Integrand size = 40, antiderivative size = 137 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {369609-175877 x}{154524672 \sqrt {3-x+2 x^2}}-\frac {3667 \sqrt {3-x+2 x^2}}{31104 (5+2 x)^3}+\frac {152885 \sqrt {3-x+2 x^2}}{4478976 (5+2 x)^2}+\frac {430799 \sqrt {3-x+2 x^2}}{107495424 (5+2 x)}-\frac {3505819 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{1289945088 \sqrt {2}} \]
-3505819/2579890176*arctanh(1/24*(17-22*x)*2^(1/2)/(2*x^2-x+3)^(1/2))*2^(1 /2)+1/154524672*(369609-175877*x)/(2*x^2-x+3)^(1/2)-3667/31104*(2*x^2-x+3) ^(1/2)/(5+2*x)^3+152885/4478976*(2*x^2-x+3)^(1/2)/(5+2*x)^2+430799/1074954 24*(2*x^2-x+3)^(1/2)/(5+2*x)
Time = 0.56 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.59 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {\frac {12 \left (1873786587+1257975811 x+441046842 x^2+572739684 x^3+56754760 x^4\right )}{(5+2 x)^3 \sqrt {3-x+2 x^2}}+80633837 \sqrt {2} \text {arctanh}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )}{29668737024} \]
((12*(1873786587 + 1257975811*x + 441046842*x^2 + 572739684*x^3 + 56754760 *x^4))/((5 + 2*x)^3*Sqrt[3 - x + 2*x^2]) + 80633837*Sqrt[2]*ArcTanh[(5 + 2 *x - Sqrt[6 - 2*x + 4*x^2])/6])/29668737024
Time = 0.48 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {2177, 27, 2181, 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.
\(\displaystyle \int \frac {5 x^4-x^3+3 x^2+x+2}{(2 x+5)^4 \left (2 x^2-x+3\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2177 |
\(\displaystyle \frac {2}{23} \int \frac {23 \left (453064 x^3+38587980 x^2+31270710 x+15168577\right )}{26873856 (2 x+5)^4 \sqrt {2 x^2-x+3}}dx+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {453064 x^3+38587980 x^2+31270710 x+15168577}{(2 x+5)^4 \sqrt {2 x^2-x+3}}dx}{13436928}+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\) |
\(\Big \downarrow \) 2181 |
\(\displaystyle \frac {-\frac {1}{216} \int \frac {216 \left (-226532 x^2-12391084 x+3461275\right )}{(2 x+5)^3 \sqrt {2 x^2-x+3}}dx-\frac {1584144 \sqrt {2 x^2-x+3}}{(2 x+5)^3}}{13436928}+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\int \frac {-226532 x^2-12391084 x+3461275}{(2 x+5)^3 \sqrt {2 x^2-x+3}}dx-\frac {1584144 \sqrt {2 x^2-x+3}}{(2 x+5)^3}}{13436928}+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\) |
\(\Big \downarrow \) 2181 |
\(\displaystyle \frac {\frac {1}{144} \int \frac {72 (2061152 x+1275689)}{(2 x+5)^2 \sqrt {2 x^2-x+3}}dx+\frac {458655 \sqrt {2 x^2-x+3}}{(2 x+5)^2}-\frac {1584144 \sqrt {2 x^2-x+3}}{(2 x+5)^3}}{13436928}+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{2} \int \frac {2061152 x+1275689}{(2 x+5)^2 \sqrt {2 x^2-x+3}}dx+\frac {458655 \sqrt {2 x^2-x+3}}{(2 x+5)^2}-\frac {1584144 \sqrt {2 x^2-x+3}}{(2 x+5)^3}}{13436928}+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\) |
\(\Big \downarrow \) 1228 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3505819}{8} \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx+\frac {430799 \sqrt {2 x^2-x+3}}{4 (2 x+5)}\right )+\frac {458655 \sqrt {2 x^2-x+3}}{(2 x+5)^2}-\frac {1584144 \sqrt {2 x^2-x+3}}{(2 x+5)^3}}{13436928}+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {430799 \sqrt {2 x^2-x+3}}{4 (2 x+5)}-\frac {3505819}{4} \int \frac {1}{288-\frac {(17-22 x)^2}{2 x^2-x+3}}d\frac {17-22 x}{\sqrt {2 x^2-x+3}}\right )+\frac {458655 \sqrt {2 x^2-x+3}}{(2 x+5)^2}-\frac {1584144 \sqrt {2 x^2-x+3}}{(2 x+5)^3}}{13436928}+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {430799 \sqrt {2 x^2-x+3}}{4 (2 x+5)}-\frac {3505819 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{48 \sqrt {2}}\right )+\frac {458655 \sqrt {2 x^2-x+3}}{(2 x+5)^2}-\frac {1584144 \sqrt {2 x^2-x+3}}{(2 x+5)^3}}{13436928}+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\) |
(369609 - 175877*x)/(154524672*Sqrt[3 - x + 2*x^2]) + ((-1584144*Sqrt[3 - x + 2*x^2])/(5 + 2*x)^3 + (458655*Sqrt[3 - x + 2*x^2])/(5 + 2*x)^2 + ((430 799*Sqrt[3 - x + 2*x^2])/(4*(5 + 2*x)) - (3505819*ArcTanh[(17 - 22*x)/(12* Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(48*Sqrt[2]))/2)/13436928
3.4.57.3.1 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 = 141, normalized size of antiderivative = 1.03 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {80633837 \, \sqrt {2} {\left (16 \, x^{5} + 112 \, x^{4} + 264 \, x^{3} + 280 \, x^{2} + 325 \, x + 375\right )} \log \left (-\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (56754760 \, x^{4} + 572739684 \, x^{3} + 441046842 \, x^{2} + 1257975811 \, x + 1873786587\right )} \sqrt {2 \, x^{2} - x + 3}}{118674948096 \, {\left (16 \, x^{5} + 112 \, x^{4} + 264 \, x^{3} + 280 \, x^{2} + 325 \, x + 375\right )}} \]
1/118674948096*(80633837*sqrt(2)*(16*x^5 + 112*x^4 + 264*x^3 + 280*x^2 + 3 25*x + 375)*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*(56754760*x^4 + 572739684*x^3 + 4 41046842*x^2 + 1257975811*x + 1873786587)*sqrt(2*x^2 - x + 3))/(16*x^5 + 1 12*x^4 + 264*x^3 + 280*x^2 + 325*x + 375)
\[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx=\int \frac {5 x^{4} - x^{3} + 3 x^{2} + x + 2}{\left (2 x + 5\right )^{4} \left (2 x^{2} - x + 3\right )^{\frac {3}{2}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.58 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {3505819}{2579890176} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) + \frac {7094345 \, x}{2472394752 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {6128291}{824131584 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {3667}{1728 \, {\left (8 \, \sqrt {2 \, x^{2} - x + 3} x^{3} + 60 \, \sqrt {2 \, x^{2} - x + 3} x^{2} + 150 \, \sqrt {2 \, x^{2} - x + 3} x + 125 \, \sqrt {2 \, x^{2} - x + 3}\right )}} + \frac {314233}{248832 \, {\left (4 \, \sqrt {2 \, x^{2} - x + 3} x^{2} + 20 \, \sqrt {2 \, x^{2} - x + 3} x + 25 \, \sqrt {2 \, x^{2} - x + 3}\right )}} - \frac {3127169}{17915904 \, {\left (2 \, \sqrt {2 \, x^{2} - x + 3} x + 5 \, \sqrt {2 \, x^{2} - x + 3}\right )}} \]
3505819/2579890176*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*s qrt(23)/abs(2*x + 5)) + 7094345/2472394752*x/sqrt(2*x^2 - x + 3) + 6128291 /824131584/sqrt(2*x^2 - x + 3) - 3667/1728/(8*sqrt(2*x^2 - x + 3)*x^3 + 60 *sqrt(2*x^2 - x + 3)*x^2 + 150*sqrt(2*x^2 - x + 3)*x + 125*sqrt(2*x^2 - x + 3)) + 314233/248832/(4*sqrt(2*x^2 - x + 3)*x^2 + 20*sqrt(2*x^2 - x + 3)* x + 25*sqrt(2*x^2 - x + 3)) - 3127169/17915904/(2*sqrt(2*x^2 - x + 3)*x + 5*sqrt(2*x^2 - x + 3))
Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (111) = 222\).
Time = 0.28 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.98 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx=-\frac {3505819}{2579890176} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {3505819}{2579890176} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {175877 \, x - 369609}{154524672 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {\sqrt {2} {\left (10398764 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{5} - 303070900 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{4} - 529738052 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} + 3644644652 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} - 2612608649 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1052284471\right )}}{214990848 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{3}} \]
-3505819/2579890176*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) + 3505819/2579890176*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 1/154524672*(175877*x - 369609)/sqrt(2*x^2 - x + 3) - 1/214990848*sqrt(2)*(10398764*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^5 - 303070900*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^4 - 529738052*sqrt(2 )*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^3 + 3644644652*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 - 2612608649*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1052 284471)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 10*sqrt(2)*(sqrt(2)*x - s qrt(2*x^2 - x + 3)) - 11)^3
Timed out. \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx=\int \frac {5\,x^4-x^3+3\,x^2+x+2}{{\left (2\,x+5\right )}^4\,{\left (2\,x^2-x+3\right )}^{3/2}} \,d x \]