3.2533 \(\int \frac{(5-x) (2+5 x+3 x^2)}{(3+2 x)^{7/2}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{3}{8} \sqrt{2 x+3}-\frac{47}{8 \sqrt{2 x+3}}+\frac{109}{24 (2 x+3)^{3/2}}-\frac{13}{8 (2 x+3)^{5/2}} \]

[Out]

-13/(8*(3 + 2*x)^(5/2)) + 109/(24*(3 + 2*x)^(3/2)) - 47/(8*Sqrt[3 + 2*x]) - (3*Sqrt[3 + 2*x])/8

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Rubi [A]  time = 0.0152661, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {771} \[ -\frac{3}{8} \sqrt{2 x+3}-\frac{47}{8 \sqrt{2 x+3}}+\frac{109}{24 (2 x+3)^{3/2}}-\frac{13}{8 (2 x+3)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-13/(8*(3 + 2*x)^(5/2)) + 109/(24*(3 + 2*x)^(3/2)) - 47/(8*Sqrt[3 + 2*x]) - (3*Sqrt[3 + 2*x])/8

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+2 x)^{7/2}} \, dx &=\int \left (\frac{65}{8 (3+2 x)^{7/2}}-\frac{109}{8 (3+2 x)^{5/2}}+\frac{47}{8 (3+2 x)^{3/2}}-\frac{3}{8 \sqrt{3+2 x}}\right ) \, dx\\ &=-\frac{13}{8 (3+2 x)^{5/2}}+\frac{109}{24 (3+2 x)^{3/2}}-\frac{47}{8 \sqrt{3+2 x}}-\frac{3}{8} \sqrt{3+2 x}\\ \end{align*}

Mathematica [A]  time = 0.0122195, size = 28, normalized size = 0.53 \[ -\frac{9 x^3+111 x^2+245 x+153}{3 (2 x+3)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(153 + 245*x + 111*x^2 + 9*x^3)/(3*(3 + 2*x)^(5/2))

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Maple [A]  time = 0.004, size = 25, normalized size = 0.5 \begin{align*} -{\frac{9\,{x}^{3}+111\,{x}^{2}+245\,x+153}{3} \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+2*x)^(7/2),x)

[Out]

-1/3*(9*x^3+111*x^2+245*x+153)/(3+2*x)^(5/2)

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Maxima [A]  time = 1.00066, size = 45, normalized size = 0.85 \begin{align*} -\frac{3}{8} \, \sqrt{2 \, x + 3} - \frac{141 \,{\left (2 \, x + 3\right )}^{2} - 218 \, x - 288}{24 \,{\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+2*x)^(7/2),x, algorithm="maxima")

[Out]

-3/8*sqrt(2*x + 3) - 1/24*(141*(2*x + 3)^2 - 218*x - 288)/(2*x + 3)^(5/2)

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Fricas [A]  time = 1.86114, size = 109, normalized size = 2.06 \begin{align*} -\frac{{\left (9 \, x^{3} + 111 \, x^{2} + 245 \, x + 153\right )} \sqrt{2 \, x + 3}}{3 \,{\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+2*x)^(7/2),x, algorithm="fricas")

[Out]

-1/3*(9*x^3 + 111*x^2 + 245*x + 153)*sqrt(2*x + 3)/(8*x^3 + 36*x^2 + 54*x + 27)

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Sympy [B]  time = 1.55327, size = 158, normalized size = 2.98 \begin{align*} - \frac{9 x^{3}}{12 x^{2} \sqrt{2 x + 3} + 36 x \sqrt{2 x + 3} + 27 \sqrt{2 x + 3}} - \frac{111 x^{2}}{12 x^{2} \sqrt{2 x + 3} + 36 x \sqrt{2 x + 3} + 27 \sqrt{2 x + 3}} - \frac{245 x}{12 x^{2} \sqrt{2 x + 3} + 36 x \sqrt{2 x + 3} + 27 \sqrt{2 x + 3}} - \frac{153}{12 x^{2} \sqrt{2 x + 3} + 36 x \sqrt{2 x + 3} + 27 \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)/(3+2*x)**(7/2),x)

[Out]

-9*x**3/(12*x**2*sqrt(2*x + 3) + 36*x*sqrt(2*x + 3) + 27*sqrt(2*x + 3)) - 111*x**2/(12*x**2*sqrt(2*x + 3) + 36
*x*sqrt(2*x + 3) + 27*sqrt(2*x + 3)) - 245*x/(12*x**2*sqrt(2*x + 3) + 36*x*sqrt(2*x + 3) + 27*sqrt(2*x + 3)) -
153/(12*x**2*sqrt(2*x + 3) + 36*x*sqrt(2*x + 3) + 27*sqrt(2*x + 3))

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Giac [A]  time = 1.08181, size = 45, normalized size = 0.85 \begin{align*} -\frac{3}{8} \, \sqrt{2 \, x + 3} - \frac{141 \,{\left (2 \, x + 3\right )}^{2} - 218 \, x - 288}{24 \,{\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+2*x)^(7/2),x, algorithm="giac")

[Out]

-3/8*sqrt(2*x + 3) - 1/24*(141*(2*x + 3)^2 - 218*x - 288)/(2*x + 3)^(5/2)