### 3.341 $$\int \frac{(3-x+2 x^2)^{3/2} (2+x+3 x^2-x^3+5 x^4)}{(5+2 x)^6} \, dx$$

Optimal. Leaf size=195 $-\frac{3730507 \left (2 x^2-x+3\right )^{5/2}}{11943936 (2 x+5)^3}+\frac{158527 \left (2 x^2-x+3\right )^{5/2}}{165888 (2 x+5)^4}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{2880 (2 x+5)^5}+\frac{(44773976 x+246012435) \left (2 x^2-x+3\right )^{3/2}}{95551488 (2 x+5)^2}-\frac{(1028823716 x+5658774871) \sqrt{2 x^2-x+3}}{127401984 (2 x+5)}+\frac{70991525167 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{1528823808 \sqrt{2}}-\frac{23775 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{512 \sqrt{2}}$

[Out]

-((5658774871 + 1028823716*x)*Sqrt[3 - x + 2*x^2])/(127401984*(5 + 2*x)) + ((246012435 + 44773976*x)*(3 - x +
2*x^2)^(3/2))/(95551488*(5 + 2*x)^2) - (3667*(3 - x + 2*x^2)^(5/2))/(2880*(5 + 2*x)^5) + (158527*(3 - x + 2*x^
2)^(5/2))/(165888*(5 + 2*x)^4) - (3730507*(3 - x + 2*x^2)^(5/2))/(11943936*(5 + 2*x)^3) - (23775*ArcSinh[(1 -
4*x)/Sqrt[23]])/(512*Sqrt[2]) + (70991525167*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(152882380
8*Sqrt[2])

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Rubi [A]  time = 0.262875, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 40, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.175, Rules used = {1650, 812, 843, 619, 215, 724, 206} $-\frac{3730507 \left (2 x^2-x+3\right )^{5/2}}{11943936 (2 x+5)^3}+\frac{158527 \left (2 x^2-x+3\right )^{5/2}}{165888 (2 x+5)^4}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{2880 (2 x+5)^5}+\frac{(44773976 x+246012435) \left (2 x^2-x+3\right )^{3/2}}{95551488 (2 x+5)^2}-\frac{(1028823716 x+5658774871) \sqrt{2 x^2-x+3}}{127401984 (2 x+5)}+\frac{70991525167 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{1528823808 \sqrt{2}}-\frac{23775 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{512 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^6,x]

[Out]

-((5658774871 + 1028823716*x)*Sqrt[3 - x + 2*x^2])/(127401984*(5 + 2*x)) + ((246012435 + 44773976*x)*(3 - x +
2*x^2)^(3/2))/(95551488*(5 + 2*x)^2) - (3667*(3 - x + 2*x^2)^(5/2))/(2880*(5 + 2*x)^5) + (158527*(3 - x + 2*x^
2)^(5/2))/(165888*(5 + 2*x)^4) - (3730507*(3 - x + 2*x^2)^(5/2))/(11943936*(5 + 2*x)^3) - (23775*ArcSinh[(1 -
4*x)/Sqrt[23]])/(512*Sqrt[2]) + (70991525167*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(152882380
8*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 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 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

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{\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^6} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}-\frac{1}{360} \int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{60035}{16}-6615 x+2430 x^2-900 x^3\right )}{(5+2 x)^5} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac{158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}+\frac{\int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{8114455}{16}-\frac{3488315 x}{4}+129600 x^2\right )}{(5+2 x)^4} \, dx}{103680}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac{158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac{3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}-\frac{\int \frac{\left (\frac{332138325}{16}-\frac{83951205 x}{2}\right ) \left (3-x+2 x^2\right )^{3/2}}{(5+2 x)^3} \, dx}{22394880}\\ &=\frac{(246012435+44773976 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac{158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac{3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}+\frac{\int \frac{\left (\frac{7719844365}{4}-3858088935 x\right ) \sqrt{3-x+2 x^2}}{(5+2 x)^2} \, dx}{238878720}\\ &=-\frac{(5658774871+1028823716 x) \sqrt{3-x+2 x^2}}{127401984 (5+2 x)}+\frac{(246012435+44773976 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac{158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac{3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}-\frac{\int \frac{\frac{177475757505}{2}-177479424000 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{1911029760}\\ &=-\frac{(5658774871+1028823716 x) \sqrt{3-x+2 x^2}}{127401984 (5+2 x)}+\frac{(246012435+44773976 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac{158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac{3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}+\frac{23775}{512} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx-\frac{70991525167 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{254803968}\\ &=-\frac{(5658774871+1028823716 x) \sqrt{3-x+2 x^2}}{127401984 (5+2 x)}+\frac{(246012435+44773976 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac{158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac{3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}+\frac{70991525167 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{127401984}+\frac{23775 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{512 \sqrt{46}}\\ &=-\frac{(5658774871+1028823716 x) \sqrt{3-x+2 x^2}}{127401984 (5+2 x)}+\frac{(246012435+44773976 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac{158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac{3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}-\frac{23775 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{512 \sqrt{2}}+\frac{70991525167 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{1528823808 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.247036, size = 108, normalized size = 0.55 $\frac{\frac{24 \sqrt{2 x^2-x+3} \left (1592524800 x^6-30496849920 x^5-1023534029552 x^4-7117092892448 x^3-21590439797064 x^2-30872393829992 x-17093312738327\right )}{(2 x+5)^5}+354957625835 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )-354958848000 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{15288238080}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^6,x]

[Out]

((24*Sqrt[3 - x + 2*x^2]*(-17093312738327 - 30872393829992*x - 21590439797064*x^2 - 7117092892448*x^3 - 102353
4029552*x^4 - 30496849920*x^5 + 1592524800*x^6))/(5 + 2*x)^5 - 354958848000*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]
] + 354957625835*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/15288238080

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Maple [A]  time = 0.066, size = 225, normalized size = 1.2 \begin{align*} -{\frac{3667}{92160} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-5}}+{\frac{-3086715581+12346862324\,x}{9172942848}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{134077495}{6879707136} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-2}}+{\frac{70991525167\,\sqrt{2}}{3057647616}{\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{23775\,\sqrt{2}}{1024}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{70991525167}{495338913792} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}-{\frac{70991525167}{9172942848}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{3730507}{95551488} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-3}}+{\frac{-4698578717+18794314868\,x}{495338913792} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}-{\frac{4698578717}{247669456896} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}}+{\frac{158527}{2654208} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^6,x)

[Out]

-3667/92160/(x+5/2)^5*(2*(x+5/2)^2-11*x-19/2)^(5/2)+3086715581/9172942848*(-1+4*x)*(2*(x+5/2)^2-11*x-19/2)^(1/
2)+134077495/6879707136/(x+5/2)^2*(2*(x+5/2)^2-11*x-19/2)^(5/2)+70991525167/3057647616*2^(1/2)*arctanh(1/12*(1
7/2-11*x)*2^(1/2)/(2*(x+5/2)^2-11*x-19/2)^(1/2))+23775/1024*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))-70991525167
/495338913792*(2*(x+5/2)^2-11*x-19/2)^(3/2)-70991525167/9172942848*(2*(x+5/2)^2-11*x-19/2)^(1/2)-3730507/95551
488/(x+5/2)^3*(2*(x+5/2)^2-11*x-19/2)^(5/2)+4698578717/495338913792*(-1+4*x)*(2*(x+5/2)^2-11*x-19/2)^(3/2)-469
8578717/247669456896/(x+5/2)*(2*(x+5/2)^2-11*x-19/2)^(5/2)+158527/2654208/(x+5/2)^4*(2*(x+5/2)^2-11*x-19/2)^(5
/2)

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Maxima [A]  time = 1.65331, size = 339, normalized size = 1.74 \begin{align*} -\frac{134077495}{3439853568} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{2880 \,{\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} + \frac{158527 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{165888 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} - \frac{3730507 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{11943936 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} + \frac{134077495 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1719926784 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac{3086715581}{2293235712} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{23775}{1024} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) - \frac{70991525167}{3057647616} \, \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{6173186729}{764411904} \, \sqrt{2 \, x^{2} - x + 3} - \frac{4698578717 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{6879707136 \,{\left (2 \, x + 5\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^6,x, algorithm="maxima")

[Out]

-134077495/3439853568*(2*x^2 - x + 3)^(3/2) - 3667/2880*(2*x^2 - x + 3)^(5/2)/(32*x^5 + 400*x^4 + 2000*x^3 + 5
000*x^2 + 6250*x + 3125) + 158527/165888*(2*x^2 - x + 3)^(5/2)/(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625) - 3
730507/11943936*(2*x^2 - x + 3)^(5/2)/(8*x^3 + 60*x^2 + 150*x + 125) + 134077495/1719926784*(2*x^2 - x + 3)^(5
/2)/(4*x^2 + 20*x + 25) + 3086715581/2293235712*sqrt(2*x^2 - x + 3)*x + 23775/1024*sqrt(2)*arcsinh(4/23*sqrt(2
3)*x - 1/23*sqrt(23)) - 70991525167/3057647616*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/
abs(2*x + 5)) - 6173186729/764411904*sqrt(2*x^2 - x + 3) - 4698578717/6879707136*(2*x^2 - x + 3)^(3/2)/(2*x +
5)

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Fricas [A]  time = 1.52147, size = 747, normalized size = 3.83 \begin{align*} \frac{354958848000 \, \sqrt{2}{\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 354957625835 \, \sqrt{2}{\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\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 (1592524800 \, x^{6} - 30496849920 \, x^{5} - 1023534029552 \, x^{4} - 7117092892448 \, x^{3} - 21590439797064 \, x^{2} - 30872393829992 \, x - 17093312738327\right )} \sqrt{2 \, x^{2} - x + 3}}{30576476160 \,{\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^6,x, algorithm="fricas")

[Out]

1/30576476160*(354958848000*sqrt(2)*(32*x^5 + 400*x^4 + 2000*x^3 + 5000*x^2 + 6250*x + 3125)*log(-4*sqrt(2)*sq
rt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 354957625835*sqrt(2)*(32*x^5 + 400*x^4 + 2000*x^3 + 5000*x
^2 + 6250*x + 3125)*log((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*(1592524800*x^6 - 30496849920*x^5 - 1023534029552*x^4 - 7117092892448*x^3 - 21590439797064*x^2 - 3
0872393829992*x - 17093312738327)*sqrt(2*x^2 - x + 3))/(32*x^5 + 400*x^4 + 2000*x^3 + 5000*x^2 + 6250*x + 3125
)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{6}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2-x+3)**(3/2)*(5*x**4-x**3+3*x**2+x+2)/(5+2*x)**6,x)

[Out]

Integral((2*x**2 - x + 3)**(3/2)*(5*x**4 - x**3 + 3*x**2 + x + 2)/(2*x + 5)**6, x)

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Giac [B]  time = 1.30233, size = 548, normalized size = 2.81 \begin{align*} \frac{1}{256} \, \sqrt{2 \, x^{2} - x + 3}{\left (20 \, x - 633\right )} - \frac{23775}{1024} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{70991525167}{3057647616} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{70991525167}{3057647616} \, \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 (8281387393360 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{9} + 275661428628240 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{8} + 1560382703345760 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{7} + 4938646760855520 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{6} - 9673562837036232 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{5} - 30647310393849000 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{4} + 70060241036847960 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} - 97730658088823880 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 30180638363071845 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 7096913381268319\right )}}{1274019840 \,{\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 )}^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^6,x, algorithm="giac")

[Out]

1/256*sqrt(2*x^2 - x + 3)*(20*x - 633) - 23775/1024*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) +
1) + 70991525167/3057647616*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 70991525167/30
57647616*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 1/1274019840*sqrt(2)*(828138739
3360*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^9 + 275661428628240*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^8 + 15603
82703345760*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^7 + 4938646760855520*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^6
- 9673562837036232*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^5 - 30647310393849000*(sqrt(2)*x - sqrt(2*x^2 -
x + 3))^4 + 70060241036847960*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^3 - 97730658088823880*(sqrt(2)*x - sqr
t(2*x^2 - x + 3))^2 + 30180638363071845*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 7096913381268319)/(2*(sqrt
(2)*x - sqrt(2*x^2 - x + 3))^2 + 10*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 11)^5