∫ x___________________________√ −44375b4 + 576000b3cx + 576000b2c2x2 + 5308416c4x4 dx [1024]

3.11.24 \(\int \frac {x}{\sqrt {-44375 b^4+576000 b^3 c x+576000 b^2 c^2 x^2+5308416 c^4 x^4}} \, dx\) [1024]

Optimal. Leaf size=177 \[ \frac {\log \left (20738073600000000 b^8 c^4+597005697024000000 b^6 c^6 x^2+2583100705996800000 b^5 c^7 x^3+951050714480640000 b^4 c^8 x^4+21641687369515008000 b^3 c^9 x^5+32462531054272512000 b^2 c^{10} x^6+149587343098087735296 c^{12} x^8+5308416 \sqrt {-44375 b^4+576000 b^3 c x+576000 b^2 c^2 x^2+5308416 c^4 x^4} \left (12203125 b^6 c^4+79200000 b^5 c^5 x+38880000 b^4 c^6 x^2+1105920000 b^3 c^7 x^3+1990656000 b^2 c^8 x^4+12230590464 c^{10} x^6\right )\right )}{18432 c^2} \]

[Out]

1/18432*ln(20738073600000000*b^8*c^4+597005697024000000*b^6*c^6*x^2+2583100705996800000*b^5*c^7*x^3+9510507144

80640000*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*(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+7920

0000*b^5*c^5*x+12203125*b^6*c^4)*(5308416*c^4*x^4+576000*b^2*c^2*x^2+576000*b^3*c*x-44375*b^4)^(1/2))/c^2

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Rubi [A]
time = 0.06, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {2107} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[x/Sqrt[-44375*b^4 + 576000*b^3*c*x + 576000*b^2*c^2*x^2 + 5308416*c^4*x^4],x]

[Out]

Log[20738073600000000*b^8*c^4 + 597005697024000000*b^6*c^6*x^2 + 2583100705996800000*b^5*c^7*x^3 + 95105071448

0640000*b^4*c^8*x^4 + 21641687369515008000*b^3*c^9*x^5 + 32462531054272512000*b^2*c^10*x^6 + 14958734309808773

5296*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 + 122305904

64*c^10*x^6)]/(18432*c^2)

Rule 2107

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 + Dist[1/(8*Rt[e, 2]*x), D[Px, x], 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]

Rubi steps

\begin {align*} \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}\\ \end {align*}

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Mathematica [A]
time = 6.52, size = 182, normalized size = 1.03 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[x/Sqrt[-44375*b^4 + 576000*b^3*c*x + 576000*b^2*c^2*x^2 + 5308416*c^4*x^4],x]

[Out]

-1/18432*Log[-20386355871744000000000000000*b^13*c^7 - 586880480402472960000000000000*b^11*c^9*x^2 - 253929131

8023094272000000000000*b^10*c^10*x^3 - 934920894363048345600000000000*b^9*c^11*x^4 - 2127464435172803346432000

0000000*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]*(63680607682560000000000000

*b^11*c^7 + 413296112959488000000000000*b^10*c^8*x + 202890819089203200000000000*b^9*c^9*x^2 + 577111663187066

8800000000000*b^8*c^10*x^3 + 10388009937367203840000000000*b^7*c^11*x^4 + 63823933055184100392960000000*b^5*c^

13*x^6)]/c^2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.68, size = 1597, normalized size = 9.02


method	result	size

default	\(\text {Expression too large to display}\)	\(1597\)
elliptic	\(\text {Expression too large to display}\)	\(1597\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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-7

1,index=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*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*Roo

tOf(_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*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)/(5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71,ind

ex=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*Ro

otOf(_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*RootOf(_Z^4+10*_Z^2+96*_Z-71,index=2)*b/c*EllipticF(((5/48*RootO

f(_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,in

dex=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,inde

x=2)*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-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-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=2)*b/c-5/48*RootOf(_Z^4+10*_Z^2+96*_Z-71

,index=4)*b/c))^(1/2))+(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

=2)*b/c)*EllipticPi(((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),(5/48*Ro

otOf(_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)/(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=3)*b/c)*(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-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=2)*b/c-5/48*RootOf(_Z^4+10*_Z^2+96*_

Z-71,index=4)*b/c))^(1/2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

integrate(x/sqrt(5308416*c^4*x^4 + 576000*b^2*c^2*x^2 + 576000*b^3*c*x - 44375*b^4), x)

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Fricas [A]
time = 0.45, size = 164, normalized size = 0.93 \begin {gather*} \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}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

1/18432*log(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 + (12230590464*c^6*x^6 + 199065

6000*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(53084

16*c^4*x^4 + 576000*b^2*c^2*x^2 + 576000*b^3*c*x - 44375*b^4))/c^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

Integral(x/sqrt(-44375*b**4 + 576000*b**3*c*x + 576000*b**2*c**2*x**2 + 5308416*c**4*x**4), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

integrate(x/sqrt(5308416*c^4*x^4 + 576000*b^2*c^2*x^2 + 576000*b^3*c*x - 44375*b^4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x/(5308416*c^4*x^4 - 44375*b^4 + 576000*b^2*c^2*x^2 + 576000*b^3*c*x)^(1/2),x)

[Out]

int(x/(5308416*c^4*x^4 - 44375*b^4 + 576000*b^2*c^2*x^2 + 576000*b^3*c*x)^(1/2), x)

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