3.2455 \(\int \frac{(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^4} \, dx\)

Optimal. Leaf size=190 \[ -\frac{(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}+\frac{7 (414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{960 (2 x+3)^2}-\frac{7 (1652 x+5713) \left (3 x^2+5 x+2\right )^{3/2}}{768 (2 x+3)}-\frac{7 (37375-78054 x) \sqrt{3 x^2+5 x+2}}{6144}+\frac{2776697 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{12288 \sqrt{3}}-\frac{59745 \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1024} \]

[Out]

(-7*(37375 - 78054*x)*Sqrt[2 + 5*x + 3*x^2])/6144 - (7*(5713 + 1652*x)*(2 + 5*x + 3*x^2)^(3/2))/(768*(3 + 2*x)
) + (7*(1171 + 414*x)*(2 + 5*x + 3*x^2)^(5/2))/(960*(3 + 2*x)^2) - ((37 + 3*x)*(2 + 5*x + 3*x^2)^(7/2))/(30*(3
+ 2*x)^3) + (2776697*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(12288*Sqrt[3]) - (59745*Sqrt[5]*A
rcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/1024

________________________________________________________________________________________

Rubi [A]  time = 0.129434, antiderivative size = 190, 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, 814, 843, 621, 206, 724} \[ -\frac{(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}+\frac{7 (414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{960 (2 x+3)^2}-\frac{7 (1652 x+5713) \left (3 x^2+5 x+2\right )^{3/2}}{768 (2 x+3)}-\frac{7 (37375-78054 x) \sqrt{3 x^2+5 x+2}}{6144}+\frac{2776697 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{12288 \sqrt{3}}-\frac{59745 \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1024} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^4,x]

[Out]

(-7*(37375 - 78054*x)*Sqrt[2 + 5*x + 3*x^2])/6144 - (7*(5713 + 1652*x)*(2 + 5*x + 3*x^2)^(3/2))/(768*(3 + 2*x)
) + (7*(1171 + 414*x)*(2 + 5*x + 3*x^2)^(5/2))/(960*(3 + 2*x)^2) - ((37 + 3*x)*(2 + 5*x + 3*x^2)^(7/2))/(30*(3
+ 2*x)^3) + (2776697*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(12288*Sqrt[3]) - (59745*Sqrt[5]*A
rcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/1024

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 814

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

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)^4} \, dx &=-\frac{(37+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^3}-\frac{7}{120} \int \frac{(-346-414 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx\\ &=\frac{7 (1171+414 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^2}-\frac{(37+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^3}+\frac{7 \int \frac{(-16796-19824 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx}{1536}\\ &=-\frac{7 (5713+1652 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)}+\frac{7 (1171+414 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^2}-\frac{(37+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^3}-\frac{7 \int \frac{(-526968-624432 x) \sqrt{2+5 x+3 x^2}}{3+2 x} \, dx}{12288}\\ &=-\frac{7 (37375-78054 x) \sqrt{2+5 x+3 x^2}}{6144}-\frac{7 (5713+1652 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)}+\frac{7 (1171+414 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^2}-\frac{(37+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^3}+\frac{7 \int \frac{32539824+38080416 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{589824}\\ &=-\frac{7 (37375-78054 x) \sqrt{2+5 x+3 x^2}}{6144}-\frac{7 (5713+1652 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)}+\frac{7 (1171+414 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^2}-\frac{(37+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^3}+\frac{2776697 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{12288}-\frac{298725 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{1024}\\ &=-\frac{7 (37375-78054 x) \sqrt{2+5 x+3 x^2}}{6144}-\frac{7 (5713+1652 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)}+\frac{7 (1171+414 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^2}-\frac{(37+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^3}+\frac{2776697 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{6144}+\frac{298725}{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{7 (37375-78054 x) \sqrt{2+5 x+3 x^2}}{6144}-\frac{7 (5713+1652 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)}+\frac{7 (1171+414 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^2}-\frac{(37+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^3}+\frac{2776697 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{12288 \sqrt{3}}-\frac{59745 \sqrt{5} \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{1024}\\ \end{align*}

Mathematica [A]  time = 0.124197, size = 130, normalized size = 0.68 \[ \frac{-\frac{6 \sqrt{3 x^2+5 x+2} \left (82944 x^7-231552 x^6-1266816 x^5-3277520 x^4+746240 x^3+44770416 x^2+98927312 x+61268351\right )}{(2 x+3)^3}[phone] \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )[phone] \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{184320} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^4,x]

[Out]

((-6*Sqrt[2 + 5*x + 3*x^2]*(61268351 + 98927312*x + 44770416*x^2 + 746240*x^3 - 3277520*x^4 - 1266816*x^5 - 23
1552*x^6 + 82944*x^7))/(3 + 2*x)^3 + 10754100*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] +
13883485*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/184320

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Maple [A]  time = 0.013, size = 274, normalized size = 1.4 \begin{align*} -{\frac{13}{120} \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{57}{200} \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{240+288\,x}{25} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}+{\frac{6265+7518\,x}{400} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{96}{25} \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{22645+27174\,x}{768} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{455315+546378\,x}{6144}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{2776697\,\sqrt{3}}{36864}\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{59745\,\sqrt{5}}{1024}{\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{1707}{200} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}-{\frac{11949}{800} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{3983}{128} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{59745}{1024}\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)^4,x)

[Out]

-13/120/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(9/2)+57/200/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(9/2)+48/25*(5+6*x)*(3*
(x+3/2)^2-4*x-19/4)^(7/2)+1253/400*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)-96/25/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(
9/2)+4529/768*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)+91063/6144*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)+2776697/368
64*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)+59745/1024*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^
(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))-1707/200*(3*(x+3/2)^2-4*x-19/4)^(7/2)-11949/800*(3*(x+3/2)^2-4*x-19/4)^(5/
2)-3983/128*(3*(x+3/2)^2-4*x-19/4)^(3/2)-59745/1024*(12*(x+3/2)^2-16*x-19)^(1/2)

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Maxima [A]  time = 1.81266, size = 336, normalized size = 1.77 \begin{align*} -\frac{171}{200} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{15 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} + \frac{57 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{50 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{3759}{200} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{581}{800} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{48 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{5 \,{\left (2 \, x + 3\right )}} + \frac{4529}{128} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{1253}{768} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{91063}{1024} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{2776697}{36864} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) + \frac{59745}{1024} \, \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{261625}{6144} \, \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)^4,x, algorithm="maxima")

[Out]

-171/200*(3*x^2 + 5*x + 2)^(7/2) - 13/15*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) + 57/50*(3*x^2 +
5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) + 3759/200*(3*x^2 + 5*x + 2)^(5/2)*x + 581/800*(3*x^2 + 5*x + 2)^(5/2) - 48
/5*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 4529/128*(3*x^2 + 5*x + 2)^(3/2)*x - 1253/768*(3*x^2 + 5*x + 2)^(3/2) +
91063/1024*sqrt(3*x^2 + 5*x + 2)*x + 2776697/36864*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 5
9745/1024*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 261625/6144*sqrt(3*
x^2 + 5*x + 2)

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Fricas [A]  time = 1.4725, size = 578, normalized size = 3.04 \begin{align*} \frac{13883485 \, \sqrt{3}{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\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 ) + 10754100 \, \sqrt{5}{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\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 ) - 12 \,{\left (82944 \, x^{7} - 231552 \, x^{6} - 1266816 \, x^{5} - 3277520 \, x^{4} + 746240 \, x^{3} + 44770416 \, x^{2} + 98927312 \, x + 61268351\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{368640 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\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)^4,x, algorithm="fricas")

[Out]

1/368640*(13883485*sqrt(3)*(8*x^3 + 36*x^2 + 54*x + 27)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2
+ 120*x + 49) + 10754100*sqrt(5)*(8*x^3 + 36*x^2 + 54*x + 27)*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)) - 12*(82944*x^7 - 231552*x^6 - 1266816*x^5 - 3277520*x^4 + 746240
*x^3 + 44770416*x^2 + 98927312*x + 61268351)*sqrt(3*x^2 + 5*x + 2))/(8*x^3 + 36*x^2 + 54*x + 27)

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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)**4,x)

[Out]

Timed out

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Giac [B]  time = 1.29269, size = 439, normalized size = 2.31 \begin{align*} -\frac{1}{30720} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (24 \, x - 175\right )} x + 4661\right )} x - 218885\right )} x + 1563313\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{59745}{1024} \, \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{2776697}{36864} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac{5 \,{\left (424596 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 2828550 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 21565510 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 26086815 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 45375675 \, \sqrt{3} x + 10164786 \, \sqrt{3} - 45375675 \, \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{1536 \,{\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{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)^4,x, algorithm="giac")

[Out]

-1/30720*(2*(12*(18*(24*x - 175)*x + 4661)*x - 218885)*x + 1563313)*sqrt(3*x^2 + 5*x + 2) - 59745/1024*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))) - 2776697/36864*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))
- 5)) - 5/1536*(424596*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 2828550*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x
+ 2))^4 + 21565510*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 26086815*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)
)^2 + 45375675*sqrt(3)*x + 10164786*sqrt(3) - 45375675*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