3.886 \(\int \frac{1}{x^5 (a+b x^6)^2 \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=64 \[ -\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left (-\frac{2}{3};2,\frac{1}{2};\frac{1}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{4 a^2 x^4 \sqrt{c+d x^6}} \]

[Out]

-(Sqrt[1 + (d*x^6)/c]*AppellF1[-2/3, 2, 1/2, 1/3, -((b*x^6)/a), -((d*x^6)/c)])/(4*a^2*x^4*Sqrt[c + d*x^6])

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Rubi [A]  time = 0.082107, antiderivative size = 64, 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 = {465, 511, 510} \[ -\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left (-\frac{2}{3};2,\frac{1}{2};\frac{1}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{4 a^2 x^4 \sqrt{c+d x^6}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

-(Sqrt[1 + (d*x^6)/c]*AppellF1[-2/3, 2, 1/2, 1/3, -((b*x^6)/a), -((d*x^6)/c)])/(4*a^2*x^4*Sqrt[c + d*x^6])

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 511

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

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^5 \left (a+b x^6\right )^2 \sqrt{c+d x^6}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^3 \left (a+b x^3\right )^2 \sqrt{c+d x^3}} \, dx,x,x^2\right )\\ &=\frac{\sqrt{1+\frac{d x^6}{c}} \operatorname{Subst}\left (\int \frac{1}{x^3 \left (a+b x^3\right )^2 \sqrt{1+\frac{d x^3}{c}}} \, dx,x,x^2\right )}{2 \sqrt{c+d x^6}}\\ &=-\frac{\sqrt{1+\frac{d x^6}{c}} F_1\left (-\frac{2}{3};2,\frac{1}{2};\frac{1}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{4 a^2 x^4 \sqrt{c+d x^6}}\\ \end{align*}

Mathematica [B]  time = 0.259602, size = 225, normalized size = 3.52 \[ \frac{4 x^6 \left (a+b x^6\right ) \sqrt{\frac{d x^6}{c}+1} \left (3 a^2 d^2+21 a b c d-20 b^2 c^2\right ) F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+8 a \left (c+d x^6\right ) \left (3 a^2 d-3 a b \left (c-d x^6\right )-5 b^2 c x^6\right )+b d x^{12} \left (a+b x^6\right ) \sqrt{\frac{d x^6}{c}+1} (3 a d-5 b c) F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{96 a^3 c x^4 \left (a+b x^6\right ) \sqrt{c+d x^6} (b c-a d)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^5*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

(8*a*(c + d*x^6)*(3*a^2*d - 5*b^2*c*x^6 - 3*a*b*(c - d*x^6)) + 4*(-20*b^2*c^2 + 21*a*b*c*d + 3*a^2*d^2)*x^6*(a
+ b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^6)/c), -((b*x^6)/a)] + b*d*(-5*b*c + 3*a*d)*x^
12*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^6)/c), -((b*x^6)/a)])/(96*a^3*c*(b*c - a*
d)*x^4*(a + b*x^6)*Sqrt[c + d*x^6])

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{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(1/x^5/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

[Out]

int(1/x^5/(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{1}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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