\(\int \frac {1+x^3}{x^7 \sqrt {-1+x^3}} \, dx\) [479]

Optimal result

Integrand size = 18, antiderivative size = 38 \[ \int \frac {1+x^3}{x^7 \sqrt {-1+x^3}} \, dx=\frac {\sqrt {-1+x^3} \left (2+7 x^3\right )}{12 x^6}+\frac {7}{12} \arctan \left (\sqrt {-1+x^3}\right ) \]

[Out]

1/12*(x^3-1)^(1/2)*(7*x^3+2)/x^6+7/12*arctan((x^3-1)^(1/2))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {457, 79, 44, 65, 209} \[ \int \frac {1+x^3}{x^7 \sqrt {-1+x^3}} \, dx=\frac {7}{12} \arctan \left (\sqrt {x^3-1}\right )+\frac {7 \sqrt {x^3-1}}{12 x^3}+\frac {\sqrt {x^3-1}}{6 x^6} \]

[In]

Int[(1 + x^3)/(x^7*Sqrt[-1 + x^3]),x]

[Out]

Sqrt[-1 + x^3]/(6*x^6) + (7*Sqrt[-1 + x^3])/(12*x^3) + (7*ArcTan[Sqrt[-1 + x^3]])/12

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1+x}{\sqrt {-1+x} x^3} \, dx,x,x^3\right ) \\ & = \frac {\sqrt {-1+x^3}}{6 x^6}+\frac {7}{12} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,x^3\right ) \\ & = \frac {\sqrt {-1+x^3}}{6 x^6}+\frac {7 \sqrt {-1+x^3}}{12 x^3}+\frac {7}{24} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^3\right ) \\ & = \frac {\sqrt {-1+x^3}}{6 x^6}+\frac {7 \sqrt {-1+x^3}}{12 x^3}+\frac {7}{12} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^3}\right ) \\ & = \frac {\sqrt {-1+x^3}}{6 x^6}+\frac {7 \sqrt {-1+x^3}}{12 x^3}+\frac {7}{12} \arctan \left (\sqrt {-1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^3}{x^7 \sqrt {-1+x^3}} \, dx=\frac {\sqrt {-1+x^3} \left (2+7 x^3\right )}{12 x^6}+\frac {7}{12} \arctan \left (\sqrt {-1+x^3}\right ) \]

[In]

Integrate[(1 + x^3)/(x^7*Sqrt[-1 + x^3]),x]

[Out]

(Sqrt[-1 + x^3]*(2 + 7*x^3))/(12*x^6) + (7*ArcTan[Sqrt[-1 + x^3]])/12

Maple [A] (verified)

Time = 2.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {7 \sqrt {x^{3}-1}}{12 x^{3}}+\frac {7 \arctan \left (\sqrt {x^{3}-1}\right )}{12}+\frac {\sqrt {x^{3}-1}}{6 x^{6}}\)	\(36\)
risch	\(\frac {7 x^{6}-5 x^{3}-2}{12 x^{6} \sqrt {x^{3}-1}}+\frac {7 \arctan \left (\sqrt {x^{3}-1}\right )}{12}\)	\(36\)
elliptic	\(\frac {7 \sqrt {x^{3}-1}}{12 x^{3}}+\frac {7 \arctan \left (\sqrt {x^{3}-1}\right )}{12}+\frac {\sqrt {x^{3}-1}}{6 x^{6}}\)	\(36\)
pseudoelliptic	\(\frac {7 \arctan \left (\sqrt {x^{3}-1}\right ) x^{6}+7 x^{3} \sqrt {x^{3}-1}+2 \sqrt {x^{3}-1}}{12 x^{6}}\)	\(41\)
trager	\(\frac {\sqrt {x^{3}-1}\, \left (7 x^{3}+2\right )}{12 x^{6}}-\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{3}-1}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{3}}\right )}{24}\)	\(63\)
meijerg	\(-\frac {\sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, \left (-\frac {\sqrt {\pi }\, \left (-4 x^{3}+8\right )}{8 x^{3}}+\frac {\sqrt {\pi }\, \sqrt {-x^{3}+1}}{x^{3}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }}{x^{3}}\right )}{3 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{3}-1\right )}}+\frac {\sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, \left (\frac {\sqrt {\pi }\, \left (-7 x^{6}+8 x^{3}+8\right )}{16 x^{6}}-\frac {\sqrt {\pi }\, \left (12 x^{3}+8\right ) \sqrt {-x^{3}+1}}{16 x^{6}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )}{4}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{8}-\frac {\sqrt {\pi }}{2 x^{6}}-\frac {\sqrt {\pi }}{2 x^{3}}\right )}{3 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{3}-1\right )}}\)	\(223\)

[In]

int((x^3+1)/x^7/(x^3-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

7/12*(x^3-1)^(1/2)/x^3+7/12*arctan((x^3-1)^(1/2))+1/6*(x^3-1)^(1/2)/x^6

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {1+x^3}{x^7 \sqrt {-1+x^3}} \, dx=\frac {7 \, x^{6} \arctan \left (\sqrt {x^{3} - 1}\right ) + {\left (7 \, x^{3} + 2\right )} \sqrt {x^{3} - 1}}{12 \, x^{6}} \]

[In]

integrate((x^3+1)/x^7/(x^3-1)^(1/2),x, algorithm="fricas")

[Out]

1/12*(7*x^6*arctan(sqrt(x^3 - 1)) + (7*x^3 + 2)*sqrt(x^3 - 1))/x^6

Sympy [A] (verification not implemented)

Time = 8.33 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.21 \[ \int \frac {1+x^3}{x^7 \sqrt {-1+x^3}} \, dx=\frac {7 \operatorname {atan}{\left (\sqrt {x^{3} - 1} \right )}}{12} + \frac {2 \sqrt {x^{3} - 1}}{3 x^{3}} + \frac {\left (2 - x^{3}\right ) \sqrt {x^{3} - 1}}{12 x^{6}} \]

[In]

integrate((x**3+1)/x**7/(x**3-1)**(1/2),x)

[Out]

7*atan(sqrt(x**3 - 1))/12 + 2*sqrt(x**3 - 1)/(3*x**3) + (2 - x**3)*sqrt(x**3 - 1)/(12*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.58 \[ \int \frac {1+x^3}{x^7 \sqrt {-1+x^3}} \, dx=\frac {3 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {x^{3} - 1}}{12 \, {\left (2 \, x^{3} + {\left (x^{3} - 1\right )}^{2} - 1\right )}} + \frac {\sqrt {x^{3} - 1}}{3 \, x^{3}} + \frac {7}{12} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]

[In]

integrate((x^3+1)/x^7/(x^3-1)^(1/2),x, algorithm="maxima")

[Out]

1/12*(3*(x^3 - 1)^(3/2) + 5*sqrt(x^3 - 1))/(2*x^3 + (x^3 - 1)^2 - 1) + 1/3*sqrt(x^3 - 1)/x^3 + 7/12*arctan(sqr
t(x^3 - 1))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \frac {1+x^3}{x^7 \sqrt {-1+x^3}} \, dx=\frac {7 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 9 \, \sqrt {x^{3} - 1}}{12 \, x^{6}} + \frac {7}{12} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]

[In]

integrate((x^3+1)/x^7/(x^3-1)^(1/2),x, algorithm="giac")

[Out]

1/12*(7*(x^3 - 1)^(3/2) + 9*sqrt(x^3 - 1))/x^6 + 7/12*arctan(sqrt(x^3 - 1))

Mupad [B] (verification not implemented)

Time = 4.92 (sec) , antiderivative size = 189, normalized size of antiderivative = 4.97 \[ \int \frac {1+x^3}{x^7 \sqrt {-1+x^3}} \, dx=\frac {7\,\sqrt {x^3-1}}{12\,x^3}+\frac {\sqrt {x^3-1}}{6\,x^6}-\frac {7\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{4\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

[In]

int((x^3 + 1)/(x^7*(x^3 - 1)^(1/2)),x)

[Out]

(7*(x^3 - 1)^(1/2))/(12*x^3) + (x^3 - 1)^(1/2)/(6*x^6) - (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 + (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)*ellipticPi((3^(1/2)*1i)/2 + 3/2, 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)))/(4*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2
- 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2))