Integrand size = 29, antiderivative size = 256 \[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {26}{25 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}-\frac {538}{125 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}-\frac {21534}{625 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}-\frac {2 \sqrt {3+2 x} (33946+102381 x)}{3125 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 \sqrt {3+2 x} (255821+80661 x)}{15625 \sqrt {2+5 x+3 x^2}}-\frac {107548 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{15625 \sqrt {2+5 x+3 x^2}}+\frac {129268 \sqrt {3} \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{3125 \sqrt {2+5 x+3 x^2}} \] Output:
-26/25/(3+2*x)^(5/2)/(3*x^2+5*x+2)^(3/2)-538/125/(3+2*x)^(3/2)/(3*x^2+5*x+ 2)^(3/2)-21534/625/(3+2*x)^(1/2)/(3*x^2+5*x+2)^(3/2)-2/3125*(3+2*x)^(1/2)* (33946+102381*x)/(3*x^2+5*x+2)^(3/2)+4/15625*(3+2*x)^(1/2)*(255821+80661*x )/(3*x^2+5*x+2)^(1/2)-107548/15625*(-3*x^2-5*x-2)^(1/2)*EllipticE((1+x)^(1 /2)*3^(1/2),1/3*I*6^(1/2))*3^(1/2)/(3*x^2+5*x+2)^(1/2)+129268/3125*(-3*x^2 -5*x-2)^(1/2)*EllipticF((1+x)^(1/2)*3^(1/2),1/3*I*6^(1/2))*3^(1/2)/(3*x^2+ 5*x+2)^(1/2)
Time = 31.66 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.89 \[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \left (20514383+106756189 x+220795962 x^2+231620622 x^3+129381052 x^4+36155064 x^5+3871728 x^6-2 (3+2 x)^2 \left (2+5 x+3 x^2\right ) \left (53774 \left (2+5 x+3 x^2\right )+26887 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )+70064 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )\right )\right )}{15625 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}} \] Input:
Integrate[(5 - x)/((3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2)^(5/2)),x]
Output:
(2*(20514383 + 106756189*x + 220795962*x^2 + 231620622*x^3 + 129381052*x^4 + 36155064*x^5 + 3871728*x^6 - 2*(3 + 2*x)^2*(2 + 5*x + 3*x^2)*(53774*(2 + 5*x + 3*x^2) + 26887*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqr t[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] + 7 0064*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2 *x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])))/(15625*(3 + 2*x)^( 5/2)*(2 + 5*x + 3*x^2)^(3/2))
Time = 0.83 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.08, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {1235, 27, 1235, 27, 1237, 27, 1237, 27, 1237, 27, 1269, 1172, 27, 321, 327}
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.
| \(\displaystyle \int \frac {5-x}{(2 x+3)^{7/2} \left (3 x^2+5 x+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {2}{15} \int \frac {3 (423 x+454)}{(2 x+3)^{7/2} \left (3 x^2+5 x+2\right )^{3/2}}dx-\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \int \frac {423 x+454}{(2 x+3)^{7/2} \left (3 x^2+5 x+2\right )^{3/2}}dx-\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {2}{5} \int \frac {3 (3355 x+3217)}{(2 x+3)^{7/2} \sqrt {3 x^2+5 x+2}}dx-\frac {2 (2013 x+1858)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {6}{5} \int \frac {3355 x+3217}{(2 x+3)^{7/2} \sqrt {3 x^2+5 x+2}}dx-\frac {2 (2013 x+1858)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {6}{5} \left (\frac {7262 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}-\frac {2}{25} \int -\frac {32679 x+32860}{2 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {2 (2013 x+1858)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {6}{5} \left (\frac {1}{25} \int \frac {32679 x+32860}{(2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}dx+\frac {7262 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {2 (2013 x+1858)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {6}{5} \left (\frac {1}{25} \left (\frac {64634 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}-\frac {2}{15} \int -\frac {96951 x+131983}{2 (2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {7262 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {2 (2013 x+1858)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {6}{5} \left (\frac {1}{25} \left (\frac {1}{15} \int \frac {96951 x+131983}{(2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}}dx+\frac {64634 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )+\frac {7262 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {2 (2013 x+1858)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {6}{5} \left (\frac {1}{25} \left (\frac {1}{15} \left (\frac {53774 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {2}{5} \int -\frac {3 (40462-26887 x)}{2 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {64634 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )+\frac {7262 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {2 (2013 x+1858)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {6}{5} \left (\frac {1}{25} \left (\frac {1}{15} \left (\frac {3}{5} \int \frac {40462-26887 x}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx+\frac {53774 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )+\frac {64634 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )+\frac {7262 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {2 (2013 x+1858)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {6}{5} \left (\frac {1}{25} \left (\frac {1}{15} \left (\frac {3}{5} \left (\frac {161585}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {26887}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {53774 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )+\frac {64634 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )+\frac {7262 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {2 (2013 x+1858)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {6}{5} \left (\frac {1}{25} \left (\frac {1}{15} \left (\frac {3}{5} \left (\frac {161585 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {26887 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {53774 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )+\frac {64634 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )+\frac {7262 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {2 (2013 x+1858)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {6}{5} \left (\frac {1}{25} \left (\frac {1}{15} \left (\frac {3}{5} \left (\frac {161585 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {26887 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )+\frac {53774 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )+\frac {64634 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )+\frac {7262 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {2 (2013 x+1858)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {6}{5} \left (\frac {1}{25} \left (\frac {1}{15} \left (\frac {3}{5} \left (\frac {161585 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {26887 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )+\frac {53774 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )+\frac {64634 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )+\frac {7262 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {2 (2013 x+1858)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {6}{5} \left (\frac {1}{25} \left (\frac {1}{15} \left (\frac {3}{5} \left (\frac {161585 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {26887 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {53774 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )+\frac {64634 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )+\frac {7262 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {2 (2013 x+1858)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
Input:
Int[(5 - x)/((3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2)^(5/2)),x]
Output:
(-2*(37 + 47*x))/(5*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^(3/2)) - (2*((-2*(18 58 + 2013*x))/(5*(3 + 2*x)^(5/2)*Sqrt[2 + 5*x + 3*x^2]) - (6*((7262*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)^(5/2)) + ((64634*Sqrt[2 + 5*x + 3*x^2])/(15 *(3 + 2*x)^(3/2)) + ((53774*Sqrt[2 + 5*x + 3*x^2])/(5*Sqrt[3 + 2*x]) + (3* ((-26887*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/ 3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (161585*Sqrt[-2 - 5*x - 3*x^2]*Ellip ticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]))) /5)/15)/25))/5))/5
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (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[(e*f - d*g)*(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*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (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]
Time = 2.14 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.20
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}\, \left (-\frac {52 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{625 \left (x +\frac {3}{2}\right )^{3}}-\frac {3816 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{3125 \left (x +\frac {3}{2}\right )^{2}}-\frac {274832 \left (6 x^{2}+10 x +4\right )}{15625 \sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}+\frac {\left (-\frac {5918}{5625}-\frac {2806 x}{1875}\right ) \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 \left (6 x +9\right ) \left (-\frac {40598}{3125}-\frac {38238 x}{3125}\right )}{\sqrt {\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right ) \left (6 x +9\right )}}-\frac {161848 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{78125 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {107548 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{78125 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\) | \(307\) |
| default | \(-\frac {2 \sqrt {3 x^{2}+5 x +2}\, \left (4849128 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{4} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-645288 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{4} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+22629264 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {3+3 x}\, \sqrt {2 x +3}\, \sqrt {-30 x -20}-3011344 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {3+3 x}\, \sqrt {2 x +3}\, \sqrt {-30 x -20}+38388930 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-5108530 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+27882486 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-3710406 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-58075920 x^{6}+7273692 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-967932 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-542325960 x^{5}-1940715780 x^{4}-3474309330 x^{3}-3311939430 x^{2}-1601342835 x -307715745\right )}{234375 \left (2 x +3\right )^{\frac {5}{2}} \left (x +1\right )^{2} \left (3 x +2\right )^{2}}\) | \(494\) |
Input:
int((5-x)/(2*x+3)^(7/2)/(3*x^2+5*x+2)^(5/2),x,method=_RETURNVERBOSE)
Output:
((3*x^2+5*x+2)*(2*x+3))^(1/2)/(2*x+3)^(1/2)/(3*x^2+5*x+2)^(1/2)*(-52/625*( 6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^3-3816/3125*(6*x^3+19*x^2+19*x+6)^(1/2) /(x+3/2)^2-274832/15625*(6*x^2+10*x+4)/((x+3/2)*(6*x^2+10*x+4))^(1/2)+(-59 18/5625-2806/1875*x)*(6*x^3+19*x^2+19*x+6)^(1/2)/(x^2+5/3*x+2/3)^2-2*(6*x+ 9)*(-40598/3125-38238/3125*x)/((x^2+5/3*x+2/3)*(6*x+9))^(1/2)-161848/78125 *(-30*x-20)^(1/2)*(3+3*x)^(1/2)*(30*x+45)^(1/2)/(6*x^3+19*x^2+19*x+6)^(1/2 )*EllipticF(1/5*(-30*x-20)^(1/2),1/2*10^(1/2))+107548/78125*(-30*x-20)^(1/ 2)*(3+3*x)^(1/2)*(30*x+45)^(1/2)/(6*x^3+19*x^2+19*x+6)^(1/2)*(1/3*Elliptic E(1/5*(-30*x-20)^(1/2),1/2*10^(1/2))-EllipticF(1/5*(-30*x-20)^(1/2),1/2*10 ^(1/2))))
Time = 0.12 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.73 \[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (1239169 \, \sqrt {6} {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 483966 \, \sqrt {6} {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 9 \, {\left (3871728 \, x^{6} + 36155064 \, x^{5} + 129381052 \, x^{4} + 231620622 \, x^{3} + 220795962 \, x^{2} + 106756189 \, x + 20514383\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}\right )}}{140625 \, {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )}} \] Input:
integrate((5-x)/(3+2*x)^(7/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")
Output:
2/140625*(1239169*sqrt(6)*(72*x^7 + 564*x^6 + 1862*x^5 + 3355*x^4 + 3560*x ^3 + 2223*x^2 + 756*x + 108)*weierstrassPInverse(19/27, -28/729, x + 19/18 ) + 483966*sqrt(6)*(72*x^7 + 564*x^6 + 1862*x^5 + 3355*x^4 + 3560*x^3 + 22 23*x^2 + 756*x + 108)*weierstrassZeta(19/27, -28/729, weierstrassPInverse( 19/27, -28/729, x + 19/18)) + 9*(3871728*x^6 + 36155064*x^5 + 129381052*x^ 4 + 231620622*x^3 + 220795962*x^2 + 106756189*x + 20514383)*sqrt(3*x^2 + 5 *x + 2)*sqrt(2*x + 3))/(72*x^7 + 564*x^6 + 1862*x^5 + 3355*x^4 + 3560*x^3 + 2223*x^2 + 756*x + 108)
\[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \frac {x}{72 x^{7} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 564 x^{6} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 1862 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 3355 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 3560 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 2223 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 756 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 108 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{72 x^{7} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 564 x^{6} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 1862 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 3355 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 3560 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 2223 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 756 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 108 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \] Input:
integrate((5-x)/(3+2*x)**(7/2)/(3*x**2+5*x+2)**(5/2),x)
Output:
-Integral(x/(72*x**7*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 564*x**6*sqrt( 2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 1862*x**5*sqrt(2*x + 3)*sqrt(3*x**2 + 5* x + 2) + 3355*x**4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 3560*x**3*sqrt(2 *x + 3)*sqrt(3*x**2 + 5*x + 2) + 2223*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 756*x*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 108*sqrt(2*x + 3)*sqr t(3*x**2 + 5*x + 2)), x) - Integral(-5/(72*x**7*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 564*x**6*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 1862*x**5*sqr t(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 3355*x**4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 3560*x**3*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 2223*x**2*sqrt (2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 756*x*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 108*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)), x)
\[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (2 \, x + 3\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((5-x)/(3+2*x)^(7/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")
Output:
-integrate((x - 5)/((3*x^2 + 5*x + 2)^(5/2)*(2*x + 3)^(7/2)), x)
\[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (2 \, x + 3\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((5-x)/(3+2*x)^(7/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="giac")
Output:
integrate(-(x - 5)/((3*x^2 + 5*x + 2)^(5/2)*(2*x + 3)^(7/2)), x)
Timed out. \[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {x-5}{{\left (2\,x+3\right )}^{7/2}\,{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \] Input:
int(-(x - 5)/((2*x + 3)^(7/2)*(5*x + 3*x^2 + 2)^(5/2)),x)
Output:
-int((x - 5)/((2*x + 3)^(7/2)*(5*x + 3*x^2 + 2)^(5/2)), x)
\[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x}{432 x^{10}+4752 x^{9}+23256 x^{8}+66656 x^{7}+123867 x^{6}+155895 x^{5}+134543 x^{4}+78609 x^{3}+29754 x^{2}+6588 x +648}d x \right )+5 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{432 x^{10}+4752 x^{9}+23256 x^{8}+66656 x^{7}+123867 x^{6}+155895 x^{5}+134543 x^{4}+78609 x^{3}+29754 x^{2}+6588 x +648}d x \right ) \] Input:
int((5-x)/(3+2*x)^(7/2)/(3*x^2+5*x+2)^(5/2),x)
Output:
- int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x)/(432*x**10 + 4752*x**9 + 2 3256*x**8 + 66656*x**7 + 123867*x**6 + 155895*x**5 + 134543*x**4 + 78609*x **3 + 29754*x**2 + 6588*x + 648),x) + 5*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5 *x + 2))/(432*x**10 + 4752*x**9 + 23256*x**8 + 66656*x**7 + 123867*x**6 + 155895*x**5 + 134543*x**4 + 78609*x**3 + 29754*x**2 + 6588*x + 648),x)