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

Optimal. Leaf size=169 \[ \frac{87677717 \left (2 x^2-x+3\right )^{3/2}}{8599633920 (2 x+5)^3}-\frac{5703277 \left (2 x^2-x+3\right )^{3/2}}{39813120 (2 x+5)^4}+\frac{92239 \left (2 x^2-x+3\right )^{3/2}}{138240 (2 x+5)^5}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{3456 (2 x+5)^6}-\frac{1172725 (17-22 x) \sqrt{2 x^2-x+3}}{330225942528 (2 x+5)^2}-\frac{26972675 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{3962711310336 \sqrt{2}} \]

[Out]

(-1172725*(17 - 22*x)*Sqrt[3 - x + 2*x^2])/(330225942528*(5 + 2*x)^2) - (3667*(3 - x + 2*x^2)^(3/2))/(3456*(5
+ 2*x)^6) + (92239*(3 - x + 2*x^2)^(3/2))/(138240*(5 + 2*x)^5) - (5703277*(3 - x + 2*x^2)^(3/2))/(39813120*(5
+ 2*x)^4) + (87677717*(3 - x + 2*x^2)^(3/2))/(8599633920*(5 + 2*x)^3) - (26972675*ArcTanh[(17 - 22*x)/(12*Sqrt
[2]*Sqrt[3 - x + 2*x^2])])/(3962711310336*Sqrt[2])

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Rubi [A]  time = 0.218244, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1650, 806, 720, 724, 206} \[ \frac{87677717 \left (2 x^2-x+3\right )^{3/2}}{8599633920 (2 x+5)^3}-\frac{5703277 \left (2 x^2-x+3\right )^{3/2}}{39813120 (2 x+5)^4}+\frac{92239 \left (2 x^2-x+3\right )^{3/2}}{138240 (2 x+5)^5}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{3456 (2 x+5)^6}-\frac{1172725 (17-22 x) \sqrt{2 x^2-x+3}}{330225942528 (2 x+5)^2}-\frac{26972675 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{3962711310336 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1172725*(17 - 22*x)*Sqrt[3 - x + 2*x^2])/(330225942528*(5 + 2*x)^2) - (3667*(3 - x + 2*x^2)^(3/2))/(3456*(5
+ 2*x)^6) + (92239*(3 - x + 2*x^2)^(3/2))/(138240*(5 + 2*x)^5) - (5703277*(3 - x + 2*x^2)^(3/2))/(39813120*(5
+ 2*x)^4) + (87677717*(3 - x + 2*x^2)^(3/2))/(8599633920*(5 + 2*x)^3) - (26972675*ArcTanh[(17 - 22*x)/(12*Sqrt
[2]*Sqrt[3 - x + 2*x^2])])/(3962711310336*Sqrt[2])

Rule 1650

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*
x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^
(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(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] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 720

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

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])

Rubi steps

\begin{align*} \int \frac{\sqrt{3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^7} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}-\frac{1}{432} \int \frac{\sqrt{3-x+2 x^2} \left (\frac{61041}{16}-\frac{20751 x}{4}+2916 x^2-1080 x^3\right )}{(5+2 x)^6} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac{92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}+\frac{\int \frac{\sqrt{3-x+2 x^2} \left (\frac{8057313}{16}-\frac{1191609 x}{2}+194400 x^2\right )}{(5+2 x)^5} \, dx}{155520}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac{92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac{5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}-\frac{\int \frac{\left (\frac{182650383}{16}-\frac{60644907 x}{4}\right ) \sqrt{3-x+2 x^2}}{(5+2 x)^4} \, dx}{44789760}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac{92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac{5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac{87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}+\frac{1172725 \int \frac{\sqrt{3-x+2 x^2}}{(5+2 x)^3} \, dx}{1146617856}\\ &=-\frac{1172725 (17-22 x) \sqrt{3-x+2 x^2}}{330225942528 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac{92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac{5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac{87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}+\frac{26972675 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{660451885056}\\ &=-\frac{1172725 (17-22 x) \sqrt{3-x+2 x^2}}{330225942528 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac{92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac{5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac{87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}-\frac{26972675 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{330225942528}\\ &=-\frac{1172725 (17-22 x) \sqrt{3-x+2 x^2}}{330225942528 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac{92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac{5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac{87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}-\frac{26972675 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{3962711310336 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.172825, size = 91, normalized size = 0.54 \[ \frac{24 \sqrt{2 x^2-x+3} \left (271409942624 x^5+12256250416 x^4+397498825328 x^3+158340720344 x^2+27245373694 x-219337079305\right )-134863375 \sqrt{2} (2 x+5)^6 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )}{39627113103360 (2 x+5)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(24*Sqrt[3 - x + 2*x^2]*(-219337079305 + 27245373694*x + 158340720344*x^2 + 397498825328*x^3 + 12256250416*x^4
+ 271409942624*x^5) - 134863375*Sqrt[2]*(5 + 2*x)^6*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/(3962711
3103360*(5 + 2*x)^6)

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Maple [A]  time = 0.089, size = 195, normalized size = 1.2 \begin{align*} -{\frac{3667}{221184} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-6}}-{\frac{12899975}{11888133931008} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}}+{\frac{-12899975+51599900\,x}{23776267862016}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{1172725}{330225942528} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-2}}-{\frac{26972675\,\sqrt{2}}{7925422620672}{\it Artanh} \left ({\frac{\sqrt{2}}{12} \left ({\frac{17}{2}}-11\,x \right ){\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}} \right ) }+{\frac{26972675}{23776267862016}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{5703277}{637009920} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-4}}+{\frac{87677717}{68797071360} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-3}}+{\frac{92239}{4423680} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^7,x)

[Out]

-3667/221184/(x+5/2)^6*(2*(x+5/2)^2-11*x-19/2)^(3/2)-12899975/11888133931008/(x+5/2)*(2*(x+5/2)^2-11*x-19/2)^(
3/2)[phone]/23776267862016*(-1+4*x)*(2*(x+5/2)^2-11*x-19/2)^(1/2)-1172725/330225942528/(x+5/2)^2*(2*(x+5/2)^
2-11*x-19/2)^(3/2)-26972675/7925422620672*2^(1/2)*arctanh(1/12*(17/2-11*x)*2^(1/2)/(2*(x+5/2)^2-11*x-19/2)^(1/
2))[phone]/23776267862016*(2*(x+5/2)^2-11*x-19/2)^(1/2)-5703277/637009920/(x+5/2)^4*(2*(x+5/2)^2-11*x-19/2)^
(3/2)[phone]/68797071360/(x+5/2)^3*(2*(x+5/2)^2-11*x-19/2)^(3/2)+92239/4423680/(x+5/2)^5*(2*(x+5/2)^2-11*x-1
9/2)^(3/2)

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Maxima [A]  time = 1.60522, size = 338, normalized size = 2. \begin{align*} \frac{26972675}{7925422620672} \, \sqrt{2} \operatorname{arsinh}\left (\frac{22 \, \sqrt{23} x}{23 \,{\left | 2 \, x + 5 \right |}} - \frac{17 \, \sqrt{23}}{23 \,{\left | 2 \, x + 5 \right |}}\right ) + \frac{1172725}{165112971264} \, \sqrt{2 \, x^{2} - x + 3} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{3456 \,{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} + \frac{92239 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{138240 \,{\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} - \frac{5703277 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{39813120 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac{87677717 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{8599633920 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac{1172725 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{82556485632 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac{12899975 \, \sqrt{2 \, x^{2} - x + 3}}{330225942528 \,{\left (2 \, x + 5\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^7,x, algorithm="maxima")

[Out]

26972675/7925422620672*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) + 1172725/
165112971264*sqrt(2*x^2 - x + 3) - 3667/3456*(2*x^2 - x + 3)^(3/2)/(64*x^6 + 960*x^5 + 6000*x^4 + 20000*x^3 +
37500*x^2 + 37500*x + 15625) + 92239/138240*(2*x^2 - x + 3)^(3/2)/(32*x^5 + 400*x^4 + 2000*x^3 + 5000*x^2 + 62
50*x + 3125) - 5703277/39813120*(2*x^2 - x + 3)^(3/2)/(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625) + 87677717/8
599633920*(2*x^2 - x + 3)^(3/2)/(8*x^3 + 60*x^2 + 150*x + 125) - 1172725/82556485632*(2*x^2 - x + 3)^(3/2)/(4*
x^2 + 20*x + 25) - 12899975/330225942528*sqrt(2*x^2 - x + 3)/(2*x + 5)

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Fricas [A]  time = 1.35957, size = 555, normalized size = 3.28 \begin{align*} \frac{134863375 \, \sqrt{2}{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )} \log \left (-\frac{24 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \,{\left (271409942624 \, x^{5} + 12256250416 \, x^{4} + 397498825328 \, x^{3} + 158340720344 \, x^{2} + 27245373694 \, x - 219337079305\right )} \sqrt{2 \, x^{2} - x + 3}}{79254226206720 \,{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^7,x, algorithm="fricas")

[Out]

1/79254226206720*(134863375*sqrt(2)*(64*x^6 + 960*x^5 + 6000*x^4 + 20000*x^3 + 37500*x^2 + 37500*x + 15625)*lo
g(-(24*sqrt(2)*sqrt(2*x^2 - x + 3)*(22*x - 17) + 1060*x^2 - 1036*x + 1153)/(4*x^2 + 20*x + 25)) + 48*(27140994
2624*x^5 + 12256250416*x^4 + 397498825328*x^3 + 158340720344*x^2 + 27245373694*x - 219337079305)*sqrt(2*x^2 -
x + 3))/(64*x^6 + 960*x^5 + 6000*x^4 + 20000*x^3 + 37500*x^2 + 37500*x + 15625)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 x^{2} - x + 3} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**4-x**3+3*x**2+x+2)*(2*x**2-x+3)**(1/2)/(5+2*x)**7,x)

[Out]

Integral(sqrt(2*x**2 - x + 3)*(5*x**4 - x**3 + 3*x**2 + x + 2)/(2*x + 5)**7, x)

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Giac [B]  time = 1.30763, size = 547, normalized size = 3.24 \begin{align*} -\frac{26972675}{7925422620672} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{26972675}{7925422620672} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x - 11 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{\sqrt{2}{\left (16506981498400 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{11} + 389429252643040 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{10} + 2263923918689840 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{9} + 11663651054548560 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{8} + 902212326134736 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{7} - 84192729519861840 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{6} - 4317200555009448 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{5} + 351543414066518760 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{4} - 376787166452923830 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} + 356306707647610982 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} - 82348353128195465 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 15499394004553969\right )}}{3302259425280 \,{\left (2 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 11\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^7,x, algorithm="giac")

[Out]

-26972675/7925422620672*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) + 26972675/7925422620
672*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) + 1/3302259425280*sqrt(2)*(16506981498
400*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^11 + 389429252643040*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^10 + 2263
923918689840*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^9 + 11663651054548560*(sqrt(2)*x - sqrt(2*x^2 - x + 3))
^8 + 902212326134736*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^7 - 84192729519861840*(sqrt(2)*x - sqrt(2*x^2 -
x + 3))^6 - 4317200555009448*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^5 + 351543414066518760*(sqrt(2)*x - sq
rt(2*x^2 - x + 3))^4 - 376787166452923830*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^3 + 356306707647610982*(sq
rt(2)*x - sqrt(2*x^2 - x + 3))^2 - 82348353128195465*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1549939400455
3969)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 10*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 11)^6