Integrand size = 38, antiderivative size = 177 \[ \int \frac {x}{\sqrt {-44375 b^4+576000 b^3 c x+576000 b^2 c^2 x^2+5308416 c^4 x^4}} \, dx=\frac {\log \left (20738073600000000 b^8 c^4+597005697024000000 b^6 c^6 x^2+2583100705996800000 b^5 c^7 x^3+951050714480640000 b^4 c^8 x^4+21641687369515008000 b^3 c^9 x^5+32462531054272512000 b^2 c^{10} x^6+149587343098087735296 c^{12} x^8+5308416 \sqrt {-44375 b^4+576000 b^3 c x+576000 b^2 c^2 x^2+5308416 c^4 x^4} \left (12203125 b^6 c^4+79200000 b^5 c^5 x+38880000 b^4 c^6 x^2+1105920000 b^3 c^7 x^3+1990656000 b^2 c^8 x^4+12230590464 c^{10} x^6\right )\right )}{18432 c^2} \]
1/18432*ln(20738073600000000*b^8*c^4+597005697024000000*b^6*c^6*x^2+258310 0705996800000*b^5*c^7*x^3+951050714480640000*b^4*c^8*x^4+21641687369515008 000*b^3*c^9*x^5+32462531054272512000*b^2*c^10*x^6+149587343098087735296*c^ 12*x^8+5308416*(12230590464*c^10*x^6+1990656000*b^2*c^8*x^4+1105920000*b^3 *c^7*x^3+38880000*b^4*c^6*x^2+79200000*b^5*c^5*x+12203125*b^6*c^4)*(530841 6*c^4*x^4+576000*b^2*c^2*x^2+576000*b^3*c*x-44375*b^4)^(1/2))/c^2
Time = 5.60 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.03 \[ \int \frac {x}{\sqrt {-44375 b^4+576000 b^3 c x+576000 b^2 c^2 x^2+5308416 c^4 x^4}} \, dx=-\frac {\log \left (-20386355871744000000000000000 b^{13} c^7-586880480402472960000000000000 b^{11} c^9 x^2-2539291318023094272000000000000 b^{10} c^{10} x^3-934920894363048345600000000000 b^9 c^{11} x^4-21274644351728033464320000000000 b^8 c^{12} x^5-31911966527592050196480000000000 b^7 c^{13} x^6-147050341759144167305379840000000 b^5 c^{15} x^8+\sqrt {-44375 b^4+576000 b^3 c x+576000 b^2 c^2 x^2+5308416 c^4 x^4} \left (63680607682560000000000000 b^{11} c^7+413296112959488000000000000 b^{10} c^8 x+202890819089203200000000000 b^9 c^9 x^2+5771116631870668800000000000 b^8 c^{10} x^3+10388009937367203840000000000 b^7 c^{11} x^4+63823933055184100392960000000 b^5 c^{13} x^6\right )\right )}{18432 c^2} \]
-1/18432*Log[-20386355871744000000000000000*b^13*c^7 - 5868804804024729600 00000000000*b^11*c^9*x^2 - 2539291318023094272000000000000*b^10*c^10*x^3 - 934920894363048345600000000000*b^9*c^11*x^4 - 212746443517280334643200000 00000*b^8*c^12*x^5 - 31911966527592050196480000000000*b^7*c^13*x^6 - 14705 0341759144167305379840000000*b^5*c^15*x^8 + Sqrt[-44375*b^4 + 576000*b^3*c *x + 576000*b^2*c^2*x^2 + 5308416*c^4*x^4]*(63680607682560000000000000*b^1 1*c^7 + 413296112959488000000000000*b^10*c^8*x + 2028908190892032000000000 00*b^9*c^9*x^2 + 5771116631870668800000000000*b^8*c^10*x^3 + 1038800993736 7203840000000000*b^7*c^11*x^4 + 63823933055184100392960000000*b^5*c^13*x^6 )]/c^2
Time = 0.30 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {2505}
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.
| \(\displaystyle \int \frac {x}{\sqrt {-44375 b^4+576000 b^3 c x+576000 b^2 c^2 x^2+5308416 c^4 x^4}} \, dx\) |
| \(\Big \downarrow \) 2505 |
| \(\displaystyle \frac {\log \left (20738073600000000 b^8 c^4+597005697024000000 b^6 c^6 x^2+2583100705996800000 b^5 c^7 x^3+951050714480640000 b^4 c^8 x^4+21641687369515008000 b^3 c^9 x^5+32462531054272512000 b^2 c^{10} x^6+5308416 \sqrt {-44375 b^4+576000 b^3 c x+576000 b^2 c^2 x^2+5308416 c^4 x^4} \left (12203125 b^6 c^4+79200000 b^5 c^5 x+38880000 b^4 c^6 x^2+1105920000 b^3 c^7 x^3+1990656000 b^2 c^8 x^4+12230590464 c^{10} x^6\right )+149587343098087735296 c^{12} x^8\right )}{18432 c^2}\) |
Log[20738073600000000*b^8*c^4 + 597005697024000000*b^6*c^6*x^2 + 258310070 5996800000*b^5*c^7*x^3 + 951050714480640000*b^4*c^8*x^4 + 2164168736951500 8000*b^3*c^9*x^5 + 32462531054272512000*b^2*c^10*x^6 + 1495873430980877352 96*c^12*x^8 + 5308416*Sqrt[-44375*b^4 + 576000*b^3*c*x + 576000*b^2*c^2*x^ 2 + 5308416*c^4*x^4]*(12203125*b^6*c^4 + 79200000*b^5*c^5*x + 38880000*b^4 *c^6*x^2 + 1105920000*b^3*c^7*x^3 + 1990656000*b^2*c^8*x^4 + 12230590464*c ^10*x^6)]/(18432*c^2)
3.11.24.3.1 Defintions of rubi rules used
Int[(x_)/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (e_.)*(x_)^4], x_Symbol] : > With[{Px = (1/320)*(33*b^2*c + 6*a*c^2 + 40*a^2*e) - (22/5)*a*c*e*x^2 + ( 22/15)*b*c*e*x^3 + (1/4)*e*(5*c^2 + 4*a*e)*x^4 + (4/3)*b*e^2*x^5 + 2*c*e^2* x^6 + e^3*x^8}, Simp[(1/(8*Rt[e, 2]))*Log[Px + (1/(8*Rt[e, 2]*x) D[Px, x] )*Sqrt[a + b*x + c*x^2 + e*x^4]], x]] /; FreeQ[{a, b, c, e}, x] && EqQ[71*c ^2 + 100*a*e, 0] && EqQ[1152*c^3 - 125*b^2*e, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.54 (sec) , antiderivative size = 1597, normalized size of antiderivative = 9.02
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1597\) |
| elliptic | \(\text {Expression too large to display}\) | \(1597\) |
int(x/(5308416*c^4*x^4+576000*b^2*c^2*x^2+576000*b^3*c*x-44375*b^4)^(1/2), x,method=_RETURNVERBOSE)
1/1152*(5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,index=1)*b/c-5/48*RootOf(_Z^4+10 *_Z^2+96*_Z-71,index=4)*b/c)*((5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,index=4)* b/c-5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,index=2)*b/c)*(x-5/48*RootOf(_Z^4+10 *_Z^2+96*_Z-71,index=1)*b/c)/(5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,index=4)*b /c-5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,index=1)*b/c)/(x-5/48*RootOf(_Z^4+10* _Z^2+96*_Z-71,index=2)*b/c))^(1/2)*(x-5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,in dex=2)*b/c)^2*((5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,index=2)*b/c-5/48*RootOf (_Z^4+10*_Z^2+96*_Z-71,index=1)*b/c)*(x-5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71, index=3)*b/c)/(5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,index=3)*b/c-5/48*RootOf( _Z^4+10*_Z^2+96*_Z-71,index=1)*b/c)/(x-5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,i ndex=2)*b/c))^(1/2)*((5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,index=2)*b/c-5/48* RootOf(_Z^4+10*_Z^2+96*_Z-71,index=1)*b/c)*(x-5/48*RootOf(_Z^4+10*_Z^2+96* _Z-71,index=4)*b/c)/(5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,index=4)*b/c-5/48*R ootOf(_Z^4+10*_Z^2+96*_Z-71,index=1)*b/c)/(x-5/48*RootOf(_Z^4+10*_Z^2+96*_ Z-71,index=2)*b/c))^(1/2)/(5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,index=4)*b/c- 5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,index=2)*b/c)/(5/48*RootOf(_Z^4+10*_Z^2+ 96*_Z-71,index=2)*b/c-5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,index=1)*b/c)/(c^4 *(x-5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,index=1)*b/c)*(x-5/48*RootOf(_Z^4+10 *_Z^2+96*_Z-71,index=2)*b/c)*(x-5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,index=3) *b/c)*(x-5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,index=4)*b/c))^(1/2)*(5/48*R...
Time = 0.35 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.93 \[ \int \frac {x}{\sqrt {-44375 b^4+576000 b^3 c x+576000 b^2 c^2 x^2+5308416 c^4 x^4}} \, dx=\frac {\log \left (28179280429056 \, c^{8} x^{8} + 6115295232000 \, b^{2} c^{6} x^{6} + 4076863488000 \, b^{3} c^{5} x^{5} + 179159040000 \, b^{4} c^{4} x^{4} + 486604800000 \, b^{5} c^{3} x^{3} + 112464000000 \, b^{6} c^{2} x^{2} + 3906640625 \, b^{8} + {\left (12230590464 \, c^{6} x^{6} + 1990656000 \, b^{2} c^{4} x^{4} + 1105920000 \, b^{3} c^{3} x^{3} + 38880000 \, b^{4} c^{2} x^{2} + 79200000 \, b^{5} c x + 12203125 \, b^{6}\right )} \sqrt {5308416 \, c^{4} x^{4} + 576000 \, b^{2} c^{2} x^{2} + 576000 \, b^{3} c x - 44375 \, b^{4}}\right )}{18432 \, c^{2}} \]
integrate(x/(5308416*c^4*x^4+576000*b^2*c^2*x^2+576000*b^3*c*x-44375*b^4)^ (1/2),x, algorithm="fricas")
1/18432*log(28179280429056*c^8*x^8 + 6115295232000*b^2*c^6*x^6 + 407686348 8000*b^3*c^5*x^5 + 179159040000*b^4*c^4*x^4 + 486604800000*b^5*c^3*x^3 + 1 12464000000*b^6*c^2*x^2 + 3906640625*b^8 + (12230590464*c^6*x^6 + 19906560 00*b^2*c^4*x^4 + 1105920000*b^3*c^3*x^3 + 38880000*b^4*c^2*x^2 + 79200000* b^5*c*x + 12203125*b^6)*sqrt(5308416*c^4*x^4 + 576000*b^2*c^2*x^2 + 576000 *b^3*c*x - 44375*b^4))/c^2
\[ \int \frac {x}{\sqrt {-44375 b^4+576000 b^3 c x+576000 b^2 c^2 x^2+5308416 c^4 x^4}} \, dx=\int \frac {x}{\sqrt {- 44375 b^{4} + 576000 b^{3} c x + 576000 b^{2} c^{2} x^{2} + 5308416 c^{4} x^{4}}}\, dx \]
Integral(x/sqrt(-44375*b**4 + 576000*b**3*c*x + 576000*b**2*c**2*x**2 + 53 08416*c**4*x**4), x)
\[ \int \frac {x}{\sqrt {-44375 b^4+576000 b^3 c x+576000 b^2 c^2 x^2+5308416 c^4 x^4}} \, dx=\int { \frac {x}{\sqrt {5308416 \, c^{4} x^{4} + 576000 \, b^{2} c^{2} x^{2} + 576000 \, b^{3} c x - 44375 \, b^{4}}} \,d x } \]
integrate(x/(5308416*c^4*x^4+576000*b^2*c^2*x^2+576000*b^3*c*x-44375*b^4)^ (1/2),x, algorithm="maxima")
\[ \int \frac {x}{\sqrt {-44375 b^4+576000 b^3 c x+576000 b^2 c^2 x^2+5308416 c^4 x^4}} \, dx=\int { \frac {x}{\sqrt {5308416 \, c^{4} x^{4} + 576000 \, b^{2} c^{2} x^{2} + 576000 \, b^{3} c x - 44375 \, b^{4}}} \,d x } \]
integrate(x/(5308416*c^4*x^4+576000*b^2*c^2*x^2+576000*b^3*c*x-44375*b^4)^ (1/2),x, algorithm="giac")
Timed out. \[ \int \frac {x}{\sqrt {-44375 b^4+576000 b^3 c x+576000 b^2 c^2 x^2+5308416 c^4 x^4}} \, dx=\int \frac {x}{\sqrt {-44375\,b^4+576000\,b^3\,c\,x+576000\,b^2\,c^2\,x^2+5308416\,c^4\,x^4}} \,d x \]