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

Optimal. Leaf size=59 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b-a x^8}}\right )}{2 \sqrt [4]{c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b-a x^8}}\right )}{2 \sqrt [4]{c}} \]

________________________________________________________________________________________

Rubi [C]  time = 1.50, antiderivative size = 460, normalized size of antiderivative = 7.80, 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 = {6728, 246, 245, 1438, 430, 429, 511, 510} \begin {gather*} -\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}{c^2-\sqrt {c^2+4 a b} c+2 a b},\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 {c^2+4 a b}\right )},\frac {a x^8}{b}\right )}{\sqrt [4]{b-a x^8}}+\frac {a x^5 \left (c-\sqrt {4 a b+c^2}\right ) \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}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^8}{b}\right )}{5 \left (c \left (c-\sqrt {4 a b+c^2}\right )+2 a b\right ) \sqrt [4]{b-a x^8}}+\frac {a x^5 \left (\sqrt {4 a b+c^2}+c\right ) \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 {c^2+4 a b}\right )},\frac {a x^8}{b}\right )}{5 \left (c \left (\sqrt {4 a b+c^2}+c\right )+2 a b\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 {gather*}

Warning: Unable to verify antiderivative.

[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^2 - 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^2 - 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 245

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 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[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 429

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 430

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 510

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)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), 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 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[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 1438

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 6728

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^2-c \sqrt {4 a b+c^2}},\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^2-c \sqrt {4 a b+c^2}},\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*}

________________________________________________________________________________________

Mathematica [F]  time = 1.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b+a x^8}{\sqrt [4]{b-a x^8} \left (-b+c x^4+a x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 12.31, size = 59, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

________________________________________________________________________________________

fricas [F(-1)]  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

________________________________________________________________________________________

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

________________________________________________________________________________________

maple [F]  time = 0.02, 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)

________________________________________________________________________________________

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  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

________________________________________________________________________________________