\(\int \frac {\sqrt {a+b x^3+c x^6}}{x^7} \, dx\) [192]

Optimal result

Integrand size = 20, antiderivative size = 88 \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^7} \, dx=-\frac {\left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{12 a x^6}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{24 a^{3/2}} \]

[Out]

1/24*(-4*a*c+b^2)*arctanh(1/2*(b*x^3+2*a)/a^(1/2)/(c*x^6+b*x^3+a)^(1/2))/a^(3/2)-1/12*(b*x^3+2*a)*(c*x^6+b*x^3
+a)^(1/2)/a/x^6

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1371, 734, 738, 212} \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^7} \, dx=\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{24 a^{3/2}}-\frac {\left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{12 a x^6} \]

[In]

Int[Sqrt[a + b*x^3 + c*x^6]/x^7,x]

[Out]

-1/12*((2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(a*x^6) + ((b^2 - 4*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a
+ b*x^3 + c*x^6])])/(24*a^(3/2))

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 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))
*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[p*((b^2
- 4*a*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; Free
Q[{a, b, c, d, e}, 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] && EqQ[m
+ 2*p + 2, 0] && GtQ[p, 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 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^3} \, dx,x,x^3\right ) \\ & = -\frac {\left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{12 a x^6}-\frac {\left (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} \\ & = -\frac {\left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{12 a x^6}+\frac {\left (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} \\ & = -\frac {\left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{12 a x^6}+\frac {\left (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^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^7} \, dx=\frac {\left (-2 a-b x^3\right ) \sqrt {a+b x^3+c x^6}}{12 a x^6}+\frac {\left (-b^2+4 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{12 a^{3/2}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.44 \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^7} \, dx=\left [-\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {a} x^{6} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (a b x^{3} + 2 \, a^{2}\right )}}{48 \, a^{2} x^{6}}, -\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \, \sqrt {c x^{6} + b x^{3} + a} {\left (a b x^{3} + 2 \, a^{2}\right )}}{24 \, a^{2} x^{6}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*x^3 + c*x^6)^(1/2)/x^7,x)

[Out]

int((a + b*x^3 + c*x^6)^(1/2)/x^7, x)