3.193 \(\int \frac{\sqrt{a+b x^3+c x^6}}{x^{10}} \, dx\)

Optimal. Leaf size=116 \[ -\frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{48 a^{5/2}}+\frac{b \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{24 a^2 x^6}-\frac{\left (a+b x^3+c x^6\right )^{3/2}}{9 a x^9} \]

[Out]

(b*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(24*a^2*x^6) - (a + b*x^3 + c*x^6)^(3/2)/(9*a*x^9) - (b*(b^2 - 4*a*c
)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(48*a^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0954686, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1357, 730, 720, 724, 206} \[ -\frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{48 a^{5/2}}+\frac{b \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{24 a^2 x^6}-\frac{\left (a+b x^3+c x^6\right )^{3/2}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*x^3 + c*x^6]/x^10,x]

[Out]

(b*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(24*a^2*x^6) - (a + b*x^3 + c*x^6)^(3/2)/(9*a*x^9) - (b*(b^2 - 4*a*c
)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(48*a^(5/2))

Rule 1357

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]]

Rule 730

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[(2*c*d - b*e)/(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, 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] && EqQ[m + 2*p + 3, 0]

Rule 720

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] /; FreeQ[
{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 724

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 206

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

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b x^3+c x^6}}{x^{10}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x^4} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3+c x^6\right )^{3/2}}{9 a x^9}-\frac{b \operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x^3} \, dx,x,x^3\right )}{6 a}\\ &=\frac{b \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{24 a^2 x^6}-\frac{\left (a+b x^3+c x^6\right )^{3/2}}{9 a x^9}+\frac{\left (b \left (b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{48 a^2}\\ &=\frac{b \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{24 a^2 x^6}-\frac{\left (a+b x^3+c x^6\right )^{3/2}}{9 a x^9}-\frac{\left (b \left (b^2-4 a c\right )\right ) \operatorname{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 )}{24 a^2}\\ &=\frac{b \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{24 a^2 x^6}-\frac{\left (a+b x^3+c x^6\right )^{3/2}}{9 a x^9}-\frac{b \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 )}{48 a^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0856186, size = 108, normalized size = 0.93 \[ -\frac{\sqrt{a+b x^3+c x^6} \left (8 a^2+2 a x^3 \left (b+4 c x^3\right )-3 b^2 x^6\right )}{72 a^2 x^9}-\frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{48 a^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*x^3 + c*x^6]/x^10,x]

[Out]

-(Sqrt[a + b*x^3 + c*x^6]*(8*a^2 - 3*b^2*x^6 + 2*a*x^3*(b + 4*c*x^3)))/(72*a^2*x^9) - (b*(b^2 - 4*a*c)*ArcTanh
[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(48*a^(5/2))

________________________________________________________________________________________

Maple [F]  time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{10}}\sqrt{c{x}^{6}+b{x}^{3}+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^6+b*x^3+a)^(1/2)/x^10,x)

[Out]

int((c*x^6+b*x^3+a)^(1/2)/x^10,x)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^6+b*x^3+a)^(1/2)/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.91256, size = 595, normalized size = 5.13 \begin{align*} \left [-\frac{3 \,{\left (b^{3} - 4 \, a b c\right )} \sqrt{a} x^{9} \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 \,{\left ({\left (3 \, a b^{2} - 8 \, a^{2} c\right )} x^{6} - 2 \, a^{2} b x^{3} - 8 \, a^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{288 \, a^{3} x^{9}}, \frac{3 \,{\left (b^{3} - 4 \, a b c\right )} \sqrt{-a} x^{9} \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 \,{\left ({\left (3 \, a b^{2} - 8 \, a^{2} c\right )} x^{6} - 2 \, a^{2} b x^{3} - 8 \, a^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{144 \, a^{3} x^{9}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^6+b*x^3+a)^(1/2)/x^10,x, algorithm="fricas")

[Out]

[-1/288*(3*(b^3 - 4*a*b*c)*sqrt(a)*x^9*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*((3*a*b^2 - 8*a^2*c)*x^6 - 2*a^2*b*x^3 - 8*a^3)*sqrt(c*x^6 + b*x^3 + a))/(a^3
*x^9), 1/144*(3*(b^3 - 4*a*b*c)*sqrt(-a)*x^9*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*((3*a*b^2 - 8*a^2*c)*x^6 - 2*a^2*b*x^3 - 8*a^3)*sqrt(c*x^6 + b*x^3 + a))/(a^3*x^9)]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b x^{3} + c x^{6}}}{x^{10}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**6+b*x**3+a)**(1/2)/x**10,x)

[Out]

Integral(sqrt(a + b*x**3 + c*x**6)/x**10, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{6} + b x^{3} + a}}{x^{10}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^6+b*x^3+a)^(1/2)/x^10,x, algorithm="giac")

[Out]

integrate(sqrt(c*x^6 + b*x^3 + a)/x^10, x)