Integrand size = 26, antiderivative size = 267 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^9} \, dx=-\frac {75045071 \sqrt {1-2 x} \sqrt {3+5 x}}{85349376 (2+3 x)^4}+\frac {372439373 \sqrt {1-2 x} \sqrt {3+5 x}}{512096256 (2+3 x)^3}+\frac {64983635965 \sqrt {1-2 x} \sqrt {3+5 x}}{14338695168 (2+3 x)^2}+\frac {6796051494355 \sqrt {1-2 x} \sqrt {3+5 x}}{200741732352 (2+3 x)}-\frac {720833 \sqrt {1-2 x} (3+5 x)^{3/2}}{508032 (2+3 x)^5}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{24 (2+3 x)^8}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1008 (2+3 x)^7}+\frac {47365 \sqrt {1-2 x} (3+5 x)^{5/2}}{36288 (2+3 x)^6}-\frac {106656830005 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{275365888 \sqrt {7}} \]
-1/24*(1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^8+185/1008*(1-2*x)^(3/2)*(3+5*x) ^(5/2)/(2+3*x)^7-106656830005/1927561216*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/ (3+5*x)^(1/2))*7^(1/2)-720833/508032*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^5 +47365/36288*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^6-75045071/85349376*(1-2* x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+372439373/512096256*(1-2*x)^(1/2)*(3+5*x) ^(1/2)/(2+3*x)^3+64983635965/14338695168*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3* x)^2+6796051494355/200741732352*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
Time = 0.68 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.38 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^9} \, dx=\frac {1771561 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (3840133416192+39899303549504 x+177688060285568 x^2+439702534402320 x^3+652979564561296 x^4+581931572602156 x^5+288163475473440 x^6+61164463449195 x^7\right )}{1771561 (2+3 x)^8}-180615 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{5782683648} \]
(1771561*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3840133416192 + 39899303549504*x + 177688060285568*x^2 + 439702534402320*x^3 + 652979564561296*x^4 + 58193 1572602156*x^5 + 288163475473440*x^6 + 61164463449195*x^7))/(1771561*(2 + 3*x)^8) - 180615*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/5 782683648
Time = 0.34 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.13, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {108, 27, 166, 27, 166, 27, 166, 27, 166, 27, 168, 27, 168, 27, 168, 27, 104, 217}
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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^9} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{24} \int -\frac {5 (1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{2 (3 x+2)^8}dx-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{48} \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{(3 x+2)^8}dx-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {5}{48} \left (-\frac {1}{21} \int \frac {(2051-1660 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^7}dx-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{48} \left (-\frac {1}{42} \int \frac {(2051-1660 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{(3 x+2)^7}dx-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {5}{48} \left (\frac {1}{42} \left (\frac {1}{18} \int -\frac {(368473-528540 x) (5 x+3)^{3/2}}{2 \sqrt {1-2 x} (3 x+2)^6}dx-\frac {9473 \sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{48} \left (\frac {1}{42} \left (-\frac {1}{36} \int \frac {(368473-528540 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^6}dx-\frac {9473 \sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {5}{48} \left (\frac {1}{42} \left (\frac {1}{36} \left (\frac {720833 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}-\frac {1}{105} \int \frac {3 (12739677-18413020 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^5}dx\right )-\frac {9473 \sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{48} \left (\frac {1}{42} \left (\frac {1}{36} \left (\frac {720833 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}-\frac {1}{70} \int \frac {(12739677-18413020 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^5}dx\right )-\frac {9473 \sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {5}{48} \left (\frac {1}{42} \left (\frac {1}{36} \left (\frac {1}{70} \left (\frac {75045071 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}-\frac {1}{84} \int \frac {433730977-652941340 x}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx\right )+\frac {720833 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {9473 \sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{48} \left (\frac {1}{42} \left (\frac {1}{36} \left (\frac {1}{70} \left (\frac {75045071 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}-\frac {1}{168} \int \frac {433730977-652941340 x}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx\right )+\frac {720833 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {9473 \sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {5}{48} \left (\frac {1}{42} \left (\frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{168} \left (-\frac {1}{21} \int \frac {35 (2345899075-2979514984 x)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {372439373 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {75045071 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {720833 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {9473 \sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{48} \left (\frac {1}{42} \left (\frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{168} \left (-\frac {5}{6} \int \frac {2345899075-2979514984 x}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {372439373 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {75045071 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {720833 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {9473 \sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {5}{48} \left (\frac {1}{42} \left (\frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{168} \left (-\frac {5}{6} \left (\frac {1}{14} \int \frac {279780403717-259934543860 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {12996727193 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {372439373 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {75045071 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {720833 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {9473 \sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{48} \left (\frac {1}{42} \left (\frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{168} \left (-\frac {5}{6} \left (\frac {1}{28} \int \frac {279780403717-259934543860 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {12996727193 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {372439373 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {75045071 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {720833 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {9473 \sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {5}{48} \left (\frac {1}{42} \left (\frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{168} \left (-\frac {5}{6} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {15550565814729}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {1359210298871 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {12996727193 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {372439373 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {75045071 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {720833 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {9473 \sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{48} \left (\frac {1}{42} \left (\frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{168} \left (-\frac {5}{6} \left (\frac {1}{28} \left (\frac {15550565814729}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {1359210298871 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {12996727193 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {372439373 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {75045071 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {720833 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {9473 \sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {5}{48} \left (\frac {1}{42} \left (\frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{168} \left (-\frac {5}{6} \left (\frac {1}{28} \left (\frac {15550565814729}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {1359210298871 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {12996727193 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {372439373 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {75045071 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {720833 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {9473 \sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {5}{48} \left (\frac {1}{42} \left (\frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{168} \left (-\frac {5}{6} \left (\frac {1}{28} \left (\frac {1359210298871 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {15550565814729 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {12996727193 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {372439373 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {75045071 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {720833 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {9473 \sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}\) |
-1/24*((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^8 - (5*((-37*(1 - 2*x)^( 3/2)*(3 + 5*x)^(5/2))/(21*(2 + 3*x)^7) + ((-9473*Sqrt[1 - 2*x]*(3 + 5*x)^( 5/2))/(18*(2 + 3*x)^6) + ((720833*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(35*(2 + 3*x)^5) + ((75045071*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84*(2 + 3*x)^4) + ((-37 2439373*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) - (5*((12996727193*Sq rt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((1359210298871*Sqrt[1 - 2*x ]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (15550565814729*ArcTan[Sqrt[1 - 2*x]/(Sqr t[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/6)/168)/70)/36)/42))/48
3.25.23.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Time = 7.91 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.56
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (61164463449195 x^{7}+288163475473440 x^{6}+581931572602156 x^{5}+652979564561296 x^{4}+439702534402320 x^{3}+177688060285568 x^{2}+39899303549504 x +3840133416192\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{826097664 \left (2+3 x \right )^{8} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {106656830005 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{3855122432 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(149\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (2099326384988415 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{8}+11196407386604880 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{7}+26124950568744720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}+856302488288730 \sqrt {-10 x^{2}-x +3}\, x^{7}+34833267424992960 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+4034288656628160 \sqrt {-10 x^{2}-x +3}\, x^{6}+29027722854160800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+8147042016430184 x^{5} \sqrt {-10 x^{2}-x +3}+15481452188885760 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+9141713903858144 x^{4} \sqrt {-10 x^{2}-x +3}+5160484062961920 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+6155835481632480 x^{3} \sqrt {-10 x^{2}-x +3}+982949345326080 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +2487632843997952 x^{2} \sqrt {-10 x^{2}-x +3}+81912445443840 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+558590249693056 x \sqrt {-10 x^{2}-x +3}+53761867826688 \sqrt {-10 x^{2}-x +3}\right )}{11565367296 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{8}}\) | \(442\) |
-1/826097664*(-1+2*x)*(3+5*x)^(1/2)*(61164463449195*x^7+288163475473440*x^ 6+581931572602156*x^5+652979564561296*x^4+439702534402320*x^3+177688060285 568*x^2+39899303549504*x+3840133416192)/(2+3*x)^8/(-(-1+2*x)*(3+5*x))^(1/2 )*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+106656830005/3855122432*7^(1/2)*ar ctan(9/14*(20/3+37/3*x)*7^(1/2)/(-90*(2/3+x)^2+67+111*x)^(1/2))*((1-2*x)*( 3+5*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.25 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^9} \, dx=-\frac {319970490015 \, \sqrt {7} {\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (61164463449195 \, x^{7} + 288163475473440 \, x^{6} + 581931572602156 \, x^{5} + 652979564561296 \, x^{4} + 439702534402320 \, x^{3} + 177688060285568 \, x^{2} + 39899303549504 \, x + 3840133416192\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{11565367296 \, {\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\right )}} \]
-1/11565367296*(319970490015*sqrt(7)*(6561*x^8 + 34992*x^7 + 81648*x^6 + 1 08864*x^5 + 90720*x^4 + 48384*x^3 + 16128*x^2 + 3072*x + 256)*arctan(1/14* sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(6 1164463449195*x^7 + 288163475473440*x^6 + 581931572602156*x^5 + 6529795645 61296*x^4 + 439702534402320*x^3 + 177688060285568*x^2 + 39899303549504*x + 3840133416192)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(6561*x^8 + 34992*x^7 + 8164 8*x^6 + 108864*x^5 + 90720*x^4 + 48384*x^3 + 16128*x^2 + 3072*x + 256)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^9} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.53 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^9} \, dx=\frac {39793036595}{30359089152} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{56 \, {\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\right )}} + \frac {999 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{5488 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {12041 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{21952 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {445517 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{307328 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {52823867 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{17210368 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {984147053 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{240945152 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {7958607319 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{6746464256 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {712927441325}{20239392768} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {1368574460935}{40478785536} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {1321083986311 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{121436356608 \, {\left (3 \, x + 2\right )}} + \frac {163070359925}{963780608} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {106656830005}{3855122432} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {143678209015}{1927561216} \, \sqrt {-10 \, x^{2} - x + 3} \]
39793036595/30359089152*(-10*x^2 - x + 3)^(5/2) + 3/56*(-10*x^2 - x + 3)^( 7/2)/(6561*x^8 + 34992*x^7 + 81648*x^6 + 108864*x^5 + 90720*x^4 + 48384*x^ 3 + 16128*x^2 + 3072*x + 256) + 999/5488*(-10*x^2 - x + 3)^(7/2)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128 ) + 12041/21952*(-10*x^2 - x + 3)^(7/2)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4 320*x^3 + 2160*x^2 + 576*x + 64) + 445517/307328*(-10*x^2 - x + 3)^(7/2)/( 243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 52823867/17210368*( -10*x^2 - x + 3)^(7/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 98414705 3/240945152*(-10*x^2 - x + 3)^(7/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 7958607 319/6746464256*(-10*x^2 - x + 3)^(7/2)/(9*x^2 + 12*x + 4) - 712927441325/2 0239392768*(-10*x^2 - x + 3)^(3/2)*x + 1368574460935/40478785536*(-10*x^2 - x + 3)^(3/2) - 1321083986311/121436356608*(-10*x^2 - x + 3)^(5/2)/(3*x + 2) + 163070359925/963780608*sqrt(-10*x^2 - x + 3)*x + 106656830005/385512 2432*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 143678209 015/1927561216*sqrt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (210) = 420\).
Time = 0.99 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.25 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^9} \, dx=\frac {21331366001}{7710244864} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {8857805 \, \sqrt {10} {\left (36123 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{15} + 77544040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{13} + 72311503040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} - 37368091174400 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 10615979648512000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 1587382114734080000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 133456146460672000000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {4874050566389760000000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {19496202265559040000000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{413048832 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{8}} \]
21331366001/7710244864*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sq rt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2 )*sqrt(-10*x + 5) - sqrt(22)))) - 8857805/413048832*sqrt(10)*(36123*((sqrt (2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*s qrt(-10*x + 5) - sqrt(22)))^15 + 77544040*((sqrt(2)*sqrt(-10*x + 5) - sqrt (22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)) )^13 + 72311503040*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4 *sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^11 - 37368091174400*( (sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt (2)*sqrt(-10*x + 5) - sqrt(22)))^9 - 10615979648512000*((sqrt(2)*sqrt(-10* x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5 ) - sqrt(22)))^7 - 1587382114734080000*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22 ))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 133456146460672000000*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 48740505663 89760000000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 194962022 65559040000000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt (2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*s qrt(-10*x + 5) - sqrt(22)))^2 + 280)^8
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^9} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^9} \,d x \]