3.859 \(\int \frac{x^8}{(a+b x^6) \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=91 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^3}{\sqrt{c+d x^6}}\right )}{3 b \sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{3 b \sqrt{b c-a d}} \]

[Out]

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

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Rubi [A]  time = 0.0873107, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {465, 483, 217, 206, 377, 205} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^3}{\sqrt{c+d x^6}}\right )}{3 b \sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{3 b \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

[In]

Int[x^8/((a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

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

Rule 465

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 483

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Dist[e^n/b, Int[
(e*x)^(m - n)*(c + d*x^n)^q, x], x] - Dist[(a*e^n)/b, Int[((e*x)^(m - n)*(c + d*x^n)^q)/(a + b*x^n), x], x] /;
FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1] && IntBinomialQ[a, b
, c, d, e, m, n, -1, q, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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])

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^8}{\left (a+b x^6\right ) \sqrt{c+d x^6}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^3\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^2}} \, dx,x,x^3\right )}{3 b}-\frac{a \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^3\right )}{3 b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1-d x^2} \, dx,x,\frac{x^3}{\sqrt{c+d x^6}}\right )}{3 b}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x^3}{\sqrt{c+d x^6}}\right )}{3 b}\\ &=-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b c-a d} x^3}{\sqrt{a} \sqrt{c+d x^6}}\right )}{3 b \sqrt{b c-a d}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^3}{\sqrt{c+d x^6}}\right )}{3 b \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 0.0582949, size = 90, normalized size = 0.99 \[ \frac{\frac{\log \left (\sqrt{d} \sqrt{c+d x^6}+d x^3\right )}{\sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{\sqrt{b c-a d}}}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/((a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

(-((Sqrt[a]*ArcTan[(Sqrt[b*c - a*d]*x^3)/(Sqrt[a]*Sqrt[c + d*x^6])])/Sqrt[b*c - a*d]) + Log[d*x^3 + Sqrt[d]*Sq
rt[c + d*x^6]]/Sqrt[d])/(3*b)

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{8}}{b{x}^{6}+a}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(b*x^6+a)/(d*x^6+c)^(1/2),x)

[Out]

int(x^8/(b*x^6+a)/(d*x^6+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{8}}{{\left (b x^{6} + a\right )} \sqrt{d x^{6} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^8/((b*x^6 + a)*sqrt(d*x^6 + c)), x)

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Fricas [A]  time = 1.4812, size = 1366, normalized size = 15.01 \begin{align*} \left [\frac{d \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} - 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{9} -{\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )} \sqrt{d x^{6} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) + 2 \, \sqrt{d} \log \left (-2 \, d x^{6} - 2 \, \sqrt{d x^{6} + c} \sqrt{d} x^{3} - c\right )}{12 \, b d}, \frac{d \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} - 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{9} -{\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )} \sqrt{d x^{6} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) - 4 \, \sqrt{-d} \arctan \left (\frac{\sqrt{-d} x^{3}}{\sqrt{d x^{6} + c}}\right )}{12 \, b d}, \frac{d \sqrt{\frac{a}{b c - a d}} \arctan \left (-\frac{{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt{d x^{6} + c} \sqrt{\frac{a}{b c - a d}}}{2 \,{\left (a d x^{9} + a c x^{3}\right )}}\right ) + \sqrt{d} \log \left (-2 \, d x^{6} - 2 \, \sqrt{d x^{6} + c} \sqrt{d} x^{3} - c\right )}{6 \, b d}, \frac{d \sqrt{\frac{a}{b c - a d}} \arctan \left (-\frac{{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt{d x^{6} + c} \sqrt{\frac{a}{b c - a d}}}{2 \,{\left (a d x^{9} + a c x^{3}\right )}}\right ) - 2 \, \sqrt{-d} \arctan \left (\frac{\sqrt{-d} x^{3}}{\sqrt{d x^{6} + c}}\right )}{6 \, b d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="fricas")

[Out]

[1/12*(d*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^12 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^6 + a^
2*c^2 - 4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^9 - (a*b*c^2 - a^2*c*d)*x^3)*sqrt(d*x^6 + c)*sqrt(-a/(b*c - a*d
)))/(b^2*x^12 + 2*a*b*x^6 + a^2)) + 2*sqrt(d)*log(-2*d*x^6 - 2*sqrt(d*x^6 + c)*sqrt(d)*x^3 - c))/(b*d), 1/12*(
d*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^12 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^6 + a^2*c^2 -
4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^9 - (a*b*c^2 - a^2*c*d)*x^3)*sqrt(d*x^6 + c)*sqrt(-a/(b*c - a*d)))/(b^
2*x^12 + 2*a*b*x^6 + a^2)) - 4*sqrt(-d)*arctan(sqrt(-d)*x^3/sqrt(d*x^6 + c)))/(b*d), 1/6*(d*sqrt(a/(b*c - a*d)
)*arctan(-1/2*((b*c - 2*a*d)*x^6 - a*c)*sqrt(d*x^6 + c)*sqrt(a/(b*c - a*d))/(a*d*x^9 + a*c*x^3)) + sqrt(d)*log
(-2*d*x^6 - 2*sqrt(d*x^6 + c)*sqrt(d)*x^3 - c))/(b*d), 1/6*(d*sqrt(a/(b*c - a*d))*arctan(-1/2*((b*c - 2*a*d)*x
^6 - a*c)*sqrt(d*x^6 + c)*sqrt(a/(b*c - a*d))/(a*d*x^9 + a*c*x^3)) - 2*sqrt(-d)*arctan(sqrt(-d)*x^3/sqrt(d*x^6
+ c)))/(b*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{8}}{\left (a + b x^{6}\right ) \sqrt{c + d x^{6}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(b*x**6+a)/(d*x**6+c)**(1/2),x)

[Out]

Integral(x**8/((a + b*x**6)*sqrt(c + d*x**6)), x)

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Giac [B]  time = 1.20023, size = 223, normalized size = 2.45 \begin{align*} \frac{1}{3} \, c{\left (\frac{a \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{6}}}}{\sqrt{a b c - a^{2} d}}\right )}{\sqrt{a b c - a^{2} d} b c \mathrm{sgn}\left (x\right )} - \frac{\arctan \left (\frac{\sqrt{d + \frac{c}{x^{6}}}}{\sqrt{-d}}\right )}{b c \sqrt{-d} \mathrm{sgn}\left (x\right )}\right )} - \frac{{\left (a \sqrt{-d} \arctan \left (\frac{a \sqrt{d}}{\sqrt{a b c - a^{2} d}}\right ) - \sqrt{a b c - a^{2} d} \arctan \left (\frac{\sqrt{d}}{\sqrt{-d}}\right )\right )} \mathrm{sgn}\left (x\right )}{3 \, \sqrt{a b c - a^{2} d} b \sqrt{-d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="giac")

[Out]

1/3*c*(a*arctan(a*sqrt(d + c/x^6)/sqrt(a*b*c - a^2*d))/(sqrt(a*b*c - a^2*d)*b*c*sgn(x)) - arctan(sqrt(d + c/x^
6)/sqrt(-d))/(b*c*sqrt(-d)*sgn(x))) - 1/3*(a*sqrt(-d)*arctan(a*sqrt(d)/sqrt(a*b*c - a^2*d)) - sqrt(a*b*c - a^2
*d)*arctan(sqrt(d)/sqrt(-d)))*sgn(x)/(sqrt(a*b*c - a^2*d)*b*sqrt(-d))