\(\int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx\) [226]

Optimal result

Integrand size = 20, antiderivative size = 108 \[ \int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx=-\frac {\sqrt {a+b x^3+c x^6}}{6 a x^6}+\frac {b \sqrt {a+b x^3+c x^6}}{4 a^2 x^3}-\frac {\left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{24 a^{5/2}} \]

[Out]

-1/24*(-4*a*c+3*b^2)*arctanh(1/2*(b*x^3+2*a)/a^(1/2)/(c*x^6+b*x^3+a)^(1/2))/a^(5/2)-1/6*(c*x^6+b*x^3+a)^(1/2)/
a/x^6+1/4*b*(c*x^6+b*x^3+a)^(1/2)/a^2/x^3

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1371, 758, 820, 738, 212} \[ \int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx=-\frac {\left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{24 a^{5/2}}+\frac {b \sqrt {a+b x^3+c x^6}}{4 a^2 x^3}-\frac {\sqrt {a+b x^3+c x^6}}{6 a x^6} \]

[In]

Int[1/(x^7*Sqrt[a + b*x^3 + c*x^6]),x]

[Out]

-1/6*Sqrt[a + b*x^3 + c*x^6]/(a*x^6) + (b*Sqrt[a + b*x^3 + c*x^6])/(4*a^2*x^3) - ((3*b^2 - 4*a*c)*ArcTanh[(2*a
+ b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(24*a^(5/2))

Rule 212

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

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 758

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*
((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e
, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 1371

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{6 a x^6}-\frac {\text {Subst}\left (\int \frac {\frac {3 b}{2}+c x}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{6 a} \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{6 a x^6}+\frac {b \sqrt {a+b x^3+c x^6}}{4 a^2 x^3}+\frac {\left (3 b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{24 a^2} \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{6 a x^6}+\frac {b \sqrt {a+b x^3+c x^6}}{4 a^2 x^3}-\frac {\left (3 b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^3}{\sqrt {a+b x^3+c x^6}}\right )}{12 a^2} \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{6 a x^6}+\frac {b \sqrt {a+b x^3+c x^6}}{4 a^2 x^3}-\frac {\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{24 a^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx=\frac {\left (-2 a+3 b x^3\right ) \sqrt {a+b x^3+c x^6}}{12 a^2 x^6}+\frac {\left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{12 a^{5/2}} \]

[In]

Integrate[1/(x^7*Sqrt[a + b*x^3 + c*x^6]),x]

[Out]

((-2*a + 3*b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(12*a^2*x^6) + ((3*b^2 - 4*a*c)*ArcTanh[(Sqrt[c]*x^3 - Sqrt[a + b*x
^3 + c*x^6])/Sqrt[a]])/(12*a^(5/2))

Maple [F]

\[\int \frac {1}{x^{7} \sqrt {c \,x^{6}+b \,x^{3}+a}}d x\]

[In]

int(1/x^7/(c*x^6+b*x^3+a)^(1/2),x)

[Out]

int(1/x^7/(c*x^6+b*x^3+a)^(1/2),x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.05 \[ \int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx=\left [-\frac {{\left (3 \, b^{2} - 4 \, a c\right )} \sqrt {a} x^{6} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (3 \, a b x^{3} - 2 \, a^{2}\right )}}{48 \, a^{3} x^{6}}, \frac {{\left (3 \, b^{2} - 4 \, a c\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \, \sqrt {c x^{6} + b x^{3} + a} {\left (3 \, a b x^{3} - 2 \, a^{2}\right )}}{24 \, a^{3} x^{6}}\right ] \]

[In]

integrate(1/x^7/(c*x^6+b*x^3+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/48*((3*b^2 - 4*a*c)*sqrt(a)*x^6*log(-((b^2 + 4*a*c)*x^6 + 8*a*b*x^3 + 4*sqrt(c*x^6 + b*x^3 + a)*(b*x^3 + 2
*a)*sqrt(a) + 8*a^2)/x^6) - 4*sqrt(c*x^6 + b*x^3 + a)*(3*a*b*x^3 - 2*a^2))/(a^3*x^6), 1/24*((3*b^2 - 4*a*c)*sq
rt(-a)*x^6*arctan(1/2*sqrt(c*x^6 + b*x^3 + a)*(b*x^3 + 2*a)*sqrt(-a)/(a*c*x^6 + a*b*x^3 + a^2)) + 2*sqrt(c*x^6
+ b*x^3 + a)*(3*a*b*x^3 - 2*a^2))/(a^3*x^6)]

Sympy [F]

\[ \int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx=\int \frac {1}{x^{7} \sqrt {a + b x^{3} + c x^{6}}}\, dx \]

[In]

integrate(1/x**7/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

Integral(1/(x**7*sqrt(a + b*x**3 + c*x**6)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/x^7/(c*x^6+b*x^3+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [F]

\[ \int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx=\int { \frac {1}{\sqrt {c x^{6} + b x^{3} + a} x^{7}} \,d x } \]

[In]

integrate(1/x^7/(c*x^6+b*x^3+a)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^7), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx=\int \frac {1}{x^7\,\sqrt {c\,x^6+b\,x^3+a}} \,d x \]

[In]

int(1/(x^7*(a + b*x^3 + c*x^6)^(1/2)),x)

[Out]

int(1/(x^7*(a + b*x^3 + c*x^6)^(1/2)), x)