\(\int \frac {1}{x^6 \sqrt {1+x^3}} \, dx\) [450]

Optimal result

Integrand size = 13, antiderivative size = 138 \[ \int \frac {1}{x^6 \sqrt {1+x^3}} \, dx=-\frac {\sqrt {1+x^3}}{5 x^5}+\frac {7 \sqrt {1+x^3}}{20 x^2}+\frac {7 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{20 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \]

[Out]

-1/5*(x^3+1)^(1/2)/x^5+7/20*(x^3+1)^(1/2)/x^2+7/60*(1+x)*EllipticF((1+x-3^(1/2))/(1+x+3^(1/2)),I*3^(1/2)+2*I)*
(1/2*6^(1/2)+1/2*2^(1/2))*((x^2-x+1)/(1+x+3^(1/2))^2)^(1/2)*3^(3/4)/(x^3+1)^(1/2)/((1+x)/(1+x+3^(1/2))^2)^(1/2
)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {331, 224} \[ \int \frac {1}{x^6 \sqrt {1+x^3}} \, dx=\frac {7 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{20 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\sqrt {x^3+1}}{5 x^5}+\frac {7 \sqrt {x^3+1}}{20 x^2} \]

[In]

Int[1/(x^6*Sqrt[1 + x^3]),x]

[Out]

-1/5*Sqrt[1 + x^3]/x^5 + (7*Sqrt[1 + x^3])/(20*x^2) + (7*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqr
t[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(20*3^(1/4)*Sqrt[(1 + x)/
(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3])

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt
[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sq
rt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)
], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1+x^3}}{5 x^5}-\frac {7}{10} \int \frac {1}{x^3 \sqrt {1+x^3}} \, dx \\ & = -\frac {\sqrt {1+x^3}}{5 x^5}+\frac {7 \sqrt {1+x^3}}{20 x^2}+\frac {7}{40} \int \frac {1}{\sqrt {1+x^3}} \, dx \\ & = -\frac {\sqrt {1+x^3}}{5 x^5}+\frac {7 \sqrt {1+x^3}}{20 x^2}+\frac {7 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{20 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.16 \[ \int \frac {1}{x^6 \sqrt {1+x^3}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{3},\frac {1}{2},-\frac {2}{3},-x^3\right )}{5 x^5} \]

[In]

Integrate[1/(x^6*Sqrt[1 + x^3]),x]

[Out]

-1/5*Hypergeometric2F1[-5/3, 1/2, -2/3, -x^3]/x^5

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 4.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.12

method	result	size
meijerg	\(-\frac {{}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {5}{3},\frac {1}{2};-\frac {2}{3};-x^{3}\right )}{5 x^{5}}\)	\(17\)
default	\(-\frac {\sqrt {x^{3}+1}}{5 x^{5}}+\frac {7 \sqrt {x^{3}+1}}{20 x^{2}}+\frac {7 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{20 \sqrt {x^{3}+1}}\)	\(141\)
risch	\(\frac {7 x^{6}+3 x^{3}-4}{20 x^{5} \sqrt {x^{3}+1}}+\frac {7 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{20 \sqrt {x^{3}+1}}\)	\(141\)
elliptic	\(-\frac {\sqrt {x^{3}+1}}{5 x^{5}}+\frac {7 \sqrt {x^{3}+1}}{20 x^{2}}+\frac {7 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{20 \sqrt {x^{3}+1}}\)	\(141\)

[In]

int(1/x^6/(x^3+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/5/x^5*hypergeom([-5/3,1/2],[-2/3],-x^3)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.22 \[ \int \frac {1}{x^6 \sqrt {1+x^3}} \, dx=\frac {7 \, x^{5} {\rm weierstrassPInverse}\left (0, -4, x\right ) + {\left (7 \, x^{3} - 4\right )} \sqrt {x^{3} + 1}}{20 \, x^{5}} \]

[In]

integrate(1/x^6/(x^3+1)^(1/2),x, algorithm="fricas")

[Out]

1/20*(7*x^5*weierstrassPInverse(0, -4, x) + (7*x^3 - 4)*sqrt(x^3 + 1))/x^5

Sympy [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^6 \sqrt {1+x^3}} \, dx=\frac {\Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} \]

[In]

integrate(1/x**6/(x**3+1)**(1/2),x)

[Out]

gamma(-5/3)*hyper((-5/3, 1/2), (-2/3,), x**3*exp_polar(I*pi))/(3*x**5*gamma(-2/3))

Maxima [F]

\[ \int \frac {1}{x^6 \sqrt {1+x^3}} \, dx=\int { \frac {1}{\sqrt {x^{3} + 1} x^{6}} \,d x } \]

[In]

integrate(1/x^6/(x^3+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^3 + 1)*x^6), x)

Giac [F]

\[ \int \frac {1}{x^6 \sqrt {1+x^3}} \, dx=\int { \frac {1}{\sqrt {x^{3} + 1} x^{6}} \,d x } \]

[In]

integrate(1/x^6/(x^3+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^3 + 1)*x^6), x)

Mupad [B] (verification not implemented)

Time = 5.44 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.31 \[ \int \frac {1}{x^6 \sqrt {1+x^3}} \, dx=\frac {7\,\sqrt {x^3+1}}{20\,x^2}-\frac {\sqrt {x^3+1}}{5\,x^5}+\frac {7\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{20\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

[In]

int(1/(x^6*(x^3 + 1)^(1/2)),x)

[Out]

(7*(x^3 + 1)^(1/2))/(20*x^2) - (x^3 + 1)^(1/2)/(5*x^5) + (7*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)
/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i
)/2 + 3/2))^(1/2)*ellipticF(asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i
)/2 - 3/2)))/(20*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/
2)*1i)/2 + 1/2))^(1/2))