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

Optimal. Leaf size=59 \[ \frac {x \sqrt {\frac {d x^6}{c}+1} F_1\left (\frac {1}{6};1,\frac {1}{2};\frac {7}{6};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{a \sqrt {c+d x^6}} \]

[Out]

x*AppellF1(1/6,1,1/2,7/6,-b*x^6/a,-d*x^6/c)*(1+d*x^6/c)^(1/2)/a/(d*x^6+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {430, 429} \[ \frac {x \sqrt {\frac {d x^6}{c}+1} F_1\left (\frac {1}{6};1,\frac {1}{2};\frac {7}{6};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{a \sqrt {c+d x^6}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[1 + (d*x^6)/c]*AppellF1[1/6, 1, 1/2, 7/6, -((b*x^6)/a), -((d*x^6)/c)])/(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])

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx &=\frac {\sqrt {1+\frac {d x^6}{c}} \int \frac {1}{\left (a+b x^6\right ) \sqrt {1+\frac {d x^6}{c}}} \, dx}{\sqrt {c+d x^6}}\\ &=\frac {x \sqrt {1+\frac {d x^6}{c}} F_1\left (\frac {1}{6};1,\frac {1}{2};\frac {7}{6};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{a \sqrt {c+d x^6}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 0.20, size = 161, normalized size = 2.73 \[ -\frac {7 a c x F_1\left (\frac {1}{6};\frac {1}{2},1;\frac {7}{6};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )}{\left (a+b x^6\right ) \sqrt {c+d x^6} \left (3 x^6 \left (2 b c F_1\left (\frac {7}{6};\frac {1}{2},2;\frac {13}{6};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )+a d F_1\left (\frac {7}{6};\frac {3}{2},1;\frac {13}{6};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )\right )-7 a c F_1\left (\frac {1}{6};\frac {1}{2},1;\frac {7}{6};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-7*a*c*x*AppellF1[1/6, 1/2, 1, 7/6, -((d*x^6)/c), -((b*x^6)/a)])/((a + b*x^6)*Sqrt[c + d*x^6]*(-7*a*c*AppellF
1[1/6, 1/2, 1, 7/6, -((d*x^6)/c), -((b*x^6)/a)] + 3*x^6*(2*b*c*AppellF1[7/6, 1/2, 2, 13/6, -((d*x^6)/c), -((b*
x^6)/a)] + a*d*AppellF1[7/6, 3/2, 1, 13/6, -((d*x^6)/c), -((b*x^6)/a)])))

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   Not integrable (provided residues have n
o relations)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.62, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{6}+a \right ) \sqrt {d \,x^{6}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________