3.4.43 \(\int \frac {(3-x+2 x^2)^{3/2} (2+x+3 x^2-x^3+5 x^4)}{(5+2 x)^8} \, dx\) [343]

3.4.43.1 Optimal result

Integrand size = 40, antiderivative size = 195 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^8} \, dx=-\frac {(146583836191+101679102454 x) \sqrt {3-x+2 x^2}}{440301256704 (5+2 x)^2}-\frac {(463558457+411822458 x) \left (3-x+2 x^2\right )^{3/2}}{2293235712 (5+2 x)^4}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{4032 (5+2 x)^7}+\frac {114335 \left (3-x+2 x^2\right )^{5/2}}{193536 (5+2 x)^6}-\frac {1930441 \left (3-x+2 x^2\right )^{5/2}}{13934592 (5+2 x)^5}-\frac {5 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{64 \sqrt {2}}+\frac {412760561351 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{5283615080448 \sqrt {2}} \]

output

-1/2293235712*(463558457+411822458*x)*(2*x^2-x+3)^(3/2)/(5+2*x)^4-3667/403 
2*(2*x^2-x+3)^(5/2)/(5+2*x)^7+114335/193536*(2*x^2-x+3)^(5/2)/(5+2*x)^6-19 
30441/13934592*(2*x^2-x+3)^(5/2)/(5+2*x)^5-5/128*arcsinh(1/23*(1-4*x)*23^( 
1/2))*2^(1/2)[phone]/10567230160896*arctanh(1/24*(17-22*x)*2^(1/2)/( 
2*x^2-x+3)^(1/2))*2^(1/2)-1/440301256704*(146583836191+101679102454*x)*(2* 
x^2-x+3)^(1/2)/(5+2*x)^2
 

3.4.43.2 Mathematica [A] (verified)

Time = 1.19 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.62 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^8} \, dx=\frac {-\frac {12 \sqrt {3-x+2 x^2} \left (3479517268702637+9065154700300572 x+9976065367498188 x^2+5966329646300704 x^3+2069947287085104 x^4+402255822731712 x^5+38463671680832 x^6\right )}{(5+2 x)^7}-2889323929457 \sqrt {2} \text {arctanh}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )-1444738498560 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{36985305563136} \]

input

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

output

((-12*Sqrt[3 - x + 2*x^2]*(3479517268702637 + 9065154700300572*x + 9976065 
367498188*x^2 + 5966329646300704*x^3 + 2069947287085104*x^4 + 402255822731 
712*x^5 + 38463671680832*x^6))/(5 + 2*x)^7 - 2889323929457*Sqrt[2]*ArcTanh 
[(5 + 2*x - Sqrt[6 - 2*x + 4*x^2])/6] - 1444738498560*Sqrt[2]*Log[1 - 4*x 
+ 2*Sqrt[6 - 2*x + 4*x^2]])/36985305563136
 

3.4.43.3 Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.12, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2181, 27, 2181, 27, 2181, 27, 1229, 27, 1229, 27, 1269, 1090, 222, 1154, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (2 x^2-x+3\right )^{3/2} \left (5 x^4-x^3+3 x^2+x+2\right )}{(2 x+5)^8} \, dx\)

\(\Big \downarrow \) 2181

\(\displaystyle -\frac {1}{504} \int \frac {\left (2 x^2-x+3\right )^{3/2} \left (-20160 x^3+54432 x^2-118840 x+76715\right )}{16 (2 x+5)^7}dx-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\left (2 x^2-x+3\right )^{3/2} \left (-20160 x^3+54432 x^2-118840 x+76715\right )}{(2 x+5)^7}dx}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}\)

\(\Big \downarrow \) 2181

\(\displaystyle \frac {\frac {1}{432} \int \frac {9 \left (2 x^2-x+3\right )^{3/2} \left (483840 x^2-2058628 x+1481635\right )}{(2 x+5)^6}dx+\frac {114335 \left (2 x^2-x+3\right )^{5/2}}{24 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{48} \int \frac {\left (2 x^2-x+3\right )^{3/2} \left (483840 x^2-2058628 x+1481635\right )}{(2 x+5)^6}dx+\frac {114335 \left (2 x^2-x+3\right )^{5/2}}{24 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}\)

\(\Big \downarrow \) 2181

\(\displaystyle \frac {\frac {1}{48} \left (-\frac {1}{360} \int \frac {35 (1640279-2488320 x) \left (2 x^2-x+3\right )^{3/2}}{(2 x+5)^5}dx-\frac {1930441 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}\right )+\frac {114335 \left (2 x^2-x+3\right )^{5/2}}{24 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{48} \left (-\frac {7}{72} \int \frac {(1640279-2488320 x) \left (2 x^2-x+3\right )^{3/2}}{(2 x+5)^5}dx-\frac {1930441 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}\right )+\frac {114335 \left (2 x^2-x+3\right )^{5/2}}{24 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {\frac {1}{48} \left (-\frac {7}{72} \left (\frac {(411822458 x+463558457) \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)^4}-\frac {\int -\frac {6 (300244177-477757440 x) \sqrt {2 x^2-x+3}}{(2 x+5)^3}dx}{2304}\right )-\frac {1930441 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}\right )+\frac {114335 \left (2 x^2-x+3\right )^{5/2}}{24 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{48} \left (-\frac {7}{72} \left (\frac {1}{384} \int \frac {(300244177-477757440 x) \sqrt {2 x^2-x+3}}{(2 x+5)^3}dx+\frac {(411822458 x+463558457) \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)^4}\right )-\frac {1930441 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}\right )+\frac {114335 \left (2 x^2-x+3\right )^{5/2}}{24 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {\frac {1}{48} \left (-\frac {7}{72} \left (\frac {1}{384} \left (\frac {(101679102454 x+146583836191) \sqrt {2 x^2-x+3}}{288 (2 x+5)^2}-\frac {\int -\frac {2 (68775204551-137594142720 x)}{(2 x+5) \sqrt {2 x^2-x+3}}dx}{1152}\right )+\frac {(411822458 x+463558457) \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)^4}\right )-\frac {1930441 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}\right )+\frac {114335 \left (2 x^2-x+3\right )^{5/2}}{24 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{48} \left (-\frac {7}{72} \left (\frac {1}{384} \left (\frac {1}{576} \int \frac {68775204551-137594142720 x}{(2 x+5) \sqrt {2 x^2-x+3}}dx+\frac {\sqrt {2 x^2-x+3} (101679102454 x+146583836191)}{288 (2 x+5)^2}\right )+\frac {(411822458 x+463558457) \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)^4}\right )-\frac {1930441 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}\right )+\frac {114335 \left (2 x^2-x+3\right )^{5/2}}{24 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {1}{48} \left (-\frac {7}{72} \left (\frac {1}{384} \left (\frac {1}{576} \left (412760561351 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-68797071360 \int \frac {1}{\sqrt {2 x^2-x+3}}dx\right )+\frac {\sqrt {2 x^2-x+3} (101679102454 x+146583836191)}{288 (2 x+5)^2}\right )+\frac {(411822458 x+463558457) \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)^4}\right )-\frac {1930441 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}\right )+\frac {114335 \left (2 x^2-x+3\right )^{5/2}}{24 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {\frac {1}{48} \left (-\frac {7}{72} \left (\frac {1}{384} \left (\frac {1}{576} \left (412760561351 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-34398535680 \sqrt {\frac {2}{23}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)\right )+\frac {\sqrt {2 x^2-x+3} (101679102454 x+146583836191)}{288 (2 x+5)^2}\right )+\frac {(411822458 x+463558457) \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)^4}\right )-\frac {1930441 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}\right )+\frac {114335 \left (2 x^2-x+3\right )^{5/2}}{24 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {\frac {1}{48} \left (-\frac {7}{72} \left (\frac {1}{384} \left (\frac {1}{576} \left (412760561351 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-34398535680 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+\frac {\sqrt {2 x^2-x+3} (101679102454 x+146583836191)}{288 (2 x+5)^2}\right )+\frac {(411822458 x+463558457) \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)^4}\right )-\frac {1930441 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}\right )+\frac {114335 \left (2 x^2-x+3\right )^{5/2}}{24 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {1}{48} \left (-\frac {7}{72} \left (\frac {1}{384} \left (\frac {1}{576} \left (-825521122702 \int \frac {1}{288-\frac {(17-22 x)^2}{2 x^2-x+3}}d\frac {17-22 x}{\sqrt {2 x^2-x+3}}-34398535680 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+\frac {\sqrt {2 x^2-x+3} (101679102454 x+146583836191)}{288 (2 x+5)^2}\right )+\frac {(411822458 x+463558457) \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)^4}\right )-\frac {1930441 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}\right )+\frac {114335 \left (2 x^2-x+3\right )^{5/2}}{24 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{48} \left (-\frac {7}{72} \left (\frac {1}{384} \left (\frac {1}{576} \left (-34398535680 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )-\frac {412760561351 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{6 \sqrt {2}}\right )+\frac {\sqrt {2 x^2-x+3} (101679102454 x+146583836191)}{288 (2 x+5)^2}\right )+\frac {(411822458 x+463558457) \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)^4}\right )-\frac {1930441 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}\right )+\frac {114335 \left (2 x^2-x+3\right )^{5/2}}{24 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}\)

input

Int[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^8,x]
 

output

(-3667*(3 - x + 2*x^2)^(5/2))/(4032*(5 + 2*x)^7) + ((114335*(3 - x + 2*x^2 
)^(5/2))/(24*(5 + 2*x)^6) + ((-1930441*(3 - x + 2*x^2)^(5/2))/(36*(5 + 2*x 
)^5) - (7*(((463558457 + 411822458*x)*(3 - x + 2*x^2)^(3/2))/(576*(5 + 2*x 
)^4) + (((146583836191 + 101679102454*x)*Sqrt[3 - x + 2*x^2])/(288*(5 + 2* 
x)^2) + (-34398535680*Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[23]] - (412760561351 
*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(6*Sqrt[2]))/576)/ 
384))/72)/48)/8064
 

3.4.43.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 222

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 1090

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* 
(c/(b^2 - 4*a*c)))^p)   Subst[Int[Simp[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 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-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]
 

rule 1229

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 
)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((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), x] - Simp[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] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(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] &&  !IGtQ[m, 0]
 

rule 2181

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi 
alRemainder[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] + Simp[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)*Qx + 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]
 

3.4.43.4 Maple [F(-1)]

Timed out.

hanged

input

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

output

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

3.4.43.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.25 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^8} \, dx=\frac {2889476997120 \, \sqrt {2} {\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 2889323929457 \, \sqrt {2} {\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\right )} \log \left (\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) - 48 \, {\left (38463671680832 \, x^{6} + 402255822731712 \, x^{5} + 2069947287085104 \, x^{4} + 5966329646300704 \, x^{3} + 9976065367498188 \, x^{2} + 9065154700300572 \, x + 3479517268702637\right )} \sqrt {2 \, x^{2} - x + 3}}{147941222252544 \, {\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\right )}} \]

input

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")
 

output

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)*sq 
rt(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 + 21 
8750*x + 78125)*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*(38463671680832*x^6 + 40225582 
2731712*x^5 + 2069947287085104*x^4 + 5966329646300704*x^3 + 99760653674981 
88*x^2 + 9065154700300572*x + 3479517268702637)*sqrt(2*x^2 - x + 3))/(128* 
x^7 + 2240*x^6 + 16800*x^5 + 70000*x^4 + 175000*x^3 + 262500*x^2 + 218750* 
x + 78125)
 

3.4.43.6 Sympy [F]

\[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^8} \, dx=\int \frac {\left (2 x^{2} - x + 3\right )^{\frac {3}{2}} \cdot \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{8}}\, dx \]

input

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

output

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

3.4.43.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (160) = 320\).

Time = 0.34 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.78 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^8} \, dx=-\frac {769352975}{11888133931008} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {3667 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{4032 \, {\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\right )}} + \frac {114335 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{193536 \, {\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} - \frac {1930441 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{13934592 \, {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} + \frac {7861079 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{573308928 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} - \frac {32967491 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{41278242816 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} + \frac {769352975 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{5944066965504 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac {17957520133}{7925422620672} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {5}{128} \, \sqrt {2} \operatorname {arsinh}\left (\frac {4}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) - \frac {412760561351}{10567230160896} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) - \frac {35893173457}{2641807540224} \, \sqrt {2 \, x^{2} - x + 3} - \frac {27452157541 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{23776267862016 \, {\left (2 \, x + 5\right )}} \]

input

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")
 

output

-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 + 96 
0*x^5 + 6000*x^4 + 20000*x^3 + 37500*x^2 + 37500*x + 15625) - 1930441/1393 
4592*(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 + 60 
0*x^2 + 1000*x + 625) - 32967491/41278242816*(2*x^2 - x + 3)^(5/2)/(8*x^3 
+ 60*x^2 + 150*x + 125) + 769352975/5944066965504*(2*x^2 - x + 3)^(5/2)/(4 
*x^2 + 20*x + 25) + 17957520133/7925422620672*sqrt(2*x^2 - x + 3)*x + 5/12 
8*sqrt(2)*arcsinh(4/23*sqrt(23)*x - 1/23*sqrt(23)) - 412760561351/10567230 
160896*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs( 
2*x + 5)) - 35893173457/2641807540224*sqrt(2*x^2 - x + 3) - 27452157541/23 
776267862016*(2*x^2 - x + 3)^(3/2)/(2*x + 5)
 

3.4.43.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (160) = 320\).

Time = 0.31 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.51 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^8} \, dx=-\frac {5}{128} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac {412760561351}{10567230160896} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {412760561351}{10567230160896} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {\sqrt {2} {\left (1121897398412224 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{13} + 48260296303776704 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{12} + 444673458321712704 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{11} + 3996455936659982656 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{10} + 6725227967167489360 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{9} - 17132661028483948080 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{8} - 63713012094737246112 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{7} + 106515880136064432096 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{6} + 226947197958946260516 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{5} - 856601202771483308188 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{4} + 617998258357377713732 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} - 467121785339763351756 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 92292080735560562227 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 15161716093827501349\right )}}{6164217593856 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{7}} \]

input

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")
 

output

-5/128*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 412 
760561351/10567230160896*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)*(1121897 
398412224*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^13 + 48260296303776704 
*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^12 + 444673458321712704*sqrt(2)*(sqrt(2 
)*x - sqrt(2*x^2 - x + 3))^11 + 3996455936659982656*(sqrt(2)*x - sqrt(2*x^ 
2 - x + 3))^10 + 6725227967167489360*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 
 3))^9 - 17132661028483948080*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^8 - 637130 
12094737246112*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^7 + 1065158801360 
64432096*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^6 + 226947197958946260516*sqrt( 
2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^5 - 856601202771483308188*(sqrt(2)*x 
- sqrt(2*x^2 - x + 3))^4 + 617998258357377713732*sqrt(2)*(sqrt(2)*x - sqrt 
(2*x^2 - x + 3))^3 - 467121785339763351756*(sqrt(2)*x - sqrt(2*x^2 - x + 3 
))^2 + 92292080735560562227*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 15 
161716093827501349)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 10*sqrt(2)*(s 
qrt(2)*x - sqrt(2*x^2 - x + 3)) - 11)^7
 

3.4.43.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^8} \, dx=\int \frac {{\left (2\,x^2-x+3\right )}^{3/2}\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{{\left (2\,x+5\right )}^8} \,d x \]

input

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

output

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