3.228 \(\int \frac{1}{x^{13} \sqrt{a+b x^3+c x^6}} \, dx\)

Optimal. Leaf size=192 \[ -\frac{\left (48 a^2 c^2-120 a b^2 c+35 b^4\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{384 a^{9/2}}+\frac{5 b \left (21 b^2-44 a c\right ) \sqrt{a+b x^3+c x^6}}{576 a^4 x^3}-\frac{\left (35 b^2-36 a c\right ) \sqrt{a+b x^3+c x^6}}{288 a^3 x^6}+\frac{7 b \sqrt{a+b x^3+c x^6}}{72 a^2 x^9}-\frac{\sqrt{a+b x^3+c x^6}}{12 a x^{12}} \]

[Out]

-Sqrt[a + b*x^3 + c*x^6]/(12*a*x^12) + (7*b*Sqrt[a + b*x^3 + c*x^6])/(72*a^2*x^9) - ((35*b^2 - 36*a*c)*Sqrt[a
+ b*x^3 + c*x^6])/(288*a^3*x^6) + (5*b*(21*b^2 - 44*a*c)*Sqrt[a + b*x^3 + c*x^6])/(576*a^4*x^3) - ((35*b^4 - 1
20*a*b^2*c + 48*a^2*c^2)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(384*a^(9/2))

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Rubi [A]  time = 0.233341, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1357, 744, 834, 806, 724, 206} \[ -\frac{\left (48 a^2 c^2-120 a b^2 c+35 b^4\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{384 a^{9/2}}+\frac{5 b \left (21 b^2-44 a c\right ) \sqrt{a+b x^3+c x^6}}{576 a^4 x^3}-\frac{\left (35 b^2-36 a c\right ) \sqrt{a+b x^3+c x^6}}{288 a^3 x^6}+\frac{7 b \sqrt{a+b x^3+c x^6}}{72 a^2 x^9}-\frac{\sqrt{a+b x^3+c x^6}}{12 a x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^13*Sqrt[a + b*x^3 + c*x^6]),x]

[Out]

-Sqrt[a + b*x^3 + c*x^6]/(12*a*x^12) + (7*b*Sqrt[a + b*x^3 + c*x^6])/(72*a^2*x^9) - ((35*b^2 - 36*a*c)*Sqrt[a
+ b*x^3 + c*x^6])/(288*a^3*x^6) + (5*b*(21*b^2 - 44*a*c)*Sqrt[a + b*x^3 + c*x^6])/(576*a^4*x^3) - ((35*b^4 - 1
20*a*b^2*c + 48*a^2*c^2)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(384*a^(9/2))

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 744

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 834

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))/((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)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((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[Sim
plify[m + 2*p + 3], 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{1}{x^{13} \sqrt{a+b x^3+c x^6}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^5 \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{a+b x^3+c x^6}}{12 a x^{12}}-\frac{\operatorname{Subst}\left (\int \frac{\frac{7 b}{2}+3 c x}{x^4 \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{12 a}\\ &=-\frac{\sqrt{a+b x^3+c x^6}}{12 a x^{12}}+\frac{7 b \sqrt{a+b x^3+c x^6}}{72 a^2 x^9}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{4} \left (35 b^2-36 a c\right )+7 b c x}{x^3 \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{36 a^2}\\ &=-\frac{\sqrt{a+b x^3+c x^6}}{12 a x^{12}}+\frac{7 b \sqrt{a+b x^3+c x^6}}{72 a^2 x^9}-\frac{\left (35 b^2-36 a c\right ) \sqrt{a+b x^3+c x^6}}{288 a^3 x^6}-\frac{\operatorname{Subst}\left (\int \frac{\frac{5}{8} b \left (21 b^2-44 a c\right )+\frac{1}{4} c \left (35 b^2-36 a c\right ) x}{x^2 \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{72 a^3}\\ &=-\frac{\sqrt{a+b x^3+c x^6}}{12 a x^{12}}+\frac{7 b \sqrt{a+b x^3+c x^6}}{72 a^2 x^9}-\frac{\left (35 b^2-36 a c\right ) \sqrt{a+b x^3+c x^6}}{288 a^3 x^6}+\frac{5 b \left (21 b^2-44 a c\right ) \sqrt{a+b x^3+c x^6}}{576 a^4 x^3}+\frac{\left (35 b^4-120 a b^2 c+48 a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{384 a^4}\\ &=-\frac{\sqrt{a+b x^3+c x^6}}{12 a x^{12}}+\frac{7 b \sqrt{a+b x^3+c x^6}}{72 a^2 x^9}-\frac{\left (35 b^2-36 a c\right ) \sqrt{a+b x^3+c x^6}}{288 a^3 x^6}+\frac{5 b \left (21 b^2-44 a c\right ) \sqrt{a+b x^3+c x^6}}{576 a^4 x^3}-\frac{\left (35 b^4-120 a b^2 c+48 a^2 c^2\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 )}{192 a^4}\\ &=-\frac{\sqrt{a+b x^3+c x^6}}{12 a x^{12}}+\frac{7 b \sqrt{a+b x^3+c x^6}}{72 a^2 x^9}-\frac{\left (35 b^2-36 a c\right ) \sqrt{a+b x^3+c x^6}}{288 a^3 x^6}+\frac{5 b \left (21 b^2-44 a c\right ) \sqrt{a+b x^3+c x^6}}{576 a^4 x^3}-\frac{\left (35 b^4-120 a b^2 c+48 a^2 c^2\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{384 a^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.101818, size = 141, normalized size = 0.73 \[ \frac{\sqrt{a+b x^3+c x^6} \left (8 a^2 \left (7 b x^3+9 c x^6\right )-48 a^3-10 a b x^6 \left (7 b+22 c x^3\right )+105 b^3 x^9\right )}{576 a^4 x^{12}}-\frac{\left (48 a^2 c^2-120 a b^2 c+35 b^4\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{384 a^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^13*Sqrt[a + b*x^3 + c*x^6]),x]

[Out]

(Sqrt[a + b*x^3 + c*x^6]*(-48*a^3 + 105*b^3*x^9 - 10*a*b*x^6*(7*b + 22*c*x^3) + 8*a^2*(7*b*x^3 + 9*c*x^6)))/(5
76*a^4*x^12) - ((35*b^4 - 120*a*b^2*c + 48*a^2*c^2)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])]
)/(384*a^(9/2))

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{13}}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^13/(c*x^6+b*x^3+a)^(1/2),x)

[Out]

int(1/x^13/(c*x^6+b*x^3+a)^(1/2),x)

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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(1/x^13/(c*x^6+b*x^3+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.1326, size = 764, normalized size = 3.98 \begin{align*} \left [\frac{3 \,{\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} \sqrt{a} x^{12} \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 (5 \,{\left (21 \, a b^{3} - 44 \, a^{2} b c\right )} x^{9} + 56 \, a^{3} b x^{3} - 2 \,{\left (35 \, a^{2} b^{2} - 36 \, a^{3} c\right )} x^{6} - 48 \, a^{4}\right )} \sqrt{c x^{6} + b x^{3} + a}}{2304 \, a^{5} x^{12}}, \frac{3 \,{\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} \sqrt{-a} x^{12} \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 (5 \,{\left (21 \, a b^{3} - 44 \, a^{2} b c\right )} x^{9} + 56 \, a^{3} b x^{3} - 2 \,{\left (35 \, a^{2} b^{2} - 36 \, a^{3} c\right )} x^{6} - 48 \, a^{4}\right )} \sqrt{c x^{6} + b x^{3} + a}}{1152 \, a^{5} x^{12}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(c*x^6+b*x^3+a)^(1/2),x, algorithm="fricas")

[Out]

[1/2304*(3*(35*b^4 - 120*a*b^2*c + 48*a^2*c^2)*sqrt(a)*x^12*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*(5*(21*a*b^3 - 44*a^2*b*c)*x^9 + 56*a^3*b*x^3 - 2*(35*a^
2*b^2 - 36*a^3*c)*x^6 - 48*a^4)*sqrt(c*x^6 + b*x^3 + a))/(a^5*x^12), 1/1152*(3*(35*b^4 - 120*a*b^2*c + 48*a^2*
c^2)*sqrt(-a)*x^12*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*(5
*(21*a*b^3 - 44*a^2*b*c)*x^9 + 56*a^3*b*x^3 - 2*(35*a^2*b^2 - 36*a^3*c)*x^6 - 48*a^4)*sqrt(c*x^6 + b*x^3 + a))
/(a^5*x^12)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{13} \sqrt{a + b x^{3} + c x^{6}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**13/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

Integral(1/(x**13*sqrt(a + b*x**3 + c*x**6)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{6} + b x^{3} + a} x^{13}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(c*x^6+b*x^3+a)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^13), x)