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

Optimal. Leaf size=79 \[ -\frac {1}{32} (2 x+3)^{9/2}+\frac {165}{224} (2 x+3)^{7/2}-\frac {359}{80} (2 x+3)^{5/2}+\frac {217}{16} (2 x+3)^{3/2}-\frac {1065}{32} \sqrt {2 x+3}-\frac {325}{32 \sqrt {2 x+3}} \]

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Rubi [A]  time = 0.03, 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 {1}{32} (2 x+3)^{9/2}+\frac {165}{224} (2 x+3)^{7/2}-\frac {359}{80} (2 x+3)^{5/2}+\frac {217}{16} (2 x+3)^{3/2}-\frac {1065}{32} \sqrt {2 x+3}-\frac {325}{32 \sqrt {2 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-325/(32*Sqrt[3 + 2*x]) - (1065*Sqrt[3 + 2*x])/32 + (217*(3 + 2*x)^(3/2))/16 - (359*(3 + 2*x)^(5/2))/80 + (165
*(3 + 2*x)^(7/2))/224 - (3 + 2*x)^(9/2)/32

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)^{3/2}} \, dx &=\int \left (\frac {325}{32 (3+2 x)^{3/2}}-\frac {1065}{32 \sqrt {3+2 x}}+\frac {651}{16} \sqrt {3+2 x}-\frac {359}{16} (3+2 x)^{3/2}+\frac {165}{32} (3+2 x)^{5/2}-\frac {9}{32} (3+2 x)^{7/2}\right ) \, dx\\ &=-\frac {325}{32 \sqrt {3+2 x}}-\frac {1065}{32} \sqrt {3+2 x}+\frac {217}{16} (3+2 x)^{3/2}-\frac {359}{80} (3+2 x)^{5/2}+\frac {165}{224} (3+2 x)^{7/2}-\frac {1}{32} (3+2 x)^{9/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {35 x^5-150 x^4-431 x^3-632 x^2+432 x+1996}{35 \sqrt {2 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/35*(1996 + 432*x - 632*x^2 - 431*x^3 - 150*x^4 + 35*x^5)/Sqrt[3 + 2*x]

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IntegrateAlgebraic [A]  time = 0.04, size = 58, normalized size = 0.73 \begin {gather*} \frac {-35 (2 x+3)^5+825 (2 x+3)^4-5026 (2 x+3)^3+15190 (2 x+3)^2-37275 (2 x+3)-11375}{1120 \sqrt {2 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-11375 - 37275*(3 + 2*x) + 15190*(3 + 2*x)^2 - 5026*(3 + 2*x)^3 + 825*(3 + 2*x)^4 - 35*(3 + 2*x)^5)/(1120*Sqr
t[3 + 2*x])

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fricas [A]  time = 0.39, size = 34, normalized size = 0.43 \begin {gather*} -\frac {35 \, x^{5} - 150 \, x^{4} - 431 \, x^{3} - 632 \, x^{2} + 432 \, x + 1996}{35 \, \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)^(3/2),x, algorithm="fricas")

[Out]

-1/35*(35*x^5 - 150*x^4 - 431*x^3 - 632*x^2 + 432*x + 1996)/sqrt(2*x + 3)

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giac [A]  time = 0.17, size = 55, normalized size = 0.70 \begin {gather*} -\frac {1}{32} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {165}{224} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {359}{80} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {217}{16} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {1065}{32} \, \sqrt {2 \, x + 3} - \frac {325}{32 \, \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)^(3/2),x, algorithm="giac")

[Out]

-1/32*(2*x + 3)^(9/2) + 165/224*(2*x + 3)^(7/2) - 359/80*(2*x + 3)^(5/2) + 217/16*(2*x + 3)^(3/2) - 1065/32*sq
rt(2*x + 3) - 325/32/sqrt(2*x + 3)

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maple [A]  time = 0.01, size = 35, normalized size = 0.44 \begin {gather*} -\frac {35 x^{5}-150 x^{4}-431 x^{3}-632 x^{2}+432 x +1996}{35 \sqrt {2 x +3}} \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)^(3/2),x)

[Out]

-1/35*(35*x^5-150*x^4-431*x^3-632*x^2+432*x+1996)/(2*x+3)^(1/2)

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maxima [A]  time = 0.52, size = 55, normalized size = 0.70 \begin {gather*} -\frac {1}{32} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {165}{224} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {359}{80} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {217}{16} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {1065}{32} \, \sqrt {2 \, x + 3} - \frac {325}{32 \, \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)^(3/2),x, algorithm="maxima")

[Out]

-1/32*(2*x + 3)^(9/2) + 165/224*(2*x + 3)^(7/2) - 359/80*(2*x + 3)^(5/2) + 217/16*(2*x + 3)^(3/2) - 1065/32*sq
rt(2*x + 3) - 325/32/sqrt(2*x + 3)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {217\,{\left (2\,x+3\right )}^{3/2}}{16}-\frac {1065\,\sqrt {2\,x+3}}{32}-\frac {325}{32\,\sqrt {2\,x+3}}-\frac {359\,{\left (2\,x+3\right )}^{5/2}}{80}+\frac {165\,{\left (2\,x+3\right )}^{7/2}}{224}-\frac {{\left (2\,x+3\right )}^{9/2}}{32} \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)^(3/2),x)

[Out]

(217*(2*x + 3)^(3/2))/16 - (1065*(2*x + 3)^(1/2))/32 - 325/(32*(2*x + 3)^(1/2)) - (359*(2*x + 3)^(5/2))/80 + (
165*(2*x + 3)^(7/2))/224 - (2*x + 3)^(9/2)/32

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sympy [A]  time = 35.24, size = 68, normalized size = 0.86 \begin {gather*} - \frac {\left (2 x + 3\right )^{\frac {9}{2}}}{32} + \frac {165 \left (2 x + 3\right )^{\frac {7}{2}}}{224} - \frac {359 \left (2 x + 3\right )^{\frac {5}{2}}}{80} + \frac {217 \left (2 x + 3\right )^{\frac {3}{2}}}{16} - \frac {1065 \sqrt {2 x + 3}}{32} - \frac {325}{32 \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)**(3/2),x)

[Out]

-(2*x + 3)**(9/2)/32 + 165*(2*x + 3)**(7/2)/224 - 359*(2*x + 3)**(5/2)/80 + 217*(2*x + 3)**(3/2)/16 - 1065*sqr
t(2*x + 3)/32 - 325/(32*sqrt(2*x + 3))

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