3.10.26 \(\int \frac {1}{(a+b x^8)^2 \sqrt {c+d x^8}} \, dx\) [926]

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

[Out]

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

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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 = {441, 440} \begin {gather*} \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}} \end {gather*}

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 440

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 441

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^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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(328\) vs. \(2(59)=118\).
time = 10.23, size = 328, normalized size = 5.56 \begin {gather*} -\frac {x \left (b d x^8 \sqrt {1+\frac {d x^8}{c}} 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 (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 )+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 )\right )}{\left (a+b x^8\right ) \left (-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 )+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 )\right )}\right )}{24 a^2 (-b c+a d) \sqrt {c+d x^8}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/24*(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*Ap
pellF1[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)])))
))/(a^2*(-(b*c) + a*d)*Sqrt[c + d*x^8])

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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x^{8}\right )^{2} \sqrt {c + d x^{8}}}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (b\,x^8+a\right )}^2\,\sqrt {d\,x^8+c}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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