3.2441 \(\int \frac{(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^5} \, dx\)

Optimal. Leaf size=167 \[ -\frac{(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac{(2898 x+3727) \left (3 x^2+5 x+2\right )^{3/2}}{384 (2 x+3)^3}+\frac{(5718 x+12265) \sqrt{3 x^2+5 x+2}}{512 (2 x+3)}-\frac{1875}{256} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{29047 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1024 \sqrt{5}} \]

[Out]

((12265 + 5718*x)*Sqrt[2 + 5*x + 3*x^2])/(512*(3 + 2*x)) - ((3727 + 2898*x)*(2 + 5*x + 3*x^2)^(3/2))/(384*(3 +
2*x)^3) - ((19 + 4*x)*(2 + 5*x + 3*x^2)^(5/2))/(16*(3 + 2*x)^4) - (1875*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*
Sqrt[2 + 5*x + 3*x^2])])/256 + (29047*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(1024*Sqrt[5])

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Rubi [A]  time = 0.103596, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {812, 810, 843, 621, 206, 724} \[ -\frac{(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac{(2898 x+3727) \left (3 x^2+5 x+2\right )^{3/2}}{384 (2 x+3)^3}+\frac{(5718 x+12265) \sqrt{3 x^2+5 x+2}}{512 (2 x+3)}-\frac{1875}{256} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{29047 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1024 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((12265 + 5718*x)*Sqrt[2 + 5*x + 3*x^2])/(512*(3 + 2*x)) - ((3727 + 2898*x)*(2 + 5*x + 3*x^2)^(3/2))/(384*(3 +
2*x)^3) - ((19 + 4*x)*(2 + 5*x + 3*x^2)^(5/2))/(16*(3 + 2*x)^4) - (1875*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*
Sqrt[2 + 5*x + 3*x^2])])/256 + (29047*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(1024*Sqrt[5])

Rule 812

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

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 )^{5/2}}{(3+2 x)^5} \, dx &=-\frac{(19+4 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac{5}{64} \int \frac{(-158-188 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx\\ &=-\frac{(3727+2898 x) \left (2+5 x+3 x^2\right )^{3/2}}{384 (3+2 x)^3}-\frac{(19+4 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^4}+\frac{\int \frac{(19556+22872 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^2} \, dx}{1024}\\ &=\frac{(12265+5718 x) \sqrt{2+5 x+3 x^2}}{512 (3+2 x)}-\frac{(3727+2898 x) \left (2+5 x+3 x^2\right )^{3/2}}{384 (3+2 x)^3}-\frac{(19+4 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac{\int \frac{307624+360000 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{8192}\\ &=\frac{(12265+5718 x) \sqrt{2+5 x+3 x^2}}{512 (3+2 x)}-\frac{(3727+2898 x) \left (2+5 x+3 x^2\right )^{3/2}}{384 (3+2 x)^3}-\frac{(19+4 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac{5625}{256} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx+\frac{29047 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{1024}\\ &=\frac{(12265+5718 x) \sqrt{2+5 x+3 x^2}}{512 (3+2 x)}-\frac{(3727+2898 x) \left (2+5 x+3 x^2\right )^{3/2}}{384 (3+2 x)^3}-\frac{(19+4 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac{5625}{128} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )-\frac{29047}{512} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=\frac{(12265+5718 x) \sqrt{2+5 x+3 x^2}}{512 (3+2 x)}-\frac{(3727+2898 x) \left (2+5 x+3 x^2\right )^{3/2}}{384 (3+2 x)^3}-\frac{(19+4 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac{1875}{256} \sqrt{3} \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )+\frac{29047 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{1024 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.14012, size = 120, normalized size = 0.72 \[ \frac{-\frac{10 \sqrt{3 x^2+5 x+2} \left (3456 x^5-39744 x^4-533280 x^3-1672268 x^2-2059268 x-896721\right )}{(2 x+3)^4}-87141 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )-112500 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{15360} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(-896721 - 2059268*x - 1672268*x^2 - 533280*x^3 - 39744*x^4 + 3456*x^5))/(3 + 2*x)
^4 - 87141*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] - 112500*Sqrt[3]*ArcTanh[(5 + 6*x)/(2
*Sqrt[6 + 15*x + 9*x^2])])/15360

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Maple [A]  time = 0.012, size = 258, normalized size = 1.5 \begin{align*} -{\frac{13}{320} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{1}{75} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1627}{12000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{1307}{2500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{29047}{20000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{6935+8322\,x}{2400} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{2305+2766\,x}{320}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{1875\,\sqrt{3}}{256}\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 ) }+{\frac{29047}{9600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{29047}{5120}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{29047\,\sqrt{5}}{5120}{\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{6535+7842\,x}{5000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/320/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(7/2)-1/75/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(7/2)-1627/12000/(x+3/2)^
2*(3*(x+3/2)^2-4*x-19/4)^(7/2)+1307/2500/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(7/2)+29047/20000*(3*(x+3/2)^2-4*x-19/
4)^(5/2)-1387/2400*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-461/320*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-1875/256*
ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)+29047/9600*(3*(x+3/2)^2-4*x-19/4)^(3/2)+29047/5
120*(12*(x+3/2)^2-16*x-19)^(1/2)-29047/5120*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2
))-1307/5000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)

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Maxima [A]  time = 1.54738, size = 346, normalized size = 2.07 \begin{align*} \frac{1627}{4000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{20 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{8 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{75 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{1627 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{3000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{1387}{400} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{1307}{9600} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{1307 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{1000 \,{\left (2 \, x + 3\right )}} - \frac{1383}{160} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{1875}{256} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{29047}{5120} \, \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{10607}{2560} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1627/4000*(3*x^2 + 5*x + 2)^(5/2) - 13/20*(3*x^2 + 5*x + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 8
/75*(3*x^2 + 5*x + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1627/3000*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9
) - 1387/400*(3*x^2 + 5*x + 2)^(3/2)*x + 1307/9600*(3*x^2 + 5*x + 2)^(3/2) + 1307/1000*(3*x^2 + 5*x + 2)^(5/2)
/(2*x + 3) - 1383/160*sqrt(3*x^2 + 5*x + 2)*x - 1875/256*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2
) - 29047/5120*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 10607/2560*sqr
t(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.49171, size = 572, normalized size = 3.43 \begin{align*} \frac{112500 \, \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 ) + 87141 \, \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 (3456 \, x^{5} - 39744 \, x^{4} - 533280 \, x^{3} - 1672268 \, x^{2} - 2059268 \, x - 896721\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{30720 \,{\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)^(5/2)/(3+2*x)^5,x, algorithm="fricas")

[Out]

1/30720*(112500*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) + 87141*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*(3456*x^5 - 39744*x^4 - 533280*x^3 - 16722
68*x^2 - 2059268*x - 896721)*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{20 \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{96 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{165 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{113 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 - \int - \frac{15 x^{4} \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{9 x^{5} \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)**(5/2)/(3+2*x)**5,x)

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(
-96*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(-165*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(-113*x**3*sq
rt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-15*x**4*sqrt(3*
x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(9*x**5*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 = 1.7734, size = 601, normalized size = 3.6 \begin{align*} \frac{1875}{256} \, \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{29047}{5120} \, \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 ) \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - \frac{1}{3072} \,{\left (\frac{\frac{10 \,{\left (\frac{195 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )}{2 \, x + 3} - 904 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 18577 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )}{2 \, x + 3} - 27132 \, \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} - \frac{9 \,{\left (157 \,{\left (\sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{5}}{2 \, x + 3}\right )}^{3} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - 126 \, \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 )}^{2} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - 409 \,{\left (\sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{5}}{2 \, x + 3}\right )} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) + 330 \, \sqrt{5} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )}}{128 \,{\left ({\left (\sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{5}}{2 \, x + 3}\right )}^{2} - 3\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1875/256*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*sq
rt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3)))*sgn(1/(2*x + 3)) - 29047/5120*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)) - 1/3072*
((10*(195*sgn(1/(2*x + 3))/(2*x + 3) - 904*sgn(1/(2*x + 3)))/(2*x + 3) + 18577*sgn(1/(2*x + 3)))/(2*x + 3) - 2
7132*sgn(1/(2*x + 3)))*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) - 9/128*(157*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2
+ 3) + sqrt(5)/(2*x + 3))^3*sgn(1/(2*x + 3)) - 126*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(
2*x + 3))^2*sgn(1/(2*x + 3)) - 409*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))*sgn(1/(2*x + 3
)) + 330*sqrt(5)*sgn(1/(2*x + 3)))/((sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^2 - 3)^2