\(\int (3-x+2 x^2)^{3/2} (2+3 x+5 x^2)^4 \, dx\) [99]

Optimal result

Integrand size = 27, antiderivative size = 231 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^4 \, dx=-\frac {26366414481 (1-4 x) \sqrt {3-x+2 x^2}}{2147483648}-\frac {382121949 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{134217728}+\frac {2124689283 \left (3-x+2 x^2\right )^{5/2}}{146800640}+\frac {48669967 x \left (3-x+2 x^2\right )^{5/2}}{22020096}-\frac {56422489 x^2 \left (3-x+2 x^2\right )^{5/2}}{8257536}+\frac {10444117 x^3 \left (3-x+2 x^2\right )^{5/2}}{294912}+\frac {941905 x^4 \left (3-x+2 x^2\right )^{5/2}}{9216}+\frac {95165}{768} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac {7625}{96} x^6 \left (3-x+2 x^2\right )^{5/2}+\frac {625}{24} x^7 \left (3-x+2 x^2\right )^{5/2}-\frac {606427533063 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4294967296 \sqrt {2}} \] Output:

-26366414481/2147483648*(1-4*x)*(2*x^2-x+3)^(1/2)-382121949/134217728*(1-4 
*x)*(2*x^2-x+3)^(3/2)[phone]/146800640*(2*x^2-x+3)^(5/2)[phone]/2202 
0096*x*(2*x^2-x+3)^(5/2)-56422489/8257536*x^2*(2*x^2-x+3)^(5/2)[phone]/2 
94912*x^3*(2*x^2-x+3)^(5/2)+941905/9216*x^4*(2*x^2-x+3)^(5/2)+95165/768*x^ 
5*(2*x^2-x+3)^(5/2)+7625/96*x^6*(2*x^2-x+3)^(5/2)+625/24*x^7*(2*x^2-x+3)^( 
5/2)-606427533063/8589934592*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)
 

Mathematica [A] (verified)

Time = 1.57 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.45 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^4 \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (74032009514181+12971175524316 x+65151998063712 x^2+239021184223104 x^3+451581382260736 x^4+675479464714240 x^5+765087080448000 x^6+745133229998080 x^7+534038708224000 x^8+349379651174400 x^9+144451829760000 x^{10}[phone] x^{11}\right )-191024672914845 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{2705829396480} \] Input:

Integrate[(3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^4,x]
 

Output:

(4*Sqrt[3 - x + 2*x^2]*(74032009514181 + 12971175524316*x + 65151998063712 
*x^2 + 239021184223104*x^3 + 451581382260736*x^4 + 675479464714240*x^5 + 7 
65087080448000*x^6 + 745133229998080*x^7 + 534038708224000*x^8 + 349379651 
174400*x^9 + 144451829760000*x^10 + 70464307200000*x^11) - 191024672914845 
*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/2705829396480
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.19, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.704, Rules used = {2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 1160, 1087, 1087, 1090, 222}

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.

\(\displaystyle \int \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^4 \, dx\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{24} \int \frac {1}{2} \left (2 x^2-x+3\right )^{3/2} \left (83875 x^7+86550 x^6+112320 x^5+84528 x^4+44928 x^3+18048 x^2+4608 x+768\right )dx+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{48} \int \left (2 x^2-x+3\right )^{3/2} \left (83875 x^7+86550 x^6+112320 x^5+84528 x^4+44928 x^3+18048 x^2+4608 x+768\right )dx+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{48} \left (\frac {1}{22} \int \frac {11}{2} \left (2 x^2-x+3\right )^{3/2} \left (475825 x^6+174780 x^5+338112 x^4+179712 x^3+72192 x^2+18432 x+3072\right )dx+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{48} \left (\frac {1}{4} \int \left (2 x^2-x+3\right )^{3/2} \left (475825 x^6+174780 x^5+338112 x^4+179712 x^3+72192 x^2+18432 x+3072\right )dx+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{48} \left (\frac {1}{4} \left (\frac {1}{20} \int \frac {15}{2} \left (2 x^2-x+3\right )^{3/2} \left (941905 x^5-50018 x^4+479232 x^3+192512 x^2+49152 x+8192\right )dx+\frac {95165}{4} \left (2 x^2-x+3\right )^{5/2} x^5\right )+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{48} \left (\frac {1}{4} \left (\frac {3}{8} \int \left (2 x^2-x+3\right )^{3/2} \left (941905 x^5-50018 x^4+479232 x^3+192512 x^2+49152 x+8192\right )dx+\frac {95165}{4} \left (2 x^2-x+3\right )^{5/2} x^5\right )+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{48} \left (\frac {1}{4} \left (\frac {3}{8} \left (\frac {1}{18} \int \frac {1}{2} \left (2 x^2-x+3\right )^{3/2} \left (10444117 x^4-5353368 x^3+6930432 x^2+1769472 x+294912\right )dx+\frac {941905}{18} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {95165}{4} \left (2 x^2-x+3\right )^{5/2} x^5\right )+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{48} \left (\frac {1}{4} \left (\frac {3}{8} \left (\frac {1}{36} \int \left (2 x^2-x+3\right )^{3/2} \left (10444117 x^4-5353368 x^3+6930432 x^2+1769472 x+294912\right )dx+\frac {941905}{18} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {95165}{4} \left (2 x^2-x+3\right )^{5/2} x^5\right )+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{48} \left (\frac {1}{4} \left (\frac {3}{8} \left (\frac {1}{36} \left (\frac {1}{16} \int \frac {1}{2} \left (2 x^2-x+3\right )^{3/2} \left (-56422489 x^3+33779718 x^2+56623104 x+9437184\right )dx+\frac {10444117}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {941905}{18} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {95165}{4} \left (2 x^2-x+3\right )^{5/2} x^5\right )+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{48} \left (\frac {1}{4} \left (\frac {3}{8} \left (\frac {1}{36} \left (\frac {1}{32} \int \left (2 x^2-x+3\right )^{3/2} \left (-56422489 x^3+33779718 x^2+56623104 x+9437184\right )dx+\frac {10444117}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {941905}{18} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {95165}{4} \left (2 x^2-x+3\right )^{5/2} x^5\right )+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{48} \left (\frac {1}{4} \left (\frac {3}{8} \left (\frac {1}{36} \left (\frac {1}{32} \left (\frac {1}{14} \int \frac {3}{2} \left (2 x^2-x+3\right )^{3/2} \left (146009901 x^2+754172260 x+88080384\right )dx-\frac {56422489}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {10444117}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {941905}{18} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {95165}{4} \left (2 x^2-x+3\right )^{5/2} x^5\right )+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{48} \left (\frac {1}{4} \left (\frac {3}{8} \left (\frac {1}{36} \left (\frac {1}{32} \left (\frac {3}{28} \int \left (2 x^2-x+3\right )^{3/2} \left (146009901 x^2+754172260 x+88080384\right )dx-\frac {56422489}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {10444117}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {941905}{18} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {95165}{4} \left (2 x^2-x+3\right )^{5/2} x^5\right )+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{48} \left (\frac {1}{4} \left (\frac {3}{8} \left (\frac {1}{36} \left (\frac {1}{32} \left (\frac {3}{28} \left (\frac {1}{12} \int \frac {27}{2} (708229761 x+45847030) \left (2 x^2-x+3\right )^{3/2}dx+\frac {48669967}{4} x \left (2 x^2-x+3\right )^{5/2}\right )-\frac {56422489}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {10444117}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {941905}{18} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {95165}{4} \left (2 x^2-x+3\right )^{5/2} x^5\right )+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{48} \left (\frac {1}{4} \left (\frac {3}{8} \left (\frac {1}{36} \left (\frac {1}{32} \left (\frac {3}{28} \left (\frac {9}{8} \int (708229761 x+45847030) \left (2 x^2-x+3\right )^{3/2}dx+\frac {48669967}{4} x \left (2 x^2-x+3\right )^{5/2}\right )-\frac {56422489}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {10444117}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {941905}{18} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {95165}{4} \left (2 x^2-x+3\right )^{5/2} x^5\right )+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{48} \left (\frac {1}{4} \left (\frac {3}{8} \left (\frac {1}{36} \left (\frac {1}{32} \left (\frac {3}{28} \left (\frac {9}{8} \left (\frac {891617881}{4} \int \left (2 x^2-x+3\right )^{3/2}dx+\frac {708229761}{10} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {48669967}{4} x \left (2 x^2-x+3\right )^{5/2}\right )-\frac {56422489}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {10444117}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {941905}{18} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {95165}{4} \left (2 x^2-x+3\right )^{5/2} x^5\right )+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{48} \left (\frac {1}{4} \left (\frac {3}{8} \left (\frac {1}{36} \left (\frac {1}{32} \left (\frac {3}{28} \left (\frac {9}{8} \left (\frac {891617881}{4} \left (\frac {69}{32} \int \sqrt {2 x^2-x+3}dx-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {708229761}{10} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {48669967}{4} x \left (2 x^2-x+3\right )^{5/2}\right )-\frac {56422489}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {10444117}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {941905}{18} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {95165}{4} \left (2 x^2-x+3\right )^{5/2} x^5\right )+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{48} \left (\frac {1}{4} \left (\frac {3}{8} \left (\frac {1}{36} \left (\frac {1}{32} \left (\frac {3}{28} \left (\frac {9}{8} \left (\frac {891617881}{4} \left (\frac {69}{32} \left (\frac {23}{16} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {708229761}{10} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {48669967}{4} x \left (2 x^2-x+3\right )^{5/2}\right )-\frac {56422489}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {10444117}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {941905}{18} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {95165}{4} \left (2 x^2-x+3\right )^{5/2} x^5\right )+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{48} \left (\frac {1}{4} \left (\frac {3}{8} \left (\frac {1}{36} \left (\frac {1}{32} \left (\frac {3}{28} \left (\frac {9}{8} \left (\frac {891617881}{4} \left (\frac {69}{32} \left (\frac {1}{16} \sqrt {\frac {23}{2}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {708229761}{10} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {48669967}{4} x \left (2 x^2-x+3\right )^{5/2}\right )-\frac {56422489}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {10444117}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {941905}{18} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {95165}{4} \left (2 x^2-x+3\right )^{5/2} x^5\right )+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{48} \left (\frac {1}{4} \left (\frac {3}{8} \left (\frac {1}{36} \left (\frac {1}{32} \left (\frac {3}{28} \left (\frac {9}{8} \left (\frac {891617881}{4} \left (\frac {69}{32} \left (\frac {23 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{16 \sqrt {2}}-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {708229761}{10} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {48669967}{4} x \left (2 x^2-x+3\right )^{5/2}\right )-\frac {56422489}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {10444117}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {941905}{18} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {95165}{4} \left (2 x^2-x+3\right )^{5/2} x^5\right )+\frac {7625}{2} \left (2 x^2-x+3\right )^{5/2} x^6\right )+\frac {625}{24} \left (2 x^2-x+3\right )^{5/2} x^7\)

Input:

Int[(3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^4,x]
 

Output:

(625*x^7*(3 - x + 2*x^2)^(5/2))/24 + ((7625*x^6*(3 - x + 2*x^2)^(5/2))/2 + 
 ((95165*x^5*(3 - x + 2*x^2)^(5/2))/4 + (3*((941905*x^4*(3 - x + 2*x^2)^(5 
/2))/18 + ((10444117*x^3*(3 - x + 2*x^2)^(5/2))/16 + ((-56422489*x^2*(3 - 
x + 2*x^2)^(5/2))/14 + (3*((48669967*x*(3 - x + 2*x^2)^(5/2))/4 + (9*((708 
229761*(3 - x + 2*x^2)^(5/2))/10 + (891617881*(-1/16*((1 - 4*x)*(3 - x + 2 
*x^2)^(3/2)) + (69*(-1/8*((1 - 4*x)*Sqrt[3 - x + 2*x^2]) + (23*ArcSinh[(-1 
 + 4*x)/Sqrt[23]])/(16*Sqrt[2])))/32))/4))/8))/28)/32)/36))/8)/4)/48
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 222

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]
 

rule 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1090

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]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 2.43 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.37

method	result	size
risch	\(\frac {\left (70464307200000 x^{11}[phone] x^{10}[phone] x^{9}[phone] x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {2 x^{2}-x +3}}{676457349120}+\frac {606427533063 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8589934592}\)	\(85\)
trager	\(\left (\frac {625}{6} x^{11}+\frac {5125}{24} x^{10}+\frac {33055}{64} x^{9}+\frac {1818925}{2304} x^{8}+\frac {81213077}{73728} x^{7}+\frac {778286825}{688128} x^{6}+\frac {16491197869}{16515072} x^{5}+\frac {31499817401}{47185920} x^{4}+\frac {622451000581}{1761607680} x^{3}+\frac {32317459357}{335544320} x^{2}+\frac {360310431231}{18790481920} x +\frac {8225778834909}{75161927680}\right ) \sqrt {2 x^{2}-x +3}+\frac {606427533063 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{8589934592}\)	\(111\)
default	\(\frac {382121949 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{134217728}+\frac {26366414481 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{2147483648}+\frac {606427533063 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8589934592}+\frac {2124689283 \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{146800640}+\frac {48669967 x \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{22020096}-\frac {56422489 x^{2} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{8257536}+\frac {10444117 x^{3} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{294912}+\frac {941905 x^{4} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{9216}+\frac {95165 x^{5} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{768}+\frac {7625 x^{6} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{96}+\frac {625 x^{7} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{24}\)	\(185\)

Input:

int((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^4,x,method=_RETURNVERBOSE)
 

Output:

1/676457349120*(70464307200000*x^11+144451829760000*x^10+349379651174400*x 
^9+534038708224000*x^8+745133229998080*x^7+765087080448000*x^6+67547946471 
4240*x^5+451581382260736*x^4+239021184223104*x^3+65151998063712*x^2+129711 
75524316*x[phone])*(2*x^2-x+3)^(1/2)[phone]/8589934592*2^(1/ 
2)*arcsinh(4/23*23^(1/2)*(x-1/4))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.47 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^4 \, dx=\frac {1}{676457349120} \, {\left (70464307200000 \, x^{11} + 144451829760000 \, x^{10} + 349379651174400 \, x^{9} + 534038708224000 \, x^{8} + 745133229998080 \, x^{7} + 765087080448000 \, x^{6} + 675479464714240 \, x^{5} + 451581382260736 \, x^{4} + 239021184223104 \, x^{3} + 65151998063712 \, x^{2} + 12971175524316 \, x + 74032009514181\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {606427533063}{17179869184} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \] Input:

integrate((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^4,x, algorithm="fricas")
 

Output:

1/676457349120*(70464307200000*x^11 + 144451829760000*x^10 + 3493796511744 
00*x^9 + 534038708224000*x^8 + 745133229998080*x^7 + 765087080448000*x^6 + 
 675479464714240*x^5 + 451581382260736*x^4 + 239021184223104*x^3 + 6515199 
8063712*x^2 + 12971175524316*x + 74032009514181)*sqrt(2*x^2 - x + 3) + 606 
427533063/17179869184*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) 
 - 32*x^2 + 16*x - 25)
 

Sympy [A] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.48 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^4 \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (\frac {625 x^{11}}{6} + \frac {5125 x^{10}}{24} + \frac {33055 x^{9}}{64} + \frac {1818925 x^{8}}{2304} + \frac {81213077 x^{7}}{73728} + \frac {778286825 x^{6}}{688128} + \frac {16491197869 x^{5}}{16515072} + \frac {31499817401 x^{4}}{47185920} + \frac {622451000581 x^{3}}{1761607680} + \frac {32317459357 x^{2}}{335544320} + \frac {360310431231 x}{18790481920} + \frac {8225778834909}{75161927680}\right ) + \frac {606427533063 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{8589934592} \] Input:

integrate((2*x**2-x+3)**(3/2)*(5*x**2+3*x+2)**4,x)
 

Output:

sqrt(2*x**2 - x + 3)*(625*x**11/6 + 5125*x**10/24 + 33055*x**9/64 + 181892 
5*x**8/2304 + 81213077*x**7/73728 + 778286825*x**6/688128 + 16491197869*x* 
*5/16515072 + 31499817401*x**4/47185920 + 622451000581*x**3/1761607680 + 3 
2317459357*x**2/335544320 + 360310431231*x/18790481920 + 8225778834909/751 
61927680) + 606427533063*sqrt(2)*asinh(4*sqrt(23)*(x - 1/4)/23)/8589934592
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.89 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^4 \, dx=\frac {625}{24} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{7} + \frac {7625}{96} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{6} + \frac {95165}{768} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{5} + \frac {941905}{9216} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{4} + \frac {10444117}{294912} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{3} - \frac {56422489}{8257536} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} + \frac {48669967}{22020096} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {2124689283}{146800640} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {382121949}{33554432} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {382121949}{134217728} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {26366414481}{536870912} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {606427533063}{8589934592} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {26366414481}{2147483648} \, \sqrt {2 \, x^{2} - x + 3} \] Input:

integrate((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^4,x, algorithm="maxima")
 

Output:

625/24*(2*x^2 - x + 3)^(5/2)*x^7 + 7625/96*(2*x^2 - x + 3)^(5/2)*x^6 + 951 
65/768*(2*x^2 - x + 3)^(5/2)*x^5 + 941905/9216*(2*x^2 - x + 3)^(5/2)*x^4 + 
 10444117/294912*(2*x^2 - x + 3)^(5/2)*x^3 - 56422489/8257536*(2*x^2 - x + 
 3)^(5/2)*x^2 + 48669967/22020096*(2*x^2 - x + 3)^(5/2)*x + 2124689283/146 
800640*(2*x^2 - x + 3)^(5/2) + 382121949/33554432*(2*x^2 - x + 3)^(3/2)*x 
- 382121949/134217728*(2*x^2 - x + 3)^(3/2) + 26366414481/536870912*sqrt(2 
*x^2 - x + 3)*x + 606427533063/8589934592*sqrt(2)*arcsinh(1/23*sqrt(23)*(4 
*x - 1)) - 26366414481/2147483648*sqrt(2*x^2 - x + 3)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.45 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^4 \, dx=\frac {1}{676457349120} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (8 \, {\left (28 \, {\left (160 \, {\left (12 \, {\left (200 \, {\left (20 \, x + 41\right )} x + 19833\right )} x + 363785\right )} x + 81213077\right )} x + 2334860475\right )} x + 16491197869\right )} x + 220498721807\right )} x + 1867353001743\right )} x + 2035999939491\right )} x + 3242793881079\right )} x + 74032009514181\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {606427533063}{8589934592} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \] Input:

integrate((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^4,x, algorithm="giac")
 

Output:

1/676457349120*(4*(8*(4*(16*(20*(8*(28*(160*(12*(200*(20*x + 41)*x + 19833 
)*x + 363785)*x + 81213077)*x + 2334860475)*x + 16491197869)*x + 220498721 
807)*x + 1867353001743)*x + 2035999939491)*x + 3242793881079)*x + 74032009 
514181)*sqrt(2*x^2 - x + 3) - 606427533063/8589934592*sqrt(2)*log(-2*sqrt( 
2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)
 

Mupad [F(-1)]

Timed out. \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^4 \, dx=\int {\left (2\,x^2-x+3\right )}^{3/2}\,{\left (5\,x^2+3\,x+2\right )}^4 \,d x \] Input:

int((2*x^2 - x + 3)^(3/2)*(3*x + 5*x^2 + 2)^4,x)
 

Output:

int((2*x^2 - x + 3)^(3/2)*(3*x + 5*x^2 + 2)^4, x)
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.94 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^4 \, dx=\frac {625 \sqrt {2 x^{2}-x +3}\, x^{11}}{6}+\frac {5125 \sqrt {2 x^{2}-x +3}\, x^{10}}{24}+\frac {33055 \sqrt {2 x^{2}-x +3}\, x^{9}}{64}+\frac {1818925 \sqrt {2 x^{2}-x +3}\, x^{8}}{2304}+\frac {81213077 \sqrt {2 x^{2}-x +3}\, x^{7}}{73728}+\frac {778286825 \sqrt {2 x^{2}-x +3}\, x^{6}}{688128}+\frac {16491197869 \sqrt {2 x^{2}-x +3}\, x^{5}}{16515072}+\frac {31499817401 \sqrt {2 x^{2}-x +3}\, x^{4}}{47185920}+\frac {622451000581 \sqrt {2 x^{2}-x +3}\, x^{3}}{1761607680}+\frac {32317459357 \sqrt {2 x^{2}-x +3}\, x^{2}}{335544320}+\frac {360310431231 \sqrt {2 x^{2}-x +3}\, x}{18790481920}+\frac {8225778834909 \sqrt {2 x^{2}-x +3}}{75161927680}+\frac {606427533063 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right )}{8589934592} \] Input:

int((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^4,x)
 

Output:

(281857228800000*sqrt(2*x**2 - x + 3)*x**11 + 577807319040000*sqrt(2*x**2 
- x + 3)*x**10 + 1397518604697600*sqrt(2*x**2 - x + 3)*x**9 + 213615483289 
6000*sqrt(2*x**2 - x + 3)*x**8 + 2980532919992320*sqrt(2*x**2 - x + 3)*x** 
7 + 3060348321792000*sqrt(2*x**2 - x + 3)*x**6 + 2701917858856960*sqrt(2*x 
**2 - x + 3)*x**5 + 1806325529042944*sqrt(2*x**2 - x + 3)*x**4 + 956084736 
892416*sqrt(2*x**2 - x + 3)*x**3 + 260607992254848*sqrt(2*x**2 - x + 3)*x* 
*2 + 51884702097264*sqrt(2*x**2 - x + 3)*x + 296128038056724*sqrt(2*x**2 - 
 x + 3) + 191024672914845*sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4* 
x - 1)/sqrt(23)))/2705829396480