fricas")

[Out]

-1/15*(45*x^7 - 135*x^6 - 702*x^5 - 1940*x^4 + 3195*x^3 + 41805*x^2 + 85070*x + 51162)*sqrt(2*x + 3)/(8*x^3 +
36*x^2 + 54*x + 27)

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giac [A]  time = 0.18, size = 69, normalized size = 0.66 \begin {gather*} -\frac {3}{128} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {81}{128} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {3519}{640} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {10475}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {17201}{128} \, \sqrt {2 \, x + 3} - \frac {5 \, {\left (9603 \, {\left (2 \, x + 3\right )}^{2} - 3170 \, x - 4560\right )}}{384 \, {\left (2 \, x + 3\right )}^{\frac {5}{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)^(7/2),x, algorithm="giac")

[Out]

-3/128*(2*x + 3)^(9/2) + 81/128*(2*x + 3)^(7/2) - 3519/640*(2*x + 3)^(5/2) + 10475/384*(2*x + 3)^(3/2) - 17201
/128*sqrt(2*x + 3) - 5/384*(9603*(2*x + 3)^2 - 3170*x - 4560)/(2*x + 3)^(5/2)

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maple [A]  time = 0.00, size = 45, normalized size = 0.43 \begin {gather*} -\frac {45 x^{7}-135 x^{6}-702 x^{5}-1940 x^{4}+3195 x^{3}+41805 x^{2}+85070 x +51162}{15 \left (2 x +3\right )^{\frac {5}{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)^(7/2),x)

[Out]

-1/15*(45*x^7-135*x^6-702*x^5-1940*x^4+3195*x^3+41805*x^2+85070*x+51162)/(2*x+3)^(5/2)

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maxima [A]  time = 0.49, size = 69, normalized size = 0.66 \begin {gather*} -\frac {3}{128} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {81}{128} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {3519}{640} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {10475}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {17201}{128} \, \sqrt {2 \, x + 3} - \frac {5 \, {\left (9603 \, {\left (2 \, x + 3\right )}^{2} - 3170 \, x - 4560\right )}}{384 \, {\left (2 \, x + 3\right )}^{\frac {5}{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)^(7/2),x, algorithm="maxima")

[Out]

-3/128*(2*x + 3)^(9/2) + 81/128*(2*x + 3)^(7/2) - 3519/640*(2*x + 3)^(5/2) + 10475/384*(2*x + 3)^(3/2) - 17201
/128*sqrt(2*x + 3) - 5/384*(9603*(2*x + 3)^2 - 3170*x - 4560)/(2*x + 3)^(5/2)

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mupad [B]  time = 0.03, size = 68, normalized size = 0.65 \begin {gather*} \frac {\frac {7925\,x}{192}-\frac {16005\,{\left (2\,x+3\right )}^2}{128}+\frac {475}{8}}{{\left (2\,x+3\right )}^{5/2}}-\frac {17201\,\sqrt {2\,x+3}}{128}+\frac {10475\,{\left (2\,x+3\right )}^{3/2}}{384}-\frac {3519\,{\left (2\,x+3\right )}^{5/2}}{640}+\frac {81\,{\left (2\,x+3\right )}^{7/2}}{128}-\frac {3\,{\left (2\,x+3\right )}^{9/2}}{128} \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)^(7/2),x)

[Out]

((7925*x)/192 - (16005*(2*x + 3)^2)/128 + 475/8)/(2*x + 3)^(5/2) - (17201*(2*x + 3)^(1/2))/128 + (10475*(2*x +
3)^(3/2))/384 - (3519*(2*x + 3)^(5/2))/640 + (81*(2*x + 3)^(7/2))/128 - (3*(2*x + 3)^(9/2))/128

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sympy [A]  time = 62.14, size = 94, normalized size = 0.90 \begin {gather*} - \frac {3 \left (2 x + 3\right )^{\frac {9}{2}}}{128} + \frac {81 \left (2 x + 3\right )^{\frac {7}{2}}}{128} - \frac {3519 \left (2 x + 3\right )^{\frac {5}{2}}}{640} + \frac {10475 \left (2 x + 3\right )^{\frac {3}{2}}}{384} - \frac {17201 \sqrt {2 x + 3}}{128} - \frac {16005}{128 \sqrt {2 x + 3}} + \frac {7925}{384 \left (2 x + 3\right )^{\frac {3}{2}}} - \frac {325}{128 \left (2 x + 3\right )^{\frac {5}{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)**(7/2),x)

[Out]

-3*(2*x + 3)**(9/2)/128 + 81*(2*x + 3)**(7/2)/128 - 3519*(2*x + 3)**(5/2)/640 + 10475*(2*x + 3)**(3/2)/384 - 1
7201*sqrt(2*x + 3)/128 - 16005/(128*sqrt(2*x + 3)) + 7925/(384*(2*x + 3)**(3/2)) - 325/(128*(2*x + 3)**(5/2))

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