3.876 \(\int \frac{x^8}{(a+b x^6)^2 \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=93 \[ \frac{c \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{6 \sqrt{a} (b c-a d)^{3/2}}-\frac{x^3 \sqrt{c+d x^6}}{6 \left (a+b x^6\right ) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.09026, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {465, 471, 12, 377, 205} \[ \frac{c \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{6 \sqrt{a} (b c-a d)^{3/2}}-\frac{x^3 \sqrt{c+d x^6}}{6 \left (a+b x^6\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[x^8/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

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

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 471

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

Mathematica [A]  time = 0.443684, size = 124, normalized size = 1.33 \[ \frac{\sqrt{c+d x^6} \left (-\frac{x^6 (b c-a d)}{a+b x^6}-\frac{c \sqrt{x^6 \left (\frac{d}{c}-\frac{b}{a}\right )} \tanh ^{-1}\left (\frac{\sqrt{x^6 \left (\frac{d}{c}-\frac{b}{a}\right )}}{\sqrt{\frac{d x^6}{c}+1}}\right )}{\sqrt{\frac{d x^6}{c}+1}}\right )}{6 x^3 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

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

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{8}}{ \left ( b{x}^{6}+a \right ) ^{2}}{\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)^2/(d*x^6+c)^(1/2),x)

[Out]

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

[Out]

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

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Fricas [B]  time = 2.44086, size = 887, normalized size = 9.54 \begin{align*} \left [-\frac{4 \, \sqrt{d x^{6} + c}{\left (a b c - a^{2} d\right )} x^{3} -{\left (b c x^{6} + a c\right )} \sqrt{-a b c + a^{2} 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 c - 2 \, a d\right )} x^{9} - a c x^{3}\right )} \sqrt{d x^{6} + c} \sqrt{-a b c + a^{2} d}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right )}{24 \,{\left ({\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{6} + a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}}, -\frac{2 \, \sqrt{d x^{6} + c}{\left (a b c - a^{2} d\right )} x^{3} -{\left (b c x^{6} + a c\right )} \sqrt{a b c - a^{2} d} \arctan \left (\frac{{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt{d x^{6} + c} \sqrt{a b c - a^{2} d}}{2 \,{\left ({\left (a b c d - a^{2} d^{2}\right )} x^{9} +{\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )}}\right )}{12 \,{\left ({\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{6} + a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/24*(4*sqrt(d*x^6 + c)*(a*b*c - a^2*d)*x^3 - (b*c*x^6 + a*c)*sqrt(-a*b*c + a^2*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*c - 2*a*d)*x^9 - a*c*x^3)*sqrt(d*x^6 + c)
*sqrt(-a*b*c + a^2*d))/(b^2*x^12 + 2*a*b*x^6 + a^2)))/((a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x^6 + a^2*b^2*c
^2 - 2*a^3*b*c*d + a^4*d^2), -1/12*(2*sqrt(d*x^6 + c)*(a*b*c - a^2*d)*x^3 - (b*c*x^6 + a*c)*sqrt(a*b*c - a^2*d
)*arctan(1/2*((b*c - 2*a*d)*x^6 - a*c)*sqrt(d*x^6 + c)*sqrt(a*b*c - a^2*d)/((a*b*c*d - a^2*d^2)*x^9 + (a*b*c^2
- a^2*c*d)*x^3)))/((a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x^6 + a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.27726, size = 248, normalized size = 2.67 \begin{align*} -\frac{1}{6} \, c{\left (\frac{\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}{\left (b c \mathrm{sgn}\left (x\right ) - a d \mathrm{sgn}\left (x\right )\right )}} + \frac{\sqrt{d + \frac{c}{x^{6}}}}{{\left (b c \mathrm{sgn}\left (x\right ) - a d \mathrm{sgn}\left (x\right )\right )}{\left (b c + a{\left (d + \frac{c}{x^{6}}\right )} - a d\right )}}\right )} + \frac{{\left (b c \arctan \left (\frac{a \sqrt{d}}{\sqrt{a b c - a^{2} d}}\right ) + \sqrt{a b c - a^{2} d} \sqrt{d}\right )} \mathrm{sgn}\left (x\right )}{6 \,{\left (\sqrt{a b c - a^{2} d} b^{2} c - \sqrt{a b c - a^{2} d} a b d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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