3.928 \(\int \frac{1}{x^4 (a+b x^8)^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=64 \[ -\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left (-\frac{3}{8};2,\frac{1}{2};\frac{5}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{3 a^2 x^3 \sqrt{c+d x^8}} \]

[Out]

-(Sqrt[1 + (d*x^8)/c]*AppellF1[-3/8, 2, 1/2, 5/8, -((b*x^8)/a), -((d*x^8)/c)])/(3*a^2*x^3*Sqrt[c + d*x^8])

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Rubi [A]  time = 0.0561598, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {511, 510} \[ -\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left (-\frac{3}{8};2,\frac{1}{2};\frac{5}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{3 a^2 x^3 \sqrt{c+d x^8}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

-(Sqrt[1 + (d*x^8)/c]*AppellF1[-3/8, 2, 1/2, 5/8, -((b*x^8)/a), -((d*x^8)/c)])/(3*a^2*x^3*Sqrt[c + d*x^8])

Rule 511

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

Rule 510

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

Rubi steps

\begin{align*} \int \frac{1}{x^4 \left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx &=\frac{\sqrt{1+\frac{d x^8}{c}} \int \frac{1}{x^4 \left (a+b x^8\right )^2 \sqrt{1+\frac{d x^8}{c}}} \, dx}{\sqrt{c+d x^8}}\\ &=-\frac{\sqrt{1+\frac{d x^8}{c}} F_1\left (-\frac{3}{8};2,\frac{1}{2};\frac{5}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{3 a^2 x^3 \sqrt{c+d x^8}}\\ \end{align*}

Mathematica [B]  time = 0.270698, size = 226, normalized size = 3.53 \[ \frac{-13 x^8 \left (a+b x^8\right ) \sqrt{\frac{d x^8}{c}+1} \left (8 a^2 d^2-56 a b c d+33 b^2 c^2\right ) F_1\left (\frac{5}{8};\frac{1}{2},1;\frac{13}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+65 a \left (c+d x^8\right ) \left (8 a^2 d-8 a b \left (c-d x^8\right )-11 b^2 c x^8\right )+5 b d x^{16} \left (a+b x^8\right ) \sqrt{\frac{d x^8}{c}+1} (11 b c-8 a d) F_1\left (\frac{13}{8};\frac{1}{2},1;\frac{21}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{1560 a^3 c x^3 \left (a+b x^8\right ) \sqrt{c+d x^8} (b c-a d)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^4*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

(65*a*(c + d*x^8)*(8*a^2*d - 11*b^2*c*x^8 - 8*a*b*(c - d*x^8)) - 13*(33*b^2*c^2 - 56*a*b*c*d + 8*a^2*d^2)*x^8*
(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[5/8, 1/2, 1, 13/8, -((d*x^8)/c), -((b*x^8)/a)] + 5*b*d*(11*b*c - 8*a*
d)*x^16*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[13/8, 1/2, 1, 21/8, -((d*x^8)/c), -((b*x^8)/a)])/(1560*a^3*c*
(b*c - a*d)*x^3*(a + b*x^8)*Sqrt[c + d*x^8])

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \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/x^4/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(1/x^4/(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} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(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^4), 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/x^4/(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/x**4/(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} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(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^4), x)