3.8.79 \(\int \frac {\sqrt {a+c x^4}}{x^8} \, dx\) [779]

Optimal. Leaf size=129 \[ -\frac {\sqrt {a+c x^4}}{7 x^7}-\frac {2 c \sqrt {a+c x^4}}{21 a x^3}-\frac {c^{7/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{21 a^{5/4} \sqrt {a+c x^4}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {283, 331, 226} \begin {gather*} -\frac {c^{7/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{21 a^{5/4} \sqrt {a+c x^4}}-\frac {\sqrt {a+c x^4}}{7 x^7}-\frac {2 c \sqrt {a+c x^4}}{21 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + c*x^4]/x^8,x]

[Out]

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

Rule 226

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

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.01, size = 51, normalized size = 0.40 \begin {gather*} -\frac {\sqrt {a+c x^4} \, _2F_1\left (-\frac {7}{4},-\frac {1}{2};-\frac {3}{4};-\frac {c x^4}{a}\right )}{7 x^7 \sqrt {1+\frac {c x^4}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + c*x^4]/x^8,x]

[Out]

-1/7*(Sqrt[a + c*x^4]*Hypergeometric2F1[-7/4, -1/2, -3/4, -((c*x^4)/a)])/(x^7*Sqrt[1 + (c*x^4)/a])

________________________________________________________________________________________

Maple [C] Result contains complex when optimal does not.
time = 0.15, size = 110, normalized size = 0.85

method	result	size
risch	\(-\frac {\sqrt {x^{4} c +a}\, \left (2 x^{4} c +3 a \right )}{21 x^{7} a}-\frac {2 c^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{21 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\)	\(105\)
default	\(-\frac {\sqrt {x^{4} c +a}}{7 x^{7}}-\frac {2 c \sqrt {x^{4} c +a}}{21 a \,x^{3}}-\frac {2 c^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{21 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\)	\(110\)
elliptic	\(-\frac {\sqrt {x^{4} c +a}}{7 x^{7}}-\frac {2 c \sqrt {x^{4} c +a}}{21 a \,x^{3}}-\frac {2 c^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{21 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\)	\(110\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^(1/2)/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/7*(c*x^4+a)^(1/2)/x^7-2/21*c*(c*x^4+a)^(1/2)/a/x^3-2/21*c^2/a/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2
)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)

________________________________________________________________________________________

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((c*x^4+a)^(1/2)/x^8,x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^4 + a)/x^8, x)

________________________________________________________________________________________

Fricas [A]
time = 0.07, size = 60, normalized size = 0.47 \begin {gather*} \frac {2 \, \sqrt {a} c x^{7} \left (-\frac {c}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left (2 \, c x^{4} + 3 \, a\right )} \sqrt {c x^{4} + a}}{21 \, a x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(1/2)/x^8,x, algorithm="fricas")

[Out]

1/21*(2*sqrt(a)*c*x^7*(-c/a)^(3/4)*elliptic_f(arcsin(x*(-c/a)^(1/4)), -1) - (2*c*x^4 + 3*a)*sqrt(c*x^4 + a))/(
a*x^7)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.53, size = 46, normalized size = 0.36 \begin {gather*} \frac {\sqrt {a} \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**(1/2)/x**8,x)

[Out]

sqrt(a)*gamma(-7/4)*hyper((-7/4, -1/2), (-3/4,), c*x**4*exp_polar(I*pi)/a)/(4*x**7*gamma(-3/4))

________________________________________________________________________________________

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((c*x^4+a)^(1/2)/x^8,x, algorithm="giac")

[Out]

integrate(sqrt(c*x^4 + a)/x^8, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^4+a}}{x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^(1/2)/x^8,x)

[Out]

int((a + c*x^4)^(1/2)/x^8, x)

________________________________________________________________________________________