3.9.86 \(\int \frac {1}{x^5 (a+b x^6)^2 \sqrt {c+d x^6}} \, dx\) [886]

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

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 64, normalized size of antiderivative = 1.00, 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 = {476, 525, 524} \begin {gather*} -\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 476

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 524

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)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], 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])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[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])

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} \text {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}} \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(225\) vs. \(2(64)=128\).
time = 10.21, size = 225, normalized size = 3.52 \begin {gather*} \frac {8 a \left (c+d x^6\right ) \left (3 a^2 d-5 b^2 c x^6-3 a b \left (c-d x^6\right )\right )+4 \left (-20 b^2 c^2+21 a b c d+3 a^2 d^2\right ) x^6 \left (a+b x^6\right ) \sqrt {1+\frac {d x^6}{c}} F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )+b d (-5 b c+3 a d) x^{12} \left (a+b x^6\right ) \sqrt {1+\frac {d x^6}{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 (b c-a d) x^4 \left (a+b x^6\right ) \sqrt {c+d x^6}} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{5} \left (b \,x^{6}+a \right )^{2} \sqrt {d \,x^{6}+c}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{5} \left (a + b x^{6}\right )^{2} \sqrt {c + d x^{6}}}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^5\,{\left (b\,x^6+a\right )}^2\,\sqrt {d\,x^6+c}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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