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

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

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Rubi [A]  time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {771} \begin {gather*} -\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}} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.53 \begin {gather*} -\frac {9 x^3+111 x^2+245 x+153}{3 (2 x+3)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.05, size = 40, normalized size = 0.75 \begin {gather*} \frac {-9 (2 x+3)^3-141 (2 x+3)^2+109 (2 x+3)-39}{24 (2 x+3)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-39 + 109*(3 + 2*x) - 141*(3 + 2*x)^2 - 9*(3 + 2*x)^3)/(24*(3 + 2*x)^(5/2))

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fricas [A]  time = 0.39, size = 41, normalized size = 0.77 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.21, size = 33, normalized size = 0.62 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.00, size = 25, normalized size = 0.47 \begin {gather*} -\frac {9 x^{3}+111 x^{2}+245 x +153}{3 \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)/(2*x+3)^(7/2),x)

[Out]

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

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maxima [A]  time = 0.46, size = 33, normalized size = 0.62 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 2.37, size = 24, normalized size = 0.45 \begin {gather*} -\frac {9\,x^3+111\,x^2+245\,x+153}{3\,{\left (2\,x+3\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 1.37, size = 158, normalized size = 2.98 \begin {gather*} - \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 {gather*}

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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