3.2457 \(\int \frac{(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^6} \, dx\)

Optimal. Leaf size=197 \[ -\frac{(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}+\frac{7 (548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{960 (2 x+3)^4}+\frac{7 (33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{7680 (2 x+3)^3}-\frac{21 (21974 x+47145) \sqrt{3 x^2+5 x+2}}{10240 (2 x+3)}+\frac{30275 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1024}-\frac{2345091 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{20480 \sqrt{5}} \]

[Out]

(-21*(47145 + 21974*x)*Sqrt[2 + 5*x + 3*x^2])/(10240*(3 + 2*x)) + (7*(42733 + 33142*x)*(2 + 5*x + 3*x^2)^(3/2)
)/(7680*(3 + 2*x)^3) + (7*(1003 + 548*x)*(2 + 5*x + 3*x^2)^(5/2))/(960*(3 + 2*x)^4) - ((27 + 5*x)*(2 + 5*x + 3
*x^2)^(7/2))/(30*(3 + 2*x)^5) + (30275*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/1024 - (2
345091*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(20480*Sqrt[5])

________________________________________________________________________________________

Rubi [A]  time = 0.126487, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}+\frac{7 (548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{960 (2 x+3)^4}+\frac{7 (33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{7680 (2 x+3)^3}-\frac{21 (21974 x+47145) \sqrt{3 x^2+5 x+2}}{10240 (2 x+3)}+\frac{30275 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1024}-\frac{2345091 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{20480 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^6,x]

[Out]

(-21*(47145 + 21974*x)*Sqrt[2 + 5*x + 3*x^2])/(10240*(3 + 2*x)) + (7*(42733 + 33142*x)*(2 + 5*x + 3*x^2)^(3/2)
)/(7680*(3 + 2*x)^3) + (7*(1003 + 548*x)*(2 + 5*x + 3*x^2)^(5/2))/(960*(3 + 2*x)^4) - ((27 + 5*x)*(2 + 5*x + 3
*x^2)^(7/2))/(30*(3 + 2*x)^5) + (30275*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/1024 - (2
345091*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(20480*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 )^{7/2}}{(3+2 x)^6} \, dx &=-\frac{(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}-\frac{7}{120} \int \frac{(-230-274 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx\\ &=\frac{7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac{(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}+\frac{7 \int \frac{(-11292-13112 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx}{1536}\\ &=\frac{7 (42733+33142 x) \left (2+5 x+3 x^2\right )^{3/2}}{7680 (3+2 x)^3}+\frac{7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac{(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}-\frac{7 \int \frac{(1351944+1582128 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^2} \, dx}{122880}\\ &=-\frac{21 (47145+21974 x) \sqrt{2+5 x+3 x^2}}{10240 (3+2 x)}+\frac{7 (42733+33142 x) \left (2+5 x+3 x^2\right )^{3/2}}{7680 (3+2 x)^3}+\frac{7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac{(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}+\frac{7 \int \frac{21287376+24912000 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{983040}\\ &=-\frac{21 (47145+21974 x) \sqrt{2+5 x+3 x^2}}{10240 (3+2 x)}+\frac{7 (42733+33142 x) \left (2+5 x+3 x^2\right )^{3/2}}{7680 (3+2 x)^3}+\frac{7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac{(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}+\frac{90825 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{1024}-\frac{2345091 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{20480}\\ &=-\frac{21 (47145+21974 x) \sqrt{2+5 x+3 x^2}}{10240 (3+2 x)}+\frac{7 (42733+33142 x) \left (2+5 x+3 x^2\right )^{3/2}}{7680 (3+2 x)^3}+\frac{7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac{(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}+\frac{90825}{512} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )+\frac{2345091 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{10240}\\ &=-\frac{21 (47145+21974 x) \sqrt{2+5 x+3 x^2}}{10240 (3+2 x)}+\frac{7 (42733+33142 x) \left (2+5 x+3 x^2\right )^{3/2}}{7680 (3+2 x)^3}+\frac{7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac{(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}+\frac{30275 \sqrt{3} \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{1024}-\frac{2345091 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{20480 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.167127, size = 130, normalized size = 0.66 \[ \frac{-\frac{10 \sqrt{3 x^2+5 x+2} \left (46080 x^7-257280 x^6+483840 x^5+27897856 x^4+127665096 x^3+242016116 x^2+213122626 x+72189541\right )}{(2 x+3)^5}+2345091 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )+3027500 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{102400} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^6,x]

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(72189541 + 213122626*x + 242016116*x^2 + 127665096*x^3 + 27897856*x^4 + 483840*x^
5 - 257280*x^6 + 46080*x^7))/(3 + 2*x)^5 + 2345091*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]
)] + 3027500*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/102400

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 316, normalized size = 1.6 \begin{align*} -{\frac{13}{800} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{27}{8000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{251}{5000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}+{\frac{10023}{100000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{95295+114354\,x}{25000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}+{\frac{614355+737226\,x}{100000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{19059}{12500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{185185+222222\,x}{16000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{186165+223398\,x}{6400}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{30275\,\sqrt{3}}{1024}\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{2345091\,\sqrt{5}}{102400}{\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{335013}{100000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}-{\frac{2345091}{400000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{781697}{64000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{2345091}{102400}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^6,x)

[Out]

-13/800/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(9/2)-27/8000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(9/2)-251/5000/(x+3/2)
^3*(3*(x+3/2)^2-4*x-19/4)^(9/2)+10023/100000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(9/2)+19059/25000*(5+6*x)*(3*(x+
3/2)^2-4*x-19/4)^(7/2)+122871/100000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)-19059/12500/(x+3/2)*(3*(x+3/2)^2-4*x
-19/4)^(9/2)+37037/16000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)+37233/6400*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)+
30275/1024*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)+2345091/102400*5^(1/2)*arctanh(2/5*(
-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))-335013/100000*(3*(x+3/2)^2-4*x-19/4)^(7/2)-2345091/400000*(3*(
x+3/2)^2-4*x-19/4)^(5/2)-781697/64000*(3*(x+3/2)^2-4*x-19/4)^(3/2)-2345091/102400*(12*(x+3/2)^2-16*x-19)^(1/2)

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Maxima [B]  time = 2.21978, size = 440, normalized size = 2.23 \begin{align*} -\frac{30069}{100000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{25 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{500 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{251 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{625 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} + \frac{10023 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{25000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{368613}{50000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{112329}{400000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{19059 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{5000 \,{\left (2 \, x + 3\right )}} + \frac{111111}{8000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{40957}{64000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{111699}{3200} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{30275}{1024} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) + \frac{2345091}{102400} \, \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{855771}{51200} \, \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)^(7/2)/(3+2*x)^6,x, algorithm="maxima")

[Out]

-30069/100000*(3*x^2 + 5*x + 2)^(7/2) - 13/25*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 +
 810*x + 243) - 27/500*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 251/625*(3*x^2 + 5*x
 + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) + 10023/25000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) + 368613/500
00*(3*x^2 + 5*x + 2)^(5/2)*x + 112329/400000*(3*x^2 + 5*x + 2)^(5/2) - 19059/5000*(3*x^2 + 5*x + 2)^(7/2)/(2*x
 + 3) + 111111/8000*(3*x^2 + 5*x + 2)^(3/2)*x - 40957/64000*(3*x^2 + 5*x + 2)^(3/2) + 111699/3200*sqrt(3*x^2 +
 5*x + 2)*x + 30275/1024*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 2345091/102400*sqrt(5)*log(s
qrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 855771/51200*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.65764, size = 683, normalized size = 3.47 \begin{align*} \frac{3027500 \, \sqrt{3}{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\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 ) + 2345091 \, \sqrt{5}{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\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 (46080 \, x^{7} - 257280 \, x^{6} + 483840 \, x^{5} + 27897856 \, x^{4} + 127665096 \, x^{3} + 242016116 \, x^{2} + 213122626 \, x + 72189541\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{204800 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^6,x, algorithm="fricas")

[Out]

1/204800*(3027500*sqrt(3)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x
 + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 2345091*sqrt(5)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)
*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*(46080*x^7 -
 257280*x^6 + 483840*x^5 + 27897856*x^4 + 127665096*x^3 + 242016116*x^2 + 213122626*x + 72189541)*sqrt(3*x^2 +
 5*x + 2))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+5*x+2)**(7/2)/(3+2*x)**6,x)

[Out]

Timed out

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Giac [B]  time = 1.27964, size = 563, normalized size = 2.86 \begin{align*} -\frac{3}{512} \,{\left (2 \,{\left (12 \, x - 157\right )} x + 2067\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{2345091}{102400} \, \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{30275}{1024} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac{60397264 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 739203704 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 11836231432 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 36096211012 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 207702455456 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 259725515674 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 635418284542 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 326158305587 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 287216072451 \, \sqrt{3} x + 36785380096 \, \sqrt{3} - 287216072451 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{10240 \,{\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 )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^6,x, algorithm="giac")

[Out]

-3/512*(2*(12*x - 157)*x + 2067)*sqrt(3*x^2 + 5*x + 2) - 2345091/102400*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)))
 - 30275/1024*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 1/10240*(60397264*(sqrt(3
)*x - sqrt(3*x^2 + 5*x + 2))^9 + 739203704*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 11836231432*(sqrt(3
)*x - sqrt(3*x^2 + 5*x + 2))^7 + 36096211012*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 207702455456*(sqr
t(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 259725515674*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 635418284542*
(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 326158305587*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 287216072
451*sqrt(3)*x + 36785380096*sqrt(3) - 287216072451*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)^5