3.8.42 \(\int \frac {-b+a x^8}{\sqrt [4]{b+a x^8} (b-c x^4+a x^8)} \, dx\) [742]

Optimal. Leaf size=57 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^8}}\right )}{2 \sqrt [4]{c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^8}}\right )}{2 \sqrt [4]{c}} \]

[Out]

-1/2*arctan(c^(1/4)*x/(a*x^8+b)^(1/4))/c^(1/4)-1/2*arctanh(c^(1/4)*x/(a*x^8+b)^(1/4))/c^(1/4)

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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 1.03, antiderivative size = 461, normalized size of antiderivative = 8.09, number of steps used = 18, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6860, 252, 251, 1452, 441, 440, 525, 524} \begin {gather*} -\frac {x \sqrt [4]{\frac {a x^8}{b}+1} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};-\frac {2 a^2 x^8}{2 a b-c \left (c-\sqrt {c^2-4 a b}\right )},-\frac {a x^8}{b}\right )}{\sqrt [4]{a x^8+b}}-\frac {x \sqrt [4]{\frac {a x^8}{b}+1} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};-\frac {2 a^2 x^8}{2 a b-c \left (c+\sqrt {c^2-4 a b}\right )},-\frac {a x^8}{b}\right )}{\sqrt [4]{a x^8+b}}+\frac {a x^5 \left (c-\sqrt {c^2-4 a b}\right ) \sqrt [4]{\frac {a x^8}{b}+1} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};-\frac {2 a^2 x^8}{2 a b-c \left (c-\sqrt {c^2-4 a b}\right )},-\frac {a x^8}{b}\right )}{5 \left (2 a b-c \left (c-\sqrt {c^2-4 a b}\right )\right ) \sqrt [4]{a x^8+b}}+\frac {a x^5 \left (\sqrt {c^2-4 a b}+c\right ) \sqrt [4]{\frac {a x^8}{b}+1} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};-\frac {2 a^2 x^8}{2 a b-c \left (c+\sqrt {c^2-4 a b}\right )},-\frac {a x^8}{b}\right )}{5 \left (2 a b-c \left (\sqrt {c^2-4 a b}+c\right )\right ) \sqrt [4]{a x^8+b}}+\frac {x \sqrt [4]{\frac {a x^8}{b}+1} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^8}{b}\right )}{\sqrt [4]{a x^8+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((x*(1 + (a*x^8)/b)^(1/4)*AppellF1[1/8, 1, 1/4, 9/8, (-2*a^2*x^8)/(2*a*b - c*(c - Sqrt[-4*a*b + c^2])), -((a*
x^8)/b)])/(b + a*x^8)^(1/4)) - (x*(1 + (a*x^8)/b)^(1/4)*AppellF1[1/8, 1, 1/4, 9/8, (-2*a^2*x^8)/(2*a*b - c*(c
+ Sqrt[-4*a*b + c^2])), -((a*x^8)/b)])/(b + a*x^8)^(1/4) + (a*(c - Sqrt[-4*a*b + c^2])*x^5*(1 + (a*x^8)/b)^(1/
4)*AppellF1[5/8, 1, 1/4, 13/8, (-2*a^2*x^8)/(2*a*b - c*(c - Sqrt[-4*a*b + c^2])), -((a*x^8)/b)])/(5*(2*a*b - c
*(c - Sqrt[-4*a*b + c^2]))*(b + a*x^8)^(1/4)) + (a*(c + Sqrt[-4*a*b + c^2])*x^5*(1 + (a*x^8)/b)^(1/4)*AppellF1
[5/8, 1, 1/4, 13/8, (-2*a^2*x^8)/(2*a*b - c*(c + Sqrt[-4*a*b + c^2])), -((a*x^8)/b)])/(5*(2*a*b - c*(c + Sqrt[
-4*a*b + c^2]))*(b + a*x^8)^(1/4)) + (x*(1 + (a*x^8)/b)^(1/4)*Hypergeometric2F1[1/8, 1/4, 9/8, -((a*x^8)/b)])/
(b + a*x^8)^(1/4)

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[p]), Int[(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

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])

Rule 524

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

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[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 1452

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^(2
*n))^p, (d/(d^2 - e^2*x^(2*n)) - e*(x^n/(d^2 - e^2*x^(2*n))))^(-q), x], x] /; FreeQ[{a, c, d, e, n, p}, x] &&
EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[q, 0]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {-b+a x^8}{\sqrt [4]{b+a x^8} \left (b-c x^4+a x^8\right )} \, dx &=\int \left (\frac {1}{\sqrt [4]{b+a x^8}}-\frac {2 b-c x^4}{\sqrt [4]{b+a x^8} \left (b-c x^4+a x^8\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt [4]{b+a x^8}} \, dx-\int \frac {2 b-c x^4}{\sqrt [4]{b+a x^8} \left (b-c x^4+a x^8\right )} \, dx\\ &=\frac {\sqrt [4]{1+\frac {a x^8}{b}} \int \frac {1}{\sqrt [4]{1+\frac {a x^8}{b}}} \, dx}{\sqrt [4]{b+a x^8}}-\int \left (\frac {-c-\sqrt {-4 a b+c^2}}{\left (-c-\sqrt {-4 a b+c^2}+2 a x^4\right ) \sqrt [4]{b+a x^8}}+\frac {-c+\sqrt {-4 a b+c^2}}{\left (-c+\sqrt {-4 a b+c^2}+2 a x^4\right ) \sqrt [4]{b+a x^8}}\right ) \, dx\\ &=\frac {x \sqrt [4]{1+\frac {a x^8}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^8}{b}\right )}{\sqrt [4]{b+a x^8}}-\left (-c-\sqrt {-4 a b+c^2}\right ) \int \frac {1}{\left (-c-\sqrt {-4 a b+c^2}+2 a x^4\right ) \sqrt [4]{b+a x^8}} \, dx-\left (-c+\sqrt {-4 a b+c^2}\right ) \int \frac {1}{\left (-c+\sqrt {-4 a b+c^2}+2 a x^4\right ) \sqrt [4]{b+a x^8}} \, dx\\ &=\frac {x \sqrt [4]{1+\frac {a x^8}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^8}{b}\right )}{\sqrt [4]{b+a x^8}}-\left (-c-\sqrt {-4 a b+c^2}\right ) \int \left (\frac {-c-\sqrt {-4 a b+c^2}}{2 \sqrt [4]{b+a x^8} \left (-2 a b+c^2+c \sqrt {-4 a b+c^2}-2 a^2 x^8\right )}+\frac {a x^4}{\sqrt [4]{b+a x^8} \left (2 a b-c^2-c \sqrt {-4 a b+c^2}+2 a^2 x^8\right )}\right ) \, dx-\left (-c+\sqrt {-4 a b+c^2}\right ) \int \left (\frac {c-\sqrt {-4 a b+c^2}}{2 \sqrt [4]{b+a x^8} \left (2 a b-c^2+c \sqrt {-4 a b+c^2}+2 a^2 x^8\right )}+\frac {a x^4}{\sqrt [4]{b+a x^8} \left (2 a b-c^2+c \sqrt {-4 a b+c^2}+2 a^2 x^8\right )}\right ) \, dx\\ &=\frac {x \sqrt [4]{1+\frac {a x^8}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^8}{b}\right )}{\sqrt [4]{b+a x^8}}+\left (a \left (c-\sqrt {-4 a b+c^2}\right )\right ) \int \frac {x^4}{\sqrt [4]{b+a x^8} \left (2 a b-c^2+c \sqrt {-4 a b+c^2}+2 a^2 x^8\right )} \, dx+\frac {1}{2} \left (c-\sqrt {-4 a b+c^2}\right )^2 \int \frac {1}{\sqrt [4]{b+a x^8} \left (2 a b-c^2+c \sqrt {-4 a b+c^2}+2 a^2 x^8\right )} \, dx+\left (a \left (c+\sqrt {-4 a b+c^2}\right )\right ) \int \frac {x^4}{\sqrt [4]{b+a x^8} \left (2 a b-c^2-c \sqrt {-4 a b+c^2}+2 a^2 x^8\right )} \, dx-\frac {1}{2} \left (c+\sqrt {-4 a b+c^2}\right )^2 \int \frac {1}{\sqrt [4]{b+a x^8} \left (-2 a b+c^2+c \sqrt {-4 a b+c^2}-2 a^2 x^8\right )} \, dx\\ &=\frac {x \sqrt [4]{1+\frac {a x^8}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^8}{b}\right )}{\sqrt [4]{b+a x^8}}+\frac {\left (a \left (c-\sqrt {-4 a b+c^2}\right ) \sqrt [4]{1+\frac {a x^8}{b}}\right ) \int \frac {x^4}{\left (2 a b-c^2+c \sqrt {-4 a b+c^2}+2 a^2 x^8\right ) \sqrt [4]{1+\frac {a x^8}{b}}} \, dx}{\sqrt [4]{b+a x^8}}+\frac {\left (\left (c-\sqrt {-4 a b+c^2}\right )^2 \sqrt [4]{1+\frac {a x^8}{b}}\right ) \int \frac {1}{\left (2 a b-c^2+c \sqrt {-4 a b+c^2}+2 a^2 x^8\right ) \sqrt [4]{1+\frac {a x^8}{b}}} \, dx}{2 \sqrt [4]{b+a x^8}}+\frac {\left (a \left (c+\sqrt {-4 a b+c^2}\right ) \sqrt [4]{1+\frac {a x^8}{b}}\right ) \int \frac {x^4}{\left (2 a b-c^2-c \sqrt {-4 a b+c^2}+2 a^2 x^8\right ) \sqrt [4]{1+\frac {a x^8}{b}}} \, dx}{\sqrt [4]{b+a x^8}}-\frac {\left (\left (c+\sqrt {-4 a b+c^2}\right )^2 \sqrt [4]{1+\frac {a x^8}{b}}\right ) \int \frac {1}{\left (-2 a b+c^2+c \sqrt {-4 a b+c^2}-2 a^2 x^8\right ) \sqrt [4]{1+\frac {a x^8}{b}}} \, dx}{2 \sqrt [4]{b+a x^8}}\\ &=-\frac {x \sqrt [4]{1+\frac {a x^8}{b}} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};-\frac {2 a^2 x^8}{2 a b-c \left (c-\sqrt {-4 a b+c^2}\right )},-\frac {a x^8}{b}\right )}{\sqrt [4]{b+a x^8}}-\frac {x \sqrt [4]{1+\frac {a x^8}{b}} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};-\frac {2 a^2 x^8}{2 a b-c \left (c+\sqrt {-4 a b+c^2}\right )},-\frac {a x^8}{b}\right )}{\sqrt [4]{b+a x^8}}+\frac {a \left (c-\sqrt {-4 a b+c^2}\right ) x^5 \sqrt [4]{1+\frac {a x^8}{b}} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};-\frac {2 a^2 x^8}{2 a b-c \left (c-\sqrt {-4 a b+c^2}\right )},-\frac {a x^8}{b}\right )}{5 \left (2 a b-c \left (c-\sqrt {-4 a b+c^2}\right )\right ) \sqrt [4]{b+a x^8}}+\frac {a \left (c+\sqrt {-4 a b+c^2}\right ) x^5 \sqrt [4]{1+\frac {a x^8}{b}} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};-\frac {2 a^2 x^8}{2 a b-c \left (c+\sqrt {-4 a b+c^2}\right )},-\frac {a x^8}{b}\right )}{5 \left (2 a b-c \left (c+\sqrt {-4 a b+c^2}\right )\right ) \sqrt [4]{b+a x^8}}+\frac {x \sqrt [4]{1+\frac {a x^8}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^8}{b}\right )}{\sqrt [4]{b+a x^8}}\\ \end {align*}

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Mathematica [A]
time = 7.48, size = 48, normalized size = 0.84 \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^8}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^8}}\right )}{2 \sqrt [4]{c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(ArcTan[(c^(1/4)*x)/(b + a*x^8)^(1/4)] + ArcTanh[(c^(1/4)*x)/(b + a*x^8)^(1/4)])/c^(1/4)

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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{8}-b}{\left (a \,x^{8}+b \right )^{\frac {1}{4}} \left (a \,x^{8}-c \,x^{4}+b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

Timed out

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Sympy [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((a*x**8-b)/(a*x**8+b)**(1/4)/(a*x**8-c*x**4+b),x)

[Out]

Timed out

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int -\frac {b-a\,x^8}{{\left (a\,x^8+b\right )}^{1/4}\,\left (a\,x^8-c\,x^4+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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