Integrand size = 40, antiderivative size = 181 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^4} \, dx=-\frac {(135068604-22512089 x) \sqrt {3-x+2 x^2}}{331776}-\frac {(138006843-34265045 x) \left (3-x+2 x^2\right )^{3/2}}{17915904}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}+\frac {556255 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^2}-\frac {32865365 \left (3-x+2 x^2\right )^{5/2}}{17915904 (5+2 x)}-\frac {19176431 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8192 \sqrt {2}}+\frac {517762327 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{221184 \sqrt {2}} \] Output:
-1/331776*(135068604-22512089*x)*(2*x^2-x+3)^(1/2)-1/17915904*(138006843-3 4265045*x)*(2*x^2-x+3)^(3/2)-3667/1728*(2*x^2-x+3)^(5/2)/(5+2*x)^3+556255/ 248832*(2*x^2-x+3)^(5/2)/(5+2*x)^2-32865365*(2*x^2-x+3)^(5/2)/(89579520+35 831808*x)-19176431/16384*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)+517762327/ 442368*arctanh(1/24*(17-22*x)*2^(1/2)/(2*x^2-x+3)^(1/2))*2^(1/2)
Time = 1.24 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.66 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^4} \, dx=\frac {\frac {12 \sqrt {3-x+2 x^2} \left (-1994650739-2006873194 x-594798908 x^2-33595416 x^3+2626848 x^4-315648 x^5+46080 x^6\right )}{(5+2 x)^3}-1035524654 \sqrt {2} \text {arctanh}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )-517763637 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{442368} \] Input:
Integrate[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^ 4,x]
Output:
((12*Sqrt[3 - x + 2*x^2]*(-1994650739 - 2006873194*x - 594798908*x^2 - 335 95416*x^3 + 2626848*x^4 - 315648*x^5 + 46080*x^6))/(5 + 2*x)^3 - 103552465 4*Sqrt[2]*ArcTanh[(5 + 2*x - Sqrt[6 - 2*x + 4*x^2])/6] - 517763637*Sqrt[2] *Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/442368
Time = 1.02 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {2181, 27, 2181, 2181, 1231, 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.
\(\displaystyle \int \frac {\left (2 x^2-x+3\right )^{3/2} \left (5 x^4-x^3+3 x^2+x+2\right )}{(2 x+5)^4} \, dx\) |
\(\Big \downarrow \) 2181 |
\(\displaystyle -\frac {1}{216} \int \frac {\left (2 x^2-x+3\right )^{3/2} \left (-8640 x^3+23328 x^2-92840 x+43355\right )}{16 (2 x+5)^3}dx-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {\left (2 x^2-x+3\right )^{3/2} \left (-8640 x^3+23328 x^2-92840 x+43355\right )}{(2 x+5)^3}dx}{3456}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}\) |
\(\Big \downarrow \) 2181 |
\(\displaystyle \frac {\frac {1}{144} \int \frac {\left (2 x^2-x+3\right )^{3/2} \left (622080 x^2-9909876 x+4202675\right )}{(2 x+5)^2}dx+\frac {556255 \left (2 x^2-x+3\right )^{5/2}}{72 (2 x+5)^2}}{3456}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}\) |
\(\Big \downarrow \) 2181 |
\(\displaystyle \frac {\frac {1}{144} \left (-\frac {1}{72} \int \frac {(182685181-548240720 x) \left (2 x^2-x+3\right )^{3/2}}{2 x+5}dx-\frac {32865365 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)}\right )+\frac {556255 \left (2 x^2-x+3\right )^{5/2}}{72 (2 x+5)^2}}{3456}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}\) |
\(\Big \downarrow \) 1231 |
\(\displaystyle \frac {\frac {1}{144} \left (\frac {1}{72} \left (\frac {1}{64} \int -\frac {3456 (67520547-180096712 x) \sqrt {2 x^2-x+3}}{2 x+5}dx-2 (138006843-34265045 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {32865365 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)}\right )+\frac {556255 \left (2 x^2-x+3\right )^{5/2}}{72 (2 x+5)^2}}{3456}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{144} \left (\frac {1}{72} \left (-54 \int \frac {(67520547-180096712 x) \sqrt {2 x^2-x+3}}{2 x+5}dx-2 (138006843-34265045 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {32865365 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)}\right )+\frac {556255 \left (2 x^2-x+3\right )^{5/2}}{72 (2 x+5)^2}}{3456}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}\) |
\(\Big \downarrow \) 1231 |
\(\displaystyle \frac {\frac {1}{144} \left (\frac {1}{72} \left (-54 \left (2 (135068604-22512089 x) \sqrt {2 x^2-x+3}-\frac {1}{32} \int -\frac {288 (172585259-345175758 x)}{(2 x+5) \sqrt {2 x^2-x+3}}dx\right )-2 (138006843-34265045 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {32865365 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)}\right )+\frac {556255 \left (2 x^2-x+3\right )^{5/2}}{72 (2 x+5)^2}}{3456}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{144} \left (\frac {1}{72} \left (-54 \left (9 \int \frac {172585259-345175758 x}{(2 x+5) \sqrt {2 x^2-x+3}}dx+2 \sqrt {2 x^2-x+3} (135068604-22512089 x)\right )-2 (138006843-34265045 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {32865365 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)}\right )+\frac {556255 \left (2 x^2-x+3\right )^{5/2}}{72 (2 x+5)^2}}{3456}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {\frac {1}{144} \left (\frac {1}{72} \left (-54 \left (9 \left (1035524654 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-172587879 \int \frac {1}{\sqrt {2 x^2-x+3}}dx\right )+2 \sqrt {2 x^2-x+3} (135068604-22512089 x)\right )-2 (138006843-34265045 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {32865365 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)}\right )+\frac {556255 \left (2 x^2-x+3\right )^{5/2}}{72 (2 x+5)^2}}{3456}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {\frac {1}{144} \left (\frac {1}{72} \left (-54 \left (9 \left (1035524654 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-\frac {172587879 \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)}{\sqrt {46}}\right )+2 \sqrt {2 x^2-x+3} (135068604-22512089 x)\right )-2 (138006843-34265045 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {32865365 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)}\right )+\frac {556255 \left (2 x^2-x+3\right )^{5/2}}{72 (2 x+5)^2}}{3456}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {\frac {1}{144} \left (\frac {1}{72} \left (-54 \left (9 \left (1035524654 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-\frac {172587879 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {2}}\right )+2 \sqrt {2 x^2-x+3} (135068604-22512089 x)\right )-2 (138006843-34265045 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {32865365 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)}\right )+\frac {556255 \left (2 x^2-x+3\right )^{5/2}}{72 (2 x+5)^2}}{3456}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {\frac {1}{144} \left (\frac {1}{72} \left (-54 \left (9 \left (-2071049308 \int \frac {1}{288-\frac {(17-22 x)^2}{2 x^2-x+3}}d\frac {17-22 x}{\sqrt {2 x^2-x+3}}-\frac {172587879 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {2}}\right )+2 \sqrt {2 x^2-x+3} (135068604-22512089 x)\right )-2 (138006843-34265045 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {32865365 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)}\right )+\frac {556255 \left (2 x^2-x+3\right )^{5/2}}{72 (2 x+5)^2}}{3456}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {1}{144} \left (\frac {1}{72} \left (-54 \left (9 \left (-\frac {172587879 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {2}}-\frac {517762327 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{3 \sqrt {2}}\right )+2 \sqrt {2 x^2-x+3} (135068604-22512089 x)\right )-2 (138006843-34265045 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {32865365 \left (2 x^2-x+3\right )^{5/2}}{36 (2 x+5)}\right )+\frac {556255 \left (2 x^2-x+3\right )^{5/2}}{72 (2 x+5)^2}}{3456}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}\) |
Input:
Int[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^4,x]
Output:
(-3667*(3 - x + 2*x^2)^(5/2))/(1728*(5 + 2*x)^3) + ((556255*(3 - x + 2*x^2 )^(5/2))/(72*(5 + 2*x)^2) + ((-32865365*(3 - x + 2*x^2)^(5/2))/(36*(5 + 2* x)) + (-2*(138006843 - 34265045*x)*(3 - x + 2*x^2)^(3/2) - 54*(2*(13506860 4 - 22512089*x)*Sqrt[3 - x + 2*x^2] + 9*((-172587879*ArcSinh[(-1 + 4*x)/Sq rt[23]])/Sqrt[2] - (517762327*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(3*Sqrt[2]))))/72)/144)/3456
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]
Timed out.
hanged
Input:
int((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^4,x)
Output:
int((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^4,x)
Time = 0.09 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.01 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^4} \, dx=\frac {517763637 \, \sqrt {2} {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 517762327 \, \sqrt {2} {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )} \log \left (\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 24 \, {\left (46080 \, x^{6} - 315648 \, x^{5} + 2626848 \, x^{4} - 33595416 \, x^{3} - 594798908 \, x^{2} - 2006873194 \, x - 1994650739\right )} \sqrt {2 \, x^{2} - x + 3}}{884736 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} \] Input:
integrate((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^4,x, algorithm=" fricas")
Output:
1/884736*(517763637*sqrt(2)*(8*x^3 + 60*x^2 + 150*x + 125)*log(-4*sqrt(2)* sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 517762327*sqrt(2)*(8 *x^3 + 60*x^2 + 150*x + 125)*log((24*sqrt(2)*sqrt(2*x^2 - x + 3)*(22*x - 1 7) - 1060*x^2 + 1036*x - 1153)/(4*x^2 + 20*x + 25)) + 24*(46080*x^6 - 3156 48*x^5 + 2626848*x^4 - 33595416*x^3 - 594798908*x^2 - 2006873194*x - 19946 50739)*sqrt(2*x^2 - x + 3))/(8*x^3 + 60*x^2 + 150*x + 125)
\[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^4} \, dx=\int \frac {\left (2 x^{2} - x + 3\right )^{\frac {3}{2}} \cdot \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{4}}\, dx \] Input:
integrate((2*x**2-x+3)**(3/2)*(5*x**4-x**3+3*x**2+x+2)/(5+2*x)**4,x)
Output:
Integral((2*x**2 - x + 3)**(3/2)*(5*x**4 - x**3 + 3*x**2 + x + 2)/(2*x + 5 )**4, x)
Time = 0.15 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.04 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^4} \, dx=\frac {5}{64} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {1094743}{497664} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {3667 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1728 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} + \frac {556255 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{248832 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac {22512089}{331776} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {19176431}{16384} \, \sqrt {2} \operatorname {arsinh}\left (\frac {4}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) - \frac {517762327}{442368} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) - \frac {11255717}{27648} \, \sqrt {2 \, x^{2} - x + 3} - \frac {32865365 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{995328 \, {\left (2 \, x + 5\right )}} \] Input:
integrate((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^4,x, algorithm=" maxima")
Output:
5/64*(2*x^2 - x + 3)^(3/2)*x - 1094743/497664*(2*x^2 - x + 3)^(3/2) - 3667 /1728*(2*x^2 - x + 3)^(5/2)/(8*x^3 + 60*x^2 + 150*x + 125) + 556255/248832 *(2*x^2 - x + 3)^(5/2)/(4*x^2 + 20*x + 25) + 22512089/331776*sqrt(2*x^2 - x + 3)*x + 19176431/16384*sqrt(2)*arcsinh(4/23*sqrt(23)*x - 1/23*sqrt(23)) - 517762327/442368*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23* sqrt(23)/abs(2*x + 5)) - 11255717/27648*sqrt(2*x^2 - x + 3) - 32865365/995 328*(2*x^2 - x + 3)^(3/2)/(2*x + 5)
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (146) = 292\).
Time = 0.21 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.73 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^4} \, dx=\frac {1}{4096} \, {\left (4 \, {\left (8 \, {\left (20 \, x - 287\right )} x + 23341\right )} x - 1004633\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {19176431}{16384} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac {517762327}{442368} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {517762327}{442368} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {\sqrt {2} {\left (1092794276 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{5} + 18284336132 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{4} + 20314214356 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} - 151449344092 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 102529692109 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 41882448755\right )}}{36864 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{3}} \] Input:
integrate((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^4,x, algorithm=" giac")
Output:
1/4096*(4*(8*(20*x - 287)*x + 23341)*x - 1004633)*sqrt(2*x^2 - x + 3) - 19 176431/16384*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 517762327/442368*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 517762327/442368*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 1/36864*sqrt(2)*(1092794276*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^5 + 18284336132*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^4 + 20314214356*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^3 - 151449344092*( sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 102529692109*sqrt(2)*(sqrt(2)*x - sqr t(2*x^2 - x + 3)) - 41882448755)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 10*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 11)^3
Timed out. \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^4} \, dx=\int \frac {{\left (2\,x^2-x+3\right )}^{3/2}\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{{\left (2\,x+5\right )}^4} \,d x \] Input:
int(((2*x^2 - x + 3)^(3/2)*(x + 3*x^2 - x^3 + 5*x^4 + 2))/(2*x + 5)^4,x)
Output:
int(((2*x^2 - x + 3)^(3/2)*(x + 3*x^2 - x^3 + 5*x^4 + 2))/(2*x + 5)^4, x)
Time = 0.17 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.14 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^4} \, dx=\frac {552960 \sqrt {2 x^{2}-x +3}\, x^{6}-3787776 \sqrt {2 x^{2}-x +3}\, x^{5}+31522176 \sqrt {2 x^{2}-x +3}\, x^{4}-403144992 \sqrt {2 x^{2}-x +3}\, x^{3}-7137586896 \sqrt {2 x^{2}-x +3}\, x^{2}-24082478328 \sqrt {2 x^{2}-x +3}\, x -23935808868 \sqrt {2 x^{2}-x +3}+4142098616 \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x^{3}+31065739620 \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x^{2}+77664349050 \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x +64720290875 \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right )+4142109096 \sqrt {2}\, \mathrm {log}\left (-2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}-4 x +1\right ) x^{3}+31065818220 \sqrt {2}\, \mathrm {log}\left (-2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}-4 x +1\right ) x^{2}+77664545550 \sqrt {2}\, \mathrm {log}\left (-2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}-4 x +1\right ) x +64720454625 \sqrt {2}\, \mathrm {log}\left (-2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}-4 x +1\right )-4142098616 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x^{3}-31065739620 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x^{2}-77664349050 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x -64720290875 \sqrt {2}\, \mathrm {log}\left (2 x +5\right )}{3538944 x^{3}+26542080 x^{2}+66355200 x +55296000} \] Input:
int((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^4,x)
Output:
(552960*sqrt(2*x**2 - x + 3)*x**6 - 3787776*sqrt(2*x**2 - x + 3)*x**5 + 31 522176*sqrt(2*x**2 - x + 3)*x**4 - 403144992*sqrt(2*x**2 - x + 3)*x**3 - 7 137586896*sqrt(2*x**2 - x + 3)*x**2 - 24082478328*sqrt(2*x**2 - x + 3)*x - 23935808868*sqrt(2*x**2 - x + 3) + 4142098616*sqrt(2)*log( - 12*sqrt(2*x* *2 - x + 3)*sqrt(2) + 22*x - 17)*x**3 + 31065739620*sqrt(2)*log( - 12*sqrt (2*x**2 - x + 3)*sqrt(2) + 22*x - 17)*x**2 + 77664349050*sqrt(2)*log( - 12 *sqrt(2*x**2 - x + 3)*sqrt(2) + 22*x - 17)*x + 64720290875*sqrt(2)*log( - 12*sqrt(2*x**2 - x + 3)*sqrt(2) + 22*x - 17) + 4142109096*sqrt(2)*log( - 2 *sqrt(2*x**2 - x + 3)*sqrt(2) - 4*x + 1)*x**3 + 31065818220*sqrt(2)*log( - 2*sqrt(2*x**2 - x + 3)*sqrt(2) - 4*x + 1)*x**2 + 77664545550*sqrt(2)*log( - 2*sqrt(2*x**2 - x + 3)*sqrt(2) - 4*x + 1)*x + 64720454625*sqrt(2)*log( - 2*sqrt(2*x**2 - x + 3)*sqrt(2) - 4*x + 1) - 4142098616*sqrt(2)*log(2*x + 5)*x**3 - 31065739620*sqrt(2)*log(2*x + 5)*x**2 - 77664349050*sqrt(2)*log (2*x + 5)*x - 64720290875*sqrt(2)*log(2*x + 5))/(442368*(8*x**3 + 60*x**2 + 150*x + 125))