3.2549 \(\int \frac{(5-x) (2+5 x+3 x^2)^3}{(3+2 x)^{7/2}} \, dx\)

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.0282774, antiderivative size = 105, normalized size of antiderivative = 1., 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} \[ -\frac{3}{128} (2 x+3)^{9/2}+\frac{81}{128} (2 x+3)^{7/2}-\frac{3519}{640} (2 x+3)^{5/2}+\frac{10475}{384} (2 x+3)^{3/2}-\frac{17201}{128} \sqrt{2 x+3}-\frac{16005}{128 \sqrt{2 x+3}}+\frac{7925}{384 (2 x+3)^{3/2}}-\frac{325}{128 (2 x+3)^{5/2}} \]

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*}

Mathematica [A]  time = 0.0212726, size = 48, normalized size = 0.46 \[ -\frac{45 x^7-135 x^6-702 x^5-1940 x^4+3195 x^3+41805 x^2+85070 x+51162}{15 (2 x+3)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 45, normalized size = 0.4 \begin{align*} -{\frac{45\,{x}^{7}-135\,{x}^{6}-702\,{x}^{5}-1940\,{x}^{4}+3195\,{x}^{3}+41805\,{x}^{2}+85070\,x+51162}{15} \left ( 3+2\,x \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02203, size = 93, normalized size = 0.89 \begin{align*} -\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{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.90673, size = 177, normalized size = 1.69 \begin{align*} -\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{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 48.7825, size = 94, normalized size = 0.9 \begin{align*} - \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{align*}

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))

________________________________________________________________________________________

Giac [A]  time = 1.09548, size = 93, normalized size = 0.89 \begin{align*} -\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{align*}

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)