Integrand size = 40, antiderivative size = 149 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx=-\frac {(1996953-333380 x) \sqrt {3-x+2 x^2}}{18432}-\frac {541}{384} \left (3-x+2 x^2\right )^{3/2}-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{576 (5+2 x)}+\frac {5}{64} (5+2 x) \left (3-x+2 x^2\right )^{3/2}-\frac {2551847 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4096 \sqrt {2}}+\frac {239201 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{384 \sqrt {2}} \] Output:
-1/18432*(1996953-333380*x)*(2*x^2-x+3)^(1/2)-541/384*(2*x^2-x+3)^(3/2)-36 67*(2*x^2-x+3)^(3/2)/(2880+1152*x)+5/64*(5+2*x)*(2*x^2-x+3)^(3/2)-2551847/ 8192*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)+239201/768*arctanh(1/24*(17-22 *x)*2^(1/2)/(2*x^2-x+3)^(1/2))*2^(1/2)
Time = 0.79 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx=\frac {\frac {4 \sqrt {3-x+2 x^2} \left (-3539439-728410 x+94936 x^2-17344 x^3+3840 x^4\right )}{5+2 x}-15308864 \sqrt {2} \text {arctanh}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )-7655541 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{24576} \] Input:
Integrate[(Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^2, x]
Output:
((4*Sqrt[3 - x + 2*x^2]*(-3539439 - 728410*x + 94936*x^2 - 17344*x^3 + 384 0*x^4))/(5 + 2*x) - 15308864*Sqrt[2]*ArcTanh[(5 + 2*x - Sqrt[6 - 2*x + 4*x ^2])/6] - 7655541*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/24576
Time = 0.91 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {2181, 27, 2184, 27, 2184, 27, 1231, 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.
| \(\displaystyle \int \frac {\sqrt {2 x^2-x+3} \left (5 x^4-x^3+3 x^2+x+2\right )}{(2 x+5)^2} \, dx\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle -\frac {1}{72} \int \frac {\sqrt {2 x^2-x+3} \left (-2880 x^3+7776 x^2-50504 x+19341\right )}{16 (2 x+5)}dx-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\sqrt {2 x^2-x+3} \left (-2880 x^3+7776 x^2-50504 x+19341\right )}{2 x+5}dx}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {90 (2 x+5) \left (2 x^2-x+3\right )^{3/2}-\frac {1}{64} \int \frac {128 \sqrt {2 x^2-x+3} \left (9738 x^2-19762 x+9333\right )}{2 x+5}dx}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {90 (2 x+5) \left (2 x^2-x+3\right )^{3/2}-2 \int \frac {\sqrt {2 x^2-x+3} \left (9738 x^2-19762 x+9333\right )}{2 x+5}dx}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {90 (2 x+5) \left (2 x^2-x+3\right )^{3/2}-2 \left (\frac {1}{24} \int \frac {6 (61677-166690 x) \sqrt {2 x^2-x+3}}{2 x+5}dx+\frac {1623}{2} \left (2 x^2-x+3\right )^{3/2}\right )}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {90 (2 x+5) \left (2 x^2-x+3\right )^{3/2}-2 \left (\frac {1}{4} \int \frac {(61677-166690 x) \sqrt {2 x^2-x+3}}{2 x+5}dx+\frac {1623}{2} \left (2 x^2-x+3\right )^{3/2}\right )}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {90 (2 x+5) \left (2 x^2-x+3\right )^{3/2}-2 \left (\frac {1}{4} \left (\frac {1}{8} (1996953-333380 x) \sqrt {2 x^2-x+3}-\frac {1}{32} \int -\frac {18 (2549629-5103694 x)}{(2 x+5) \sqrt {2 x^2-x+3}}dx\right )+\frac {1623}{2} \left (2 x^2-x+3\right )^{3/2}\right )}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {90 (2 x+5) \left (2 x^2-x+3\right )^{3/2}-2 \left (\frac {1}{4} \left (\frac {9}{16} \int \frac {2549629-5103694 x}{(2 x+5) \sqrt {2 x^2-x+3}}dx+\frac {1}{8} \sqrt {2 x^2-x+3} (1996953-333380 x)\right )+\frac {1623}{2} \left (2 x^2-x+3\right )^{3/2}\right )}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {90 (2 x+5) \left (2 x^2-x+3\right )^{3/2}-2 \left (\frac {1}{4} \left (\frac {9}{16} \left (15308864 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-2551847 \int \frac {1}{\sqrt {2 x^2-x+3}}dx\right )+\frac {1}{8} \sqrt {2 x^2-x+3} (1996953-333380 x)\right )+\frac {1623}{2} \left (2 x^2-x+3\right )^{3/2}\right )}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {90 (2 x+5) \left (2 x^2-x+3\right )^{3/2}-2 \left (\frac {1}{4} \left (\frac {9}{16} \left (15308864 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-\frac {2551847 \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)}{\sqrt {46}}\right )+\frac {1}{8} \sqrt {2 x^2-x+3} (1996953-333380 x)\right )+\frac {1623}{2} \left (2 x^2-x+3\right )^{3/2}\right )}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {90 (2 x+5) \left (2 x^2-x+3\right )^{3/2}-2 \left (\frac {1}{4} \left (\frac {9}{16} \left (15308864 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-\frac {2551847 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {2}}\right )+\frac {1}{8} \sqrt {2 x^2-x+3} (1996953-333380 x)\right )+\frac {1623}{2} \left (2 x^2-x+3\right )^{3/2}\right )}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {90 (2 x+5) \left (2 x^2-x+3\right )^{3/2}-2 \left (\frac {1}{4} \left (\frac {9}{16} \left (-30617728 \int \frac {1}{288-\frac {(17-22 x)^2}{2 x^2-x+3}}d\frac {17-22 x}{\sqrt {2 x^2-x+3}}-\frac {2551847 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {2}}\right )+\frac {1}{8} \sqrt {2 x^2-x+3} (1996953-333380 x)\right )+\frac {1623}{2} \left (2 x^2-x+3\right )^{3/2}\right )}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {90 (2 x+5) \left (2 x^2-x+3\right )^{3/2}-2 \left (\frac {1}{4} \left (\frac {9}{16} \left (-\frac {2551847 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {2}}-\frac {3827216}{3} \sqrt {2} \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )\right )+\frac {1}{8} \sqrt {2 x^2-x+3} (1996953-333380 x)\right )+\frac {1623}{2} \left (2 x^2-x+3\right )^{3/2}\right )}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)}\) |
Input:
Int[(Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^2,x]
Output:
(-3667*(3 - x + 2*x^2)^(3/2))/(576*(5 + 2*x)) + (90*(5 + 2*x)*(3 - x + 2*x ^2)^(3/2) - 2*((1623*(3 - x + 2*x^2)^(3/2))/2 + (((1996953 - 333380*x)*Sqr t[3 - x + 2*x^2])/8 + (9*((-2551847*ArcSinh[(-1 + 4*x)/Sqrt[23]])/Sqrt[2] - (3827216*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/ 3))/16)/4))/1152
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)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[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]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c *d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !(IGt Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Timed out.
hanged
Input:
int((2*x^2-x+3)^(1/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^2,x)
Output:
int((2*x^2-x+3)^(1/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^2,x)
Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx=\frac {7655541 \, \sqrt {2} {\left (2 \, x + 5\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 7654432 \, \sqrt {2} {\left (2 \, x + 5\right )} \log \left (\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 8 \, {\left (3840 \, x^{4} - 17344 \, x^{3} + 94936 \, x^{2} - 728410 \, x - 3539439\right )} \sqrt {2 \, x^{2} - x + 3}}{49152 \, {\left (2 \, x + 5\right )}} \] Input:
integrate((2*x^2-x+3)^(1/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^2,x, algorithm=" fricas")
Output:
1/49152*(7655541*sqrt(2)*(2*x + 5)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 7654432*sqrt(2)*(2*x + 5)*log((24*sqrt(2)*sq rt(2*x^2 - x + 3)*(22*x - 17) - 1060*x^2 + 1036*x - 1153)/(4*x^2 + 20*x + 25)) + 8*(3840*x^4 - 17344*x^3 + 94936*x^2 - 728410*x - 3539439)*sqrt(2*x^ 2 - x + 3))/(2*x + 5)
\[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx=\int \frac {\sqrt {2 x^{2} - x + 3} \cdot \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{2}}\, dx \] Input:
integrate((2*x**2-x+3)**(1/2)*(5*x**4-x**3+3*x**2+x+2)/(5+2*x)**2,x)
Output:
Integral(sqrt(2*x**2 - x + 3)*(5*x**4 - x**3 + 3*x**2 + x + 2)/(2*x + 5)** 2, x)
Time = 0.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx=\frac {5}{32} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {391}{384} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {6001}{512} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {2551847}{8192} \, \sqrt {2} \operatorname {arsinh}\left (\frac {4}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) - \frac {239201}{768} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) - \frac {182769}{2048} \, \sqrt {2 \, x^{2} - x + 3} - \frac {3667 \, \sqrt {2 \, x^{2} - x + 3}}{32 \, {\left (2 \, x + 5\right )}} \] Input:
integrate((2*x^2-x+3)^(1/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^2,x, algorithm=" maxima")
Output:
5/32*(2*x^2 - x + 3)^(3/2)*x - 391/384*(2*x^2 - x + 3)^(3/2) + 6001/512*sq rt(2*x^2 - x + 3)*x + 2551847/8192*sqrt(2)*arcsinh(4/23*sqrt(23)*x - 1/23* sqrt(23)) - 239201/768*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/ 23*sqrt(23)/abs(2*x + 5)) - 182769/2048*sqrt(2*x^2 - x + 3) - 3667/32*sqrt (2*x^2 - x + 3)/(2*x + 5)
Leaf count of result is larger than twice the leaf count of optimal. 531 vs. \(2 (118) = 236\).
Time = 0.23 (sec) , antiderivative size = 531, normalized size of antiderivative = 3.56 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx =\text {Too large to display} \] Input:
integrate((2*x^2-x+3)^(1/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^2,x, algorithm=" giac")
Output:
1/24576*sqrt(2)*(7654432*log(12*sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 72/(2*x + 5) - 11)*sgn(1/(2*x + 5)) + 7655541*log(abs(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5) + 1))*sgn(1/(2*x + 5)) - 7655541*log(a bs(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5) - 1))*sgn(1/(2*x + 5)) - 1408128*sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1)*sgn(1/(2*x + 5)) + 2*(16367883*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^7* sgn(1/(2*x + 5)) - 34896384*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/ (2*x + 5))^6*sgn(1/(2*x + 5)) - 93395*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^5*sgn(1/(2*x + 5)) + 25574400*(sqrt(-11/(2*x + 5) + 3 6/(2*x + 5)^2 + 1) + 6/(2*x + 5))^4*sgn(1/(2*x + 5)) + 19752365*(sqrt(-11/ (2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^3*sgn(1/(2*x + 5)) - 319219 20*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^2*sgn(1/(2*x + 5)) - 2445813*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))*sg n(1/(2*x + 5)) + 7663104*sgn(1/(2*x + 5)))/((sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^2 - 1)^4)
Timed out. \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx=\int \frac {\sqrt {2\,x^2-x+3}\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{{\left (2\,x+5\right )}^2} \,d x \] Input:
int(((2*x^2 - x + 3)^(1/2)*(x + 3*x^2 - x^3 + 5*x^4 + 2))/(2*x + 5)^2,x)
Output:
int(((2*x^2 - x + 3)^(1/2)*(x + 3*x^2 - x^3 + 5*x^4 + 2))/(2*x + 5)^2, x)
Time = 0.19 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx=\frac {15360 \sqrt {2 x^{2}-x +3}\, x^{4}-69376 \sqrt {2 x^{2}-x +3}\, x^{3}+379744 \sqrt {2 x^{2}-x +3}\, x^{2}-2913640 \sqrt {2 x^{2}-x +3}\, x -14157756 \sqrt {2 x^{2}-x +3}+15308864 \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x +38272160 \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right )+15311082 \sqrt {2}\, \mathrm {log}\left (-2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}-4 x +1\right ) x +38277705 \sqrt {2}\, \mathrm {log}\left (-2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}-4 x +1\right )-15308864 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x -38272160 \sqrt {2}\, \mathrm {log}\left (2 x +5\right )}{49152 x +122880} \] Input:
int((2*x^2-x+3)^(1/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^2,x)
Output:
(15360*sqrt(2*x**2 - x + 3)*x**4 - 69376*sqrt(2*x**2 - x + 3)*x**3 + 37974 4*sqrt(2*x**2 - x + 3)*x**2 - 2913640*sqrt(2*x**2 - x + 3)*x - 14157756*sq rt(2*x**2 - x + 3) + 15308864*sqrt(2)*log( - 12*sqrt(2*x**2 - x + 3)*sqrt( 2) + 22*x - 17)*x + 38272160*sqrt(2)*log( - 12*sqrt(2*x**2 - x + 3)*sqrt(2 ) + 22*x - 17) + 15311082*sqrt(2)*log( - 2*sqrt(2*x**2 - x + 3)*sqrt(2) - 4*x + 1)*x + 38277705*sqrt(2)*log( - 2*sqrt(2*x**2 - x + 3)*sqrt(2) - 4*x + 1) - 15308864*sqrt(2)*log(2*x + 5)*x - 38272160*sqrt(2)*log(2*x + 5))/(2 4576*(2*x + 5))