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

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

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {465, 430, 429} \[ \frac {x^2 \sqrt {\frac {d x^6}{c}+1} F_1\left (\frac {1}{3};1,\frac {1}{2};\frac {4}{3};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{2 a \sqrt {c+d x^6}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 429

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

Rule 430

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 65, normalized size = 1.02 \[ \frac {x^2 \sqrt {\frac {c+d x^6}{c}} F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )}{2 a \sqrt {c+d x^6}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.67, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (b \,x^{6}+a \right ) \sqrt {d \,x^{6}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x}{\left (b\,x^6+a\right )\,\sqrt {d\,x^6+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a + b x^{6}\right ) \sqrt {c + d x^{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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