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

Optimal. Leaf size=124 \[ -\frac {2 (47 x+37)}{5 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4416 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)}+\frac {4 (462 x+401)}{5 (2 x+3) \sqrt {3 x^2+5 x+2}}+\frac {408 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{25 \sqrt {5}} \]

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Rubi [A]  time = 0.07, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {822, 806, 724, 206} \begin {gather*} -\frac {2 (47 x+37)}{5 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4416 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)}+\frac {4 (462 x+401)}{5 (2 x+3) \sqrt {3 x^2+5 x+2}}+\frac {408 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{25 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(37 + 47*x))/(5*(3 + 2*x)*(2 + 5*x + 3*x^2)^(3/2)) + (4*(401 + 462*x))/(5*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])
+ (4416*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)) + (408*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(2
5*Sqrt[5])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}-\frac {2}{15} \int \frac {1029+846 x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (401+462 x)}{5 (3+2 x) \sqrt {2+5 x+3 x^2}}+\frac {4}{75} \int \frac {12510+13860 x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (401+462 x)}{5 (3+2 x) \sqrt {2+5 x+3 x^2}}+\frac {4416 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)}+\frac {408}{25} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (401+462 x)}{5 (3+2 x) \sqrt {2+5 x+3 x^2}}+\frac {4416 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)}-\frac {816}{25} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (401+462 x)}{5 (3+2 x) \sqrt {2+5 x+3 x^2}}+\frac {4416 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)}+\frac {408 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{25 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 113, normalized size = 0.91 \begin {gather*} \frac {10 \left (19872 x^4+80100 x^3+116826 x^2+73215 x+16667\right )-408 \sqrt {5} \sqrt {3 x^2+5 x+2} \left (6 x^3+19 x^2+19 x+6\right ) \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{125 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*(16667 + 73215*x + 116826*x^2 + 80100*x^3 + 19872*x^4) - 408*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]*(6 + 19*x + 19*
x^2 + 6*x^3)*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(125*(3 + 2*x)*(2 + 5*x + 3*x^2)^(3/2))

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IntegrateAlgebraic [A]  time = 0.47, size = 93, normalized size = 0.75 \begin {gather*} \frac {816 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{25 \sqrt {5}}+\frac {2 \sqrt {3 x^2+5 x+2} \left (19872 x^4+80100 x^3+116826 x^2+73215 x+16667\right )}{25 (x+1)^2 (2 x+3) (3 x+2)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[2 + 5*x + 3*x^2]*(16667 + 73215*x + 116826*x^2 + 80100*x^3 + 19872*x^4))/(25*(1 + x)^2*(3 + 2*x)*(2 +
3*x)^2) + (816*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/(25*Sqrt[5])

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fricas [A]  time = 0.41, size = 140, normalized size = 1.13 \begin {gather*} \frac {2 \, {\left (102 \, \sqrt {5} {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 5 \, {\left (19872 \, x^{4} + 80100 \, x^{3} + 116826 \, x^{2} + 73215 \, x + 16667\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{125 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^2/(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")

[Out]

2/125*(102*sqrt(5)*(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)*log((4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x
+ 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9)) + 5*(19872*x^4 + 80100*x^3 + 116826*x^2 + 73215*x + 16667)*s
qrt(3*x^2 + 5*x + 2))/(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)

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giac [B]  time = 0.39, size = 235, normalized size = 1.90 \begin {gather*} -\frac {24}{125} \, \sqrt {5} {\left (92 \, \sqrt {5} \sqrt {3} - 17 \, \log \left (-\sqrt {5} \sqrt {3} + 4\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {408 \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right )}{125 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {8 \, {\left (\frac {\frac {\frac {5 \, {\left (\frac {972}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {13}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} - \frac {12324}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {9783}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} - \frac {2484}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{25 \, {\left (\frac {8}{2 \, x + 3} - \frac {5}{{\left (2 \, x + 3\right )}^{2}} - 3\right )} \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^2/(3*x^2+5*x+2)^(5/2),x, algorithm="giac")

[Out]

-24/125*sqrt(5)*(92*sqrt(5)*sqrt(3) - 17*log(-sqrt(5)*sqrt(3) + 4))*sgn(1/(2*x + 3)) - 408/125*sqrt(5)*log(abs
(sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)) - 4))/sgn(1/(2*x + 3)) + 8/25*(((5*(972/
sgn(1/(2*x + 3)) + 13/((2*x + 3)*sgn(1/(2*x + 3))))/(2*x + 3) - 12324/sgn(1/(2*x + 3)))/(2*x + 3) + 9783/sgn(1
/(2*x + 3)))/(2*x + 3) - 2484/sgn(1/(2*x + 3)))/((8/(2*x + 3) - 5/(2*x + 3)^2 - 3)*sqrt(-8/(2*x + 3) + 5/(2*x
+ 3)^2 + 3))

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maple [A]  time = 0.01, size = 127, normalized size = 1.02 \begin {gather*} -\frac {408 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{125}-\frac {13}{10 \left (x +\frac {3}{2}\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {17}{5 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {16 \left (6 x +5\right )}{5 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {\frac {6624 x}{25}+\frac {1104}{5}}{\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}+\frac {204}{25 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/10/(x+3/2)/(-4*x+3*(x+3/2)^2-19/4)^(3/2)+17/5/(-4*x+3*(x+3/2)^2-19/4)^(3/2)-16/5*(6*x+5)/(-4*x+3*(x+3/2)^2
-19/4)^(3/2)+1104/25*(6*x+5)/(-4*x+3*(x+3/2)^2-19/4)^(1/2)+204/25/(-4*x+3*(x+3/2)^2-19/4)^(1/2)-408/125*5^(1/2
)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))

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maxima [A]  time = 1.31, size = 135, normalized size = 1.09 \begin {gather*} -\frac {408}{125} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {6624 \, x}{25 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {5724}{25 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {96 \, x}{5 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {13}{5 \, {\left (2 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + 3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}\right )}} - \frac {63}{5 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^2/(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")

[Out]

-408/125*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 6624/25*x/sqrt(3*x^2
+ 5*x + 2) + 5724/25/sqrt(3*x^2 + 5*x + 2) - 96/5*x/(3*x^2 + 5*x + 2)^(3/2) - 13/5/(2*(3*x^2 + 5*x + 2)^(3/2)
*x + 3*(3*x^2 + 5*x + 2)^(3/2)) - 63/5/(3*x^2 + 5*x + 2)^(3/2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x-5}{{\left (2\,x+3\right )}^2\,{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{36 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 228 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 589 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 794 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 589 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 228 x \sqrt {3 x^{2} + 5 x + 2} + 36 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{36 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 228 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 589 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 794 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 589 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 228 x \sqrt {3 x^{2} + 5 x + 2} + 36 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x/(36*x**6*sqrt(3*x**2 + 5*x + 2) + 228*x**5*sqrt(3*x**2 + 5*x + 2) + 589*x**4*sqrt(3*x**2 + 5*x + 2
) + 794*x**3*sqrt(3*x**2 + 5*x + 2) + 589*x**2*sqrt(3*x**2 + 5*x + 2) + 228*x*sqrt(3*x**2 + 5*x + 2) + 36*sqrt
(3*x**2 + 5*x + 2)), x) - Integral(-5/(36*x**6*sqrt(3*x**2 + 5*x + 2) + 228*x**5*sqrt(3*x**2 + 5*x + 2) + 589*
x**4*sqrt(3*x**2 + 5*x + 2) + 794*x**3*sqrt(3*x**2 + 5*x + 2) + 589*x**2*sqrt(3*x**2 + 5*x + 2) + 228*x*sqrt(3
*x**2 + 5*x + 2) + 36*sqrt(3*x**2 + 5*x + 2)), x)

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