\(\int \frac {1}{x^7 (a+b x^8)} \, dx\) [1459]

Optimal result

Integrand size = 13, antiderivative size = 203 \[ \int \frac {1}{x^7 \left (a+b x^8\right )} \, dx=-\frac {1}{6 a x^6}+\frac {b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4}}-\frac {b^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4}}+\frac {b^{3/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} a^{7/4}}-\frac {b^{3/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} a^{7/4}} \]

[Out]

-1/6/a/x^6-1/8*b^(3/4)*arctan(-1+b^(1/4)*x^2*2^(1/2)/a^(1/4))/a^(7/4)*2^(1/2)-1/8*b^(3/4)*arctan(1+b^(1/4)*x^2
*2^(1/2)/a^(1/4))/a^(7/4)*2^(1/2)+1/16*b^(3/4)*ln(-a^(1/4)*b^(1/4)*x^2*2^(1/2)+a^(1/2)+x^4*b^(1/2))/a^(7/4)*2^
(1/2)-1/16*b^(3/4)*ln(a^(1/4)*b^(1/4)*x^2*2^(1/2)+a^(1/2)+x^4*b^(1/2))/a^(7/4)*2^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {281, 331, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{x^7 \left (a+b x^8\right )} \, dx=\frac {b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4}}-\frac {b^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{7/4}}+\frac {b^{3/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {a}+\sqrt {b} x^4\right )}{8 \sqrt {2} a^{7/4}}-\frac {b^{3/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {a}+\sqrt {b} x^4\right )}{8 \sqrt {2} a^{7/4}}-\frac {1}{6 a x^6} \]

[In]

Int[1/(x^7*(a + b*x^8)),x]

[Out]

-1/6*1/(a*x^6) + (b^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*a^(7/4)) - (b^(3/4)*ArcTan[1 +
(Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*a^(7/4)) + (b^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sq
rt[b]*x^4])/(8*Sqrt[2]*a^(7/4)) - (b^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4])/(8*Sqrt[2
]*a^(7/4))

Rule 210

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

Rule 217

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

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 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 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^4 \left (a+b x^4\right )} \, dx,x,x^2\right ) \\ & = -\frac {1}{6 a x^6}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,x^2\right )}{2 a} \\ & = -\frac {1}{6 a x^6}-\frac {b \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,x^2\right )}{4 a^{3/2}}-\frac {b \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,x^2\right )}{4 a^{3/2}} \\ & = -\frac {1}{6 a x^6}-\frac {\sqrt {b} \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,x^2\right )}{8 a^{3/2}}-\frac {\sqrt {b} \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,x^2\right )}{8 a^{3/2}}+\frac {b^{3/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,x^2\right )}{8 \sqrt {2} a^{7/4}}+\frac {b^{3/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,x^2\right )}{8 \sqrt {2} a^{7/4}} \\ & = -\frac {1}{6 a x^6}+\frac {b^{3/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} a^{7/4}}-\frac {b^{3/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} a^{7/4}}-\frac {b^{3/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4}}+\frac {b^{3/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4}} \\ & = -\frac {1}{6 a x^6}+\frac {b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4}}-\frac {b^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4}}+\frac {b^{3/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} a^{7/4}}-\frac {b^{3/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} a^{7/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.91 \[ \int \frac {1}{x^7 \left (a+b x^8\right )} \, dx=\frac {-8 a^{3/4}+6 \sqrt {2} b^{3/4} x^6 \arctan \left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right )+6 \sqrt {2} b^{3/4} x^6 \arctan \left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right )-6 \sqrt {2} b^{3/4} x^6 \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right )+6 \sqrt {2} b^{3/4} x^6 \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right )+3 \sqrt {2} b^{3/4} x^6 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )+3 \sqrt {2} b^{3/4} x^6 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )-3 \sqrt {2} b^{3/4} x^6 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right )-3 \sqrt {2} b^{3/4} x^6 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right )}{48 a^{7/4} x^6} \]

[In]

Integrate[1/(x^7*(a + b*x^8)),x]

[Out]

(-8*a^(3/4) + 6*Sqrt[2]*b^(3/4)*x^6*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8)] + 6*Sqrt[2]*b^(3/4)*x^6*
ArcTan[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/a^(1/8)] - 6*Sqrt[2]*b^(3/4)*x^6*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8)
- Tan[Pi/8]] + 6*Sqrt[2]*b^(3/4)*x^6*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]] + 3*Sqrt[2]*b^(3/4)*x^
6*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] + 3*Sqrt[2]*b^(3/4)*x^6*Log[a^(1/4) + b^(1/4)*x^2
+ 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] - 3*Sqrt[2]*b^(3/4)*x^6*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Sin[
Pi/8]] - 3*Sqrt[2]*b^(3/4)*x^6*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]])/(48*a^(7/4)*x^6)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.29

method	result	size
risch	\(-\frac {1}{6 a \,x^{6}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{7} \textit {\_Z}^{4}+b^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-9 \textit {\_R}^{4} a^{7}-8 b^{3}\right ) x^{2}-a^{2} b^{2} \textit {\_R} \right )\right )}{8}\)	\(58\)
default	\(-\frac {1}{6 a \,x^{6}}-\frac {b \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{4}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x^{2} \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x^{2} \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a^{2}}\)	\(120\)

[In]

int(1/x^7/(b*x^8+a),x,method=_RETURNVERBOSE)

[Out]

-1/6/a/x^6+1/8*sum(_R*ln((-9*_R^4*a^7-8*b^3)*x^2-a^2*b^2*_R),_R=RootOf(_Z^4*a^7+b^3))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^7 \left (a+b x^8\right )} \, dx=-\frac {3 \, a x^{6} \left (-\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} \log \left (b x^{2} + a^{2} \left (-\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}}\right ) + 3 i \, a x^{6} \left (-\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} \log \left (b x^{2} + i \, a^{2} \left (-\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}}\right ) - 3 i \, a x^{6} \left (-\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} \log \left (b x^{2} - i \, a^{2} \left (-\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}}\right ) - 3 \, a x^{6} \left (-\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} \log \left (b x^{2} - a^{2} \left (-\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}}\right ) + 4}{24 \, a x^{6}} \]

[In]

integrate(1/x^7/(b*x^8+a),x, algorithm="fricas")

[Out]

-1/24*(3*a*x^6*(-b^3/a^7)^(1/4)*log(b*x^2 + a^2*(-b^3/a^7)^(1/4)) + 3*I*a*x^6*(-b^3/a^7)^(1/4)*log(b*x^2 + I*a
^2*(-b^3/a^7)^(1/4)) - 3*I*a*x^6*(-b^3/a^7)^(1/4)*log(b*x^2 - I*a^2*(-b^3/a^7)^(1/4)) - 3*a*x^6*(-b^3/a^7)^(1/
4)*log(b*x^2 - a^2*(-b^3/a^7)^(1/4)) + 4)/(a*x^6)

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.17 \[ \int \frac {1}{x^7 \left (a+b x^8\right )} \, dx=\operatorname {RootSum} {\left (4096 t^{4} a^{7} + b^{3}, \left ( t \mapsto t \log {\left (- \frac {8 t a^{2}}{b} + x^{2} \right )} \right )\right )} - \frac {1}{6 a x^{6}} \]

[In]

integrate(1/x**7/(b*x**8+a),x)

[Out]

RootSum(4096*_t**4*a**7 + b**3, Lambda(_t, _t*log(-8*_t*a**2/b + x**2))) - 1/(6*a*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^7 \left (a+b x^8\right )} \, dx=-\frac {\frac {2 \, \sqrt {2} b \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} b \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {b} x^{4} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x^{2} + \sqrt {a}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {b} x^{4} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x^{2} + \sqrt {a}\right )}{a^{\frac {3}{4}}}}{16 \, a} - \frac {1}{6 \, a x^{6}} \]

[In]

integrate(1/x^7/(b*x^8+a),x, algorithm="maxima")

[Out]

-1/16*(2*sqrt(2)*b*arctan(1/2*sqrt(2)*(2*sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a
)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*b*arctan(1/2*sqrt(2)*(2*sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(
a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*b^(3/4)*log(sqrt(b)*x^4 + sqrt(2)*a^(1/4)*b^(1/4)*x^2 +
sqrt(a))/a^(3/4) - sqrt(2)*b^(3/4)*log(sqrt(b)*x^4 - sqrt(2)*a^(1/4)*b^(1/4)*x^2 + sqrt(a))/a^(3/4))/a - 1/6/
(a*x^6)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^7 \left (a+b x^8\right )} \, dx=-\frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x^{2} + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2}} - \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x^{2} - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2}} - \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (x^{4} + \sqrt {2} x^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2}} + \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (x^{4} - \sqrt {2} x^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2}} - \frac {1}{6 \, a x^{6}} \]

[In]

integrate(1/x^7/(b*x^8+a),x, algorithm="giac")

[Out]

-1/8*sqrt(2)*(a*b^3)^(1/4)*arctan(1/2*sqrt(2)*(2*x^2 + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/a^2 - 1/8*sqrt(2)*(a*
b^3)^(1/4)*arctan(1/2*sqrt(2)*(2*x^2 - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/a^2 - 1/16*sqrt(2)*(a*b^3)^(1/4)*log(
x^4 + sqrt(2)*x^2*(a/b)^(1/4) + sqrt(a/b))/a^2 + 1/16*sqrt(2)*(a*b^3)^(1/4)*log(x^4 - sqrt(2)*x^2*(a/b)^(1/4)
+ sqrt(a/b))/a^2 - 1/6/(a*x^6)

Mupad [B] (verification not implemented)

Time = 5.71 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.27 \[ \int \frac {1}{x^7 \left (a+b x^8\right )} \, dx=\frac {{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,x^2}{a^{1/4}}\right )}{4\,a^{7/4}}-\frac {1}{6\,a\,x^6}+\frac {{\left (-b\right )}^{3/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,x^2}{a^{1/4}}\right )}{4\,a^{7/4}} \]

[In]

int(1/(x^7*(a + b*x^8)),x)

[Out]

((-b)^(3/4)*atan(((-b)^(1/4)*x^2)/a^(1/4)))/(4*a^(7/4)) - 1/(6*a*x^6) + ((-b)^(3/4)*atanh(((-b)^(1/4)*x^2)/a^(
1/4)))/(4*a^(7/4))