3.23.73 \(\int \frac {5-x}{(3+2 x)^4 (2+5 x+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=144 \[ -\frac {6 (47 x+37)}{5 (2 x+3)^3 \sqrt {3 x^2+5 x+2}}-\frac {4632 \sqrt {3 x^2+5 x+2}}{125 (2 x+3)}-\frac {478 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^2}-\frac {2464 \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^3}+\frac {3289 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{125 \sqrt {5}} \]

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Rubi [A]  time = 0.10, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {822, 834, 806, 724, 206} \begin {gather*} -\frac {6 (47 x+37)}{5 (2 x+3)^3 \sqrt {3 x^2+5 x+2}}-\frac {4632 \sqrt {3 x^2+5 x+2}}{125 (2 x+3)}-\frac {478 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^2}-\frac {2464 \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^3}+\frac {3289 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{125 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 - x)/((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(3/2)),x]

[Out]

(-6*(37 + 47*x))/(5*(3 + 2*x)^3*Sqrt[2 + 5*x + 3*x^2]) - (2464*Sqrt[2 + 5*x + 3*x^2])/(75*(3 + 2*x)^3) - (478*
Sqrt[2 + 5*x + 3*x^2])/(15*(3 + 2*x)^2) - (4632*Sqrt[2 + 5*x + 3*x^2])/(125*(3 + 2*x)) + (3289*ArcTanh[(7 + 8*
x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(125*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])

Rule 834

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

Rubi steps

\begin {align*} \int \frac {5-x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac {6 (37+47 x)}{5 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}-\frac {2}{5} \int \frac {653+846 x}{(3+2 x)^4 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}-\frac {2464 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}+\frac {2}{75} \int \frac {-5113-7392 x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}-\frac {2464 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {478 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^2}-\frac {1}{375} \int \frac {19035+35850 x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}-\frac {2464 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {478 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^2}-\frac {4632 \sqrt {2+5 x+3 x^2}}{125 (3+2 x)}+\frac {3289}{125} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}-\frac {2464 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {478 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^2}-\frac {4632 \sqrt {2+5 x+3 x^2}}{125 (3+2 x)}-\frac {6578}{125} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}-\frac {2464 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {478 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^2}-\frac {4632 \sqrt {2+5 x+3 x^2}}{125 (3+2 x)}+\frac {3289 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{125 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 84, normalized size = 0.58 \begin {gather*} \frac {-9867 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-\frac {10 \left (83376 x^4+424938 x^3+792065 x^2+634312 x+181559\right )}{(2 x+3)^3 \sqrt {3 x^2+5 x+2}}}{1875} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)/((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(3/2)),x]

[Out]

((-10*(181559 + 634312*x + 792065*x^2 + 424938*x^3 + 83376*x^4))/((3 + 2*x)^3*Sqrt[2 + 5*x + 3*x^2]) - 9867*Sq
rt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/1875

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IntegrateAlgebraic [A]  time = 0.55, size = 93, normalized size = 0.65 \begin {gather*} \frac {6578 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{125 \sqrt {5}}-\frac {2 \sqrt {3 x^2+5 x+2} \left (83376 x^4+424938 x^3+792065 x^2+634312 x+181559\right )}{375 (x+1) (2 x+3)^3 (3 x+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(5 - x)/((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(3/2)),x]

[Out]

(-2*Sqrt[2 + 5*x + 3*x^2]*(181559 + 634312*x + 792065*x^2 + 424938*x^3 + 83376*x^4))/(375*(1 + x)*(3 + 2*x)^3*
(2 + 3*x)) + (6578*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/(125*Sqrt[5])

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fricas [A]  time = 0.40, size = 140, normalized size = 0.97 \begin {gather*} \frac {9867 \, \sqrt {5} {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\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 ) - 20 \, {\left (83376 \, x^{4} + 424938 \, x^{3} + 792065 \, x^{2} + 634312 \, x + 181559\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{3750 \, {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+5*x+2)^(3/2),x, algorithm="fricas")

[Out]

1/3750*(9867*sqrt(5)*(24*x^5 + 148*x^4 + 358*x^3 + 423*x^2 + 243*x + 54)*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)) - 20*(83376*x^4 + 424938*x^3 + 792065*x^2 + 634312*x + 1
81559)*sqrt(3*x^2 + 5*x + 2))/(24*x^5 + 148*x^4 + 358*x^3 + 423*x^2 + 243*x + 54)

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giac [B]  time = 0.32, size = 276, normalized size = 1.92 \begin {gather*} \frac {3289}{625} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {6 \, {\left (4209 \, x + 2959\right )}}{625 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {2 \, {\left (118356 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 851850 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 6938110 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 8824815 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 15944775 \, \sqrt {3} x + 3678471 \, \sqrt {3} - 15944775 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{1875 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+5*x+2)^(3/2),x, algorithm="giac")

[Out]

3289/625*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*
sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) - 6/625*(4209*x + 2959)/sqrt(3*x^2 + 5*x + 2) - 2/1875*(118356
*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 851850*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 6938110*(sqrt(
3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 8824815*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 15944775*sqrt(3)*x +
3678471*sqrt(3) - 15944775*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(
3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^3

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maple [A]  time = 0.01, size = 132, normalized size = 0.92 \begin {gather*} -\frac {3289 \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 )}{625}-\frac {349}{600 \left (x +\frac {3}{2}\right )^{2} \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {271}{75 \left (x +\frac {3}{2}\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}+\frac {3289}{250 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {1158 \left (6 x +5\right )}{125 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {13}{120 \left (x +\frac {3}{2}\right )^{3} \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)^4/(3*x^2+5*x+2)^(3/2),x)

[Out]

-349/600/(x+3/2)^2/(-4*x+3*(x+3/2)^2-19/4)^(1/2)-271/75/(x+3/2)/(-4*x+3*(x+3/2)^2-19/4)^(1/2)+3289/250/(-4*x+3
*(x+3/2)^2-19/4)^(1/2)-1158/125*(6*x+5)/(-4*x+3*(x+3/2)^2-19/4)^(1/2)-3289/625*5^(1/2)*arctanh(2/5*(-4*x-7/2)*
5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))-13/120/(x+3/2)^3/(-4*x+3*(x+3/2)^2-19/4)^(1/2)

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maxima [A]  time = 1.30, size = 225, normalized size = 1.56 \begin {gather*} -\frac {3289}{625} \, \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 {6948 \, x}{125 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {8291}{250 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {13}{15 \, {\left (8 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{3} + 36 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + 54 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 27 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} - \frac {349}{150 \, {\left (4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + 12 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 9 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} - \frac {542}{75 \, {\left (2 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+5*x+2)^(3/2),x, algorithm="maxima")

[Out]

-3289/625*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 6948/125*x/sqrt(3*x
^2 + 5*x + 2) - 8291/250/sqrt(3*x^2 + 5*x + 2) - 13/15/(8*sqrt(3*x^2 + 5*x + 2)*x^3 + 36*sqrt(3*x^2 + 5*x + 2)
*x^2 + 54*sqrt(3*x^2 + 5*x + 2)*x + 27*sqrt(3*x^2 + 5*x + 2)) - 349/150/(4*sqrt(3*x^2 + 5*x + 2)*x^2 + 12*sqrt
(3*x^2 + 5*x + 2)*x + 9*sqrt(3*x^2 + 5*x + 2)) - 542/75/(2*sqrt(3*x^2 + 5*x + 2)*x + 3*sqrt(3*x^2 + 5*x + 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 )}^4\,{\left (3\,x^2+5\,x+2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 5)/((2*x + 3)^4*(5*x + 3*x^2 + 2)^(3/2)),x)

[Out]

-int((x - 5)/((2*x + 3)^4*(5*x + 3*x^2 + 2)^(3/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{48 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 368 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 1160 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 1920 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 1755 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 837 x \sqrt {3 x^{2} + 5 x + 2} + 162 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{48 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 368 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 1160 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 1920 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 1755 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 837 x \sqrt {3 x^{2} + 5 x + 2} + 162 \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)**4/(3*x**2+5*x+2)**(3/2),x)

[Out]

-Integral(x/(48*x**6*sqrt(3*x**2 + 5*x + 2) + 368*x**5*sqrt(3*x**2 + 5*x + 2) + 1160*x**4*sqrt(3*x**2 + 5*x +
2) + 1920*x**3*sqrt(3*x**2 + 5*x + 2) + 1755*x**2*sqrt(3*x**2 + 5*x + 2) + 837*x*sqrt(3*x**2 + 5*x + 2) + 162*
sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(48*x**6*sqrt(3*x**2 + 5*x + 2) + 368*x**5*sqrt(3*x**2 + 5*x + 2) +
1160*x**4*sqrt(3*x**2 + 5*x + 2) + 1920*x**3*sqrt(3*x**2 + 5*x + 2) + 1755*x**2*sqrt(3*x**2 + 5*x + 2) + 837*x
*sqrt(3*x**2 + 5*x + 2) + 162*sqrt(3*x**2 + 5*x + 2)), x)

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