3.14.41 \(\int \frac {1}{x^2 (a+b x^6)^2} \, dx\) [1341]

Optimal. Leaf size=244 \[ -\frac {7}{6 a^2 x}+\frac {1}{6 a x \left (a+b x^6\right )}-\frac {7 \sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{13/6}}+\frac {7 \sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac {7 \sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac {7 \sqrt [6]{b} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{13/6}}+\frac {7 \sqrt [6]{b} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{13/6}} \]

[Out]

-7/6/a^2/x+1/6/a/x/(b*x^6+a)-7/18*b^(1/6)*arctan(b^(1/6)*x/a^(1/6))/a^(13/6)+7/36*b^(1/6)*arctan((-2*b^(1/6)*x
+a^(1/6)*3^(1/2))/a^(1/6))/a^(13/6)-7/36*b^(1/6)*arctan((2*b^(1/6)*x+a^(1/6)*3^(1/2))/a^(1/6))/a^(13/6)-7/72*b
^(1/6)*ln(a^(1/3)+b^(1/3)*x^2-a^(1/6)*b^(1/6)*x*3^(1/2))/a^(13/6)*3^(1/2)+7/72*b^(1/6)*ln(a^(1/3)+b^(1/3)*x^2+
a^(1/6)*b^(1/6)*x*3^(1/2))/a^(13/6)*3^(1/2)

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Rubi [A]
time = 0.37, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {296, 331, 301, 648, 632, 210, 642, 211} \begin {gather*} -\frac {7 \sqrt [6]{b} \text {ArcTan}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{13/6}}+\frac {7 \sqrt [6]{b} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac {7 \sqrt [6]{b} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac {7 \sqrt [6]{b} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{13/6}}+\frac {7 \sqrt [6]{b} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{13/6}}-\frac {7}{6 a^2 x}+\frac {1}{6 a x \left (a+b x^6\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x^6)^2),x]

[Out]

-7/(6*a^2*x) + 1/(6*a*x*(a + b*x^6)) - (7*b^(1/6)*ArcTan[(b^(1/6)*x)/a^(1/6)])/(18*a^(13/6)) + (7*b^(1/6)*ArcT
an[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*x)/a^(1/6)])/(36*a^(13/6)) - (7*b^(1/6)*ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)
/a^(1/6)])/(36*a^(13/6)) - (7*b^(1/6)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(24*Sqrt[3]*a^(1
3/6)) + (7*b^(1/6)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(24*Sqrt[3]*a^(13/6))

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 211

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

Rule 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 301

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(
2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 +
2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m))*Int[1/(r^2 + s^2*x^2), x] +
Dist[2*(r^(m + 1)/(a*n*s^m)), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] &&
IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]

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 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (a+b x^6\right )^2} \, dx &=\frac {1}{6 a x \left (a+b x^6\right )}+\frac {7 \int \frac {1}{x^2 \left (a+b x^6\right )} \, dx}{6 a}\\ &=-\frac {7}{6 a^2 x}+\frac {1}{6 a x \left (a+b x^6\right )}-\frac {(7 b) \int \frac {x^4}{a+b x^6} \, dx}{6 a^2}\\ &=-\frac {7}{6 a^2 x}+\frac {1}{6 a x \left (a+b x^6\right )}-\frac {\left (7 \sqrt [3]{b}\right ) \int \frac {-\frac {\sqrt [6]{a}}{2}+\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{18 a^{13/6}}-\frac {\left (7 \sqrt [3]{b}\right ) \int \frac {-\frac {\sqrt [6]{a}}{2}-\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{18 a^{13/6}}-\frac {\left (7 \sqrt [3]{b}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx}{18 a^2}\\ &=-\frac {7}{6 a^2 x}+\frac {1}{6 a x \left (a+b x^6\right )}-\frac {7 \sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{13/6}}-\frac {\left (7 \sqrt [6]{b}\right ) \int \frac {-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{24 \sqrt {3} a^{13/6}}+\frac {\left (7 \sqrt [6]{b}\right ) \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{24 \sqrt {3} a^{13/6}}-\frac {\left (7 \sqrt [3]{b}\right ) \int \frac {1}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a^2}-\frac {\left (7 \sqrt [3]{b}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a^2}\\ &=-\frac {7}{6 a^2 x}+\frac {1}{6 a x \left (a+b x^6\right )}-\frac {7 \sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{13/6}}-\frac {7 \sqrt [6]{b} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{13/6}}+\frac {7 \sqrt [6]{b} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{13/6}}-\frac {\left (7 \sqrt [6]{b}\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{36 \sqrt {3} a^{13/6}}+\frac {\left (7 \sqrt [6]{b}\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{36 \sqrt {3} a^{13/6}}\\ &=-\frac {7}{6 a^2 x}+\frac {1}{6 a x \left (a+b x^6\right )}-\frac {7 \sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{13/6}}+\frac {7 \sqrt [6]{b} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac {7 \sqrt [6]{b} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac {7 \sqrt [6]{b} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{13/6}}+\frac {7 \sqrt [6]{b} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{13/6}}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 205, normalized size = 0.84 \begin {gather*} \frac {-\frac {72 \sqrt [6]{a}}{x}-\frac {12 \sqrt [6]{a} b x^5}{a+b x^6}-28 \sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )+14 \sqrt [6]{b} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-14 \sqrt [6]{b} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-7 \sqrt {3} \sqrt [6]{b} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )+7 \sqrt {3} \sqrt [6]{b} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{72 a^{13/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x^6)^2),x]

[Out]

((-72*a^(1/6))/x - (12*a^(1/6)*b*x^5)/(a + b*x^6) - 28*b^(1/6)*ArcTan[(b^(1/6)*x)/a^(1/6)] + 14*b^(1/6)*ArcTan
[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] - 14*b^(1/6)*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)] - 7*Sqrt[3]*b^(1/6)*Log
[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2] + 7*Sqrt[3]*b^(1/6)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*
x + b^(1/3)*x^2])/(72*a^(13/6))

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Maple [A]
time = 0.18, size = 188, normalized size = 0.77

method	result	size
risch	\(\frac {-\frac {7 b \,x^{6}}{6 a^{2}}-\frac {1}{a}}{x \left (b \,x^{6}+a \right )}+\frac {7 \left (\munderset {\textit {\_R} =\RootOf \left (a^{13} \textit {\_Z}^{6}+b \right )}{\sum }\textit {\_R} \ln \left (\left (7 \textit {\_R}^{6} a^{13}+6 b \right ) x +a^{11} \textit {\_R}^{5}\right )\right )}{36}\)	\(70\)
default	\(-\frac {b \left (\frac {x^{5}}{6 b \,x^{6}+6 a}-\frac {7 \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 a}+\frac {7 \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{36 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {7 \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 a}+\frac {7 \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{36 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {7 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{a^{2}}-\frac {1}{a^{2} x}\)	\(188\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b*x^6+a)^2,x,method=_RETURNVERBOSE)

[Out]

-b/a^2*(1/6*x^5/(b*x^6+a)-7/72/a*3^(1/2)*(a/b)^(5/6)*ln(x^2+3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))+7/36/b/(a/b)^(1
/6)*arctan(2*x/(a/b)^(1/6)+3^(1/2))+7/72/a*3^(1/2)*(a/b)^(5/6)*ln(x^2-3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))+7/36/
b/(a/b)^(1/6)*arctan(2*x/(a/b)^(1/6)-3^(1/2))+7/18/b/(a/b)^(1/6)*arctan(x/(a/b)^(1/6)))-1/a^2/x

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Maxima [A]
time = 0.51, size = 218, normalized size = 0.89 \begin {gather*} -\frac {7 \, b x^{6} + 6 \, a}{6 \, {\left (a^{2} b x^{7} + a^{3} x\right )}} + \frac {7 \, b {\left (\frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {4 \, \arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}}{72 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^6+a)^2,x, algorithm="maxima")

[Out]

-1/6*(7*b*x^6 + 6*a)/(a^2*b*x^7 + a^3*x) + 7/72*b*(sqrt(3)*log(b^(1/3)*x^2 + sqrt(3)*a^(1/6)*b^(1/6)*x + a^(1/
3))/(a^(1/6)*b^(5/6)) - sqrt(3)*log(b^(1/3)*x^2 - sqrt(3)*a^(1/6)*b^(1/6)*x + a^(1/3))/(a^(1/6)*b^(5/6)) - 4*a
rctan(b^(1/3)*x/sqrt(a^(1/3)*b^(1/3)))/(b^(2/3)*sqrt(a^(1/3)*b^(1/3))) - 2*arctan((2*b^(1/3)*x + sqrt(3)*a^(1/
6)*b^(1/6))/sqrt(a^(1/3)*b^(1/3)))/(b^(2/3)*sqrt(a^(1/3)*b^(1/3))) - 2*arctan((2*b^(1/3)*x - sqrt(3)*a^(1/6)*b
^(1/6))/sqrt(a^(1/3)*b^(1/3)))/(b^(2/3)*sqrt(a^(1/3)*b^(1/3))))/a^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (168) = 336\).
time = 0.42, size = 466, normalized size = 1.91 \begin {gather*} -\frac {84 \, b x^{6} - 28 \, \sqrt {3} {\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{6}} \arctan \left (-\frac {33614 \, \sqrt {3} a^{2} b x \left (-\frac {b}{a^{13}}\right )^{\frac {1}{6}} - 2 \, \sqrt {3} \sqrt {-282475249 \, a^{11} b x \left (-\frac {b}{a^{13}}\right )^{\frac {5}{6}} - 282475249 \, a^{9} b \left (-\frac {b}{a^{13}}\right )^{\frac {2}{3}} + 282475249 \, b^{2} x^{2}} a^{2} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{6}} + 16807 \, \sqrt {3} b}{50421 \, b}\right ) - 28 \, \sqrt {3} {\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \sqrt {3} a^{2} b x \left (-\frac {b}{a^{13}}\right )^{\frac {1}{6}} - 2 \, \sqrt {3} \sqrt {a^{11} b x \left (-\frac {b}{a^{13}}\right )^{\frac {5}{6}} - a^{9} b \left (-\frac {b}{a^{13}}\right )^{\frac {2}{3}} + b^{2} x^{2}} a^{2} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{6}} - \sqrt {3} b}{3 \, b}\right ) + 7 \, {\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{6}} \log \left (282475249 \, a^{11} b x \left (-\frac {b}{a^{13}}\right )^{\frac {5}{6}} - 282475249 \, a^{9} b \left (-\frac {b}{a^{13}}\right )^{\frac {2}{3}} + 282475249 \, b^{2} x^{2}\right ) - 7 \, {\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{6}} \log \left (-282475249 \, a^{11} b x \left (-\frac {b}{a^{13}}\right )^{\frac {5}{6}} - 282475249 \, a^{9} b \left (-\frac {b}{a^{13}}\right )^{\frac {2}{3}} + 282475249 \, b^{2} x^{2}\right ) + 14 \, {\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{6}} \log \left (16807 \, a^{11} \left (-\frac {b}{a^{13}}\right )^{\frac {5}{6}} + 16807 \, b x\right ) - 14 \, {\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{6}} \log \left (-16807 \, a^{11} \left (-\frac {b}{a^{13}}\right )^{\frac {5}{6}} + 16807 \, b x\right ) + 72 \, a}{72 \, {\left (a^{2} b x^{7} + a^{3} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^6+a)^2,x, algorithm="fricas")

[Out]

-1/72*(84*b*x^6 - 28*sqrt(3)*(a^2*b*x^7 + a^3*x)*(-b/a^13)^(1/6)*arctan(-1/50421*(33614*sqrt(3)*a^2*b*x*(-b/a^
13)^(1/6) - 2*sqrt(3)*sqrt(-282475249*a^11*b*x*(-b/a^13)^(5/6) - 282475249*a^9*b*(-b/a^13)^(2/3) + 282475249*b
^2*x^2)*a^2*(-b/a^13)^(1/6) + 16807*sqrt(3)*b)/b) - 28*sqrt(3)*(a^2*b*x^7 + a^3*x)*(-b/a^13)^(1/6)*arctan(-1/3
*(2*sqrt(3)*a^2*b*x*(-b/a^13)^(1/6) - 2*sqrt(3)*sqrt(a^11*b*x*(-b/a^13)^(5/6) - a^9*b*(-b/a^13)^(2/3) + b^2*x^
2)*a^2*(-b/a^13)^(1/6) - sqrt(3)*b)/b) + 7*(a^2*b*x^7 + a^3*x)*(-b/a^13)^(1/6)*log(282475249*a^11*b*x*(-b/a^13
)^(5/6) - 282475249*a^9*b*(-b/a^13)^(2/3) + 282475249*b^2*x^2) - 7*(a^2*b*x^7 + a^3*x)*(-b/a^13)^(1/6)*log(-28
2475249*a^11*b*x*(-b/a^13)^(5/6) - 282475249*a^9*b*(-b/a^13)^(2/3) + 282475249*b^2*x^2) + 14*(a^2*b*x^7 + a^3*
x)*(-b/a^13)^(1/6)*log(16807*a^11*(-b/a^13)^(5/6) + 16807*b*x) - 14*(a^2*b*x^7 + a^3*x)*(-b/a^13)^(1/6)*log(-1
6807*a^11*(-b/a^13)^(5/6) + 16807*b*x) + 72*a)/(a^2*b*x^7 + a^3*x)

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Sympy [A]
time = 0.20, size = 56, normalized size = 0.23 \begin {gather*} \frac {- 6 a - 7 b x^{6}}{6 a^{3} x + 6 a^{2} b x^{7}} + \operatorname {RootSum} {\left (2176782336 t^{6} a^{13} + 117649 b, \left ( t \mapsto t \log {\left (- \frac {60466176 t^{5} a^{11}}{16807 b} + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x**6+a)**2,x)

[Out]

(-6*a - 7*b*x**6)/(6*a**3*x + 6*a**2*b*x**7) + RootSum(2176782336*_t**6*a**13 + 117649*b, Lambda(_t, _t*log(-6
0466176*_t**5*a**11/(16807*b) + x)))

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Giac [A]
time = 3.82, size = 214, normalized size = 0.88 \begin {gather*} -\frac {7 \, b \left (\frac {a}{b}\right )^{\frac {5}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 \, a^{3}} - \frac {7 \, b x^{6} + 6 \, a}{6 \, {\left (b x^{7} + a x\right )} a^{2}} + \frac {7 \, \sqrt {3} \left (a b^{5}\right )^{\frac {5}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 \, a^{3} b^{4}} - \frac {7 \, \sqrt {3} \left (a b^{5}\right )^{\frac {5}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 \, a^{3} b^{4}} - \frac {7 \, \left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 \, a^{3} b^{4}} - \frac {7 \, \left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 \, a^{3} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^6+a)^2,x, algorithm="giac")

[Out]

-7/18*b*(a/b)^(5/6)*arctan(x/(a/b)^(1/6))/a^3 - 1/6*(7*b*x^6 + 6*a)/((b*x^7 + a*x)*a^2) + 7/72*sqrt(3)*(a*b^5)
^(5/6)*log(x^2 + sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/3))/(a^3*b^4) - 7/72*sqrt(3)*(a*b^5)^(5/6)*log(x^2 - sqrt(3)
*x*(a/b)^(1/6) + (a/b)^(1/3))/(a^3*b^4) - 7/36*(a*b^5)^(5/6)*arctan((2*x + sqrt(3)*(a/b)^(1/6))/(a/b)^(1/6))/(
a^3*b^4) - 7/36*(a*b^5)^(5/6)*arctan((2*x - sqrt(3)*(a/b)^(1/6))/(a/b)^(1/6))/(a^3*b^4)

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Mupad [B]
time = 1.18, size = 169, normalized size = 0.69 \begin {gather*} -\frac {\frac {1}{a}+\frac {7\,b\,x^6}{6\,a^2}}{b\,x^7+a\,x}-\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/6}\,x\,1{}\mathrm {i}}{a^{1/6}}\right )\,7{}\mathrm {i}}{18\,a^{13/6}}-\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {a^{21/2}\,{\left (-b\right )}^{13/2}\,x\,43563744{}\mathrm {i}}{21781872\,a^{32/3}\,{\left (-b\right )}^{19/3}-\sqrt {3}\,a^{32/3}\,{\left (-b\right )}^{19/3}\,21781872{}\mathrm {i}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,7{}\mathrm {i}}{18\,a^{13/6}}+\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {a^{21/2}\,{\left (-b\right )}^{13/2}\,x\,43563744{}\mathrm {i}}{21781872\,a^{32/3}\,{\left (-b\right )}^{19/3}+\sqrt {3}\,a^{32/3}\,{\left (-b\right )}^{19/3}\,21781872{}\mathrm {i}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,7{}\mathrm {i}}{18\,a^{13/6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x^6)^2),x)

[Out]

((-b)^(1/6)*atan((a^(21/2)*(-b)^(13/2)*x*43563744i)/(21781872*a^(32/3)*(-b)^(19/3) + 3^(1/2)*a^(32/3)*(-b)^(19
/3)*21781872i))*((3^(1/2)*1i)/2 - 1/2)*7i)/(18*a^(13/6)) - ((-b)^(1/6)*atan(((-b)^(1/6)*x*1i)/a^(1/6))*7i)/(18
*a^(13/6)) - ((-b)^(1/6)*atan((a^(21/2)*(-b)^(13/2)*x*43563744i)/(21781872*a^(32/3)*(-b)^(19/3) - 3^(1/2)*a^(3
2/3)*(-b)^(19/3)*21781872i))*((3^(1/2)*1i)/2 + 1/2)*7i)/(18*a^(13/6)) - (1/a + (7*b*x^6)/(6*a^2))/(a*x + b*x^7
)

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