\(\int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx\) [191]

Optimal result

Integrand size = 20, antiderivative size = 112 \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx=-\frac {\sqrt {a+b x^3+c x^6}}{3 x^3}-\frac {b \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{6 \sqrt {a}}+\frac {1}{3} \sqrt {c} \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1371, 746, 857, 635, 212, 738} \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx=-\frac {b \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{6 \sqrt {a}}+\frac {1}{3} \sqrt {c} \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )-\frac {\sqrt {a+b x^3+c x^6}}{3 x^3} \]

[In]

Int[Sqrt[a + b*x^3 + c*x^6]/x^4,x]

[Out]

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

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 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 746

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

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 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 {\sqrt {a+b x+c x^2}}{x^2} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{3 x^3}+\frac {1}{6} \text {Subst}\left (\int \frac {b+2 c x}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{3 x^3}+\frac {1}{6} b \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )+\frac {1}{3} c \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{3 x^3}-\frac {1}{3} b \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 )+\frac {1}{3} (2 c) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^3}{\sqrt {a+b x^3+c x^6}}\right ) \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{3 x^3}-\frac {b \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{6 \sqrt {a}}+\frac {1}{3} \sqrt {c} \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx=\frac {1}{3} \left (-\frac {\sqrt {a+b x^3+c x^6}}{x^3}+\frac {b \text {arctanh}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{\sqrt {a}}-\sqrt {c} \log \left (b+2 c x^3-2 \sqrt {c} \sqrt {a+b x^3+c x^6}\right )\right ) \]

[In]

Integrate[Sqrt[a + b*x^3 + c*x^6]/x^4,x]

[Out]

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

Maple [F]

\[\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{x^{4}}d x\]

[In]

int((c*x^6+b*x^3+a)^(1/2)/x^4,x)

[Out]

int((c*x^6+b*x^3+a)^(1/2)/x^4,x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 601, normalized size of antiderivative = 5.37 \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx=\left [\frac {2 \, a \sqrt {c} x^{3} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) + \sqrt {a} b x^{3} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 4 \, \sqrt {c x^{6} + b x^{3} + a} a}{12 \, a x^{3}}, -\frac {4 \, a \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) - \sqrt {a} b x^{3} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 4 \, \sqrt {c x^{6} + b x^{3} + a} a}{12 \, a x^{3}}, \frac {\sqrt {-a} b x^{3} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + a \sqrt {c} x^{3} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) - 2 \, \sqrt {c x^{6} + b x^{3} + a} a}{6 \, a x^{3}}, \frac {\sqrt {-a} b x^{3} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) - 2 \, a \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) - 2 \, \sqrt {c x^{6} + b x^{3} + a} a}{6 \, a x^{3}}\right ] \]

[In]

integrate((c*x^6+b*x^3+a)^(1/2)/x^4,x, algorithm="fricas")

[Out]

[1/12*(2*a*sqrt(c)*x^3*log(-8*c^2*x^6 - 8*b*c*x^3 - b^2 - 4*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)*sqrt(c) - 4*
a*c) + sqrt(a)*b*x^3*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)*a)/(a*x^3), -1/12*(4*a*sqrt(-c)*x^3*arctan(1/2*sqrt(c*x^6 + b*x^3 + a)*
(2*c*x^3 + b)*sqrt(-c)/(c^2*x^6 + b*c*x^3 + a*c)) - sqrt(a)*b*x^3*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)*a)/(a*x^3), 1/6*(sqrt(-a)*
b*x^3*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)) + a*sqrt(c)*x^3*log
(-8*c^2*x^6 - 8*b*c*x^3 - b^2 - 4*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)*sqrt(c) - 4*a*c) - 2*sqrt(c*x^6 + b*x^
3 + a)*a)/(a*x^3), 1/6*(sqrt(-a)*b*x^3*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*a*sqrt(-c)*x^3*arctan(1/2*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)*sqrt(-c)/(c^2*x^6 + b*c*x^3
+ a*c)) - 2*sqrt(c*x^6 + b*x^3 + a)*a)/(a*x^3)]

Sympy [F]

\[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx=\int \frac {\sqrt {a + b x^{3} + c x^{6}}}{x^{4}}\, dx \]

[In]

integrate((c*x**6+b*x**3+a)**(1/2)/x**4,x)

[Out]

Integral(sqrt(a + b*x**3 + c*x**6)/x**4, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^6+b*x^3+a)^(1/2)/x^4,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]

\[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx=\int { \frac {\sqrt {c x^{6} + b x^{3} + a}}{x^{4}} \,d x } \]

[In]

integrate((c*x^6+b*x^3+a)^(1/2)/x^4,x, algorithm="giac")

[Out]

integrate(sqrt(c*x^6 + b*x^3 + a)/x^4, x)

Mupad [B] (verification not implemented)

Time = 8.57 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx=\frac {\sqrt {c}\,\ln \left (\sqrt {c\,x^6+b\,x^3+a}+\frac {c\,x^3+\frac {b}{2}}{\sqrt {c}}\right )}{3}-\frac {\sqrt {c\,x^6+b\,x^3+a}}{3\,x^3}-\frac {b\,\ln \left (\frac {b}{2}+\frac {a}{x^3}+\frac {\sqrt {a}\,\sqrt {c\,x^6+b\,x^3+a}}{x^3}\right )}{6\,\sqrt {a}} \]

[In]

int((a + b*x^3 + c*x^6)^(1/2)/x^4,x)

[Out]

(c^(1/2)*log((a + b*x^3 + c*x^6)^(1/2) + (b/2 + c*x^3)/c^(1/2)))/3 - (a + b*x^3 + c*x^6)^(1/2)/(3*x^3) - (b*lo
g(b/2 + a/x^3 + (a^(1/2)*(a + b*x^3 + c*x^6)^(1/2))/x^3))/(6*a^(1/2))