∫ 1__________√ 8 + 8x − x3 + 8x4 dx [796]

3.8.96 $\int \frac {1}{\sqrt {8+8 x-x^3+8 x^4}} \, dx$ [796]

Optimal. Leaf size=129 \[ -\frac {x^2 \sqrt {\frac {261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4}{\left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right )^2}} \left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {4+x}{\sqrt {3} \sqrt [4]{29} x}\right )|\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{8 \sqrt {3} \sqrt [4]{29} \sqrt {8+8 x-x^3+8 x^4}} \]

[Out]

-1/696*x^2*(cos(2*arctan(1/87*(4+x)*29^(3/4)/x*3^(1/2)))^2)^(1/2)/cos(2*arctan(1/87*(4+x)*29^(3/4)/x*3^(1/2)))

*EllipticF(sin(2*arctan(1/87*(4+x)*29^(3/4)/x*3^(1/2))),1/58*(1682+58*29^(1/2))^(1/2))*(87+(4+x)^2*29^(1/2)/x^

2)*((261-6*(1+4/x)^2+(1+4/x)^4)/(87+(4+x)^2*29^(1/2)/x^2)^2)^(1/2)*29^(3/4)*3^(1/2)/(8*x^4-x^3+8*x+8)^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.210, Rules used = {2094, 12, 6851, 1117} \begin {gather*} -\frac {x^2 \sqrt {\frac {\left (\frac {4}{x}+1\right )^4-6 \left (\frac {4}{x}+1\right )^2+261}{\left (\frac {\sqrt {29} (x+4)^2}{x^2}+87\right )^2}} \left (\frac {\sqrt {29} (x+4)^2}{x^2}+87\right ) F\left (2 \text {ArcTan}\left (\frac {x+4}{\sqrt {3} \sqrt [4]{29} x}\right )|\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{8 \sqrt {3} \sqrt [4]{29} \sqrt {8 x^4-x^3+8 x+8}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[1/Sqrt[8 + 8*x - x^3 + 8*x^4],x]

[Out]

-1/8*(x^2*Sqrt[(261 - 6*(1 + 4/x)^2 + (1 + 4/x)^4)/(87 + (Sqrt[29]*(4 + x)^2)/x^2)^2]*(87 + (Sqrt[29]*(4 + x)^

2)/x^2)*EllipticF[2*ArcTan[(4 + x)/(Sqrt[3]*29^(1/4)*x)], (29 + Sqrt[29])/58])/(Sqrt[3]*29^(1/4)*Sqrt[8 + 8*x

- x^3 + 8*x^4])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1117

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]

Rule 2094

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4

, x, 3], e = Coeff[P4, x, 4]}, Dist[-16*a^2, Subst[Int[(1/(b - 4*a*x)^2)*(a*((-3*b^4 + 16*a*b^2*c - 64*a^2*b*d

+ 256*a^3*e - 32*a^2*(3*b^2 - 8*a*c)*x^2 + 256*a^4*x^4)/(b - 4*a*x)^4))^p, x], x, b/(4*a) + 1/x], x] /; NeQ[a

, 0] && NeQ[b, 0] && EqQ[b^3 - 4*a*b*c + 8*a^2*d, 0]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && IntegerQ[2*p] && !

IGtQ[p, 0]

Rule 6851

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n)^FracPart[p]/(v^(m*Fr

acPart[p])*w^(n*FracPart[p]))), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !

FreeQ[v, x] && !FreeQ[w, x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {8+8 x-x^3+8 x^4}} \, dx &=-\left (1024 \text {Subst}\left (\int \frac {1}{2 \sqrt {2} (8-32 x)^2 \sqrt {\frac {1069056-393216 x^2+1048576 x^4}{(8-32 x)^4}}} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )\right )\\ &=-\left (\left (256 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{(8-32 x)^2 \sqrt {\frac {1069056-393216 x^2+1048576 x^4}{(8-32 x)^4}}} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )\right )\\ &=-\frac {\left (\sqrt {1069056-393216 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1048576 \left (\frac {1}{4}+\frac {1}{x}\right )^4} x^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1069056-393216 x^2+1048576 x^4}} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{\sqrt {8+8 x-x^3+8 x^4}}\\ &=-\frac {x^2 \sqrt {\frac {261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4}{\left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right )^2}} \left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {4+x}{\sqrt {3} \sqrt [4]{29} x}\right )|\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{8 \sqrt {3} \sqrt [4]{29} \sqrt {8+8 x-x^3+8 x^4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
time = 10.58, size = 927, normalized size = 7.19 \begin {gather*} -\frac {2 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (x-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,1\right ]\right ) \left (\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,2\right ]-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,4\right ]\right )}{\left (x-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,2\right ]\right ) \left (\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,1\right ]-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,4\right ]\right )}}\right )|\frac {\left (\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,2\right ]-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,3\right ]\right ) \left (\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,1\right ]-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,4\right ]\right )}{\left (\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,1\right ]-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,3\right ]\right ) \left (\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,2\right ]-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,4\right ]\right )}\right ) \left (x-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,2\right ]\right )^2 \sqrt {\frac {\left (\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,1\right ]-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,2\right ]\right ) \left (x-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,3\right ]\right )}{\left (x-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,2\right ]\right ) \left (\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,1\right ]-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,3\right ]\right )}} \left (\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,1\right ]-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,4\right ]\right ) \sqrt {\frac {\left (x-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,1\right ]\right ) \left (\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,1\right ]-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,2\right ]\right ) \left (x-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,4\right ]\right ) \left (\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,2\right ]-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,4\right ]\right )}{\left (x-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,2\right ]\right )^2 \left (\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,1\right ]-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,4\right ]\right )^2}}}{\sqrt {8+8 x-x^3+8 x^4} \left (-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,1\right ]+\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,2\right ]\right ) \left (\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,2\right ]-\text {Root}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\&,4\right ]\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[8 + 8*x - x^3 + 8*x^4],x]

[Out]

(-2*EllipticF[ArcSin[Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2,

0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1

- #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]], ((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2

, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^

3 + 8*#1^4 & , 4, 0]))/((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(R

oot[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]*(x - Root[8 + 8*#1 - #1^3

+ 8*#1^4 & , 2, 0])^2*Sqrt[((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0]

)*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1

- #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))]*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1,

0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])*Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*

#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & ,

4, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*

#1 - #1^3 + 8*#1^4 & , 2, 0])^2*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4

, 0])^2)])/(Sqrt[8 + 8*x - x^3 + 8*x^4]*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#

1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.89, size = 965, normalized size = 7.48


method	result	size

default	$\text {Expression too large to display}$	$965$
elliptic	$\text {Expression too large to display}$	$965$

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(8*x^4-x^3+8*x+8)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4))*((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=

4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index

=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2)*(x-RootOf(8*_Z^4-_Z^3+8*

_Z+8,index=2))^2*((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x-RootOf(8*_Z^4-_Z^

3+8*_Z+8,index=3))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z

^3+8*_Z+8,index=2)))^(1/2)*((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x-RootOf(

8*_Z^4-_Z^3+8*_Z+8,index=4))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf

(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2)/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))/(

RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*2^(1/2)/((x-RootOf(8*_Z^4-_Z^3+8*_Z+8,i

ndex=1))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3))*(x-RootOf(8*_Z^4-_Z^3+8

*_Z+8,index=4)))^(1/2)*EllipticF(((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))*(x-R

ootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-

RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2),((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,inde

x=3))*(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index

=1)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)

))^(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(1/(8*x^4-x^3+8*x+8)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(8*x^4 - x^3 + 8*x + 8), x)

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Fricas [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(1/(8*x^4-x^3+8*x+8)^(1/2),x, algorithm="fricas")

[Out]

integral(1/sqrt(8*x^4 - x^3 + 8*x + 8), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {8 x^{4} - x^{3} + 8 x + 8}}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*x**4-x**3+8*x+8)**(1/2),x)

[Out]

Integral(1/sqrt(8*x**4 - x**3 + 8*x + 8), 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(1/(8*x^4-x^3+8*x+8)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(8*x^4 - x^3 + 8*x + 8), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {8\,x^4-x^3+8\,x+8}} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(8*x - x^3 + 8*x^4 + 8)^(1/2),x)

[Out]

int(1/(8*x - x^3 + 8*x^4 + 8)^(1/2), x)

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3.8.96 \(\int \frac {1}{\sqrt {8+8 x-x^3+8 x^4}} \, dx\) [796]