Integrand size = 40, antiderivative size = 85 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x) \left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {1191+917 x}{9936 \left (3-x+2 x^2\right )^{3/2}}-\frac {335337+146729 x}{1371168 \sqrt {3-x+2 x^2}}-\frac {3667 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{31104 \sqrt {2}} \] Output:
1/9936*(1191+917*x)/(2*x^2-x+3)^(3/2)-1/1371168*(335337+146729*x)/(2*x^2-x +3)^(1/2)-3667/62208*arctanh(1/24*(17-22*x)*2^(1/2)/(2*x^2-x+3)^(1/2))*2^( 1/2)
Time = 0.70 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.81 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x) \left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {-841653+21696 x-523945 x^2-293458 x^3}{1371168 \left (3-x+2 x^2\right )^{3/2}}+\frac {3667 \text {arctanh}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )}{15552 \sqrt {2}} \] Input:
Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)*(3 - x + 2*x^2)^(5/2)), x]
Output:
(-841653 + 21696*x - 523945*x^2 - 293458*x^3)/(1371168*(3 - x + 2*x^2)^(3/ 2)) + (3667*ArcTanh[(5 + 2*x - Sqrt[6 - 2*x + 4*x^2])/6])/(15552*Sqrt[2])
Time = 0.57 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2177, 27, 2177, 27, 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.
| \(\displaystyle \int \frac {5 x^4-x^3+3 x^2+x+2}{(2 x+5) \left (2 x^2-x+3\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2177 |
| \(\displaystyle \frac {2}{69} \int -\frac {-49680 x^2-22240 x+1877}{576 (2 x+5) \left (2 x^2-x+3\right )^{3/2}}dx+\frac {917 x+1191}{9936 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {917 x+1191}{9936 \left (2 x^2-x+3\right )^{3/2}}-\frac {\int \frac {-49680 x^2-22240 x+1877}{(2 x+5) \left (2 x^2-x+3\right )^{3/2}}dx}{19872}\) |
| \(\Big \downarrow \) 2177 |
| \(\displaystyle \frac {-\frac {2}{23} \int -\frac {1939843}{12 (2 x+5) \sqrt {2 x^2-x+3}}dx-\frac {146729 x+335337}{69 \sqrt {2 x^2-x+3}}}{19872}+\frac {917 x+1191}{9936 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {84341}{6} \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-\frac {146729 x+335337}{69 \sqrt {2 x^2-x+3}}}{19872}+\frac {917 x+1191}{9936 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {-\frac {84341}{3} \int \frac {1}{288-\frac {(17-22 x)^2}{2 x^2-x+3}}d\frac {17-22 x}{\sqrt {2 x^2-x+3}}-\frac {146729 x+335337}{69 \sqrt {2 x^2-x+3}}}{19872}+\frac {917 x+1191}{9936 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {84341 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{36 \sqrt {2}}-\frac {146729 x+335337}{69 \sqrt {2 x^2-x+3}}}{19872}+\frac {917 x+1191}{9936 \left (2 x^2-x+3\right )^{3/2}}\) |
Input:
Int[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)*(3 - x + 2*x^2)^(5/2)),x]
Output:
(1191 + 917*x)/(9936*(3 - x + 2*x^2)^(3/2)) + (-1/69*(335337 + 146729*x)/S qrt[3 - x + 2*x^2] - (84341*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2 *x^2])])/(36*Sqrt[2]))/19872
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[(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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.14 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95
| method | result | size |
| trager | \(-\frac {293458 x^{3}+523945 x^{2}-21696 x +841653}{1371168 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {3667 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {22 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x -17 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )-24 \sqrt {2 x^{2}-x +3}}{5+2 x}\right )}{62208}\) | \(81\) |
| default | \(-\frac {3817 \left (4 x -1\right )}{2944 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {3817 \left (4 x -1\right )}{4232 \sqrt {2 x^{2}-x +3}}-\frac {1597}{384 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {59 x}{32 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {5 x^{2}}{4 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {3667}{1728 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}+\frac {\frac {40337 x}{9936}-\frac {40337}{39744}}{\left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}+\frac {\frac {4800103 x}{1371168}-\frac {4800103}{5484672}}{\sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}+\frac {3667}{10368 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}-\frac {3667 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (\frac {17}{2}-11 x \right ) \sqrt {2}}{12 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}\right )}{62208}\) | \(190\) |
Input:
int((5*x^4-x^3+3*x^2+x+2)/(5+2*x)/(2*x^2-x+3)^(5/2),x,method=_RETURNVERBOS E)
Output:
-1/1371168*(293458*x^3+523945*x^2-21696*x+841653)/(2*x^2-x+3)^(3/2)-3667/6 2208*RootOf(_Z^2-2)*ln(-(22*RootOf(_Z^2-2)*x-17*RootOf(_Z^2-2)-24*(2*x^2-x +3)^(1/2))/(5+2*x))
Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.48 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x) \left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {1939843 \, \sqrt {2} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (-\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) - 48 \, {\left (293458 \, x^{3} + 523945 \, x^{2} - 21696 \, x + 841653\right )} \sqrt {2 \, x^{2} - x + 3}}{65816064 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \] Input:
integrate((5*x^4-x^3+3*x^2+x+2)/(5+2*x)/(2*x^2-x+3)^(5/2),x, algorithm="fr icas")
Output:
1/65816064*(1939843*sqrt(2)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*log(-(24*sq rt(2)*sqrt(2*x^2 - x + 3)*(22*x - 17) + 1060*x^2 - 1036*x + 1153)/(4*x^2 + 20*x + 25)) - 48*(293458*x^3 + 523945*x^2 - 21696*x + 841653)*sqrt(2*x^2 - x + 3))/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)
\[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x) \left (3-x+2 x^2\right )^{5/2}} \, dx=\int \frac {5 x^{4} - x^{3} + 3 x^{2} + x + 2}{\left (2 x + 5\right ) \left (2 x^{2} - x + 3\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((5*x**4-x**3+3*x**2+x+2)/(5+2*x)/(2*x**2-x+3)**(5/2),x)
Output:
Integral((5*x**4 - x**3 + 3*x**2 + x + 2)/((2*x + 5)*(2*x**2 - x + 3)**(5/ 2)), x)
Time = 0.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.29 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x) \left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {3667}{62208} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) - \frac {146729 \, x}{1371168 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {5 \, x^{2}}{4 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {173881}{457056 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {7127 \, x}{9936 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {5813}{3312 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \] Input:
integrate((5*x^4-x^3+3*x^2+x+2)/(5+2*x)/(2*x^2-x+3)^(5/2),x, algorithm="ma xima")
Output:
3667/62208*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/ abs(2*x + 5)) - 146729/1371168*x/sqrt(2*x^2 - x + 3) - 5/4*x^2/(2*x^2 - x + 3)^(3/2) + 173881/457056/sqrt(2*x^2 - x + 3) + 7127/9936*x/(2*x^2 - x + 3)^(3/2) - 5813/3312/(2*x^2 - x + 3)^(3/2)
Time = 0.14 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.08 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x) \left (3-x+2 x^2\right )^{5/2}} \, dx=-\frac {3667}{62208} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {3667}{62208} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {{\left ({\left (293458 \, x + 523945\right )} x - 21696\right )} x + 841653}{1371168 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \] Input:
integrate((5*x^4-x^3+3*x^2+x+2)/(5+2*x)/(2*x^2-x+3)^(5/2),x, algorithm="gi ac")
Output:
-3667/62208*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3) )) + 3667/62208*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 1/1371168*(((293458*x + 523945)*x - 21696)*x + 841653)/(2*x^2 - x + 3)^(3/2)
Timed out. \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x) \left (3-x+2 x^2\right )^{5/2}} \, dx=\int \frac {5\,x^4-x^3+3\,x^2+x+2}{\left (2\,x+5\right )\,{\left (2\,x^2-x+3\right )}^{5/2}} \,d x \] Input:
int((x + 3*x^2 - x^3 + 5*x^4 + 2)/((2*x + 5)*(2*x^2 - x + 3)^(5/2)),x)
Output:
int((x + 3*x^2 - x^3 + 5*x^4 + 2)/((2*x + 5)*(2*x^2 - x + 3)^(5/2)), x)
Time = 0.18 (sec) , antiderivative size = 433, normalized size of antiderivative = 5.09 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x) \left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {-7042992 \sqrt {2 x^{2}-x +3}\, x^{3}-12574680 \sqrt {2 x^{2}-x +3}\, x^{2}+520704 \sqrt {2 x^{2}-x +3}\, x -20199672 \sqrt {2 x^{2}-x +3}+7759372 \sqrt {2}\, \mathrm {log}\left (\frac {46 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+92 x -46}{\sqrt {23}}\right ) x^{4}-7759372 \sqrt {2}\, \mathrm {log}\left (\frac {46 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+92 x -46}{\sqrt {23}}\right ) x^{3}+25217959 \sqrt {2}\, \mathrm {log}\left (\frac {46 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+92 x -46}{\sqrt {23}}\right ) x^{2}-11639058 \sqrt {2}\, \mathrm {log}\left (\frac {46 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+92 x -46}{\sqrt {23}}\right ) x +17458587 \sqrt {2}\, \mathrm {log}\left (\frac {46 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+92 x -46}{\sqrt {23}}\right )-7759372 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x +22}{\sqrt {23}}\right ) x^{4}+7759372 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x +22}{\sqrt {23}}\right ) x^{3}-25217959 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x +22}{\sqrt {23}}\right ) x^{2}+11639058 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x +22}{\sqrt {23}}\right ) x -17458587 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x +22}{\sqrt {23}}\right )-469360 \sqrt {2}\, x^{4}+469360 \sqrt {2}\, x^{3}-1525420 \sqrt {2}\, x^{2}+704040 \sqrt {2}\, x -1056060 \sqrt {2}}{131632128 x^{4}-131632128 x^{3}+427804416 x^{2}-197448192 x +296172288} \] Input:
int((5*x^4-x^3+3*x^2+x+2)/(5+2*x)/(2*x^2-x+3)^(5/2),x)
Output:
( - 7042992*sqrt(2*x**2 - x + 3)*x**3 - 12574680*sqrt(2*x**2 - x + 3)*x**2 + 520704*sqrt(2*x**2 - x + 3)*x - 20199672*sqrt(2*x**2 - x + 3) + 7759372 *sqrt(2)*log((46*sqrt(2*x**2 - x + 3)*sqrt(2) + 92*x - 46)/sqrt(23))*x**4 - 7759372*sqrt(2)*log((46*sqrt(2*x**2 - x + 3)*sqrt(2) + 92*x - 46)/sqrt(2 3))*x**3 + 25217959*sqrt(2)*log((46*sqrt(2*x**2 - x + 3)*sqrt(2) + 92*x - 46)/sqrt(23))*x**2 - 11639058*sqrt(2)*log((46*sqrt(2*x**2 - x + 3)*sqrt(2) + 92*x - 46)/sqrt(23))*x + 17458587*sqrt(2)*log((46*sqrt(2*x**2 - x + 3)* sqrt(2) + 92*x - 46)/sqrt(23)) - 7759372*sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x + 22)/sqrt(23))*x**4 + 7759372*sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x + 22)/sqrt(23))*x**3 - 25217959*sqrt(2)*log((2*sqr t(2*x**2 - x + 3)*sqrt(2) + 4*x + 22)/sqrt(23))*x**2 + 11639058*sqrt(2)*lo g((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x + 22)/sqrt(23))*x - 17458587*sqrt( 2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x + 22)/sqrt(23)) - 469360*sqrt (2)*x**4 + 469360*sqrt(2)*x**3 - 1525420*sqrt(2)*x**2 + 704040*sqrt(2)*x - 1056060*sqrt(2))/(32908032*(4*x**4 - 4*x**3 + 13*x**2 - 6*x + 9))