3.855 \(\int \frac{x^5}{(a+b x^6) \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=51 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{3 \sqrt{b} \sqrt{b c-a d}} \]

[Out]

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

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Rubi [A]  time = 0.0459977, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {444, 63, 208} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{3 \sqrt{b} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/((a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

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

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

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

Mathematica [A]  time = 0.0154694, size = 51, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{3 \sqrt{b} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/((a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

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

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{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^5/(b*x^6+a)/(d*x^6+c)^(1/2),x)

[Out]

int(x^5/(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^5/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.0487, size = 288, normalized size = 5.65 \begin{align*} \left [\frac{\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 )}{6 \, \sqrt{b^{2} c - a b d}}, \frac{\sqrt{-b^{2} c + a b d} \arctan \left (\frac{\sqrt{d x^{6} + c} \sqrt{-b^{2} c + a b d}}{b d x^{6} + b c}\right )}{3 \,{\left (b^{2} c - a b d\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 18.6846, size = 37, normalized size = 0.73 \begin{align*} \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x^{6}}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{3 b \sqrt{\frac{a d - b c}{b}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x**6+a)/(d*x**6+c)**(1/2),x)

[Out]

atan(sqrt(c + d*x**6)/sqrt((a*d - b*c)/b))/(3*b*sqrt((a*d - b*c)/b))

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Giac [A]  time = 1.16337, size = 54, normalized size = 1.06 \begin{align*} \frac{\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="giac")

[Out]

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