Integrand size = 40, antiderivative size = 195 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^7} \, dx=\frac {(151764102421+27596573612 x) \sqrt {3-x+2 x^2}}{55037657088 (5+2 x)}-\frac {(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac {182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac {14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}+\frac {369 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{128 \sqrt {2}}-\frac {1903976002333 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{660451885056 \sqrt {2}} \]
-1/13759414272*(9802984711+6793718806*x)*(2*x^2-x+3)^(3/2)/(5+2*x)^3-3667/ 3456*(2*x^2-x+3)^(5/2)/(5+2*x)^6+182165/248832*(2*x^2-x+3)^(5/2)/(5+2*x)^5 -14087245/71663616*(2*x^2-x+3)^(5/2)/(5+2*x)^4+369/256*arcsinh(1/23*(1-4*x )*23^(1/2))*2^(1/2)-1903976002333/1320903770112*arctanh(1/24*(17-22*x)*2^( 1/2)/(2*x^2-x+3)^(1/2))*2^(1/2)+1/55037657088*(151764102421+27596573612*x) *(2*x^2-x+3)^(1/2)/(5+2*x)
Time = 0.97 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.62 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^7} \, dx=\frac {\frac {12 \sqrt {3-x+2 x^2} \left (458411625354581+1011372787716826 x+910256842473992 x^2+422554114856528 x^3+103803827945872 x^4+11854023276320 x^5+275188285440 x^6\right )}{(5+2 x)^6}+1903976002333 \sqrt {2} \text {arctanh}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )+951979474944 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{660451885056} \]
((12*Sqrt[3 - x + 2*x^2]*(458411625354581 + 1011372787716826*x + 910256842 473992*x^2 + 422554114856528*x^3 + 103803827945872*x^4 + 11854023276320*x^ 5 + 275188285440*x^6))/(5 + 2*x)^6 + 1903976002333*Sqrt[2]*ArcTanh[(5 + 2* x - Sqrt[6 - 2*x + 4*x^2])/6] + 951979474944*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[ 6 - 2*x + 4*x^2]])/660451885056
Time = 0.63 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.12, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2181, 27, 2181, 27, 2181, 1229, 27, 1230, 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.
| \(\displaystyle \int \frac {\left (2 x^2-x+3\right )^{3/2} \left (5 x^4-x^3+3 x^2+x+2\right )}{(2 x+5)^7} \, dx\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle -\frac {1}{432} \int \frac {\left (2 x^2-x+3\right )^{3/2} \left (-17280 x^3+46656 x^2-112340 x+68375\right )}{16 (2 x+5)^6}dx-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\left (2 x^2-x+3\right )^{3/2} \left (-17280 x^3+46656 x^2-112340 x+68375\right )}{(2 x+5)^6}dx}{6912}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {\frac {1}{360} \int \frac {5 \left (2 x^2-x+3\right )^{3/2} \left (622080 x^2-3234816 x+2112205\right )}{(2 x+5)^5}dx+\frac {182165 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}}{6912}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{72} \int \frac {\left (2 x^2-x+3\right )^{3/2} \left (622080 x^2-3234816 x+2112205\right )}{(2 x+5)^5}dx+\frac {182165 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}}{6912}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {\frac {1}{72} \left (-\frac {1}{288} \int \frac {(84010769-145928500 x) \left (2 x^2-x+3\right )^{3/2}}{(2 x+5)^4}dx-\frac {14087245 \left (2 x^2-x+3\right )^{5/2}}{144 (2 x+5)^4}\right )+\frac {182165 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}}{6912}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {\frac {1}{72} \left (\frac {1}{288} \left (\frac {\int -\frac {6 (13781234361-27596573612 x) \sqrt {2 x^2-x+3}}{(2 x+5)^2}dx}{1152}-\frac {(6793718806 x+9802984711) \left (2 x^2-x+3\right )^{3/2}}{96 (2 x+5)^3}\right )-\frac {14087245 \left (2 x^2-x+3\right )^{5/2}}{144 (2 x+5)^4}\right )+\frac {182165 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}}{6912}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{72} \left (\frac {1}{288} \left (-\frac {1}{192} \int \frac {(13781234361-27596573612 x) \sqrt {2 x^2-x+3}}{(2 x+5)^2}dx-\frac {(6793718806 x+9802984711) \left (2 x^2-x+3\right )^{3/2}}{96 (2 x+5)^3}\right )-\frac {14087245 \left (2 x^2-x+3\right )^{5/2}}{144 (2 x+5)^4}\right )+\frac {182165 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}}{6912}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\frac {1}{72} \left (\frac {1}{288} \left (\frac {1}{192} \left (\frac {1}{8} \int \frac {2 (317343544093-634652983296 x)}{(2 x+5) \sqrt {2 x^2-x+3}}dx+\frac {\sqrt {2 x^2-x+3} (27596573612 x+151764102421)}{2 (2 x+5)}\right )-\frac {(6793718806 x+9802984711) \left (2 x^2-x+3\right )^{3/2}}{96 (2 x+5)^3}\right )-\frac {14087245 \left (2 x^2-x+3\right )^{5/2}}{144 (2 x+5)^4}\right )+\frac {182165 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}}{6912}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{72} \left (\frac {1}{288} \left (\frac {1}{192} \left (\frac {1}{4} \int \frac {317343544093-634652983296 x}{(2 x+5) \sqrt {2 x^2-x+3}}dx+\frac {\sqrt {2 x^2-x+3} (27596573612 x+151764102421)}{2 (2 x+5)}\right )-\frac {(6793718806 x+9802984711) \left (2 x^2-x+3\right )^{3/2}}{96 (2 x+5)^3}\right )-\frac {14087245 \left (2 x^2-x+3\right )^{5/2}}{144 (2 x+5)^4}\right )+\frac {182165 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}}{6912}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {1}{72} \left (\frac {1}{288} \left (\frac {1}{192} \left (\frac {1}{4} \left (1903976002333 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-317326491648 \int \frac {1}{\sqrt {2 x^2-x+3}}dx\right )+\frac {\sqrt {2 x^2-x+3} (27596573612 x+151764102421)}{2 (2 x+5)}\right )-\frac {(6793718806 x+9802984711) \left (2 x^2-x+3\right )^{3/2}}{96 (2 x+5)^3}\right )-\frac {14087245 \left (2 x^2-x+3\right )^{5/2}}{144 (2 x+5)^4}\right )+\frac {182165 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}}{6912}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {\frac {1}{72} \left (\frac {1}{288} \left (\frac {1}{192} \left (\frac {1}{4} \left (1903976002333 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-158663245824 \sqrt {\frac {2}{23}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)\right )+\frac {\sqrt {2 x^2-x+3} (27596573612 x+151764102421)}{2 (2 x+5)}\right )-\frac {(6793718806 x+9802984711) \left (2 x^2-x+3\right )^{3/2}}{96 (2 x+5)^3}\right )-\frac {14087245 \left (2 x^2-x+3\right )^{5/2}}{144 (2 x+5)^4}\right )+\frac {182165 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}}{6912}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {\frac {1}{72} \left (\frac {1}{288} \left (\frac {1}{192} \left (\frac {1}{4} \left (1903976002333 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-158663245824 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+\frac {\sqrt {2 x^2-x+3} (27596573612 x+151764102421)}{2 (2 x+5)}\right )-\frac {(6793718806 x+9802984711) \left (2 x^2-x+3\right )^{3/2}}{96 (2 x+5)^3}\right )-\frac {14087245 \left (2 x^2-x+3\right )^{5/2}}{144 (2 x+5)^4}\right )+\frac {182165 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}}{6912}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {1}{72} \left (\frac {1}{288} \left (\frac {1}{192} \left (\frac {1}{4} \left (-3807952004666 \int \frac {1}{288-\frac {(17-22 x)^2}{2 x^2-x+3}}d\frac {17-22 x}{\sqrt {2 x^2-x+3}}-158663245824 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+\frac {\sqrt {2 x^2-x+3} (27596573612 x+151764102421)}{2 (2 x+5)}\right )-\frac {(6793718806 x+9802984711) \left (2 x^2-x+3\right )^{3/2}}{96 (2 x+5)^3}\right )-\frac {14087245 \left (2 x^2-x+3\right )^{5/2}}{144 (2 x+5)^4}\right )+\frac {182165 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}}{6912}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{72} \left (\frac {1}{288} \left (\frac {1}{192} \left (\frac {1}{4} \left (-158663245824 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )-\frac {1903976002333 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{6 \sqrt {2}}\right )+\frac {\sqrt {2 x^2-x+3} (27596573612 x+151764102421)}{2 (2 x+5)}\right )-\frac {(6793718806 x+9802984711) \left (2 x^2-x+3\right )^{3/2}}{96 (2 x+5)^3}\right )-\frac {14087245 \left (2 x^2-x+3\right )^{5/2}}{144 (2 x+5)^4}\right )+\frac {182165 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)^5}}{6912}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}\) |
(-3667*(3 - x + 2*x^2)^(5/2))/(3456*(5 + 2*x)^6) + ((182165*(3 - x + 2*x^2 )^(5/2))/(36*(5 + 2*x)^5) + ((-14087245*(3 - x + 2*x^2)^(5/2))/(144*(5 + 2 *x)^4) + (-1/96*((9802984711 + 6793718806*x)*(3 - x + 2*x^2)^(3/2))/(5 + 2 *x)^3 + (((151764102421 + 27596573612*x)*Sqrt[3 - x + 2*x^2])/(2*(5 + 2*x) ) + (-158663245824*Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[23]] - (1903976002333*A rcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(6*Sqrt[2]))/4)/192) /288)/72)/6912
3.4.42.3.1 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/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]
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]
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[(-(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]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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]
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 = 229, normalized size of antiderivative = 1.17 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^7} \, dx=\frac {1903958949888 \, \sqrt {2} {\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )} \log \left (4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 1903976002333 \, \sqrt {2} {\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )} \log \left (-\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (275188285440 \, x^{6} + 11854023276320 \, x^{5} + 103803827945872 \, x^{4} + 422554114856528 \, x^{3} + 910256842473992 \, x^{2} + 1011372787716826 \, x + 458411625354581\right )} \sqrt {2 \, x^{2} - x + 3}}{2641807540224 \, {\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} \]
1/2641807540224*(1903958949888*sqrt(2)*(64*x^6 + 960*x^5 + 6000*x^4 + 2000 0*x^3 + 37500*x^2 + 37500*x + 15625)*log(4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4* x - 1) - 32*x^2 + 16*x - 25) + 1903976002333*sqrt(2)*(64*x^6 + 960*x^5 + 6 000*x^4 + 20000*x^3 + 37500*x^2 + 37500*x + 15625)*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*(275188285440*x^6 + 11854023276320*x^5 + 103803827945872*x^4 + 42255 4114856528*x^3 + 910256842473992*x^2 + 1011372787716826*x + 45841162535458 1)*sqrt(2*x^2 - x + 3))/(64*x^6 + 960*x^5 + 6000*x^4 + 20000*x^3 + 37500*x ^2 + 37500*x + 15625)
\[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^7} \, dx=\int \frac {\left (2 x^{2} - x + 3\right )^{\frac {3}{2}} \cdot \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{7}}\, dx \]
Time = 0.32 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.52 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^7} \, dx=\frac {3607708597}{1486016741376} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {3667 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3456 \, {\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} + \frac {182165 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{248832 \, {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} - \frac {14087245 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{71663616 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac {149610673 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{5159780352 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac {3607708597 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{743008370688 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac {82772668391}{990677827584} \, \sqrt {2 \, x^{2} - x + 3} x - \frac {369}{256} \, \sqrt {2} \operatorname {arsinh}\left (\frac {4}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) + \frac {1903976002333}{1320903770112} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) + \frac {165562389227}{330225942528} \, \sqrt {2 \, x^{2} - x + 3} + \frac {125860542215 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{2972033482752 \, {\left (2 \, x + 5\right )}} \]
3607708597/1486016741376*(2*x^2 - x + 3)^(3/2) - 3667/3456*(2*x^2 - x + 3) ^(5/2)/(64*x^6 + 960*x^5 + 6000*x^4 + 20000*x^3 + 37500*x^2 + 37500*x + 15 625) + 182165/248832*(2*x^2 - x + 3)^(5/2)/(32*x^5 + 400*x^4 + 2000*x^3 + 5000*x^2 + 6250*x + 3125) - 14087245/71663616*(2*x^2 - x + 3)^(5/2)/(16*x^ 4 + 160*x^3 + 600*x^2 + 1000*x + 625) + 149610673/5159780352*(2*x^2 - x + 3)^(5/2)/(8*x^3 + 60*x^2 + 150*x + 125) - 3607708597/743008370688*(2*x^2 - x + 3)^(5/2)/(4*x^2 + 20*x + 25) - 82772668391/990677827584*sqrt(2*x^2 - x + 3)*x - 369/256*sqrt(2)*arcsinh(4/23*sqrt(23)*x - 1/23*sqrt(23)) + 1903 976002333/1320903770112*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17 /23*sqrt(23)/abs(2*x + 5)) + 165562389227/330225942528*sqrt(2*x^2 - x + 3) + 125860542215/2972033482752*(2*x^2 - x + 3)^(3/2)/(2*x + 5)
Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (160) = 320\).
Time = 0.30 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.32 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^7} \, dx=\frac {369}{256} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) - \frac {1903976002333}{1320903770112} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {1903976002333}{1320903770112} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {5}{64} \, \sqrt {2 \, x^{2} - x + 3} + \frac {\sqrt {2} {\left (159278433934432 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{11} + 6347903280912544 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{10} + 48544526840833424 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{9} + 305716670132783088 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{8} + 88313821135911024 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{7} - 2423668581998843376 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{6} - 397211131697032056 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{5} + 11708897232532299576 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{4} - 12803484860728491138 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} + 12593033197867577234 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} - 3042533760672408875 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 589526263249780195\right )}}{110075314176 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{6}} \]
369/256*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) - 19 03976002333/1320903770112*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt( 2*x^2 - x + 3))) + 1903976002333/1320903770112*sqrt(2)*log(abs(-2*sqrt(2)* x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) + 5/64*sqrt(2*x^2 - x + 3) + 1/11 0075314176*sqrt(2)*(159278433934432*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^11 + 6347903280912544*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^10 + 485445268 40833424*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^9 + 305716670132783088* (sqrt(2)*x - sqrt(2*x^2 - x + 3))^8 + 88313821135911024*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^7 - 2423668581998843376*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^6 - 397211131697032056*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^5 + 11708897232532299576*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^4 - 128034848607 28491138*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^3 + 1259303319786757723 4*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 - 3042533760672408875*sqrt(2)*(sqrt( 2)*x - sqrt(2*x^2 - x + 3)) + 589526263249780195)/(2*(sqrt(2)*x - sqrt(2*x ^2 - x + 3))^2 + 10*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 11)^6
Timed out. \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^7} \, dx=\int \frac {{\left (2\,x^2-x+3\right )}^{3/2}\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{{\left (2\,x+5\right )}^7} \,d x \]