3.343 $$\int \frac{(3-x+2 x^2)^{3/2} (2+x+3 x^2-x^3+5 x^4)}{(5+2 x)^8} \, dx$$

Optimal. Leaf size=195 $-\frac{1930441 \left (2 x^2-x+3\right )^{5/2}}{13934592 (2 x+5)^5}+\frac{114335 \left (2 x^2-x+3\right )^{5/2}}{193536 (2 x+5)^6}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}-\frac{(411822458 x+463558457) \left (2 x^2-x+3\right )^{3/2}}{2293235712 (2 x+5)^4}-\frac{(101679102454 x+146583836191) \sqrt{2 x^2-x+3}}{440301256704 (2 x+5)^2}+\frac{412760561351 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{5283615080448 \sqrt{2}}-\frac{5 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{2}}$

[Out]

-((146583836191 + 101679102454*x)*Sqrt[3 - x + 2*x^2])/(440301256704*(5 + 2*x)^2) - ((463558457 + 411822458*x)
*(3 - x + 2*x^2)^(3/2))/(2293235712*(5 + 2*x)^4) - (3667*(3 - x + 2*x^2)^(5/2))/(4032*(5 + 2*x)^7) + (114335*(
3 - x + 2*x^2)^(5/2))/(193536*(5 + 2*x)^6) - (1930441*(3 - x + 2*x^2)^(5/2))/(13934592*(5 + 2*x)^5) - (5*ArcSi
nh[(1 - 4*x)/Sqrt[23]])/(64*Sqrt[2]) + (412760561351*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(5
283615080448*Sqrt[2])

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Rubi [A]  time = 0.261981, 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, 810, 843, 619, 215, 724, 206} $-\frac{1930441 \left (2 x^2-x+3\right )^{5/2}}{13934592 (2 x+5)^5}+\frac{114335 \left (2 x^2-x+3\right )^{5/2}}{193536 (2 x+5)^6}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}-\frac{(411822458 x+463558457) \left (2 x^2-x+3\right )^{3/2}}{2293235712 (2 x+5)^4}-\frac{(101679102454 x+146583836191) \sqrt{2 x^2-x+3}}{440301256704 (2 x+5)^2}+\frac{412760561351 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{5283615080448 \sqrt{2}}-\frac{5 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{2}}$

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)^8,x]

[Out]

-((146583836191 + 101679102454*x)*Sqrt[3 - x + 2*x^2])/(440301256704*(5 + 2*x)^2) - ((463558457 + 411822458*x)
*(3 - x + 2*x^2)^(3/2))/(2293235712*(5 + 2*x)^4) - (3667*(3 - x + 2*x^2)^(5/2))/(4032*(5 + 2*x)^7) + (114335*(
3 - x + 2*x^2)^(5/2))/(193536*(5 + 2*x)^6) - (1930441*(3 - x + 2*x^2)^(5/2))/(13934592*(5 + 2*x)^5) - (5*ArcSi
nh[(1 - 4*x)/Sqrt[23]])/(64*Sqrt[2]) + (412760561351*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(5
283615080448*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 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 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)^8} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{4032 (5+2 x)^7}-\frac{1}{504} \int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{76715}{16}-\frac{14855 x}{2}+3402 x^2-1260 x^3\right )}{(5+2 x)^7} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{4032 (5+2 x)^7}+\frac{114335 \left (3-x+2 x^2\right )^{5/2}}{193536 (5+2 x)^6}+\frac{\int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{13334715}{16}-\frac{4631913 x}{4}+272160 x^2\right )}{(5+2 x)^6} \, dx}{217728}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{4032 (5+2 x)^7}+\frac{114335 \left (3-x+2 x^2\right )^{5/2}}{193536 (5+2 x)^6}-\frac{1930441 \left (3-x+2 x^2\right )^{5/2}}{13934592 (5+2 x)^5}-\frac{\int \frac{\left (\frac{516687885}{16}-48988800 x\right ) \left (3-x+2 x^2\right )^{3/2}}{(5+2 x)^5} \, dx}{78382080}\\ &=-\frac{(463558457+411822458 x) \left (3-x+2 x^2\right )^{3/2}}{2293235712 (5+2 x)^4}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{4032 (5+2 x)^7}+\frac{114335 \left (3-x+2 x^2\right )^{5/2}}{193536 (5+2 x)^6}-\frac{1930441 \left (3-x+2 x^2\right )^{5/2}}{13934592 (5+2 x)^5}+\frac{\int \frac{\left (-\frac{283730747265}{8}+56435097600 x\right ) \sqrt{3-x+2 x^2}}{(5+2 x)^3} \, dx}{180592312320}\\ &=-\frac{(146583836191+101679102454 x) \sqrt{3-x+2 x^2}}{440301256704 (5+2 x)^2}-\frac{(463558457+411822458 x) \left (3-x+2 x^2\right )^{3/2}}{2293235712 (5+2 x)^4}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{4032 (5+2 x)^7}+\frac{114335 \left (3-x+2 x^2\right )^{5/2}}{193536 (5+2 x)^6}-\frac{1930441 \left (3-x+2 x^2\right )^{5/2}}{13934592 (5+2 x)^5}-\frac{\int \frac{\frac{64992568300695}{4}-32506616217600 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{208042343792640}\\ &=-\frac{(146583836191+101679102454 x) \sqrt{3-x+2 x^2}}{440301256704 (5+2 x)^2}-\frac{(463558457+411822458 x) \left (3-x+2 x^2\right )^{3/2}}{2293235712 (5+2 x)^4}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{4032 (5+2 x)^7}+\frac{114335 \left (3-x+2 x^2\right )^{5/2}}{193536 (5+2 x)^6}-\frac{1930441 \left (3-x+2 x^2\right )^{5/2}}{13934592 (5+2 x)^5}+\frac{5}{64} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx-\frac{412760561351 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{880602513408}\\ &=-\frac{(146583836191+101679102454 x) \sqrt{3-x+2 x^2}}{440301256704 (5+2 x)^2}-\frac{(463558457+411822458 x) \left (3-x+2 x^2\right )^{3/2}}{2293235712 (5+2 x)^4}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{4032 (5+2 x)^7}+\frac{114335 \left (3-x+2 x^2\right )^{5/2}}{193536 (5+2 x)^6}-\frac{1930441 \left (3-x+2 x^2\right )^{5/2}}{13934592 (5+2 x)^5}+\frac{412760561351 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{440301256704}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{64 \sqrt{46}}\\ &=-\frac{(146583836191+101679102454 x) \sqrt{3-x+2 x^2}}{440301256704 (5+2 x)^2}-\frac{(463558457+411822458 x) \left (3-x+2 x^2\right )^{3/2}}{2293235712 (5+2 x)^4}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{4032 (5+2 x)^7}+\frac{114335 \left (3-x+2 x^2\right )^{5/2}}{193536 (5+2 x)^6}-\frac{1930441 \left (3-x+2 x^2\right )^{5/2}}{13934592 (5+2 x)^5}-\frac{5 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{2}}+\frac{412760561351 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{5283615080448 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.272285, size = 108, normalized size = 0.55 $\frac{-\frac{24 \sqrt{2 x^2-x+3} \left (38463671680832 x^6+402255822731712 x^5+2069947287085104 x^4+5966329646300704 x^3+9976065367498188 x^2+9065154700300572 x+3479517268702637\right )}{(2 x+5)^7}+2889323929457 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )-2889476997120 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{73970611126272}$

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)^8,x]

[Out]

((-24*Sqrt[3 - x + 2*x^2]*(3479517268702637 + 9065154700300572*x + 9976065367498188*x^2 + 5966329646300704*x^3
+ 2069947287085104*x^4 + 402255822731712*x^5 + 38463671680832*x^6))/(5 + 2*x)^7 - 2889476997120*Sqrt[2]*ArcSi
nh[(1 - 4*x)/Sqrt[23]] + 2889323929457*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/73970611126272

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Maple [A]  time = 0.078, size = 267, normalized size = 1.4 \begin{align*} -{\frac{1930441}{445906944} \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{-17957520133+71830080532\,x}{31701690482688}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{769352975}{23776267862016} \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{412760561351\,\sqrt{2}}{10567230160896}{\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{114335}{12386304} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-6}}+{\frac{5\,\sqrt{2}}{128}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{412760561351}{1711891286065152} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}-{\frac{412760561351}{31701690482688}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{32967491}{330225942528} \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{3667}{516096} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-7}}+{\frac{-27452157541+109808630164\,x}{1711891286065152} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}-{\frac{27452157541}{855945643032576} \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{7861079}{9172942848} \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)^8,x)

[Out]

-1930441/445906944/(x+5/2)^5*(2*(x+5/2)^2-11*x-19/2)^(5/2)+17957520133/31701690482688*(-1+4*x)*(2*(x+5/2)^2-11
*x-19/2)^(1/2)+769352975/23776267862016/(x+5/2)^2*(2*(x+5/2)^2-11*x-19/2)^(5/2)+412760561351/10567230160896*2^
(1/2)*arctanh(1/12*(17/2-11*x)*2^(1/2)/(2*(x+5/2)^2-11*x-19/2)^(1/2))+114335/12386304/(x+5/2)^6*(2*(x+5/2)^2-1
1*x-19/2)^(5/2)+5/128*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))-412760561351/1711891286065152*(2*(x+5/2)^2-11*x-1
9/2)^(3/2)-412760561351/31701690482688*(2*(x+5/2)^2-11*x-19/2)^(1/2)-32967491/330225942528/(x+5/2)^3*(2*(x+5/2
)^2-11*x-19/2)^(5/2)-3667/516096/(x+5/2)^7*(2*(x+5/2)^2-11*x-19/2)^(5/2)+27452157541/1711891286065152*(-1+4*x)
*(2*(x+5/2)^2-11*x-19/2)^(3/2)-27452157541/855945643032576/(x+5/2)*(2*(x+5/2)^2-11*x-19/2)^(5/2)+7861079/91729
42848/(x+5/2)^4*(2*(x+5/2)^2-11*x-19/2)^(5/2)

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Maxima [B]  time = 1.61073, size = 470, normalized size = 2.41 \begin{align*} -\frac{769352975}{11888133931008} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{4032 \,{\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\right )}} + \frac{114335 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{193536 \,{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} - \frac{1930441 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{13934592 \,{\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} + \frac{7861079 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{573308928 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} - \frac{32967491 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{41278242816 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} + \frac{769352975 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{5944066965504 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac{17957520133}{7925422620672} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{5}{128} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) - \frac{412760561351}{10567230160896} \, \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{35893173457}{2641807540224} \, \sqrt{2 \, x^{2} - x + 3} - \frac{27452157541 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{23776267862016 \,{\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)^8,x, algorithm="maxima")

[Out]

-769352975/11888133931008*(2*x^2 - x + 3)^(3/2) - 3667/4032*(2*x^2 - x + 3)^(5/2)/(128*x^7 + 2240*x^6 + 16800*
x^5 + 70000*x^4 + 175000*x^3 + 262500*x^2 + 218750*x + 78125) + 114335/193536*(2*x^2 - x + 3)^(5/2)/(64*x^6 +
960*x^5 + 6000*x^4 + 20000*x^3 + 37500*x^2 + 37500*x + 15625) - 1930441/13934592*(2*x^2 - x + 3)^(5/2)/(32*x^5
+ 400*x^4 + 2000*x^3 + 5000*x^2 + 6250*x + 3125) + 7861079/573308928*(2*x^2 - x + 3)^(5/2)/(16*x^4 + 160*x^3
+ 600*x^2 + 1000*x + 625) - 32967491/41278242816*(2*x^2 - x + 3)^(5/2)/(8*x^3 + 60*x^2 + 150*x + 125) + 769352
975/5944066965504*(2*x^2 - x + 3)^(5/2)/(4*x^2 + 20*x + 25) + 17957520133/7925422620672*sqrt(2*x^2 - x + 3)*x
+ 5/128*sqrt(2)*arcsinh(4/23*sqrt(23)*x - 1/23*sqrt(23)) - 412760561351/10567230160896*sqrt(2)*arcsinh(22/23*s
qrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) - 35893173457/2641807540224*sqrt(2*x^2 - x + 3) - 274521
57541/23776267862016*(2*x^2 - x + 3)^(3/2)/(2*x + 5)

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Fricas [A]  time = 1.53481, size = 915, normalized size = 4.69 \begin{align*} \frac{2889476997120 \, \sqrt{2}{\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\right )} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 2889323929457 \, \sqrt{2}{\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\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 (38463671680832 \, x^{6} + 402255822731712 \, x^{5} + 2069947287085104 \, x^{4} + 5966329646300704 \, x^{3} + 9976065367498188 \, x^{2} + 9065154700300572 \, x + 3479517268702637\right )} \sqrt{2 \, x^{2} - x + 3}}{147941222252544 \,{\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\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)^8,x, algorithm="fricas")

[Out]

1/147941222252544*(2889476997120*sqrt(2)*(128*x^7 + 2240*x^6 + 16800*x^5 + 70000*x^4 + 175000*x^3 + 262500*x^2
+ 218750*x + 78125)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 2889323929457*sqrt(2
)*(128*x^7 + 2240*x^6 + 16800*x^5 + 70000*x^4 + 175000*x^3 + 262500*x^2 + 218750*x + 78125)*log((24*sqrt(2)*sq
rt(2*x^2 - x + 3)*(22*x - 17) - 1060*x^2 + 1036*x - 1153)/(4*x^2 + 20*x + 25)) - 48*(38463671680832*x^6 + 4022
55822731712*x^5 + 2069947287085104*x^4 + 5966329646300704*x^3 + 9976065367498188*x^2 + 9065154700300572*x + 34
79517268702637)*sqrt(2*x^2 - x + 3))/(128*x^7 + 2240*x^6 + 16800*x^5 + 70000*x^4 + 175000*x^3 + 262500*x^2 + 2
18750*x + 78125)

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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 )^{8}}\, 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)**8,x)

[Out]

Integral((2*x**2 - x + 3)**(3/2)*(5*x**4 - x**3 + 3*x**2 + x + 2)/(2*x + 5)**8, x)

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Giac [B]  time = 1.26089, size = 660, normalized size = 3.38 \begin{align*} -\frac{5}{128} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{412760561351}{10567230160896} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{412760561351}{10567230160896} \, \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 (1121897398412224 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{13} + 48260296303776704 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{12} + 444673458321712704 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{11} + 3996455936659982656 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{10} + 6725227967167489360 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{9} - 17132661028483948080 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{8} - 63713012094737246112 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{7} + 106515880136064432096 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{6} + 226947197958946260516 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{5} - 856601202771483308188 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{4} + 617998258357377713732 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} - 467121785339763351756 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 92292080735560562227 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 15161716093827501349\right )}}{6164217593856 \,{\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 )}^{7}} \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)^8,x, algorithm="giac")

[Out]

-5/128*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 412760561351/10567230160896*sqrt(2)*log
(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 412760561351/10567230160896*sqrt(2)*log(abs(-2*sqrt(2)
*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 1/6164217593856*sqrt(2)*(1121897398412224*sqrt(2)*(sqrt(2)*x - sqr
t(2*x^2 - x + 3))^13 + 48260296303776704*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^12 + 444673458321712704*sqrt(2)*(sq
rt(2)*x - sqrt(2*x^2 - x + 3))^11 + 3996455936659982656*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^10 + 672522796716748
9360*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^9 - 17132661028483948080*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^8 -
63713012094737246112*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^7 + 106515880136064432096*(sqrt(2)*x - sqrt(2*x
^2 - x + 3))^6 + 226947197958946260516*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^5 - 856601202771483308188*(sq
rt(2)*x - sqrt(2*x^2 - x + 3))^4 + 617998258357377713732*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^3 - 4671217
85339763351756*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 92292080735560562227*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x
+ 3)) - 15161716093827501349)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 10*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x
+ 3)) - 11)^7