Integrand size = 40, antiderivative size = 139 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^5 \sqrt {3-x+2 x^2}} \, dx=-\frac {3667 \sqrt {3-x+2 x^2}}{2304 (5+2 x)^4}+\frac {513097 \sqrt {3-x+2 x^2}}{497664 (5+2 x)^3}-\frac {16295969 \sqrt {3-x+2 x^2}}{71663616 (5+2 x)^2}+\frac {26800085 \sqrt {3-x+2 x^2}}{1719926784 (5+2 x)}+\frac {2053207 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{20639121408 \sqrt {2}} \]
2053207/41278242816*arctanh(1/24*(17-22*x)*2^(1/2)/(2*x^2-x+3)^(1/2))*2^(1 /2)-3667/2304*(2*x^2-x+3)^(1/2)/(5+2*x)^4+513097/497664*(2*x^2-x+3)^(1/2)/ (5+2*x)^3-16295969/71663616*(2*x^2-x+3)^(1/2)/(5+2*x)^2+26800085/171992678 4*(2*x^2-x+3)^(1/2)/(5+2*x)
Time = 0.59 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.55 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^5 \sqrt {3-x+2 x^2}} \, dx=\frac {\frac {12 \sqrt {3-x+2 x^2} \left (-298655447-255525906 x+43592076 x^2+214400680 x^3\right )}{(5+2 x)^4}-2053207 \sqrt {2} \text {arctanh}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )}{20639121408} \]
((12*Sqrt[3 - x + 2*x^2]*(-298655447 - 255525906*x + 43592076*x^2 + 214400 680*x^3))/(5 + 2*x)^4 - 2053207*Sqrt[2]*ArcTanh[(5 + 2*x - Sqrt[6 - 2*x + 4*x^2])/6])/20639121408
Time = 0.47 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2181, 27, 2181, 2181, 27, 1228, 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.
\(\displaystyle \int \frac {5 x^4-x^3+3 x^2+x+2}{(2 x+5)^5 \sqrt {2 x^2-x+3}} \, dx\) |
\(\Big \downarrow \) 2181 |
\(\displaystyle -\frac {1}{288} \int \frac {-11520 x^3+31104 x^2-40668 x+37027}{16 (2 x+5)^4 \sqrt {2 x^2-x+3}}dx-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {-11520 x^3+31104 x^2-40668 x+37027}{(2 x+5)^4 \sqrt {2 x^2-x+3}}dx}{4608}-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}\) |
\(\Big \downarrow \) 2181 |
\(\displaystyle \frac {\frac {1}{216} \int \frac {1244160 x^2-2364856 x+2607829}{(2 x+5)^3 \sqrt {2 x^2-x+3}}dx+\frac {513097 \sqrt {2 x^2-x+3}}{108 (2 x+5)^3}}{4608}-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}\) |
\(\Big \downarrow \) 2181 |
\(\displaystyle \frac {\frac {1}{216} \left (-\frac {1}{144} \int \frac {13 (1493165-1876588 x)}{(2 x+5)^2 \sqrt {2 x^2-x+3}}dx-\frac {16295969 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}\right )+\frac {513097 \sqrt {2 x^2-x+3}}{108 (2 x+5)^3}}{4608}-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{216} \left (-\frac {13}{144} \int \frac {1493165-1876588 x}{(2 x+5)^2 \sqrt {2 x^2-x+3}}dx-\frac {16295969 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}\right )+\frac {513097 \sqrt {2 x^2-x+3}}{108 (2 x+5)^3}}{4608}-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}\) |
\(\Big \downarrow \) 1228 |
\(\displaystyle \frac {\frac {1}{216} \left (-\frac {13}{144} \left (\frac {157939}{24} \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-\frac {2061545 \sqrt {2 x^2-x+3}}{12 (2 x+5)}\right )-\frac {16295969 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}\right )+\frac {513097 \sqrt {2 x^2-x+3}}{108 (2 x+5)^3}}{4608}-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {\frac {1}{216} \left (-\frac {13}{144} \left (-\frac {157939}{12} \int \frac {1}{288-\frac {(17-22 x)^2}{2 x^2-x+3}}d\frac {17-22 x}{\sqrt {2 x^2-x+3}}-\frac {2061545 \sqrt {2 x^2-x+3}}{12 (2 x+5)}\right )-\frac {16295969 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}\right )+\frac {513097 \sqrt {2 x^2-x+3}}{108 (2 x+5)^3}}{4608}-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {1}{216} \left (-\frac {13}{144} \left (-\frac {157939 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{144 \sqrt {2}}-\frac {2061545 \sqrt {2 x^2-x+3}}{12 (2 x+5)}\right )-\frac {16295969 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}\right )+\frac {513097 \sqrt {2 x^2-x+3}}{108 (2 x+5)^3}}{4608}-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}\) |
(-3667*Sqrt[3 - x + 2*x^2])/(2304*(5 + 2*x)^4) + ((513097*Sqrt[3 - x + 2*x ^2])/(108*(5 + 2*x)^3) + ((-16295969*Sqrt[3 - x + 2*x^2])/(72*(5 + 2*x)^2) - (13*((-2061545*Sqrt[3 - x + 2*x^2])/(12*(5 + 2*x)) - (157939*ArcTanh[(1 7 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(144*Sqrt[2])))/144)/216)/460 8
3.4.50.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(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] - Simp[(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 ] && EqQ[Simplify[m + 2*p + 3], 0]
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]
Timed out.
hanged
Time = 0.25 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.90 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^5 \sqrt {3-x+2 x^2}} \, dx=\frac {2053207 \, \sqrt {2} {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )} \log \left (\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (214400680 \, x^{3} + 43592076 \, x^{2} - 255525906 \, x - 298655447\right )} \sqrt {2 \, x^{2} - x + 3}}{82556485632 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} \]
1/82556485632*(2053207*sqrt(2)*(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625) *log((24*sqrt(2)*sqrt(2*x^2 - x + 3)*(22*x - 17) - 1060*x^2 + 1036*x - 115 3)/(4*x^2 + 20*x + 25)) + 48*(214400680*x^3 + 43592076*x^2 - 255525906*x - 298655447)*sqrt(2*x^2 - x + 3))/(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 62 5)
\[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^5 \sqrt {3-x+2 x^2}} \, dx=\int \frac {5 x^{4} - x^{3} + 3 x^{2} + x + 2}{\left (2 x + 5\right )^{5} \sqrt {2 x^{2} - x + 3}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.07 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^5 \sqrt {3-x+2 x^2}} \, dx=-\frac {2053207}{41278242816} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) - \frac {3667 \, \sqrt {2 \, x^{2} - x + 3}}{2304 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac {513097 \, \sqrt {2 \, x^{2} - x + 3}}{497664 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac {16295969 \, \sqrt {2 \, x^{2} - x + 3}}{71663616 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac {26800085 \, \sqrt {2 \, x^{2} - x + 3}}{1719926784 \, {\left (2 \, x + 5\right )}} \]
-2053207/41278242816*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23 *sqrt(23)/abs(2*x + 5)) - 3667/2304*sqrt(2*x^2 - x + 3)/(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625) + 513097/497664*sqrt(2*x^2 - x + 3)/(8*x^3 + 60* x^2 + 150*x + 125) - 16295969/71663616*sqrt(2*x^2 - x + 3)/(4*x^2 + 20*x + 25) + 26800085/1719926784*sqrt(2*x^2 - x + 3)/(2*x + 5)
Time = 0.31 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.18 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^5 \sqrt {3-x+2 x^2}} \, dx=\frac {1}{41278242816} \, \sqrt {2} {\left (12 \, {\left (\frac {24 \, {\left (\frac {144 \, {\left (\frac {513097}{\mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )} - \frac {792072}{{\left (2 \, x + 5\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )}\right )}}{2 \, x + 5} - \frac {16295969}{\mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )}\right )}}{2 \, x + 5} + \frac {26800085}{\mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )}\right )} \sqrt {-\frac {11}{2 \, x + 5} + \frac {36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac {2053207 \, \log \left (12 \, \sqrt {-\frac {11}{2 \, x + 5} + \frac {36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac {72}{2 \, x + 5} - 11\right )}{\mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )} - 321601020 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )\right )} \]
1/41278242816*sqrt(2)*(12*(24*(144*(513097/sgn(1/(2*x + 5)) - 792072/((2*x + 5)*sgn(1/(2*x + 5))))/(2*x + 5) - 16295969/sgn(1/(2*x + 5)))/(2*x + 5) + 26800085/sgn(1/(2*x + 5)))*sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 20 53207*log(12*sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 72/(2*x + 5) - 11) /sgn(1/(2*x + 5)) - 321601020*sgn(1/(2*x + 5)))
Timed out. \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^5 \sqrt {3-x+2 x^2}} \, dx=\int \frac {5\,x^4-x^3+3\,x^2+x+2}{{\left (2\,x+5\right )}^5\,\sqrt {2\,x^2-x+3}} \,d x \]