3.926 \(\int \frac{1}{(a+b x^8)^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=59 \[ \frac{x \sqrt{\frac{d x^8}{c}+1} F_1\left (\frac{1}{8};2,\frac{1}{2};\frac{9}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{a^2 \sqrt{c+d x^8}} \]

[Out]

(x*Sqrt[1 + (d*x^8)/c]*AppellF1[1/8, 2, 1/2, 9/8, -((b*x^8)/a), -((d*x^8)/c)])/(a^2*Sqrt[c + d*x^8])

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Rubi [A]  time = 0.0270407, 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} \[ \frac{x \sqrt{\frac{d x^8}{c}+1} F_1\left (\frac{1}{8};2,\frac{1}{2};\frac{9}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{a^2 \sqrt{c+d x^8}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

(x*Sqrt[1 + (d*x^8)/c]*AppellF1[1/8, 2, 1/2, 9/8, -((b*x^8)/a), -((d*x^8)/c)])/(a^2*Sqrt[c + d*x^8])

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^8\right )^2 \sqrt{c+d x^8}} \, dx &=\frac{\sqrt{1+\frac{d x^8}{c}} \int \frac{1}{\left (a+b x^8\right )^2 \sqrt{1+\frac{d x^8}{c}}} \, dx}{\sqrt{c+d x^8}}\\ &=\frac{x \sqrt{1+\frac{d x^8}{c}} F_1\left (\frac{1}{8};2,\frac{1}{2};\frac{9}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{a^2 \sqrt{c+d x^8}}\\ \end{align*}

Mathematica [B]  time = 0.301031, size = 328, normalized size = 5.56 \[ -\frac{x \left (b d x^8 \sqrt{\frac{d x^8}{c}+1} F_1\left (\frac{9}{8};\frac{1}{2},1;\frac{17}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+\frac{3 a \left (4 b x^8 \left (c+d x^8\right ) \left (2 b c F_1\left (\frac{9}{8};\frac{1}{2},2;\frac{17}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{9}{8};\frac{3}{2},1;\frac{17}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )+9 a c \left (8 a d-b \left (8 c+d x^8\right )\right ) F_1\left (\frac{1}{8};\frac{1}{2},1;\frac{9}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )}{\left (a+b x^8\right ) \left (4 x^8 \left (2 b c F_1\left (\frac{9}{8};\frac{1}{2},2;\frac{17}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{9}{8};\frac{3}{2},1;\frac{17}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-9 a c F_1\left (\frac{1}{8};\frac{1}{2},1;\frac{9}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )}\right )}{24 a^2 \sqrt{c+d x^8} (a d-b c)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

-(x*(b*d*x^8*Sqrt[1 + (d*x^8)/c]*AppellF1[9/8, 1/2, 1, 17/8, -((d*x^8)/c), -((b*x^8)/a)] + (3*a*(9*a*c*(8*a*d
- b*(8*c + d*x^8))*AppellF1[1/8, 1/2, 1, 9/8, -((d*x^8)/c), -((b*x^8)/a)] + 4*b*x^8*(c + d*x^8)*(2*b*c*AppellF
1[9/8, 1/2, 2, 17/8, -((d*x^8)/c), -((b*x^8)/a)] + a*d*AppellF1[9/8, 3/2, 1, 17/8, -((d*x^8)/c), -((b*x^8)/a)]
)))/((a + b*x^8)*(-9*a*c*AppellF1[1/8, 1/2, 1, 9/8, -((d*x^8)/c), -((b*x^8)/a)] + 4*x^8*(2*b*c*AppellF1[9/8, 1
/2, 2, 17/8, -((d*x^8)/c), -((b*x^8)/a)] + a*d*AppellF1[9/8, 3/2, 1, 17/8, -((d*x^8)/c), -((b*x^8)/a)])))))/(2
4*a^2*(-(b*c) + a*d)*Sqrt[c + d*x^8])

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(1/(b*x^8+a)^2/(d*x^8+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^{8} + a\right )}^{2} \sqrt{d x^{8} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + 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^8+a)^2/(d*x^8+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**8+a)**2/(d*x**8+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^{8} + a\right )}^{2} \sqrt{d x^{8} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="giac")

[Out]

integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)), x)