3.2541 \(\int \frac{(5-x) (2+5 x+3 x^2)^2}{(3+2 x)^{7/2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{9}{160} (2 x+3)^{5/2}+\frac{55}{32} (2 x+3)^{3/2}-\frac{359}{16} \sqrt{2 x+3}-\frac{651}{16 \sqrt{2 x+3}}+\frac{355}{32 (2 x+3)^{3/2}}-\frac{65}{32 (2 x+3)^{5/2}} \]

[Out]

-65/(32*(3 + 2*x)^(5/2)) + 355/(32*(3 + 2*x)^(3/2)) - 651/(16*Sqrt[3 + 2*x]) - (359*Sqrt[3 + 2*x])/16 + (55*(3
+ 2*x)^(3/2))/32 - (9*(3 + 2*x)^(5/2))/160

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Rubi [A]  time = 0.0236495, antiderivative size = 79, 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} \[ -\frac{9}{160} (2 x+3)^{5/2}+\frac{55}{32} (2 x+3)^{3/2}-\frac{359}{16} \sqrt{2 x+3}-\frac{651}{16 \sqrt{2 x+3}}+\frac{355}{32 (2 x+3)^{3/2}}-\frac{65}{32 (2 x+3)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-65/(32*(3 + 2*x)^(5/2)) + 355/(32*(3 + 2*x)^(3/2)) - 651/(16*Sqrt[3 + 2*x]) - (359*Sqrt[3 + 2*x])/16 + (55*(3
+ 2*x)^(3/2))/32 - (9*(3 + 2*x)^(5/2))/160

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 )^2}{(3+2 x)^{7/2}} \, dx &=\int \left (\frac{325}{32 (3+2 x)^{7/2}}-\frac{1065}{32 (3+2 x)^{5/2}}+\frac{651}{16 (3+2 x)^{3/2}}-\frac{359}{16 \sqrt{3+2 x}}+\frac{165}{32} \sqrt{3+2 x}-\frac{9}{32} (3+2 x)^{3/2}\right ) \, dx\\ &=-\frac{65}{32 (3+2 x)^{5/2}}+\frac{355}{32 (3+2 x)^{3/2}}-\frac{651}{16 \sqrt{3+2 x}}-\frac{359}{16} \sqrt{3+2 x}+\frac{55}{32} (3+2 x)^{3/2}-\frac{9}{160} (3+2 x)^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.016754, size = 38, normalized size = 0.48 \[ -\frac{9 x^5-70 x^4+275 x^3+3300 x^2+6760 x+4076}{5 (2 x+3)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(4076 + 6760*x + 3300*x^2 + 275*x^3 - 70*x^4 + 9*x^5)/(5*(3 + 2*x)^(5/2))

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Maple [A]  time = 0.004, size = 35, normalized size = 0.4 \begin{align*} -{\frac{9\,{x}^{5}-70\,{x}^{4}+275\,{x}^{3}+3300\,{x}^{2}+6760\,x+4076}{5} \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)^2/(3+2*x)^(7/2),x)

[Out]

-1/5*(9*x^5-70*x^4+275*x^3+3300*x^2+6760*x+4076)/(3+2*x)^(5/2)

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Maxima [A]  time = 0.99809, size = 69, normalized size = 0.87 \begin{align*} -\frac{9}{160} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{55}{32} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{359}{16} \, \sqrt{2 \, x + 3} - \frac{651 \,{\left (2 \, x + 3\right )}^{2} - 355 \, x - 500}{16 \,{\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)^2/(3+2*x)^(7/2),x, algorithm="maxima")

[Out]

-9/160*(2*x + 3)^(5/2) + 55/32*(2*x + 3)^(3/2) - 359/16*sqrt(2*x + 3) - 1/16*(651*(2*x + 3)^2 - 355*x - 500)/(
2*x + 3)^(5/2)

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Fricas [A]  time = 1.70229, size = 139, normalized size = 1.76 \begin{align*} -\frac{{\left (9 \, x^{5} - 70 \, x^{4} + 275 \, x^{3} + 3300 \, x^{2} + 6760 \, x + 4076\right )} \sqrt{2 \, x + 3}}{5 \,{\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)^2/(3+2*x)^(7/2),x, algorithm="fricas")

[Out]

-1/5*(9*x^5 - 70*x^4 + 275*x^3 + 3300*x^2 + 6760*x + 4076)*sqrt(2*x + 3)/(8*x^3 + 36*x^2 + 54*x + 27)

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Sympy [B]  time = 1.66382, size = 238, normalized size = 3.01 \begin{align*} - \frac{9 x^{5}}{20 x^{2} \sqrt{2 x + 3} + 60 x \sqrt{2 x + 3} + 45 \sqrt{2 x + 3}} + \frac{70 x^{4}}{20 x^{2} \sqrt{2 x + 3} + 60 x \sqrt{2 x + 3} + 45 \sqrt{2 x + 3}} - \frac{275 x^{3}}{20 x^{2} \sqrt{2 x + 3} + 60 x \sqrt{2 x + 3} + 45 \sqrt{2 x + 3}} - \frac{3300 x^{2}}{20 x^{2} \sqrt{2 x + 3} + 60 x \sqrt{2 x + 3} + 45 \sqrt{2 x + 3}} - \frac{6760 x}{20 x^{2} \sqrt{2 x + 3} + 60 x \sqrt{2 x + 3} + 45 \sqrt{2 x + 3}} - \frac{4076}{20 x^{2} \sqrt{2 x + 3} + 60 x \sqrt{2 x + 3} + 45 \sqrt{2 x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+5*x+2)**2/(3+2*x)**(7/2),x)

[Out]

-9*x**5/(20*x**2*sqrt(2*x + 3) + 60*x*sqrt(2*x + 3) + 45*sqrt(2*x + 3)) + 70*x**4/(20*x**2*sqrt(2*x + 3) + 60*
x*sqrt(2*x + 3) + 45*sqrt(2*x + 3)) - 275*x**3/(20*x**2*sqrt(2*x + 3) + 60*x*sqrt(2*x + 3) + 45*sqrt(2*x + 3))
- 3300*x**2/(20*x**2*sqrt(2*x + 3) + 60*x*sqrt(2*x + 3) + 45*sqrt(2*x + 3)) - 6760*x/(20*x**2*sqrt(2*x + 3) +
60*x*sqrt(2*x + 3) + 45*sqrt(2*x + 3)) - 4076/(20*x**2*sqrt(2*x + 3) + 60*x*sqrt(2*x + 3) + 45*sqrt(2*x + 3))

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Giac [A]  time = 1.10487, size = 69, normalized size = 0.87 \begin{align*} -\frac{9}{160} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{55}{32} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{359}{16} \, \sqrt{2 \, x + 3} - \frac{651 \,{\left (2 \, x + 3\right )}^{2} - 355 \, x - 500}{16 \,{\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)^2/(3+2*x)^(7/2),x, algorithm="giac")

[Out]

-9/160*(2*x + 3)^(5/2) + 55/32*(2*x + 3)^(3/2) - 359/16*sqrt(2*x + 3) - 1/16*(651*(2*x + 3)^2 - 355*x - 500)/(
2*x + 3)^(5/2)