Integrand size = 38, antiderivative size = 360 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3+4 x^2\right )^{7/2}} \, dx=-\frac {(111-670 x) \sqrt {1+2 x}}{15240 \left (3+4 x^2\right )^{5/2}}+\frac {\sqrt {1+2 x} (16257+363373 x)}{17419320 \left (3+4 x^2\right )^{3/2}}+\frac {\sqrt {1+2 x} (262436763+1353435866 x)}{106188174720 \sqrt {3+4 x^2}}-\frac {676717933 \sqrt {1+2 x} \sqrt {3+4 x^2}}{106188174720 (3+2 x)}+\frac {6156 \sqrt {\frac {3}{1651}} \text {arctanh}\left (\frac {\sqrt {\frac {127}{39}} \sqrt {1+2 x}}{\sqrt {3+4 x^2}}\right )}{2048383}+\frac {676717933 (3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} E\left (2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{53094087360 \sqrt {2} \sqrt {3+4 x^2}}-\frac {299755 (3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{167225472 \sqrt {2} \sqrt {3+4 x^2}}-\frac {567 \sqrt {2} (3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} \operatorname {EllipticPi}\left (\frac {361}{312},2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{26628979 \sqrt {3+4 x^2}} \] Output:
-1/15240*(111-670*x)*(1+2*x)^(1/2)/(4*x^2+3)^(5/2)+1/17419320*(1+2*x)^(1/2 )*(16257+363373*x)/(4*x^2+3)^(3/2)+1/106188174720*(1+2*x)^(1/2)*(262436763 +1353435866*x)/(4*x^2+3)^(1/2)-676717933*(1+2*x)^(1/2)*(4*x^2+3)^(1/2)/(31 8564524160+212376349440*x)+6156/3381880333*4953^(1/2)*arctanh(1/39*4953^(1 /2)*(1+2*x)^(1/2)/(4*x^2+3)^(1/2))+676717933/106188174720*2^(1/2)*(3+2*x)* ((4*x^2+3)/(3+2*x)^2)^(1/2)*EllipticE(sin(2*arctan(1/2*(1+2*x)^(1/2)*2^(1/ 2))),1/2*3^(1/2))/(4*x^2+3)^(1/2)-299755/334450944*2^(1/2)*(3+2*x)*((4*x^2 +3)/(3+2*x)^2)^(1/2)*InverseJacobiAM(2*arctan(1/2*(1+2*x)^(1/2)*2^(1/2)),1 /2*3^(1/2))/(4*x^2+3)^(1/2)-567/26628979*2^(1/2)*(3+2*x)*((4*x^2+3)/(3+2*x )^2)^(1/2)*EllipticPi(sin(2*arctan(1/2*(1+2*x)^(1/2)*2^(1/2))),361/312,1/2 *3^(1/2))/(4*x^2+3)^(1/2)
Result contains complex when optimal does not.
Time = 25.00 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.79 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3+4 x^2\right )^{7/2}} \, dx=\frac {\sqrt {1+2 x} \left (26 \left (1885821075+23494665978 x+6694893000 x^2+41342948016 x^3+4198988208 x^4+21654973856 x^5\right )-\frac {(1+2 x) \left (3+4 x^2\right )^2 \left (\frac {17594666258 \sqrt {-\frac {i}{i+\sqrt {3}}} \left (3+4 x^2\right )}{(1+2 x)^2}+\frac {8797333129 \left (3-i \sqrt {3}\right ) \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} E\left (i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {3+6 x}}-\frac {i \sqrt {3} \left (-8797333129 i+1349977333 \sqrt {3}\right ) \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}+\frac {638254080 i \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} \operatorname {EllipticPi}\left (\frac {13}{12} \left (1-i \sqrt {3}\right ),i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}\right )}{\sqrt {-\frac {i}{i+\sqrt {3}}}}\right )}{2760892542720 \left (3+4 x^2\right )^{5/2}} \] Input:
Integrate[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*(3 + 4*x^2)^(7/2)),x]
Output:
(Sqrt[1 + 2*x]*(26*(1885821075 + 23494665978*x + 6694893000*x^2 + 41342948 016*x^3 + 4198988208*x^4 + 21654973856*x^5) - ((1 + 2*x)*(3 + 4*x^2)^2*((1 7594666258*Sqrt[(-I)/(I + Sqrt[3])]*(3 + 4*x^2))/(1 + 2*x)^2 + (8797333129 *(3 - I*Sqrt[3])*Sqrt[(3*I + Sqrt[3] + 2*(-I + Sqrt[3])*x)/((-I + Sqrt[3]) *(1 + 2*x))]*Sqrt[(-3*I + Sqrt[3] + 2*(I + Sqrt[3])*x)/((I + Sqrt[3])*(1 + 2*x))]*EllipticE[I*ArcSinh[(2*Sqrt[(-I)/(I + Sqrt[3])])/Sqrt[1 + 2*x]], ( I + Sqrt[3])/(I - Sqrt[3])])/Sqrt[3 + 6*x] - (I*Sqrt[3]*(-8797333129*I + 1 349977333*Sqrt[3])*Sqrt[(3*I + Sqrt[3] + 2*(-I + Sqrt[3])*x)/((-I + Sqrt[3 ])*(1 + 2*x))]*Sqrt[(-3*I + Sqrt[3] + 2*(I + Sqrt[3])*x)/((I + Sqrt[3])*(1 + 2*x))]*EllipticF[I*ArcSinh[(2*Sqrt[(-I)/(I + Sqrt[3])])/Sqrt[1 + 2*x]], (I + Sqrt[3])/(I - Sqrt[3])])/Sqrt[1 + 2*x] + ((638254080*I)*Sqrt[(3*I + Sqrt[3] + 2*(-I + Sqrt[3])*x)/((-I + Sqrt[3])*(1 + 2*x))]*Sqrt[(-3*I + Sqr t[3] + 2*(I + Sqrt[3])*x)/((I + Sqrt[3])*(1 + 2*x))]*EllipticPi[(13*(1 - I *Sqrt[3]))/12, I*ArcSinh[(2*Sqrt[(-I)/(I + Sqrt[3])])/Sqrt[1 + 2*x]], (I + Sqrt[3])/(I - Sqrt[3])])/Sqrt[1 + 2*x]))/Sqrt[(-I)/(I + Sqrt[3])]))/(2760 892542720*(3 + 4*x^2)^(5/2))
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.
| \(\displaystyle \int \frac {-2 x^2+6 x+4}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{7/2}}dx+\int \frac {\frac {2 x}{3}-\frac {8}{9}}{\sqrt {2 x+1} \left (4 x^2+3\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{7/2}}dx-\frac {1}{960} \int \frac {8 (289-14 x)}{9 \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}dx-\frac {\sqrt {2 x+1} (33-2 x)}{1080 \left (4 x^2+3\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{7/2}}dx-\frac {\int \frac {289-14 x}{\sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}dx}{1080}-\frac {\sqrt {2 x+1} (33-2 x)}{1080 \left (4 x^2+3\right )^{5/2}}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{7/2}}dx+\frac {\frac {1}{576} \int -\frac {64 (201 x+689)}{\sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}}dx-\frac {\sqrt {2 x+1} (67 x+111)}{9 \left (4 x^2+3\right )^{3/2}}}{1080}-\frac {\sqrt {2 x+1} (33-2 x)}{1080 \left (4 x^2+3\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{7/2}}dx+\frac {-\frac {1}{9} \int \frac {201 x+689}{\sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}}dx-\frac {\sqrt {2 x+1} (67 x+111)}{9 \left (4 x^2+3\right )^{3/2}}}{1080}-\frac {\sqrt {2 x+1} (33-2 x)}{1080 \left (4 x^2+3\right )^{5/2}}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{7/2}}dx+\frac {\frac {1}{9} \left (\frac {1}{192} \int -\frac {4 (3531-3962 x)}{\sqrt {2 x+1} \sqrt {4 x^2+3}}dx-\frac {\sqrt {2 x+1} (3962 x+3531)}{48 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (67 x+111)}{9 \left (4 x^2+3\right )^{3/2}}}{1080}-\frac {\sqrt {2 x+1} (33-2 x)}{1080 \left (4 x^2+3\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{7/2}}dx+\frac {\frac {1}{9} \left (-\frac {1}{48} \int \frac {3531-3962 x}{\sqrt {2 x+1} \sqrt {4 x^2+3}}dx-\frac {\sqrt {2 x+1} (3962 x+3531)}{48 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (67 x+111)}{9 \left (4 x^2+3\right )^{3/2}}}{1080}-\frac {\sqrt {2 x+1} (33-2 x)}{1080 \left (4 x^2+3\right )^{5/2}}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{7/2}}dx+\frac {\frac {1}{9} \left (\frac {1}{96} \int -\frac {2 (5512-1981 (2 x+1))}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {\sqrt {2 x+1} (3962 x+3531)}{48 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (67 x+111)}{9 \left (4 x^2+3\right )^{3/2}}}{1080}-\frac {\sqrt {2 x+1} (33-2 x)}{1080 \left (4 x^2+3\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{7/2}}dx+\frac {\frac {1}{9} \left (-\frac {1}{48} \int \frac {5512-1981 (2 x+1)}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {\sqrt {2 x+1} (3962 x+3531)}{48 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (67 x+111)}{9 \left (4 x^2+3\right )^{3/2}}}{1080}-\frac {\sqrt {2 x+1} (33-2 x)}{1080 \left (4 x^2+3\right )^{5/2}}\) |
| \(\Big \downarrow \) 744 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{7/2}}dx+\frac {\frac {1}{9} \left (-\frac {1}{48} \int \frac {5512-1981 (2 x+1)}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {\sqrt {2 x+1} (3962 x+3531)}{48 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (67 x+111)}{9 \left (4 x^2+3\right )^{3/2}}}{1080}-\frac {\sqrt {2 x+1} (33-2 x)}{1080 \left (4 x^2+3\right )^{5/2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{7/2}}dx+\frac {\frac {1}{9} \left (\frac {1}{48} \left (-1550 \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-3962 \int \frac {1-2 x}{2 \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )-\frac {\sqrt {2 x+1} (3962 x+3531)}{48 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (67 x+111)}{9 \left (4 x^2+3\right )^{3/2}}}{1080}-\frac {\sqrt {2 x+1} (33-2 x)}{1080 \left (4 x^2+3\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{7/2}}dx+\frac {\frac {1}{9} \left (\frac {1}{48} \left (-1550 \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-1981 \int \frac {1-2 x}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )-\frac {\sqrt {2 x+1} (3962 x+3531)}{48 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (67 x+111)}{9 \left (4 x^2+3\right )^{3/2}}}{1080}-\frac {\sqrt {2 x+1} (33-2 x)}{1080 \left (4 x^2+3\right )^{5/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {1}{48} \left (-1981 \int \frac {1-2 x}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {775 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{\sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )-\frac {\sqrt {2 x+1} (3962 x+3531)}{48 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (67 x+111)}{9 \left (4 x^2+3\right )^{3/2}}}{1080}+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{7/2}}dx-\frac {\sqrt {2 x+1} (33-2 x)}{1080 \left (4 x^2+3\right )^{5/2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{7/2}}dx+\frac {\frac {1}{9} \left (\frac {1}{48} \left (-\frac {775 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{\sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}-1981 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )\right )-\frac {\sqrt {2 x+1} (3962 x+3531)}{48 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (67 x+111)}{9 \left (4 x^2+3\right )^{3/2}}}{1080}-\frac {\sqrt {2 x+1} (33-2 x)}{1080 \left (4 x^2+3\right )^{5/2}}\) |
Input:
Int[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*(3 + 4*x^2)^(7/2)),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[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[-2/d^2 Subst[Int[(B*c - A*d - B*x^2)/Sqrt[(b*c^2 + a *d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)], x], x, Sqrt[c + d*x]], x] /; Fr eeQ[{a, b, c, d, A, B}, x] && PosQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_), x_Symbol] :> Unintegrable[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n, p}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. )*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(c + d *x)^(m + 1)*(e + f*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c + d*x, x] Int[(c + d*x)^m*(e + f*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, 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 = 0.86 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.53
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (4 x^{2}+3\right ) \left (1+2 x \right )}\, \left (\frac {\left (-\frac {37}{325120}+\frac {67 x}{97536}\right ) \sqrt {8 x^{3}+4 x^{2}+6 x +3}}{\left (x^{2}+\frac {3}{4}\right )^{3}}+\frac {\left (\frac {5419}{92903040}+\frac {363373 x}{278709120}\right ) \sqrt {8 x^{3}+4 x^{2}+6 x +3}}{\left (x^{2}+\frac {3}{4}\right )^{2}}-\frac {2 \left (4+8 x \right ) \left (-\frac {87478921}{283168465920}-\frac {676717933 x}{424752698880}\right )}{\sqrt {\left (x^{2}+\frac {3}{4}\right ) \left (4+8 x \right )}}+\frac {87478921 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\right )}{17698029120 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}-\frac {676717933 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\right )+\frac {i \sqrt {3}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2}\right )}{26547043680 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}+\frac {24624 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}, \frac {3}{13}-\frac {3 i \sqrt {3}}{13}, \sqrt {\frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\right )}{26628979 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}\right )}{\sqrt {4 x^{2}+3}\, \sqrt {1+2 x}}\) | \(550\) |
| default | \(\text {Expression too large to display}\) | \(1786\) |
Input:
int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+3)^(7/2),x,method=_RETURNV ERBOSE)
Output:
((4*x^2+3)*(1+2*x))^(1/2)/(4*x^2+3)^(1/2)/(1+2*x)^(1/2)*((-37/325120+67/97 536*x)*(8*x^3+4*x^2+6*x+3)^(1/2)/(x^2+3/4)^3+(5419/92903040+363373/2787091 20*x)*(8*x^3+4*x^2+6*x+3)^(1/2)/(x^2+3/4)^2-2*(4+8*x)*(-87478921/283168465 920-676717933/424752698880*x)/((x^2+3/4)*(4+8*x))^(1/2)+87478921/176980291 20*(1/2-1/2*I*3^(1/2))*((x+1/2)/(1/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2*I*3^(1/ 2))/(-1/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2)))^( 1/2)/(8*x^3+4*x^2+6*x+3)^(1/2)*EllipticF(((x+1/2)/(1/2-1/2*I*3^(1/2)))^(1/ 2),((-1/2+1/2*I*3^(1/2))/(-1/2-1/2*I*3^(1/2)))^(1/2))-676717933/2654704368 0*(1/2-1/2*I*3^(1/2))*((x+1/2)/(1/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2*I*3^(1/2 ))/(-1/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2)))^(1 /2)/(8*x^3+4*x^2+6*x+3)^(1/2)*((-1/2-1/2*I*3^(1/2))*EllipticE(((x+1/2)/(1/ 2-1/2*I*3^(1/2)))^(1/2),((-1/2+1/2*I*3^(1/2))/(-1/2-1/2*I*3^(1/2)))^(1/2)) +1/2*I*3^(1/2)*EllipticF(((x+1/2)/(1/2-1/2*I*3^(1/2)))^(1/2),((-1/2+1/2*I* 3^(1/2))/(-1/2-1/2*I*3^(1/2)))^(1/2)))+24624/26628979*(1/2-1/2*I*3^(1/2))* ((x+1/2)/(1/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2*I*3^(1/2))/(-1/2-1/2*I*3^(1/2) ))^(1/2)*((x+1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2)))^(1/2)/(8*x^3+4*x^2+6*x+3 )^(1/2)*EllipticPi(((x+1/2)/(1/2-1/2*I*3^(1/2)))^(1/2),3/13-3/13*I*3^(1/2) ,((-1/2+1/2*I*3^(1/2))/(-1/2-1/2*I*3^(1/2)))^(1/2)))
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3+4 x^2\right )^{7/2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{{\left (4 \, x^{2} + 3\right )}^{\frac {7}{2}} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+3)^(7/2),x, algorith m="fricas")
Output:
integral(2*sqrt(4*x^2 + 3)*(x^2 - 3*x - 2)*sqrt(2*x + 1)/(1536*x^10 - 1792 *x^9 + 3328*x^8 - 5376*x^7 + 1344*x^6 - 6048*x^5 - 1728*x^4 - 3024*x^3 - 1 674*x^2 - 567*x - 405), x)
Timed out. \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3+4 x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((-2*x**2+6*x+4)/(5-3*x)/(1+2*x)**(1/2)/(4*x**2+3)**(7/2),x)
Output:
Timed out
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3+4 x^2\right )^{7/2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{{\left (4 \, x^{2} + 3\right )}^{\frac {7}{2}} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+3)^(7/2),x, algorith m="maxima")
Output:
2*integrate((x^2 - 3*x - 2)/((4*x^2 + 3)^(7/2)*(3*x - 5)*sqrt(2*x + 1)), x )
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3+4 x^2\right )^{7/2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{{\left (4 \, x^{2} + 3\right )}^{\frac {7}{2}} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+3)^(7/2),x, algorith m="giac")
Output:
integrate(2*(x^2 - 3*x - 2)/((4*x^2 + 3)^(7/2)*(3*x - 5)*sqrt(2*x + 1)), x )
Timed out. \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3+4 x^2\right )^{7/2}} \, dx=\int -\frac {-2\,x^2+6\,x+4}{\sqrt {2\,x+1}\,\left (3\,x-5\right )\,{\left (4\,x^2+3\right )}^{7/2}} \,d x \] Input:
int(-(6*x - 2*x^2 + 4)/((2*x + 1)^(1/2)*(3*x - 5)*(4*x^2 + 3)^(7/2)),x)
Output:
int(-(6*x - 2*x^2 + 4)/((2*x + 1)^(1/2)*(3*x - 5)*(4*x^2 + 3)^(7/2)), x)
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3+4 x^2\right )^{7/2}} \, dx=\int \frac {-2 x^{2}+6 x +4}{\left (5-3 x \right ) \sqrt {2 x +1}\, \left (4 x^{2}+3\right )^{\frac {7}{2}}}d x \] Input:
int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+3)^(7/2),x)
Output:
int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+3)^(7/2),x)