3.875 \(\int \frac{x^{14}}{(a+b x^6)^2 \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=141 \[ -\frac{\sqrt{a} (3 b c-2 a d) \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{6 b^2 (b c-a d)^{3/2}}+\frac{a x^3 \sqrt{c+d x^6}}{6 b \left (a+b x^6\right ) (b c-a d)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^3}{\sqrt{c+d x^6}}\right )}{3 b^2 \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.157584, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {465, 470, 523, 217, 206, 377, 205} \[ -\frac{\sqrt{a} (3 b c-2 a d) \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{6 b^2 (b c-a d)^{3/2}}+\frac{a x^3 \sqrt{c+d x^6}}{6 b \left (a+b x^6\right ) (b c-a d)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^3}{\sqrt{c+d x^6}}\right )}{3 b^2 \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Int[x^14/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

(a*x^3*Sqrt[c + d*x^6])/(6*b*(b*c - a*d)*(a + b*x^6)) - (Sqrt[a]*(3*b*c - 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x^3)/
(Sqrt[a]*Sqrt[c + d*x^6])])/(6*b^2*(b*c - a*d)^(3/2)) + ArcTanh[(Sqrt[d]*x^3)/Sqrt[c + d*x^6]]/(3*b^2*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 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2
*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2
*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +
(a*d*(m - n + n*q + 1) + b*c*n*(p + 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] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, 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^{14}}{\left (a+b x^6\right )^2 \sqrt{c+d x^6}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b x^2\right )^2 \sqrt{c+d x^2}} \, dx,x,x^3\right )\\ &=\frac{a x^3 \sqrt{c+d x^6}}{6 b (b c-a d) \left (a+b x^6\right )}-\frac{\operatorname{Subst}\left (\int \frac{a c-2 (b c-a d) x^2}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^3\right )}{6 b (b c-a d)}\\ &=\frac{a x^3 \sqrt{c+d x^6}}{6 b (b c-a d) \left (a+b x^6\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^2}} \, dx,x,x^3\right )}{3 b^2}-\frac{(a (3 b c-2 a d)) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^3\right )}{6 b^2 (b c-a d)}\\ &=\frac{a x^3 \sqrt{c+d x^6}}{6 b (b c-a d) \left (a+b x^6\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-d x^2} \, dx,x,\frac{x^3}{\sqrt{c+d x^6}}\right )}{3 b^2}-\frac{(a (3 b c-2 a d)) \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^2 (b c-a d)}\\ &=\frac{a x^3 \sqrt{c+d x^6}}{6 b (b c-a d) \left (a+b x^6\right )}-\frac{\sqrt{a} (3 b c-2 a d) \tan ^{-1}\left (\frac{\sqrt{b c-a d} x^3}{\sqrt{a} \sqrt{c+d x^6}}\right )}{6 b^2 (b c-a d)^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^3}{\sqrt{c+d x^6}}\right )}{3 b^2 \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 0.210232, size = 135, normalized size = 0.96 \[ \frac{\frac{a b x^3 \sqrt{c+d x^6}}{\left (a+b x^6\right ) (b c-a d)}+\frac{\sqrt{a} (2 a d-3 b c) \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{(b c-a d)^{3/2}}+\frac{2 \log \left (\sqrt{d} \sqrt{c+d x^6}+d x^3\right )}{\sqrt{d}}}{6 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^14/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

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

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{14}}{ \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^14/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

[Out]

int(x^14/(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^{14}}{{\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^14/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^14/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)

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Fricas [A]  time = 4.16804, size = 2279, normalized size = 16.16 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(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*d*x^3 + 4*((b^2*c - a*b*d)*x^6 + a*b*c - a^2*d)*sqrt(d)*log(-2*d*x^6 - 2*sqrt(d*x
^6 + c)*sqrt(d)*x^3 - c) + ((3*b^2*c*d - 2*a*b*d^2)*x^6 + 3*a*b*c*d - 2*a^2*d^2)*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)))/((b
^4*c*d - a*b^3*d^2)*x^6 + a*b^3*c*d - a^2*b^2*d^2), 1/24*(4*sqrt(d*x^6 + c)*a*b*d*x^3 - 8*((b^2*c - a*b*d)*x^6
+ a*b*c - a^2*d)*sqrt(-d)*arctan(sqrt(-d)*x^3/sqrt(d*x^6 + c)) + ((3*b^2*c*d - 2*a*b*d^2)*x^6 + 3*a*b*c*d - 2
*a^2*d^2)*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)))/((b^4*c*d - a*b^3*d^2)*x^6 + a*b^3*c*d - a^2*b^2*d^2), 1/12*(2*sqrt(d*x^6
+ c)*a*b*d*x^3 + ((3*b^2*c*d - 2*a*b*d^2)*x^6 + 3*a*b*c*d - 2*a^2*d^2)*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*((b^2*c - a*b*d)*x^6 + a*b*c -
a^2*d)*sqrt(d)*log(-2*d*x^6 - 2*sqrt(d*x^6 + c)*sqrt(d)*x^3 - c))/((b^4*c*d - a*b^3*d^2)*x^6 + a*b^3*c*d - a^
2*b^2*d^2), 1/12*(2*sqrt(d*x^6 + c)*a*b*d*x^3 - 4*((b^2*c - a*b*d)*x^6 + a*b*c - a^2*d)*sqrt(-d)*arctan(sqrt(-
d)*x^3/sqrt(d*x^6 + c)) + ((3*b^2*c*d - 2*a*b*d^2)*x^6 + 3*a*b*c*d - 2*a^2*d^2)*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)))/((b^4*c*d - a*b^3*d^2)*x
^6 + a*b^3*c*d - a^2*b^2*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**14/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.24882, size = 482, normalized size = 3.42 \begin{align*} \frac{1}{6} \, c^{2}{\left (\frac{{\left (3 \, a b c - 2 \, a^{2} d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{6}}}}{\sqrt{a b c - a^{2} d}}\right )}{{\left (b^{3} c^{3} \mathrm{sgn}\left (x\right ) - a b^{2} c^{2} d \mathrm{sgn}\left (x\right )\right )} \sqrt{a b c - a^{2} d}} + \frac{a \sqrt{d + \frac{c}{x^{6}}}}{{\left (b^{2} c^{2} \mathrm{sgn}\left (x\right ) - a b c d \mathrm{sgn}\left (x\right )\right )}{\left (b c + a{\left (d + \frac{c}{x^{6}}\right )} - a d\right )}} - \frac{2 \, \arctan \left (\frac{\sqrt{d + \frac{c}{x^{6}}}}{\sqrt{-d}}\right )}{b^{2} c^{2} \sqrt{-d} \mathrm{sgn}\left (x\right )}\right )} - \frac{{\left (3 \, a b c \sqrt{-d} \arctan \left (\frac{a \sqrt{d}}{\sqrt{a b c - a^{2} d}}\right ) - 2 \, a^{2} \sqrt{-d} d \arctan \left (\frac{a \sqrt{d}}{\sqrt{a b c - a^{2} d}}\right ) - 2 \, \sqrt{a b c - a^{2} d} b c \arctan \left (\frac{\sqrt{d}}{\sqrt{-d}}\right ) + 2 \, \sqrt{a b c - a^{2} d} a d \arctan \left (\frac{\sqrt{d}}{\sqrt{-d}}\right ) + \sqrt{a b c - a^{2} d} a \sqrt{-d} \sqrt{d}\right )} \mathrm{sgn}\left (x\right )}{6 \,{\left (\sqrt{a b c - a^{2} d} b^{3} c \sqrt{-d} - \sqrt{a b c - a^{2} d} a b^{2} \sqrt{-d} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="giac")

[Out]

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