Integrand size = 38, antiderivative size = 360 \[ \int \frac {\left (4+6 x-2 x^2\right ) \left (3+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=-\frac {1615154392 \sqrt {1+2 x} \sqrt {3+4 x^2}}{1563705 (3+2 x)}-\frac {16 \sqrt {1+2 x} (59536964+19603449 x) \sqrt {3+4 x^2}}{10945935}-\frac {2 \sqrt {1+2 x} (311111+101962 x) \left (3+4 x^2\right )^{3/2}}{243243}-\frac {2 (70-33 x) \sqrt {1+2 x} \left (3+4 x^2\right )^{5/2}}{1287}+\frac {1225804 \sqrt {\frac {127}{39}} \text {arctanh}\left (\frac {\sqrt {\frac {127}{39}} \sqrt {1+2 x}}{\sqrt {3+4 x^2}}\right )}{2187}+\frac {1615154392 \sqrt {2} (3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} E\left (2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{1563705 \sqrt {3+4 x^2}}-\frac {1576225160 \sqrt {2} (3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{2189187 \sqrt {3+4 x^2}}-\frac {14338681 \sqrt {2} (3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} \operatorname {EllipticPi}\left (\frac {361}{312},2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{85293 \sqrt {3+4 x^2}} \] Output:
-1615154392*(1+2*x)^(1/2)*(4*x^2+3)^(1/2)/(4691115+3127410*x)-16/10945935* (1+2*x)^(1/2)*(59536964+19603449*x)*(4*x^2+3)^(1/2)-2/243243*(1+2*x)^(1/2) *(311111+101962*x)*(4*x^2+3)^(3/2)-2/1287*(70-33*x)*(1+2*x)^(1/2)*(4*x^2+3 )^(5/2)+1225804/85293*4953^(1/2)*arctanh(1/39*4953^(1/2)*(1+2*x)^(1/2)/(4* x^2+3)^(1/2))+1615154392/1563705*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)-1576225160/2189187*2^(1/2)*(3+2*x)*((4*x^2+3)/(3+2*x)^2)^(1/2)*In verseJacobiAM(2*arctan(1/2*(1+2*x)^(1/2)*2^(1/2)),1/2*3^(1/2))/(4*x^2+3)^( 1/2)-14338681/85293*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 = 24.87 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.76 \[ \int \frac {\left (4+6 x-2 x^2\right ) \left (3+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\frac {2 \sqrt {1+2 x} \left (3 \left (3+4 x^2\right ) \left (-523653847-168066477 x-70288380 x^2-11617200 x^3-9525600 x^4+4490640 x^5\right )-\frac {2 (1+2 x) \left (\frac {8479560558 \sqrt {-\frac {i}{i+\sqrt {3}}} \left (3+4 x^2\right )}{(1+2 x)^2}+\frac {4239780279 \left (-i+\sqrt {3}\right ) \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} E\left (i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}-\frac {3 \left (-950047897 i+1413260093 \sqrt {3}\right ) \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}+\frac {14983921645 i \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} \operatorname {EllipticPi}\left (\frac {13}{12} \left (1-i \sqrt {3}\right ),i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}\right )}{\sqrt {-\frac {i}{i+\sqrt {3}}}}\right )}{32837805 \sqrt {3+4 x^2}} \] Input:
Integrate[((4 + 6*x - 2*x^2)*(3 + 4*x^2)^(5/2))/((5 - 3*x)*Sqrt[1 + 2*x]), x]
Output:
(2*Sqrt[1 + 2*x]*(3*(3 + 4*x^2)*(-523653847 - 168066477*x - 70288380*x^2 - 11617200*x^3 - 9525600*x^4 + 4490640*x^5) - (2*(1 + 2*x)*((8479560558*Sqr t[(-I)/(I + Sqrt[3])]*(3 + 4*x^2))/(1 + 2*x)^2 + (4239780279*(-I + Sqrt[3] )*Sqrt[(3*I + Sqrt[3] + 2*(-I + Sqrt[3])*x)/((-I + Sqrt[3])*(1 + 2*x))]*Sq rt[(-3*I + Sqrt[3] + 2*(I + Sqrt[3])*x)/((I + Sqrt[3])*(1 + 2*x))]*Ellipti cE[I*ArcSinh[(2*Sqrt[(-I)/(I + Sqrt[3])])/Sqrt[1 + 2*x]], (I + Sqrt[3])/(I - Sqrt[3])])/Sqrt[1 + 2*x] - (3*(-950047897*I + 1413260093*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*Ar cSinh[(2*Sqrt[(-I)/(I + Sqrt[3])])/Sqrt[1 + 2*x]], (I + Sqrt[3])/(I - Sqrt [3])])/Sqrt[1 + 2*x] + ((14983921645*I)*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))]*EllipticPi[(13*(1 - I*Sqrt[3]))/12, I*ArcSi nh[(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])]))/(32837805*Sqrt[3 + 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.
| \(\displaystyle \int \frac {\left (-2 x^2+6 x+4\right ) \left (4 x^2+3\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}} \, dx\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (4 x^2+3\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx+\int \frac {\left (\frac {2 x}{3}-\frac {8}{9}\right ) \left (4 x^2+3\right )^{5/2}}{\sqrt {2 x+1}}dx\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {5}{572} \int -\frac {8 (321-478 x) \left (4 x^2+3\right )^{3/2}}{9 \sqrt {2 x+1}}dx+\frac {76}{9} \int \frac {\left (4 x^2+3\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2 (70-33 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}{1287}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {10 \int \frac {(321-478 x) \left (4 x^2+3\right )^{3/2}}{\sqrt {2 x+1}}dx}{1287}+\frac {76}{9} \int \frac {\left (4 x^2+3\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2 (70-33 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}{1287}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle -\frac {10 \left (\frac {1}{84} \int \frac {64 (1173-2455 x) \sqrt {4 x^2+3}}{\sqrt {2 x+1}}dx+\frac {1}{63} (4801-3346 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}\right )}{1287}+\frac {76}{9} \int \frac {\left (4 x^2+3\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2 (70-33 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}{1287}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {10 \left (\frac {16}{21} \int \frac {(1173-2455 x) \sqrt {4 x^2+3}}{\sqrt {2 x+1}}dx+\frac {1}{63} (4801-3346 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}\right )}{1287}+\frac {76}{9} \int \frac {\left (4 x^2+3\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2 (70-33 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}{1287}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle -\frac {10 \left (\frac {16}{21} \left (\frac {1}{60} \int \frac {20 (8511-17458 x)}{\sqrt {2 x+1} \sqrt {4 x^2+3}}dx+\frac {1}{3} \sqrt {2 x+1} \sqrt {4 x^2+3} (2155-1473 x)\right )+\frac {1}{63} (4801-3346 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}\right )}{1287}+\frac {76}{9} \int \frac {\left (4 x^2+3\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2 (70-33 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}{1287}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {10 \left (\frac {16}{21} \left (\frac {1}{3} \int \frac {8511-17458 x}{\sqrt {2 x+1} \sqrt {4 x^2+3}}dx+\frac {1}{3} \sqrt {2 x+1} \sqrt {4 x^2+3} (2155-1473 x)\right )+\frac {1}{63} (4801-3346 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}\right )}{1287}+\frac {76}{9} \int \frac {\left (4 x^2+3\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2 (70-33 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}{1287}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (4 x^2+3\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {10 \left (\frac {16}{21} \left (\frac {1}{3} (2155-1473 x) \sqrt {2 x+1} \sqrt {4 x^2+3}-\frac {1}{6} \int -\frac {2 (17240-8729 (2 x+1))}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )+\frac {1}{63} (4801-3346 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}\right )}{1287}-\frac {2 (70-33 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}{1287}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (4 x^2+3\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {10 \left (\frac {16}{21} \left (\frac {1}{3} \int \frac {17240-8729 (2 x+1)}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}+\frac {1}{3} \sqrt {2 x+1} \sqrt {4 x^2+3} (2155-1473 x)\right )+\frac {1}{63} (4801-3346 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}\right )}{1287}-\frac {2 (70-33 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}{1287}\) |
| \(\Big \downarrow \) 744 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (4 x^2+3\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {10 \left (\frac {16}{21} \left (\frac {1}{3} \int \frac {17240-8729 (2 x+1)}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}+\frac {1}{3} \sqrt {2 x+1} \sqrt {4 x^2+3} (2155-1473 x)\right )+\frac {1}{63} (4801-3346 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}\right )}{1287}-\frac {2 (70-33 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}{1287}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (4 x^2+3\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {10 \left (\frac {16}{21} \left (\frac {1}{3} \left (17458 \int \frac {1-2 x}{2 \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-218 \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )+\frac {1}{3} \sqrt {2 x+1} \sqrt {4 x^2+3} (2155-1473 x)\right )+\frac {1}{63} (4801-3346 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}\right )}{1287}-\frac {2 (70-33 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}{1287}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (4 x^2+3\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {10 \left (\frac {16}{21} \left (\frac {1}{3} \left (8729 \int \frac {1-2 x}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-218 \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )+\frac {1}{3} \sqrt {2 x+1} \sqrt {4 x^2+3} (2155-1473 x)\right )+\frac {1}{63} (4801-3346 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}\right )}{1287}-\frac {2 (70-33 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}{1287}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {10 \left (\frac {16}{21} \left (\frac {1}{3} \left (8729 \int \frac {1-2 x}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {109 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{\sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )+\frac {1}{3} \sqrt {2 x+1} \sqrt {4 x^2+3} (2155-1473 x)\right )+\frac {1}{63} (4801-3346 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}\right )}{1287}+\frac {76}{9} \int \frac {\left (4 x^2+3\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2 (70-33 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}{1287}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (4 x^2+3\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {10 \left (\frac {16}{21} \left (\frac {1}{3} \left (8729 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )-\frac {109 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{\sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )+\frac {1}{3} \sqrt {2 x+1} \sqrt {4 x^2+3} (2155-1473 x)\right )+\frac {1}{63} (4801-3346 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}\right )}{1287}-\frac {2 (70-33 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}{1287}\) |
Input:
Int[((4 + 6*x - 2*x^2)*(3 + 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[((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)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[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 = 1.29 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.42
| method | result | size |
| risch | \(\frac {2 \left (4490640 x^{5}-9525600 x^{4}-11617200 x^{3}-70288380 x^{2}-168066477 x -523653847\right ) \sqrt {4 x^{2}+3}\, \sqrt {1+2 x}}{10945935}+\frac {2 \left (-\frac {142672522688 \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 )}{32837805 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}-\frac {3230308784 \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 )}{1563705 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}+\frac {311354216 \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 )}{85293 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}\right ) \sqrt {\left (4 x^{2}+3\right ) \left (1+2 x \right )}}{\sqrt {4 x^{2}+3}\, \sqrt {1+2 x}}\) | \(510\) |
| elliptic | \(\frac {\sqrt {\left (4 x^{2}+3\right ) \left (1+2 x \right )}\, \left (\frac {32 x^{5} \sqrt {8 x^{3}+4 x^{2}+6 x +3}}{39}-\frac {2240 x^{4} \sqrt {8 x^{3}+4 x^{2}+6 x +3}}{1287}-\frac {73760 x^{3} \sqrt {8 x^{3}+4 x^{2}+6 x +3}}{34749}-\frac {3123928 x^{2} \sqrt {8 x^{3}+4 x^{2}+6 x +3}}{243243}-\frac {37348106 x \sqrt {8 x^{3}+4 x^{2}+6 x +3}}{1216215}-\frac {1047307694 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}{10945935}-\frac {285345045376 \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 )}{32837805 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}-\frac {6460617568 \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 )}{1563705 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}+\frac {622708432 \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 )}{85293 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}\right )}{\sqrt {4 x^{2}+3}\, \sqrt {1+2 x}}\) | \(592\) |
| default | \(\frac {2 \sqrt {4 x^{2}+3}\, \sqrt {1+2 x}\, \left (107775360 x^{8}-174726720 x^{7}+54377140228 i \sqrt {3}\, \sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x}{1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x}{-1+i \sqrt {3}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )-59935686580 i \sqrt {3}\, \sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x}{1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x}{-1+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \frac {3}{13}-\frac {3 i \sqrt {3}}{13}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )-312288480 x^{6}-1957372560 x^{5}-122213624692 \sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x}{1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x}{-1+i \sqrt {3}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )+67836484464 \sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x}{1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x}{-1+i \sqrt {3}}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )+59935686580 \sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x}{1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x}{-1+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \frac {3}{13}-\frac {3 i \sqrt {3}}{13}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )-5171896008 x^{4}-15954235692 x^{3}-9941638170 x^{2}-10938367539 x -4712884623\right )}{32837805 \left (8 x^{3}+4 x^{2}+6 x +3\right )}\) | \(645\) |
Input:
int((-2*x^2+6*x+4)*(4*x^2+3)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x,method=_RETURNV ERBOSE)
Output:
2/10945935*(4490640*x^5-9525600*x^4-11617200*x^3-70288380*x^2-168066477*x- 523653847)*(4*x^2+3)^(1/2)*(1+2*x)^(1/2)+2*(-142672522688/32837805*(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))-3230308784/1563705*(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)))+311354216/85293*(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)*Elli pticPi(((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)))*((4*x^2+3)*(1+2*x))^(1/2)/(4*x^2+ 3)^(1/2)/(1+2*x)^(1/2)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \left (3+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (4 \, x^{2} + 3\right )}^{\frac {5}{2}} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+3)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x, algorith m="fricas")
Output:
integral(2*(16*x^6 - 48*x^5 - 8*x^4 - 72*x^3 - 39*x^2 - 27*x - 18)*sqrt(4* x^2 + 3)*sqrt(2*x + 1)/(6*x^2 - 7*x - 5), x)
Timed out. \[ \int \frac {\left (4+6 x-2 x^2\right ) \left (3+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\text {Timed out} \] Input:
integrate((-2*x**2+6*x+4)*(4*x**2+3)**(5/2)/(5-3*x)/(1+2*x)**(1/2),x)
Output:
Timed out
\[ \int \frac {\left (4+6 x-2 x^2\right ) \left (3+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (4 \, x^{2} + 3\right )}^{\frac {5}{2}} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+3)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x, algorith m="maxima")
Output:
2*integrate((4*x^2 + 3)^(5/2)*(x^2 - 3*x - 2)/((3*x - 5)*sqrt(2*x + 1)), x )
\[ \int \frac {\left (4+6 x-2 x^2\right ) \left (3+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (4 \, x^{2} + 3\right )}^{\frac {5}{2}} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+3)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x, algorith m="giac")
Output:
integrate(2*(4*x^2 + 3)^(5/2)*(x^2 - 3*x - 2)/((3*x - 5)*sqrt(2*x + 1)), x )
Timed out. \[ \int \frac {\left (4+6 x-2 x^2\right ) \left (3+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int -\frac {{\left (4\,x^2+3\right )}^{5/2}\,\left (-2\,x^2+6\,x+4\right )}{\sqrt {2\,x+1}\,\left (3\,x-5\right )} \,d x \] Input:
int(-((4*x^2 + 3)^(5/2)*(6*x - 2*x^2 + 4))/((2*x + 1)^(1/2)*(3*x - 5)),x)
Output:
int(-((4*x^2 + 3)^(5/2)*(6*x - 2*x^2 + 4))/((2*x + 1)^(1/2)*(3*x - 5)), x)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \left (3+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int \frac {\left (-2 x^{2}+6 x +4\right ) \left (4 x^{2}+3\right )^{\frac {5}{2}}}{\left (5-3 x \right ) \sqrt {2 x +1}}d x \] Input:
int((-2*x^2+6*x+4)*(4*x^2+3)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x)
Output:
int((-2*x^2+6*x+4)*(4*x^2+3)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x)