3.2427 \(\int \frac{(5-x) (2+5 x+3 x^2)^{3/2}}{(3+2 x)^5} \, dx\)

Optimal. Leaf size=137 \[ \frac{(352 x+333) \left (3 x^2+5 x+2\right )^{3/2}}{240 (2 x+3)^4}+\frac{(1528 x+2087) \sqrt{3 x^2+5 x+2}}{3200 (2 x+3)^2}-\frac{3}{32} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{2359 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{6400 \sqrt{5}} \]

[Out]

((2087 + 1528*x)*Sqrt[2 + 5*x + 3*x^2])/(3200*(3 + 2*x)^2) + ((333 + 352*x)*(2 + 5*x + 3*x^2)^(3/2))/(240*(3 +
2*x)^4) - (3*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/32 + (2359*ArcTanh[(7 + 8*x)/(2*Sq
rt[5]*Sqrt[2 + 5*x + 3*x^2])])/(6400*Sqrt[5])

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Rubi [A]  time = 0.0799698, antiderivative size = 137, 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 = {810, 843, 621, 206, 724} \[ \frac{(352 x+333) \left (3 x^2+5 x+2\right )^{3/2}}{240 (2 x+3)^4}+\frac{(1528 x+2087) \sqrt{3 x^2+5 x+2}}{3200 (2 x+3)^2}-\frac{3}{32} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{2359 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{6400 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2087 + 1528*x)*Sqrt[2 + 5*x + 3*x^2])/(3200*(3 + 2*x)^2) + ((333 + 352*x)*(2 + 5*x + 3*x^2)^(3/2))/(240*(3 +
2*x)^4) - (3*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/32 + (2359*ArcTanh[(7 + 8*x)/(2*Sq
rt[5]*Sqrt[2 + 5*x + 3*x^2])])/(6400*Sqrt[5])

Rule 810

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

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

Rule 621

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

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]

Rubi steps

\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx &=\frac{(333+352 x) \left (2+5 x+3 x^2\right )^{3/2}}{240 (3+2 x)^4}-\frac{1}{160} \int \frac{(139+120 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx\\ &=\frac{(2087+1528 x) \sqrt{2+5 x+3 x^2}}{3200 (3+2 x)^2}+\frac{(333+352 x) \left (2+5 x+3 x^2\right )^{3/2}}{240 (3+2 x)^4}+\frac{\int \frac{-6082-7200 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{12800}\\ &=\frac{(2087+1528 x) \sqrt{2+5 x+3 x^2}}{3200 (3+2 x)^2}+\frac{(333+352 x) \left (2+5 x+3 x^2\right )^{3/2}}{240 (3+2 x)^4}-\frac{9}{32} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx+\frac{2359 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{6400}\\ &=\frac{(2087+1528 x) \sqrt{2+5 x+3 x^2}}{3200 (3+2 x)^2}+\frac{(333+352 x) \left (2+5 x+3 x^2\right )^{3/2}}{240 (3+2 x)^4}-\frac{9}{16} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )-\frac{2359 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{3200}\\ &=\frac{(2087+1528 x) \sqrt{2+5 x+3 x^2}}{3200 (3+2 x)^2}+\frac{(333+352 x) \left (2+5 x+3 x^2\right )^{3/2}}{240 (3+2 x)^4}-\frac{3}{32} \sqrt{3} \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )+\frac{2359 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{6400 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.12137, size = 110, normalized size = 0.8 \[ \frac{\frac{10 \sqrt{3 x^2+5 x+2} \left (60576 x^3+190412 x^2+211148 x+82989\right )}{(2 x+3)^4}-7077 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )-9000 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{96000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((10*Sqrt[2 + 5*x + 3*x^2]*(82989 + 211148*x + 190412*x^2 + 60576*x^3))/(3 + 2*x)^4 - 7077*Sqrt[5]*ArcTanh[(-7
- 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] - 9000*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/96000

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Maple [B]  time = 0.012, size = 221, normalized size = 1.6 \begin{align*} -{\frac{13}{320} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{17}{300} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1129}{12000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{911}{7500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{2359}{60000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{545+654\,x}{4000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{2359}{32000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{2359\,\sqrt{5}}{32000}{\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 ) }+{\frac{4555+5466\,x}{15000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{3\,\sqrt{3}}{32}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/320/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(5/2)-17/300/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(5/2)-1129/12000/(x+3/2
)^2*(3*(x+3/2)^2-4*x-19/4)^(5/2)-911/7500/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(5/2)+2359/60000*(3*(x+3/2)^2-4*x-19/
4)^(3/2)-109/4000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)+2359/32000*(12*(x+3/2)^2-16*x-19)^(1/2)-2359/32000*5^(1
/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+911/15000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2
)-3/32*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)

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Maxima [B]  time = 1.51608, size = 306, normalized size = 2.23 \begin{align*} \frac{1129}{4000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{20 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{34 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{75 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{1129 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{3000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{327}{2000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{3}{32} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{2359}{32000} \, \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{179}{16000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{911 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{3000 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1129/4000*(3*x^2 + 5*x + 2)^(3/2) - 13/20*(3*x^2 + 5*x + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 3
4/75*(3*x^2 + 5*x + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1129/3000*(3*x^2 + 5*x + 2)^(5/2)/(4*x^2 + 12*x +
9) - 327/2000*sqrt(3*x^2 + 5*x + 2)*x - 3/32*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 2359/320
00*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 179/16000*sqrt(3*x^2 + 5*x
+ 2) - 911/3000*(3*x^2 + 5*x + 2)^(3/2)/(2*x + 3)

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Fricas [A]  time = 1.44054, size = 533, normalized size = 3.89 \begin{align*} \frac{9000 \, \sqrt{3}{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 7077 \, \sqrt{5}{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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 (60576 \, x^{3} + 190412 \, x^{2} + 211148 \, x + 82989\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{192000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192000*(9000*sqrt(3)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5)
+ 72*x^2 + 120*x + 49) + 7077*sqrt(5)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*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*(60576*x^3 + 190412*x^2 + 211148*x + 82989)*
sqrt(3*x^2 + 5*x + 2))/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{10 \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\, dx - \int - \frac{23 x \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\, dx - \int - \frac{10 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\, dx - \int \frac{3 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(
-23*x*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-10*x**2
*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(3*x**3*sqrt(3
*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x)

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Giac [B]  time = 1.39305, size = 342, normalized size = 2.5 \begin{align*} \frac{3}{32} \, \sqrt{3} \log \left (\frac{{\left | -2 \, \sqrt{3} + 2 \, \sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{2 \, \sqrt{5}}{2 \, x + 3} \right |}}{{\left | 2 \, \sqrt{3} + 2 \, \sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{2 \, \sqrt{5}}{2 \, x + 3} \right |}}\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - \frac{2359}{32000} \, \sqrt{5} \log \left ({\left | -4 \, \sqrt{5} + 5 \, \sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{5 \, \sqrt{5}}{2 \, x + 3} \right |}\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - \frac{1}{19200} \,{\left (\frac{5 \,{\left (\frac{10 \,{\left (\frac{195 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )}{2 \, x + 3} - 488 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 4109 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 7572 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )} \sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/32*sqrt(3)*log(abs(-2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3))/abs(2*sqrt(3
) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3)))*sgn(1/(2*x + 3)) - 2359/32000*sqrt(5)*log
(abs(-4*sqrt(5) + 5*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 5*sqrt(5)/(2*x + 3)))*sgn(1/(2*x + 3)) - 1/19200*
(5*(10*(195*sgn(1/(2*x + 3))/(2*x + 3) - 488*sgn(1/(2*x + 3)))/(2*x + 3) + 4109*sgn(1/(2*x + 3)))/(2*x + 3) -
7572*sgn(1/(2*x + 3)))*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3)