3.872 \(\int \frac{x^5}{(a+b x^6)^2 \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=87 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{6 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^6}}{6 \left (a+b x^6\right ) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0685971, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {444, 51, 63, 208} \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{6 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^6}}{6 \left (a+b x^6\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[x^5/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

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

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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 )^2 \sqrt{c+d x^6}} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{(a+b x)^2 \sqrt{c+d x}} \, dx,x,x^6\right )\\ &=-\frac{\sqrt{c+d x^6}}{6 (b c-a d) \left (a+b x^6\right )}-\frac{d \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^6\right )}{12 (b c-a d)}\\ &=-\frac{\sqrt{c+d x^6}}{6 (b c-a d) \left (a+b 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 )}{6 (b c-a d)}\\ &=-\frac{\sqrt{c+d x^6}}{6 (b c-a d) \left (a+b x^6\right )}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{6 \sqrt{b} (b c-a d)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.088276, size = 85, normalized size = 0.98 \[ \frac{1}{6} \left (\frac{\sqrt{c+d x^6}}{\left (a+b x^6\right ) (a d-b c)}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{a d-b c}}\right )}{\sqrt{b} (a d-b c)^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

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

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{ \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^5/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.53353, size = 640, normalized size = 7.36 \begin{align*} \left [-\frac{{\left (b d x^{6} + a d\right )} \sqrt{b^{2} c - a b d} \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 \, \sqrt{d x^{6} + c}{\left (b^{2} c - a b d\right )}}{12 \,{\left ({\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{6} + a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )}}, -\frac{{\left (b d x^{6} + a d\right )} \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 ) + \sqrt{d x^{6} + c}{\left (b^{2} c - a b d\right )}}{6 \,{\left ({\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{6} + a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/12*((b*d*x^6 + a*d)*sqrt(b^2*c - a*b*d)*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*sqrt(d*x^6 + c)*(b^2*c - a*b*d))/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^6 + a*b^3*c^2 - 2
*a^2*b^2*c*d + a^3*b*d^2), -1/6*((b*d*x^6 + a*d)*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)) + sqrt(d*x^6 + c)*(b^2*c - a*b*d))/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^6 + a*b^3*c^2
- 2*a^2*b^2*c*d + a^3*b*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**5/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.12672, size = 124, normalized size = 1.43 \begin{align*} -\frac{1}{6} \, d{\left (\frac{\arctan \left (\frac{\sqrt{d x^{6} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d}{\left (b c - a d\right )}} + \frac{\sqrt{d x^{6} + c}}{{\left ({\left (d x^{6} + c\right )} b - b c + a d\right )}{\left (b c - a d\right )}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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