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

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

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Rubi [A]  time = 0.07, antiderivative size = 87, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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)} \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.10, size = 85, normalized size = 0.98 \begin {gather*} \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 ) \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.11, size = 97, normalized size = 1.11 \begin {gather*} -\frac {\sqrt {c+d x^6}}{6 \left (a+b x^6\right ) (b c-a d)}-\frac {d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^6} \sqrt {a d-b c}}{b c-a d}\right )}{6 \sqrt {b} (a d-b c)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.75, size = 302, normalized size = 3.47 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.30, size = 93, normalized size = 1.07 \begin {gather*} -\frac {d \arctan \left (\frac {\sqrt {d x^{6} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{6 \, \sqrt {-b^{2} c + a b d} {\left (b c - a d\right )}} - \frac {\sqrt {d x^{6} + c} d}{6 \, {\left ({\left (d x^{6} + c\right )} b - b c + a d\right )} {\left (b c - a d\right )}} \end {gather*}

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

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maple [F]  time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5}}{\left (b \,x^{6}+a \right )^{2} \sqrt {d \,x^{6}+c}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c positive or negative?

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mupad [B]  time = 4.93, size = 84, normalized size = 0.97 \begin {gather*} \frac {d\,\sqrt {d\,x^6+c}}{2\,\left (a\,d-b\,c\right )\,\left (3\,b\,\left (d\,x^6+c\right )+3\,a\,d-3\,b\,c\right )}+\frac {d\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^6+c}}{\sqrt {a\,d-b\,c}}\right )}{6\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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