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

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

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Rubi [A]  time = 0.02, antiderivative size = 79, 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 {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}} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {9 x^5-70 x^4+275 x^3+3300 x^2+6760 x+4076}{5 (2 x+3)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.05, size = 58, normalized size = 0.73 \begin {gather*} \frac {-9 (2 x+3)^5+275 (2 x+3)^4-3590 (2 x+3)^3-6510 (2 x+3)^2+1775 (2 x+3)-325}{160 (2 x+3)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-325 + 1775*(3 + 2*x) - 6510*(3 + 2*x)^2 - 3590*(3 + 2*x)^3 + 275*(3 + 2*x)^4 - 9*(3 + 2*x)^5)/(160*(3 + 2*x)
^(5/2))

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fricas [A]  time = 0.39, size = 51, normalized size = 0.65 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.17, size = 51, normalized size = 0.65 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.00, size = 35, normalized size = 0.44 \begin {gather*} -\frac {9 x^{5}-70 x^{4}+275 x^{3}+3300 x^{2}+6760 x +4076}{5 \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/(2*x+3)^(7/2),x)

[Out]

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

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maxima [A]  time = 0.66, size = 51, normalized size = 0.65 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.04, size = 50, normalized size = 0.63 \begin {gather*} \frac {\frac {355\,x}{16}-\frac {651\,{\left (2\,x+3\right )}^2}{16}+\frac {125}{4}}{{\left (2\,x+3\right )}^{5/2}}-\frac {359\,\sqrt {2\,x+3}}{16}+\frac {55\,{\left (2\,x+3\right )}^{3/2}}{32}-\frac {9\,{\left (2\,x+3\right )}^{5/2}}{160} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((355*x)/16 - (651*(2*x + 3)^2)/16 + 125/4)/(2*x + 3)^(5/2) - (359*(2*x + 3)^(1/2))/16 + (55*(2*x + 3)^(3/2))/
32 - (9*(2*x + 3)^(5/2))/160

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sympy [B]  time = 1.56, size = 238, normalized size = 3.01 \begin {gather*} - \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 {gather*}

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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