3.2515 \(\int \frac{5-x}{(3+2 x)^5 (2+5 x+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=169 \[ -\frac{6 (47 x+37)}{5 (2 x+3)^4 \sqrt{3 x^2+5 x+2}}-\frac{25458 \sqrt{3 x^2+5 x+2}}{625 (2 x+3)}-\frac{973 \sqrt{3 x^2+5 x+2}}{30 (2 x+3)^2}-\frac{11596 \sqrt{3 x^2+5 x+2}}{375 (2 x+3)^3}-\frac{817 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)^4}+\frac{82039 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{2500 \sqrt{5}} \]

[Out]

(-6*(37 + 47*x))/(5*(3 + 2*x)^4*Sqrt[2 + 5*x + 3*x^2]) - (817*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)^4) - (11596
*Sqrt[2 + 5*x + 3*x^2])/(375*(3 + 2*x)^3) - (973*Sqrt[2 + 5*x + 3*x^2])/(30*(3 + 2*x)^2) - (25458*Sqrt[2 + 5*x
+ 3*x^2])/(625*(3 + 2*x)) + (82039*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(2500*Sqrt[5])

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Rubi [A]  time = 0.117334, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{6 (47 x+37)}{5 (2 x+3)^4 \sqrt{3 x^2+5 x+2}}-\frac{25458 \sqrt{3 x^2+5 x+2}}{625 (2 x+3)}-\frac{973 \sqrt{3 x^2+5 x+2}}{30 (2 x+3)^2}-\frac{11596 \sqrt{3 x^2+5 x+2}}{375 (2 x+3)^3}-\frac{817 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)^4}+\frac{82039 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{2500 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(37 + 47*x))/(5*(3 + 2*x)^4*Sqrt[2 + 5*x + 3*x^2]) - (817*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)^4) - (11596
*Sqrt[2 + 5*x + 3*x^2])/(375*(3 + 2*x)^3) - (973*Sqrt[2 + 5*x + 3*x^2])/(30*(3 + 2*x)^2) - (25458*Sqrt[2 + 5*x
+ 3*x^2])/(625*(3 + 2*x)) + (82039*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(2500*Sqrt[5])

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])

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 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 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])

Rubi steps

\begin{align*} \int \frac{5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac{6 (37+47 x)}{5 (3+2 x)^4 \sqrt{2+5 x+3 x^2}}-\frac{2}{5} \int \frac{875+1128 x}{(3+2 x)^5 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^4 \sqrt{2+5 x+3 x^2}}-\frac{817 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^4}+\frac{1}{50} \int \frac{-10463-14706 x}{(3+2 x)^4 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^4 \sqrt{2+5 x+3 x^2}}-\frac{817 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac{11596 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac{1}{750} \int \frac{87103+139152 x}{(3+2 x)^3 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^4 \sqrt{2+5 x+3 x^2}}-\frac{817 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac{11596 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac{973 \sqrt{2+5 x+3 x^2}}{30 (3+2 x)^2}+\frac{\int \frac{-330885-729750 x}{(3+2 x)^2 \sqrt{2+5 x+3 x^2}} \, dx}{7500}\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^4 \sqrt{2+5 x+3 x^2}}-\frac{817 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac{11596 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac{973 \sqrt{2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac{25458 \sqrt{2+5 x+3 x^2}}{625 (3+2 x)}+\frac{82039 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{2500}\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^4 \sqrt{2+5 x+3 x^2}}-\frac{817 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac{11596 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac{973 \sqrt{2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac{25458 \sqrt{2+5 x+3 x^2}}{625 (3+2 x)}-\frac{82039 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{1250}\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^4 \sqrt{2+5 x+3 x^2}}-\frac{817 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac{11596 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac{973 \sqrt{2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac{25458 \sqrt{2+5 x+3 x^2}}{625 (3+2 x)}+\frac{82039 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{2500 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0679249, size = 89, normalized size = 0.53 \[ \frac{-\frac{10 \left (3665952 x^5+24066204 x^4+62190544 x^3+78737669 x^2+48537379 x+11545002\right )}{(2 x+3)^4 \sqrt{3 x^2+5 x+2}}-246117 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{37500} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-10*(11545002 + 48537379*x + 78737669*x^2 + 62190544*x^3 + 24066204*x^4 + 3665952*x^5))/((3 + 2*x)^4*Sqrt[2
+ 5*x + 3*x^2]) - 246117*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/37500

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Maple [A]  time = 0.012, size = 153, normalized size = 0.9 \begin{align*} -{\frac{13}{320} \left ( x+{\frac{3}{2}} \right ) ^{-4}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{14}{75} \left ( x+{\frac{3}{2}} \right ) ^{-3}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{9619}{12000} \left ( x+{\frac{3}{2}} \right ) ^{-2}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{6931}{1500} \left ( x+{\frac{3}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}+{\frac{82039}{5000}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{63645+76374\,x}{1250}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{82039\,\sqrt{5}}{12500}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/320/(x+3/2)^4/(3*(x+3/2)^2-4*x-19/4)^(1/2)-14/75/(x+3/2)^3/(3*(x+3/2)^2-4*x-19/4)^(1/2)-9619/12000/(x+3/2)
^2/(3*(x+3/2)^2-4*x-19/4)^(1/2)-6931/1500/(x+3/2)/(3*(x+3/2)^2-4*x-19/4)^(1/2)+82039/5000/(3*(x+3/2)^2-4*x-19/
4)^(1/2)-12729/1250*(5+6*x)/(3*(x+3/2)^2-4*x-19/4)^(1/2)-82039/12500*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(1
2*(x+3/2)^2-16*x-19)^(1/2))

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Maxima [B]  time = 1.81761, size = 419, normalized size = 2.48 \begin{align*} -\frac{82039}{12500} \, \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{38187 \, x}{625 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{172541}{5000 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{13}{20 \,{\left (16 \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{4} + 96 \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{3} + 216 \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{2} + 216 \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + 81 \, \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}} - \frac{112}{75 \,{\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{9619}{3000 \,{\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{6931}{750 \,{\left (2 \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + 3 \, \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-82039/12500*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 38187/625*x/sqrt
(3*x^2 + 5*x + 2) - 172541/5000/sqrt(3*x^2 + 5*x + 2) - 13/20/(16*sqrt(3*x^2 + 5*x + 2)*x^4 + 96*sqrt(3*x^2 +
5*x + 2)*x^3 + 216*sqrt(3*x^2 + 5*x + 2)*x^2 + 216*sqrt(3*x^2 + 5*x + 2)*x + 81*sqrt(3*x^2 + 5*x + 2)) - 112/7
5/(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)) - 9619/3000/(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)) - 69
31/750/(2*sqrt(3*x^2 + 5*x + 2)*x + 3*sqrt(3*x^2 + 5*x + 2))

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Fricas [A]  time = 1.85769, size = 485, normalized size = 2.87 \begin{align*} \frac{246117 \, \sqrt{5}{\left (48 \, x^{6} + 368 \, x^{5} + 1160 \, x^{4} + 1920 \, x^{3} + 1755 \, x^{2} + 837 \, x + 162\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 (3665952 \, x^{5} + 24066204 \, x^{4} + 62190544 \, x^{3} + 78737669 \, x^{2} + 48537379 \, x + 11545002\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{75000 \,{\left (48 \, x^{6} + 368 \, x^{5} + 1160 \, x^{4} + 1920 \, x^{3} + 1755 \, x^{2} + 837 \, x + 162\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/75000*(246117*sqrt(5)*(48*x^6 + 368*x^5 + 1160*x^4 + 1920*x^3 + 1755*x^2 + 837*x + 162)*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*(3665952*x^5 + 24066204*x^4 + 6219
0544*x^3 + 78737669*x^2 + 48537379*x + 11545002)*sqrt(3*x^2 + 5*x + 2))/(48*x^6 + 368*x^5 + 1160*x^4 + 1920*x^
3 + 1755*x^2 + 837*x + 162)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x}{96 x^{7} \sqrt{3 x^{2} + 5 x + 2} + 880 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 3424 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 7320 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 9270 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 6939 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 2835 x \sqrt{3 x^{2} + 5 x + 2} + 486 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{96 x^{7} \sqrt{3 x^{2} + 5 x + 2} + 880 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 3424 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 7320 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 9270 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 6939 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 2835 x \sqrt{3 x^{2} + 5 x + 2} + 486 \sqrt{3 x^{2} + 5 x + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x/(96*x**7*sqrt(3*x**2 + 5*x + 2) + 880*x**6*sqrt(3*x**2 + 5*x + 2) + 3424*x**5*sqrt(3*x**2 + 5*x +
2) + 7320*x**4*sqrt(3*x**2 + 5*x + 2) + 9270*x**3*sqrt(3*x**2 + 5*x + 2) + 6939*x**2*sqrt(3*x**2 + 5*x + 2) +
2835*x*sqrt(3*x**2 + 5*x + 2) + 486*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(96*x**7*sqrt(3*x**2 + 5*x + 2)
+ 880*x**6*sqrt(3*x**2 + 5*x + 2) + 3424*x**5*sqrt(3*x**2 + 5*x + 2) + 7320*x**4*sqrt(3*x**2 + 5*x + 2) + 9270
*x**3*sqrt(3*x**2 + 5*x + 2) + 6939*x**2*sqrt(3*x**2 + 5*x + 2) + 2835*x*sqrt(3*x**2 + 5*x + 2) + 486*sqrt(3*x
**2 + 5*x + 2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (2 \, x + 3\right )}^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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