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

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

[Out]

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

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Rubi [A]  time = 0.02523, antiderivative size = 59, normalized size of antiderivative = 1., 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} \[ \frac{x \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{1}{6};2,\frac{1}{2};\frac{7}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a^2 \sqrt{c+d x^6}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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])

Rubi steps

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

Mathematica [B]  time = 0.28211, size = 329, normalized size = 5.58 \[ \frac{x \left (-2 b d x^6 \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{7}{6};\frac{1}{2},1;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )-\frac{7 a \left (3 b x^6 \left (c+d x^6\right ) \left (2 b c F_1\left (\frac{7}{6};\frac{1}{2},2;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{7}{6};\frac{3}{2},1;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )+7 a c \left (6 a d-b \left (6 c+d x^6\right )\right ) F_1\left (\frac{1}{6};\frac{1}{2},1;\frac{7}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )}{\left (a+b x^6\right ) \left (3 x^6 \left (2 b c F_1\left (\frac{7}{6};\frac{1}{2},2;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{7}{6};\frac{3}{2},1;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-7 a c F_1\left (\frac{1}{6};\frac{1}{2},1;\frac{7}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )}\right )}{42 a^2 \sqrt{c+d x^6} (a d-b c)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(-2*b*d*x^6*Sqrt[1 + (d*x^6)/c]*AppellF1[7/6, 1/2, 1, 13/6, -((d*x^6)/c), -((b*x^6)/a)] - (7*a*(7*a*c*(6*a*
d - b*(6*c + d*x^6))*AppellF1[1/6, 1/2, 1, 7/6, -((d*x^6)/c), -((b*x^6)/a)] + 3*b*x^6*(c + d*x^6)*(2*b*c*Appel
lF1[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
)])))/((a + b*x^6)*(-7*a*c*AppellF1[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)])))))/
(42*a^2*(-(b*c) + a*d)*Sqrt[c + d*x^6])

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \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/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

[Out]

int(1/(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}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)

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

[Out]

Timed out

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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/(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}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)