Integrand size = 41, antiderivative size = 418 \[ \int \frac {\left (4+6 x-2 x^2\right ) \left (2+7 x+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=-\frac {\sqrt {1+2 x} (54857259443+10865116404 x) \sqrt {2+7 x+4 x^2}}{175134960}-\frac {\sqrt {1+2 x} (5684701+1459136 x) \left (2+7 x+4 x^2\right )^{3/2}}{972972}-\frac {(175-132 x) \sqrt {1+2 x} \left (2+7 x+4 x^2\right )^{5/2}}{2574}-\frac {132358116469 \sqrt {3+\sqrt {17}} \sqrt {1+2 x} \sqrt {-2-7 x-4 x^2} E\left (\arcsin \left (\frac {\sqrt {17+\sqrt {17} (7+8 x)}}{\sqrt {34}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{70053984 \sqrt {-1-2 x} \sqrt {2+7 x+4 x^2}}-\frac {2987675052409 \sqrt {-1-2 x} \sqrt {-2-7 x-4 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {17+\sqrt {17} (7+8 x)}}{\sqrt {34}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{75057840 \sqrt {3+\sqrt {17}} \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}}+\frac {13484913472 \sqrt {-1-2 x} \sqrt {-2-7 x-4 x^2} \operatorname {EllipticPi}\left (-\frac {3 \left (51-61 \sqrt {17}\right )}{1784},\arcsin \left (\frac {\sqrt {17+\sqrt {17} (7+8 x)}}{\sqrt {34}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{6561 \sqrt {3+\sqrt {17}} \left (61+3 \sqrt {17}\right ) \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \] Output:
-1/175134960*(1+2*x)^(1/2)*(54857259443+10865116404*x)*(4*x^2+7*x+2)^(1/2) -1/972972*(1+2*x)^(1/2)*(5684701+1459136*x)*(4*x^2+7*x+2)^(3/2)-1/2574*(17 5-132*x)*(1+2*x)^(1/2)*(4*x^2+7*x+2)^(5/2)-132358116469/70053984*(3+17^(1/ 2))^(1/2)*(1+2*x)^(1/2)*(-4*x^2-7*x-2)^(1/2)*EllipticE(1/34*(17+17^(1/2)*( 7+8*x))^(1/2)*34^(1/2),1/2*(17-3*17^(1/2))^(1/2))/(-1-2*x)^(1/2)/(4*x^2+7* x+2)^(1/2)-2987675052409/75057840*(-1-2*x)^(1/2)*(-4*x^2-7*x-2)^(1/2)*Elli pticF(1/34*(17+17^(1/2)*(7+8*x))^(1/2)*34^(1/2),1/2*(17-3*17^(1/2))^(1/2)) /(3+17^(1/2))^(1/2)/(1+2*x)^(1/2)/(4*x^2+7*x+2)^(1/2)+13484913472/6561*(-1 -2*x)^(1/2)*(-4*x^2-7*x-2)^(1/2)*EllipticPi(1/34*(17+17^(1/2)*(7+8*x))^(1/ 2)*34^(1/2),-153/1784+183/1784*17^(1/2),1/2*(17-3*17^(1/2))^(1/2))/(3+17^( 1/2))^(1/2)/(61+3*17^(1/2))/(1+2*x)^(1/2)/(4*x^2+7*x+2)^(1/2)
Result contains complex when optimal does not.
Time = 23.45 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.85 \[ \int \frac {\left (4+6 x-2 x^2\right ) \left (2+7 x+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\frac {12 \sqrt {3+\sqrt {17}} \left (-718741962148-2648350226628 x-1946275855429 x^2-434091365288 x^3-139410462456 x^4-34156062720 x^5-932037120 x^6+2543063040 x^7+574801920 x^8\right )-1985371747035 i \left (3+\sqrt {17}\right ) (1+2 x)^{3/2} \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right )|-\frac {13}{4}+\frac {3 \sqrt {17}}{4}\right )+3 i \left (1887399333713+661790582345 \sqrt {17}\right ) (1+2 x)^{3/2} \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right ),-\frac {13}{4}+\frac {3 \sqrt {17}}{4}\right )-41533533493760 i (1+2 x)^{3/2} \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} \operatorname {EllipticPi}\left (-\frac {13}{6} \left (-3+\sqrt {17}\right ),i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{1050809760 \sqrt {3+\sqrt {17}} \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \] Input:
Integrate[((4 + 6*x - 2*x^2)*(2 + 7*x + 4*x^2)^(5/2))/((5 - 3*x)*Sqrt[1 + 2*x]),x]
Output:
(12*Sqrt[3 + Sqrt[17]]*(-718741962148 - 2648350226628*x - 1946275855429*x^ 2 - 434091365288*x^3 - 139410462456*x^4 - 34156062720*x^5 - 932037120*x^6 + 2543063040*x^7 + 574801920*x^8) - (1985371747035*I)*(3 + Sqrt[17])*(1 + 2*x)^(3/2)*Sqrt[(2 + 7*x + 4*x^2)/(1 + 2*x)^2]*EllipticE[I*ArcSinh[Sqrt[2/ (-3 + Sqrt[17])]/Sqrt[1 + 2*x]], -13/4 + (3*Sqrt[17])/4] + (3*I)*(18873993 33713 + 661790582345*Sqrt[17])*(1 + 2*x)^(3/2)*Sqrt[(2 + 7*x + 4*x^2)/(1 + 2*x)^2]*EllipticF[I*ArcSinh[Sqrt[2/(-3 + Sqrt[17])]/Sqrt[1 + 2*x]], -13/4 + (3*Sqrt[17])/4] - (41533533493760*I)*(1 + 2*x)^(3/2)*Sqrt[(2 + 7*x + 4* x^2)/(1 + 2*x)^2]*EllipticPi[(-13*(-3 + Sqrt[17]))/6, I*ArcSinh[Sqrt[2/(-3 + Sqrt[17])]/Sqrt[1 + 2*x]], (-13 + 3*Sqrt[17])/4])/(1050809760*Sqrt[3 + Sqrt[17]]*Sqrt[1 + 2*x]*Sqrt[2 + 7*x + 4*x^2])
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.
| \(\displaystyle \int \frac {\left (-2 x^2+6 x+4\right ) \left (4 x^2+7 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}} \, dx\) |
| \(\Big \downarrow \) 2154 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (4 x^2+7 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx+\int \frac {\left (\frac {2 x}{3}-\frac {8}{9}\right ) \left (4 x^2+7 x+2\right )^{5/2}}{\sqrt {2 x+1}}dx\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle -\frac {5 \int \frac {2 (2304 x+571) \left (4 x^2+7 x+2\right )^{3/2}}{9 \sqrt {2 x+1}}dx}{1144}+\frac {76}{9} \int \frac {\left (4 x^2+7 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {(175-132 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5 \int \frac {(2304 x+571) \left (4 x^2+7 x+2\right )^{3/2}}{\sqrt {2 x+1}}dx}{5148}+\frac {76}{9} \int \frac {\left (4 x^2+7 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {(175-132 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle -\frac {5 \left (\frac {1}{7} \sqrt {2 x+1} (1792 x+891) \left (4 x^2+7 x+2\right )^{3/2}-\frac {1}{168} \int \frac {36 (11678 x+4485) \sqrt {4 x^2+7 x+2}}{\sqrt {2 x+1}}dx\right )}{5148}+\frac {76}{9} \int \frac {\left (4 x^2+7 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {(175-132 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5 \left (\frac {1}{7} \sqrt {2 x+1} (1792 x+891) \left (4 x^2+7 x+2\right )^{3/2}-\frac {3}{14} \int \frac {(11678 x+4485) \sqrt {4 x^2+7 x+2}}{\sqrt {2 x+1}}dx\right )}{5148}+\frac {76}{9} \int \frac {\left (4 x^2+7 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {(175-132 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle -\frac {5 \left (\frac {1}{7} \sqrt {2 x+1} (1792 x+891) \left (4 x^2+7 x+2\right )^{3/2}-\frac {3}{14} \left (\frac {1}{30} \sqrt {2 x+1} (70068 x+39011) \sqrt {4 x^2+7 x+2}-\frac {1}{120} \int \frac {2 (431580 x+171193)}{\sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx\right )\right )}{5148}+\frac {76}{9} \int \frac {\left (4 x^2+7 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {(175-132 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5 \left (\frac {1}{7} \sqrt {2 x+1} (1792 x+891) \left (4 x^2+7 x+2\right )^{3/2}-\frac {3}{14} \left (\frac {1}{30} \sqrt {2 x+1} (70068 x+39011) \sqrt {4 x^2+7 x+2}-\frac {1}{60} \int \frac {431580 x+171193}{\sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx\right )\right )}{5148}+\frac {76}{9} \int \frac {\left (4 x^2+7 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {(175-132 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle -\frac {5 \left (\frac {1}{7} \sqrt {2 x+1} (1792 x+891) \left (4 x^2+7 x+2\right )^{3/2}-\frac {3}{14} \left (\frac {1}{60} \left (44597 \int \frac {1}{\sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx-215790 \int \frac {\sqrt {2 x+1}}{\sqrt {4 x^2+7 x+2}}dx\right )+\frac {1}{30} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (70068 x+39011)\right )\right )}{5148}+\frac {76}{9} \int \frac {\left (4 x^2+7 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {(175-132 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (4 x^2+7 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {5 \left (\frac {1}{7} \sqrt {2 x+1} (1792 x+891) \left (4 x^2+7 x+2\right )^{3/2}-\frac {3}{14} \left (\frac {1}{60} \left (\frac {89194 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \int \frac {1}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}} \sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {107895 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} \int \frac {\sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )+\frac {1}{30} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (70068 x+39011)\right )\right )}{5148}-\frac {(175-132 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {5 \left (\frac {1}{7} \sqrt {2 x+1} (1792 x+891) \left (4 x^2+7 x+2\right )^{3/2}-\frac {3}{14} \left (\frac {1}{60} \left (\frac {89194 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {107895 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} \int \frac {\sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )+\frac {1}{30} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (70068 x+39011)\right )\right )}{5148}+\frac {76}{9} \int \frac {\left (4 x^2+7 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {(175-132 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (4 x^2+7 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {5 \left (\frac {1}{7} \sqrt {2 x+1} (1792 x+891) \left (4 x^2+7 x+2\right )^{3/2}-\frac {3}{14} \left (\frac {1}{60} \left (\frac {89194 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {107895 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )+\frac {1}{30} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (70068 x+39011)\right )\right )}{5148}-\frac {(175-132 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 1292 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (4 x^2+7 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {5 \left (\frac {1}{7} \sqrt {2 x+1} (1792 x+891) \left (4 x^2+7 x+2\right )^{3/2}-\frac {3}{14} \left (\frac {1}{60} \left (\frac {89194 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {107895 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )+\frac {1}{30} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (70068 x+39011)\right )\right )}{5148}-\frac {(175-132 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{5/2}}{2574}\) |
Input:
Int[((4 + 6*x - 2*x^2)*(2 + 7*x + 4*x^2)^(5/2))/((5 - 3*x)*Sqrt[1 + 2*x]), x]
Output:
$Aborted
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)*(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[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Unintegrable[(d + e*x)^m*(f + g*x)^n* (a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b _.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, d + e*x, x]*(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x] + Simp[Polyn omialRemainder[Px, d + e*x, x] Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x ^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && PolynomialQ[Px, x ] && LtQ[m, 0] && !IntegerQ[n] && IntegersQ[2*m, 2*n, 2*p]
Result contains complex when optimal does not.
Time = 1.30 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.24
| method | result | size |
| risch | \(\frac {\left (143700480 x^{5}+312439680 x^{4}-1133586720 x^{3}-6453975240 x^{2}-18850599504 x -56951379803\right ) \sqrt {4 x^{2}+7 x +2}\, \sqrt {1+2 x}}{175134960}+\frac {2 \left (-\frac {2262224441903 \left (-\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {-\frac {x +\frac {7}{8}-\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {\frac {x +\frac {7}{8}+\frac {\sqrt {17}}{8}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )}{95528160 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}-\frac {132358116469 \left (-\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {-\frac {x +\frac {7}{8}-\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {\frac {x +\frac {7}{8}+\frac {\sqrt {17}}{8}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \left (\left (\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \operatorname {EllipticE}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )+\left (-\frac {7}{8}+\frac {\sqrt {17}}{8}\right ) \operatorname {EllipticF}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )\right )}{17513496 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}+\frac {1685614184 \left (-\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {-\frac {x +\frac {7}{8}-\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {\frac {x +\frac {7}{8}+\frac {\sqrt {17}}{8}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, -\frac {9}{52}-\frac {3 \sqrt {17}}{52}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )}{85293 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}\right ) \sqrt {\left (4 x^{2}+7 x +2\right ) \left (1+2 x \right )}}{\sqrt {4 x^{2}+7 x +2}\, \sqrt {1+2 x}}\) | \(517\) |
| elliptic | \(\frac {\sqrt {\left (4 x^{2}+7 x +2\right ) \left (1+2 x \right )}\, \left (\frac {32 x^{5} \sqrt {8 x^{3}+18 x^{2}+11 x +2}}{39}+\frac {2296 x^{4} \sqrt {8 x^{3}+18 x^{2}+11 x +2}}{1287}-\frac {224918 x^{3} \sqrt {8 x^{3}+18 x^{2}+11 x +2}}{34749}-\frac {5975903 x^{2} \sqrt {8 x^{3}+18 x^{2}+11 x +2}}{162162}-\frac {3966877 x \sqrt {8 x^{3}+18 x^{2}+11 x +2}}{36855}-\frac {56951379803 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}{175134960}-\frac {2262224441903 \left (-\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {-\frac {x +\frac {7}{8}-\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {\frac {x +\frac {7}{8}+\frac {\sqrt {17}}{8}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )}{47764080 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}-\frac {132358116469 \left (-\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {-\frac {x +\frac {7}{8}-\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {\frac {x +\frac {7}{8}+\frac {\sqrt {17}}{8}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \left (\left (\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \operatorname {EllipticE}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )+\left (-\frac {7}{8}+\frac {\sqrt {17}}{8}\right ) \operatorname {EllipticF}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )\right )}{8756748 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}+\frac {3371228368 \left (-\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {-\frac {x +\frac {7}{8}-\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {\frac {x +\frac {7}{8}+\frac {\sqrt {17}}{8}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, -\frac {9}{52}-\frac {3 \sqrt {17}}{52}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )}{85293 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}\right )}{\sqrt {4 x^{2}+7 x +2}\, \sqrt {1+2 x}}\) | \(596\) |
| default | \(\frac {\sqrt {4 x^{2}+7 x +2}\, \sqrt {1+2 x}\, \left (-32803994766528-191279916530208 x +20913725366863 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}\, \sqrt {\left (-8 x -7+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}\, \sqrt {\left (8 x +7+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}\, \operatorname {EllipticF}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right ) \sqrt {17}-20766766746880 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}\, \sqrt {\left (-8 x -7+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}\, \sqrt {\left (8 x +7+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}\, \operatorname {EllipticPi}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, -\frac {9}{52}-\frac {3 \sqrt {17}}{52}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right ) \sqrt {17}-10934664922176 \sqrt {17}-19673892126720 x^{5}+70682663088729 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}\, \sqrt {\left (-8 x -7+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}\, \sqrt {\left (8 x +7+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}\, \operatorname {EllipticF}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right )-62300300240640 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}\, \sqrt {\left (-8 x -7+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}\, \sqrt {\left (8 x +7+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}\, \operatorname {EllipticPi}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, -\frac {9}{52}-\frac {3 \sqrt {17}}{52}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right )-7941486988140 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}\, \sqrt {\left (-8 x -7+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}\, \sqrt {\left (8 x +7+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}\, \operatorname {EllipticE}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right )-80300426374656 x^{4}+1464804311040 x^{7}-358672141865664 x^{2}-250036626405888 x^{3}-536853381120 x^{6}+331085905920 x^{8}+110361968640 \sqrt {17}\, x^{8}+488268103680 \sqrt {17}\, x^{7}-178951127040 \sqrt {17}\, x^{6}-6557964042240 \sqrt {17}\, x^{5}-26766808791552 \sqrt {17}\, x^{4}-83345542135296 \sqrt {17}\, x^{3}-119557380621888 \sqrt {17}\, x^{2}-63759972176736 \sqrt {17}\, x \right )}{2101619520 \left (3+\sqrt {17}\right )^{2} \left (-3+\sqrt {17}\right ) \left (8 x^{3}+18 x^{2}+11 x +2\right )}\) | \(655\) |
Input:
int((-2*x^2+6*x+4)*(4*x^2+7*x+2)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x,method=_RET URNVERBOSE)
Output:
1/175134960*(143700480*x^5+312439680*x^4-1133586720*x^3-6453975240*x^2-188 50599504*x-56951379803)*(4*x^2+7*x+2)^(1/2)*(1+2*x)^(1/2)+2*(-226222444190 3/95528160*(-3/8-1/8*17^(1/2))*(-(x+1/2)/(3/8+1/8*17^(1/2)))^(1/2)*(-(x+7/ 8-1/8*17^(1/2))/(-3/8+1/8*17^(1/2)))^(1/2)*((x+7/8+1/8*17^(1/2))/(3/8+1/8* 17^(1/2)))^(1/2)/(8*x^3+18*x^2+11*x+2)^(1/2)*EllipticF((-(x+1/2)/(3/8+1/8* 17^(1/2)))^(1/2),I*((3/8+1/8*17^(1/2))/(-3/8+1/8*17^(1/2)))^(1/2))-1323581 16469/17513496*(-3/8-1/8*17^(1/2))*(-(x+1/2)/(3/8+1/8*17^(1/2)))^(1/2)*(-( x+7/8-1/8*17^(1/2))/(-3/8+1/8*17^(1/2)))^(1/2)*((x+7/8+1/8*17^(1/2))/(3/8+ 1/8*17^(1/2)))^(1/2)/(8*x^3+18*x^2+11*x+2)^(1/2)*((3/8-1/8*17^(1/2))*Ellip ticE((-(x+1/2)/(3/8+1/8*17^(1/2)))^(1/2),I*((3/8+1/8*17^(1/2))/(-3/8+1/8*1 7^(1/2)))^(1/2))+(-7/8+1/8*17^(1/2))*EllipticF((-(x+1/2)/(3/8+1/8*17^(1/2) ))^(1/2),I*((3/8+1/8*17^(1/2))/(-3/8+1/8*17^(1/2)))^(1/2)))+1685614184/852 93*(-3/8-1/8*17^(1/2))*(-(x+1/2)/(3/8+1/8*17^(1/2)))^(1/2)*(-(x+7/8-1/8*17 ^(1/2))/(-3/8+1/8*17^(1/2)))^(1/2)*((x+7/8+1/8*17^(1/2))/(3/8+1/8*17^(1/2) ))^(1/2)/(8*x^3+18*x^2+11*x+2)^(1/2)*EllipticPi((-(x+1/2)/(3/8+1/8*17^(1/2 )))^(1/2),-9/52-3/52*17^(1/2),I*((3/8+1/8*17^(1/2))/(-3/8+1/8*17^(1/2)))^( 1/2)))*((4*x^2+7*x+2)*(1+2*x))^(1/2)/(4*x^2+7*x+2)^(1/2)/(1+2*x)^(1/2)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \left (2+7 x+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (4 \, x^{2} + 7 \, x + 2\right )}^{\frac {5}{2}} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+7*x+2)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x, algo rithm="fricas")
Output:
integral(2*(16*x^6 + 8*x^5 - 135*x^4 - 279*x^3 - 210*x^2 - 68*x - 8)*sqrt( 4*x^2 + 7*x + 2)*sqrt(2*x + 1)/(6*x^2 - 7*x - 5), x)
Timed out. \[ \int \frac {\left (4+6 x-2 x^2\right ) \left (2+7 x+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\text {Timed out} \] Input:
integrate((-2*x**2+6*x+4)*(4*x**2+7*x+2)**(5/2)/(5-3*x)/(1+2*x)**(1/2),x)
Output:
Timed out
\[ \int \frac {\left (4+6 x-2 x^2\right ) \left (2+7 x+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (4 \, x^{2} + 7 \, x + 2\right )}^{\frac {5}{2}} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+7*x+2)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x, algo rithm="maxima")
Output:
2*integrate((4*x^2 + 7*x + 2)^(5/2)*(x^2 - 3*x - 2)/((3*x - 5)*sqrt(2*x + 1)), x)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \left (2+7 x+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (4 \, x^{2} + 7 \, x + 2\right )}^{\frac {5}{2}} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+7*x+2)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x, algo rithm="giac")
Output:
integrate(2*(4*x^2 + 7*x + 2)^(5/2)*(x^2 - 3*x - 2)/((3*x - 5)*sqrt(2*x + 1)), x)
Timed out. \[ \int \frac {\left (4+6 x-2 x^2\right ) \left (2+7 x+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int -\frac {\left (-2\,x^2+6\,x+4\right )\,{\left (4\,x^2+7\,x+2\right )}^{5/2}}{\sqrt {2\,x+1}\,\left (3\,x-5\right )} \,d x \] Input:
int(-((6*x - 2*x^2 + 4)*(7*x + 4*x^2 + 2)^(5/2))/((2*x + 1)^(1/2)*(3*x - 5 )),x)
Output:
int(-((6*x - 2*x^2 + 4)*(7*x + 4*x^2 + 2)^(5/2))/((2*x + 1)^(1/2)*(3*x - 5 )), x)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \left (2+7 x+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int \frac {\left (-2 x^{2}+6 x +4\right ) \left (4 x^{2}+7 x +2\right )^{\frac {5}{2}}}{\left (5-3 x \right ) \sqrt {2 x +1}}d x \] Input:
int((-2*x^2+6*x+4)*(4*x^2+7*x+2)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x)
Output:
int((-2*x^2+6*x+4)*(4*x^2+7*x+2)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x)