3.11.14 \(\int \frac {\sqrt [4]{a+b x^4}}{x^8} \, dx\) [1014]

Optimal. Leaf size=104 \[ -\frac {\sqrt [4]{a+b x^4}}{7 x^7}-\frac {b \sqrt [4]{a+b x^4}}{21 a x^3}+\frac {2 b^{5/2} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{21 a^{3/2} \left (a+b x^4\right )^{3/4}} \]

[Out]

-1/7*(b*x^4+a)^(1/4)/x^7-1/21*b*(b*x^4+a)^(1/4)/a/x^3+2/21*b^(5/2)*(1+a/b/x^4)^(3/4)*x^3*(cos(1/2*arccot(x^2*b
^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arccot(x^2*b^(1/2)/a^(1/2)))*EllipticF(sin(1/2*arccot(x^2*b^(1/2)/a^(1/2))),
2^(1/2))/a^(3/2)/(b*x^4+a)^(3/4)

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Rubi [A]
time = 0.03, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {283, 331, 243, 342, 281, 237} \begin {gather*} \frac {2 b^{5/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{21 a^{3/2} \left (a+b x^4\right )^{3/4}}-\frac {\sqrt [4]{a+b x^4}}{7 x^7}-\frac {b \sqrt [4]{a+b x^4}}{21 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(1/4)/x^8,x]

[Out]

-1/7*(a + b*x^4)^(1/4)/x^7 - (b*(a + b*x^4)^(1/4))/(21*a*x^3) + (2*b^(5/2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF
[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(21*a^(3/2)*(a + b*x^4)^(3/4))

Rule 237

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]))*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]
*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 243

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Dist[x^3*((1 + a/(b*x^4))^(3/4)/(a + b*x^4)^(3/4)), Int[1/(x^3*
(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ[{a, b}, x]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 342

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{a+b x^4}}{x^8} \, dx &=-\frac {\sqrt [4]{a+b x^4}}{7 x^7}+\frac {1}{7} b \int \frac {1}{x^4 \left (a+b x^4\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{a+b x^4}}{7 x^7}-\frac {b \sqrt [4]{a+b x^4}}{21 a x^3}-\frac {\left (2 b^2\right ) \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{21 a}\\ &=-\frac {\sqrt [4]{a+b x^4}}{7 x^7}-\frac {b \sqrt [4]{a+b x^4}}{21 a x^3}-\frac {\left (2 b^2 \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1+\frac {a}{b x^4}\right )^{3/4} x^3} \, dx}{21 a \left (a+b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{7 x^7}-\frac {b \sqrt [4]{a+b x^4}}{21 a x^3}+\frac {\left (2 b^2 \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{21 a \left (a+b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{7 x^7}-\frac {b \sqrt [4]{a+b x^4}}{21 a x^3}+\frac {\left (b^2 \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{21 a \left (a+b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{7 x^7}-\frac {b \sqrt [4]{a+b x^4}}{21 a x^3}+\frac {2 b^{5/2} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{21 a^{3/2} \left (a+b x^4\right )^{3/4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.01, size = 51, normalized size = 0.49 \begin {gather*} -\frac {\sqrt [4]{a+b x^4} \, _2F_1\left (-\frac {7}{4},-\frac {1}{4};-\frac {3}{4};-\frac {b x^4}{a}\right )}{7 x^7 \sqrt [4]{1+\frac {b x^4}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(1/4)/x^8,x]

[Out]

-1/7*((a + b*x^4)^(1/4)*Hypergeometric2F1[-7/4, -1/4, -3/4, -((b*x^4)/a)])/(x^7*(1 + (b*x^4)/a)^(1/4))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{x^{8}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(1/4)/x^8,x)

[Out]

int((b*x^4+a)^(1/4)/x^8,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((b*x^4+a)^(1/4)/x^8,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(1/4)/x^8, x)

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Fricas [F]
time = 0.09, size = 15, normalized size = 0.14 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x^{8}}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(1/4)/x^8,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(1/4)/x^8, x)

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Sympy [C] Result contains complex when optimal does not.
time = 0.61, size = 31, normalized size = 0.30 \begin {gather*} - \frac {\sqrt [4]{b} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(1/4)/x**8,x)

[Out]

-b**(1/4)*hyper((-1/4, 3/2), (5/2,), a*exp_polar(I*pi)/(b*x**4))/(6*x**6)

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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((b*x^4+a)^(1/4)/x^8,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(1/4)/x^8, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,x^4+a\right )}^{1/4}}{x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^(1/4)/x^8,x)

[Out]

int((a + b*x^4)^(1/4)/x^8, x)

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