3.853 \(\int \frac{x^{17}}{(a+b x^6) \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=104 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{3 b^{5/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^6} (a d+b c)}{3 b^2 d^2}+\frac{\left (c+d x^6\right )^{3/2}}{9 b d^2} \]

[Out]

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

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Rubi [A]  time = 0.107156, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {446, 88, 63, 208} \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{3 b^{5/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^6} (a d+b c)}{3 b^2 d^2}+\frac{\left (c+d x^6\right )^{3/2}}{9 b d^2} \]

Antiderivative was successfully verified.

[In]

Int[x^17/((a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

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

Rule 446

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{x^{17}}{\left (a+b x^6\right ) \sqrt{c+d x^6}} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x) \sqrt{c+d x}} \, dx,x,x^6\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \left (\frac{-b c-a d}{b^2 d \sqrt{c+d x}}+\frac{a^2}{b^2 (a+b x) \sqrt{c+d x}}+\frac{\sqrt{c+d x}}{b d}\right ) \, dx,x,x^6\right )\\ &=-\frac{(b c+a d) \sqrt{c+d x^6}}{3 b^2 d^2}+\frac{\left (c+d x^6\right )^{3/2}}{9 b d^2}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^6\right )}{6 b^2}\\ &=-\frac{(b c+a d) \sqrt{c+d x^6}}{3 b^2 d^2}+\frac{\left (c+d x^6\right )^{3/2}}{9 b d^2}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^6}\right )}{3 b^2 d}\\ &=-\frac{(b c+a d) \sqrt{c+d x^6}}{3 b^2 d^2}+\frac{\left (c+d x^6\right )^{3/2}}{9 b d^2}-\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{3 b^{5/2} \sqrt{b c-a d}}\\ \end{align*}

Mathematica [A]  time = 0.177068, size = 91, normalized size = 0.88 \[ \frac{\sqrt{c+d x^6} \left (-3 a d-2 b c+b d x^6\right )}{9 b^2 d^2}-\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{3 b^{5/2} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^17/((a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

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

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{17}}{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^17/(b*x^6+a)/(d*x^6+c)^(1/2),x)

[Out]

int(x^17/(b*x^6+a)/(d*x^6+c)^(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(x^17/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.0947, size = 594, normalized size = 5.71 \begin{align*} \left [\frac{3 \, \sqrt{b^{2} c - a b d} a^{2} d^{2} \log \left (\frac{b d x^{6} + 2 \, b c - a d - 2 \, \sqrt{d x^{6} + c} \sqrt{b^{2} c - a b d}}{b x^{6} + a}\right ) + 2 \,{\left ({\left (b^{3} c d - a b^{2} d^{2}\right )} x^{6} - 2 \, b^{3} c^{2} - a b^{2} c d + 3 \, a^{2} b d^{2}\right )} \sqrt{d x^{6} + c}}{18 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}, \frac{3 \, \sqrt{-b^{2} c + a b d} a^{2} d^{2} \arctan \left (\frac{\sqrt{d x^{6} + c} \sqrt{-b^{2} c + a b d}}{b d x^{6} + b c}\right ) +{\left ({\left (b^{3} c d - a b^{2} d^{2}\right )} x^{6} - 2 \, b^{3} c^{2} - a b^{2} c d + 3 \, a^{2} b d^{2}\right )} \sqrt{d x^{6} + c}}{9 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^17/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="fricas")

[Out]

[1/18*(3*sqrt(b^2*c - a*b*d)*a^2*d^2*log((b*d*x^6 + 2*b*c - a*d - 2*sqrt(d*x^6 + c)*sqrt(b^2*c - a*b*d))/(b*x^
6 + a)) + 2*((b^3*c*d - a*b^2*d^2)*x^6 - 2*b^3*c^2 - a*b^2*c*d + 3*a^2*b*d^2)*sqrt(d*x^6 + c))/(b^4*c*d^2 - a*
b^3*d^3), 1/9*(3*sqrt(-b^2*c + a*b*d)*a^2*d^2*arctan(sqrt(d*x^6 + c)*sqrt(-b^2*c + a*b*d)/(b*d*x^6 + b*c)) + (
(b^3*c*d - a*b^2*d^2)*x^6 - 2*b^3*c^2 - a*b^2*c*d + 3*a^2*b*d^2)*sqrt(d*x^6 + c))/(b^4*c*d^2 - a*b^3*d^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{17}}{\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**17/(b*x**6+a)/(d*x**6+c)**(1/2),x)

[Out]

Integral(x**17/((a + b*x**6)*sqrt(c + d*x**6)), x)

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Giac [A]  time = 1.14837, size = 143, normalized size = 1.38 \begin{align*} \frac{a^{2} \arctan \left (\frac{\sqrt{d x^{6} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{3 \, \sqrt{-b^{2} c + a b d} b^{2}} + \frac{{\left (d x^{6} + c\right )}^{\frac{3}{2}} b^{2} d^{4} - 3 \, \sqrt{d x^{6} + c} b^{2} c d^{4} - 3 \, \sqrt{d x^{6} + c} a b d^{5}}{9 \, b^{3} d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^17/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="giac")

[Out]

1/3*a^2*arctan(sqrt(d*x^6 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^2) + 1/9*((d*x^6 + c)^(3/2)*b^2
*d^4 - 3*sqrt(d*x^6 + c)*b^2*c*d^4 - 3*sqrt(d*x^6 + c)*a*b*d^5)/(b^3*d^6)