Integrand size = 40, antiderivative size = 135 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {65991-8779 x}{12877056 \left (3-x+2 x^2\right )^{3/2}}-\frac {4679797-2148263 x}{592344576 \sqrt {3-x+2 x^2}}-\frac {3667 \sqrt {3-x+2 x^2}}{373248 (5+2 x)^2}-\frac {45979 \sqrt {3-x+2 x^2}}{26873856 (5+2 x)}+\frac {774079 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{322486272 \sqrt {2}} \]
1/12877056*(65991-8779*x)/(2*x^2-x+3)^(3/2)+774079/644972544*arctanh(1/24* (17-22*x)*2^(1/2)/(2*x^2-x+3)^(1/2))*2^(1/2)+1/592344576*(-4679797+2148263 *x)/(2*x^2-x+3)^(1/2)-3667/373248*(2*x^2-x+3)^(1/2)/(5+2*x)^2-45979/268738 56*(2*x^2-x+3)^(1/2)/(5+2*x)
Time = 0.62 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.70 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {\frac {12 \sqrt {3-x+2 x^2} \left (-8953831359+2280511668 x-5919924791 x^2-1503926130 x^3+107028732 x^4+217883368 x^5\right )}{\left (15+x+8 x^2+4 x^3\right )^2}-409487791 \sqrt {2} \text {arctanh}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )}{170595237888} \]
((12*Sqrt[3 - x + 2*x^2]*(-8953831359 + 2280511668*x - 5919924791*x^2 - 15 03926130*x^3 + 107028732*x^4 + 217883368*x^5))/(15 + x + 8*x^2 + 4*x^3)^2 - 409487791*Sqrt[2]*ArcTanh[(5 + 2*x - Sqrt[6 - 2*x + 4*x^2])/6])/17059523 7888
Time = 0.49 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {2177, 27, 2177, 27, 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.
| \(\displaystyle \int \frac {5 x^4-x^3+3 x^2+x+2}{(2 x+5)^3 \left (2 x^2-x+3\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2177 |
| \(\displaystyle \frac {2}{69} \int \frac {-280928 x^3+65340540 x^2+38386140 x+11115283}{746496 (2 x+5)^3 \left (2 x^2-x+3\right )^{3/2}}dx+\frac {65991-8779 x}{12877056 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-280928 x^3+65340540 x^2+38386140 x+11115283}{(2 x+5)^3 \left (2 x^2-x+3\right )^{3/2}}dx}{25754112}+\frac {65991-8779 x}{12877056 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 2177 |
| \(\displaystyle \frac {\frac {2}{23} \int -\frac {529 \left (125300 x^2+1076692 x+324461\right )}{4 (2 x+5)^3 \sqrt {2 x^2-x+3}}dx-\frac {4679797-2148263 x}{23 \sqrt {2 x^2-x+3}}}{25754112}+\frac {65991-8779 x}{12877056 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {23}{2} \int \frac {125300 x^2+1076692 x+324461}{(2 x+5)^3 \sqrt {2 x^2-x+3}}dx-\frac {4679797-2148263 x}{23 \sqrt {2 x^2-x+3}}}{25754112}+\frac {65991-8779 x}{12877056 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {-\frac {23}{2} \left (\frac {22002 \sqrt {2 x^2-x+3}}{(2 x+5)^2}-\frac {1}{144} \int -\frac {288 (53327 x+64349)}{(2 x+5)^2 \sqrt {2 x^2-x+3}}dx\right )-\frac {4679797-2148263 x}{23 \sqrt {2 x^2-x+3}}}{25754112}+\frac {65991-8779 x}{12877056 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {23}{2} \left (2 \int \frac {53327 x+64349}{(2 x+5)^2 \sqrt {2 x^2-x+3}}dx+\frac {22002 \sqrt {2 x^2-x+3}}{(2 x+5)^2}\right )-\frac {4679797-2148263 x}{23 \sqrt {2 x^2-x+3}}}{25754112}+\frac {65991-8779 x}{12877056 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {-\frac {23}{2} \left (2 \left (\frac {774079}{48} \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx+\frac {45979 \sqrt {2 x^2-x+3}}{24 (2 x+5)}\right )+\frac {22002 \sqrt {2 x^2-x+3}}{(2 x+5)^2}\right )-\frac {4679797-2148263 x}{23 \sqrt {2 x^2-x+3}}}{25754112}+\frac {65991-8779 x}{12877056 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {-\frac {23}{2} \left (2 \left (\frac {45979 \sqrt {2 x^2-x+3}}{24 (2 x+5)}-\frac {774079}{24} \int \frac {1}{288-\frac {(17-22 x)^2}{2 x^2-x+3}}d\frac {17-22 x}{\sqrt {2 x^2-x+3}}\right )+\frac {22002 \sqrt {2 x^2-x+3}}{(2 x+5)^2}\right )-\frac {4679797-2148263 x}{23 \sqrt {2 x^2-x+3}}}{25754112}+\frac {65991-8779 x}{12877056 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {23}{2} \left (2 \left (\frac {45979 \sqrt {2 x^2-x+3}}{24 (2 x+5)}-\frac {774079 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{288 \sqrt {2}}\right )+\frac {22002 \sqrt {2 x^2-x+3}}{(2 x+5)^2}\right )-\frac {4679797-2148263 x}{23 \sqrt {2 x^2-x+3}}}{25754112}+\frac {65991-8779 x}{12877056 \left (2 x^2-x+3\right )^{3/2}}\) |
(65991 - 8779*x)/(12877056*(3 - x + 2*x^2)^(3/2)) + (-1/23*(4679797 - 2148 263*x)/Sqrt[3 - x + 2*x^2] - (23*((22002*Sqrt[3 - x + 2*x^2])/(5 + 2*x)^2 + 2*((45979*Sqrt[3 - x + 2*x^2])/(24*(5 + 2*x)) - (774079*ArcTanh[(17 - 22 *x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(288*Sqrt[2]))))/2)/25754112
3.4.63.3.1 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[(d + e*x)^m*Pq, a + b*x + c* x^2, x], R = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*R - 2*a*S + (2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^ m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Qx)/(d + e*x )^m - ((2*p + 3)*(2*c*R - b*S))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a* e^2, 0] && LtQ[p, -1] && ILtQ[m, 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 = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.15 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {409487791 \, \sqrt {2} {\left (16 \, x^{6} + 64 \, x^{5} + 72 \, x^{4} + 136 \, x^{3} + 241 \, x^{2} + 30 \, x + 225\right )} \log \left (\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (217883368 \, x^{5} + 107028732 \, x^{4} - 1503926130 \, x^{3} - 5919924791 \, x^{2} + 2280511668 \, x - 8953831359\right )} \sqrt {2 \, x^{2} - x + 3}}{682380951552 \, {\left (16 \, x^{6} + 64 \, x^{5} + 72 \, x^{4} + 136 \, x^{3} + 241 \, x^{2} + 30 \, x + 225\right )}} \]
1/682380951552*(409487791*sqrt(2)*(16*x^6 + 64*x^5 + 72*x^4 + 136*x^3 + 24 1*x^2 + 30*x + 225)*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*(217883368*x^5 + 107028732 *x^4 - 1503926130*x^3 - 5919924791*x^2 + 2280511668*x - 8953831359)*sqrt(2 *x^2 - x + 3))/(16*x^6 + 64*x^5 + 72*x^4 + 136*x^3 + 241*x^2 + 30*x + 225)
\[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}} \, dx=\int \frac {5 x^{4} - x^{3} + 3 x^{2} + x + 2}{\left (2 x + 5\right )^{3} \left (2 x^{2} - x + 3\right )^{\frac {5}{2}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.32 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}} \, dx=-\frac {774079}{644972544} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) + \frac {27235421 \, x}{14216269824 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {36393601}{4738756608 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {2323723 \, x}{103016448 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {3667}{1152 \, {\left (4 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 20 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 25 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {115369}{82944 \, {\left (2 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 5 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {5254255}{34338816 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
-774079/644972544*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sq rt(23)/abs(2*x + 5)) + 27235421/14216269824*x/sqrt(2*x^2 - x + 3) - 363936 01/4738756608/sqrt(2*x^2 - x + 3) + 2323723/103016448*x/(2*x^2 - x + 3)^(3 /2) - 3667/1152/(4*(2*x^2 - x + 3)^(3/2)*x^2 + 20*(2*x^2 - x + 3)^(3/2)*x + 25*(2*x^2 - x + 3)^(3/2)) + 115369/82944/(2*(2*x^2 - x + 3)^(3/2)*x + 5* (2*x^2 - x + 3)^(3/2)) - 5254255/34338816/(2*x^2 - x + 3)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (109) = 218\).
Time = 0.29 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.69 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {774079}{644972544} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {774079}{644972544} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {\sqrt {2} {\left (44558 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} - 10136238 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 16812201 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 10182217\right )}}{53747712 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{2}} + \frac {{\left ({\left (4296526 \, x - 11507857\right )} x + 10720752\right )} x - 11003805}{592344576 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
774079/644972544*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 774079/644972544*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*s qrt(2*x^2 - x + 3))) + 1/53747712*sqrt(2)*(44558*sqrt(2)*(sqrt(2)*x - sqrt (2*x^2 - x + 3))^3 - 10136238*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 168122 01*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 10182217)/(2*(sqrt(2)*x - s qrt(2*x^2 - x + 3))^2 + 10*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 11) ^2 + 1/592344576*(((4296526*x - 11507857)*x + 10720752)*x - 11003805)/(2*x ^2 - x + 3)^(3/2)
Timed out. \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}} \, dx=\int \frac {5\,x^4-x^3+3\,x^2+x+2}{{\left (2\,x+5\right )}^3\,{\left (2\,x^2-x+3\right )}^{5/2}} \,d x \]