3.2456 \(\int \frac{(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^5} \, dx\)

Optimal. Leaf size=195 \[ -\frac{(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}+\frac{7 (43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{96 (2 x+3)^3}-\frac{35 (343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{768 (2 x+3)^2}+\frac{35 (2701 x+5795) \sqrt{3 x^2+5 x+2}}{1024 (2 x+3)}-\frac{744275 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{4096 \sqrt{3}}+\frac{192171 \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{4096} \]

[Out]

(35*(5795 + 2701*x)*Sqrt[2 + 5*x + 3*x^2])/(1024*(3 + 2*x)) - (35*(736 + 343*x)*(2 + 5*x + 3*x^2)^(3/2))/(768*
(3 + 2*x)^2) + (7*(93 + 43*x)*(2 + 5*x + 3*x^2)^(5/2))/(96*(3 + 2*x)^3) - ((8 + x)*(2 + 5*x + 3*x^2)^(7/2))/(8
*(3 + 2*x)^4) - (744275*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(4096*Sqrt[3]) + (192171*Sqrt[5]
*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/4096

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Rubi [A]  time = 0.130179, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {812, 843, 621, 206, 724} \[ -\frac{(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}+\frac{7 (43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{96 (2 x+3)^3}-\frac{35 (343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{768 (2 x+3)^2}+\frac{35 (2701 x+5795) \sqrt{3 x^2+5 x+2}}{1024 (2 x+3)}-\frac{744275 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{4096 \sqrt{3}}+\frac{192171 \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{4096} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*(5795 + 2701*x)*Sqrt[2 + 5*x + 3*x^2])/(1024*(3 + 2*x)) - (35*(736 + 343*x)*(2 + 5*x + 3*x^2)^(3/2))/(768*
(3 + 2*x)^2) + (7*(93 + 43*x)*(2 + 5*x + 3*x^2)^(5/2))/(96*(3 + 2*x)^3) - ((8 + x)*(2 + 5*x + 3*x^2)^(7/2))/(8
*(3 + 2*x)^4) - (744275*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(4096*Sqrt[3]) + (192171*Sqrt[5]
*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/4096

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 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)^5} \, dx &=-\frac{(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}-\frac{7}{128} \int \frac{(-288-344 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^4} \, dx\\ &=\frac{7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac{(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}+\frac{35 \int \frac{(-14064-16464 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx}{9216}\\ &=-\frac{35 (736+343 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)^2}+\frac{7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac{(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}-\frac{35 \int \frac{(-443136-518592 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^2} \, dx}{98304}\\ &=\frac{35 (5795+2701 x) \sqrt{2+5 x+3 x^2}}{1024 (3+2 x)}-\frac{35 (736+343 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)^2}+\frac{7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac{(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}+\frac{35 \int \frac{-6977664-8165760 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{786432}\\ &=\frac{35 (5795+2701 x) \sqrt{2+5 x+3 x^2}}{1024 (3+2 x)}-\frac{35 (736+343 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)^2}+\frac{7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac{(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}-\frac{744275 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{4096}+\frac{960855 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{4096}\\ &=\frac{35 (5795+2701 x) \sqrt{2+5 x+3 x^2}}{1024 (3+2 x)}-\frac{35 (736+343 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)^2}+\frac{7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac{(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}-\frac{744275 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{2048}-\frac{960855 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{2048}\\ &=\frac{35 (5795+2701 x) \sqrt{2+5 x+3 x^2}}{1024 (3+2 x)}-\frac{35 (736+343 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)^2}+\frac{7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac{(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}-\frac{744275 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{4096 \sqrt{3}}+\frac{192171 \sqrt{5} \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{4096}\\ \end{align*}

Mathematica [A]  time = 0.125655, size = 130, normalized size = 0.67 \[ \frac{-\frac{12 \sqrt{3 x^2+5 x+2} \left (3456 x^7-12864 x^6-38288 x^5-253688 x^4-2869312 x^3-9107922 x^2-11295211 x-4933171\right )}{(2 x+3)^4}-576513 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )-744275 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{12288} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-12*Sqrt[2 + 5*x + 3*x^2]*(-4933171 - 11295211*x - 9107922*x^2 - 2869312*x^3 - 253688*x^4 - 38288*x^5 - 1286
4*x^6 + 3456*x^7))/(3 + 2*x)^4 - 576513*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] - 744275
*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/12288

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Maple [A]  time = 0.013, size = 295, normalized size = 1.5 \begin{align*} -{\frac{13}{320} \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{3}{100} \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{1263}{4000} \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{7395+8874\,x}{1000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}-{\frac{50505+60606\,x}{4000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{1479}{500} \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{30345+36414\,x}{1280} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{122045+146454\,x}{2048}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{744275\,\sqrt{3}}{12288}\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{192171\,\sqrt{5}}{4096}{\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{27453}{4000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}+{\frac{192171}{16000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{64057}{2560} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{192171}{4096}\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)^5,x)

[Out]

-13/320/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(9/2)+3/100/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(9/2)-1263/4000/(x+3/2)^
2*(3*(x+3/2)^2-4*x-19/4)^(9/2)-1479/1000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(7/2)-10101/4000*(5+6*x)*(3*(x+3/2)^2-
4*x-19/4)^(5/2)+1479/500/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(9/2)-6069/1280*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-2
4409/2048*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-744275/12288*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1
/2))*3^(1/2)-192171/4096*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+27453/4000*(3*(x
+3/2)^2-4*x-19/4)^(7/2)+192171/16000*(3*(x+3/2)^2-4*x-19/4)^(5/2)+64057/2560*(3*(x+3/2)^2-4*x-19/4)^(3/2)+1921
71/4096*(12*(x+3/2)^2-16*x-19)^(1/2)

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Maxima [A]  time = 2.04712, size = 385, normalized size = 1.97 \begin{align*} \frac{3789}{4000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{20 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} + \frac{6 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{25 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{1263 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{1000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{30303}{2000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{9849}{16000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{1479 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{200 \,{\left (2 \, x + 3\right )}} - \frac{18207}{640} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{3367}{2560} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{73227}{1024} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{744275}{12288} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{192171}{4096} \, \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{35063}{1024} \, \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)^5,x, algorithm="maxima")

[Out]

3789/4000*(3*x^2 + 5*x + 2)^(7/2) - 13/20*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) + 6
/25*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1263/1000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9
) - 30303/2000*(3*x^2 + 5*x + 2)^(5/2)*x - 9849/16000*(3*x^2 + 5*x + 2)^(5/2) + 1479/200*(3*x^2 + 5*x + 2)^(7/
2)/(2*x + 3) - 18207/640*(3*x^2 + 5*x + 2)^(3/2)*x + 3367/2560*(3*x^2 + 5*x + 2)^(3/2) - 73227/1024*sqrt(3*x^2
+ 5*x + 2)*x - 744275/12288*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 192171/4096*sqrt(5)*log(
sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 35063/1024*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.54887, size = 612, normalized size = 3.14 \begin{align*} \frac{744275 \, \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 ) + 576513 \, \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 ) - 24 \,{\left (3456 \, x^{7} - 12864 \, x^{6} - 38288 \, x^{5} - 253688 \, x^{4} - 2869312 \, x^{3} - 9107922 \, x^{2} - 11295211 \, x - 4933171\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{24576 \,{\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)^(7/2)/(3+2*x)^5,x, algorithm="fricas")

[Out]

1/24576*(744275*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) + 576513*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)) - 24*(3456*x^7 - 12864*x^6 - 38288*x^5 - 25368
8*x^4 - 2869312*x^3 - 9107922*x^2 - 11295211*x - 4933171)*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(-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)**5,x)

[Out]

Timed out

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Giac [B]  time = 2.03633, size = 859, normalized size = 4.41 \begin{align*} \text{result too large to display} \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)^5,x, algorithm="giac")

[Out]

744275/12288*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)) - 192171/4096*sq
rt(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/
4096*(5*(50*(13*sgn(1/(2*x + 3))/(2*x + 3) - 88*sgn(1/(2*x + 3)))/(2*x + 3) + 14343*sgn(1/(2*x + 3)))/(2*x + 3
) - 181996*sgn(1/(2*x + 3)))*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) - 1/2048*(479709*(sqrt(-8/(2*x + 3) + 5/(2
*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^7*sgn(1/(2*x + 3)) - 499296*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3
) + sqrt(5)/(2*x + 3))^6*sgn(1/(2*x + 3)) - 3133183*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3
))^5*sgn(1/(2*x + 3)) + 3365712*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^4*sgn(1/(
2*x + 3)) + 7550211*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^3*sgn(1/(2*x + 3)) - 8139744*
sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^2*sgn(1/(2*x + 3)) - 6574257*(sqrt(-8/(2*
x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))*sgn(1/(2*x + 3)) + 6966000*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)^4