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

Optimal. Leaf size=114 \[ \frac {\left (a x^4+b\right )^{3/4} \left (4 a x^4-3 b\right )}{21 b^2 x^7}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \]

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Rubi [A]  time = 0.67, antiderivative size = 126, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {6725, 271, 264, 377, 212, 206, 203} \begin {gather*} \frac {4 a \left (a x^4+b\right )^{3/4}}{21 b^2 x^3}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}-\frac {\left (a x^4+b\right )^{3/4}}{7 b x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-b + a*x^4 + x^8)/(x^8*(-b + a*x^4)*(b + a*x^4)^(1/4)),x]

[Out]

-1/7*(b + a*x^4)^(3/4)/(b*x^7) + (4*a*(b + a*x^4)^(3/4))/(21*b^2*x^3) - ArcTan[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)
^(1/4)]/(2*2^(1/4)*a^(1/4)*b) - ArcTanh[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*2^(1/4)*a^(1/4)*b)

Rule 203

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 264

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] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 271

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

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 114, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{a} \left (a x^4+b\right )^{3/4} \left (4 a x^4-3 b\right )-21\ 2^{3/4} b x^7 \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )-21\ 2^{3/4} b x^7 \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{84 \sqrt [4]{a} b^2 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-b + a*x^4 + x^8)/(x^8*(-b + a*x^4)*(b + a*x^4)^(1/4)),x]

[Out]

(4*a^(1/4)*(b + a*x^4)^(3/4)*(-3*b + 4*a*x^4) - 21*2^(3/4)*b*x^7*ArcTan[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]
- 21*2^(3/4)*b*x^7*ArcTanh[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)])/(84*a^(1/4)*b^2*x^7)

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IntegrateAlgebraic [A]  time = 0.82, size = 114, normalized size = 1.00 \begin {gather*} \frac {\left (b+a x^4\right )^{3/4} \left (-3 b+4 a x^4\right )}{21 b^2 x^7}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-b + a*x^4 + x^8)/(x^8*(-b + a*x^4)*(b + a*x^4)^(1/4)),x]

[Out]

((b + a*x^4)^(3/4)*(-3*b + 4*a*x^4))/(21*b^2*x^7) - ArcTan[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*2^(1/4)*a
^(1/4)*b) - ArcTanh[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*2^(1/4)*a^(1/4)*b)

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fricas [B]  time = 166.24, size = 461, normalized size = 4.04 \begin {gather*} \frac {84 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b^{2} x^{7} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a x^{4} + b\right )}^{\frac {3}{4}} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (a x^{4} + b\right )}^{\frac {1}{4}} a b x^{3} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + {\left (2 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {a x^{4} + b} a b x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (3 \, a^{2} b^{3} x^{4} + a b^{4}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}}\right )} \sqrt {\sqrt {\frac {1}{2}} b^{2} \sqrt {\frac {1}{a b^{4}}}}\right )}}{a x^{4} - b}\right ) - 21 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b^{2} x^{7} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} + 2 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{4} + b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} - b\right )}}\right ) + 21 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b^{2} x^{7} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} - 2 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{4} + b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} - b\right )}}\right ) + 8 \, {\left (4 \, a x^{4} - 3 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}}{168 \, b^{2} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/168*(84*(1/2)^(1/4)*b^2*x^7*(1/(a*b^4))^(1/4)*arctan(2*(2*(1/2)^(3/4)*(a*x^4 + b)^(3/4)*a*b^3*x*(1/(a*b^4))^
(3/4) + 2*(1/2)^(1/4)*(a*x^4 + b)^(1/4)*a*b*x^3*(1/(a*b^4))^(1/4) + (2*(1/2)^(1/4)*sqrt(a*x^4 + b)*a*b*x^2*(1/
(a*b^4))^(1/4) + (1/2)^(3/4)*(3*a^2*b^3*x^4 + a*b^4)*(1/(a*b^4))^(3/4))*sqrt(sqrt(1/2)*b^2*sqrt(1/(a*b^4))))/(
a*x^4 - b)) - 21*(1/2)^(1/4)*b^2*x^7*(1/(a*b^4))^(1/4)*log(1/2*(4*(1/2)^(3/4)*sqrt(a*x^4 + b)*a*b^3*x^2*(1/(a*
b^4))^(3/4) + 4*sqrt(1/2)*(a*x^4 + b)^(1/4)*a*b^2*x^3*sqrt(1/(a*b^4)) + 2*(a*x^4 + b)^(3/4)*x + (1/2)^(1/4)*(3
*a*b*x^4 + b^2)*(1/(a*b^4))^(1/4))/(a*x^4 - b)) + 21*(1/2)^(1/4)*b^2*x^7*(1/(a*b^4))^(1/4)*log(-1/2*(4*(1/2)^(
3/4)*sqrt(a*x^4 + b)*a*b^3*x^2*(1/(a*b^4))^(3/4) - 4*sqrt(1/2)*(a*x^4 + b)^(1/4)*a*b^2*x^3*sqrt(1/(a*b^4)) - 2
*(a*x^4 + b)^(3/4)*x + (1/2)^(1/4)*(3*a*b*x^4 + b^2)*(1/(a*b^4))^(1/4))/(a*x^4 - b)) + 8*(4*a*x^4 - 3*b)*(a*x^
4 + b)^(3/4))/(b^2*x^7)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} + a x^{4} - b}{{\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )} x^{8}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} + a x^{4} - b}{{\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )} x^{8}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{4} - b + x^{8}}{x^{8} \left (a x^{4} - b\right ) \sqrt [4]{a x^{4} + b}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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