3.1332 \(\int \frac {x^7}{(a+b x^6)^2} \, dx\)

Optimal. Leaf size=142 \[ \frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{2/3} b^{4/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{6 \sqrt {3} a^{2/3} b^{4/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{36 a^{2/3} b^{4/3}}-\frac {x^2}{6 b \left (a+b x^6\right )} \]

[Out]

-1/6*x^2/b/(b*x^6+a)+1/18*ln(a^(1/3)+b^(1/3)*x^2)/a^(2/3)/b^(4/3)-1/36*ln(a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*
x^4)/a^(2/3)/b^(4/3)-1/18*arctan(1/3*(a^(1/3)-2*b^(1/3)*x^2)/a^(1/3)*3^(1/2))/a^(2/3)/b^(4/3)*3^(1/2)

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Rubi [A]  time = 0.11, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {275, 288, 200, 31, 634, 617, 204, 628} \[ \frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{2/3} b^{4/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{36 a^{2/3} b^{4/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{6 \sqrt {3} a^{2/3} b^{4/3}}-\frac {x^2}{6 b \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x^6)^2,x]

[Out]

-x^2/(6*b*(a + b*x^6)) - ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))]/(6*Sqrt[3]*a^(2/3)*b^(4/3)) + Log
[a^(1/3) + b^(1/3)*x^2]/(18*a^(2/3)*b^(4/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4]/(36*a^(2/3)*b^
(4/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 204

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

Rule 275

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 288

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

Rule 617

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 628

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 634

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 {x^7}{\left (a+b x^6\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{\left (a+b x^3\right )^2} \, dx,x,x^2\right )\\ &=-\frac {x^2}{6 b \left (a+b x^6\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,x^2\right )}{6 b}\\ &=-\frac {x^2}{6 b \left (a+b x^6\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,x^2\right )}{18 a^{2/3} b}+\frac {\operatorname {Subst}\left (\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{18 a^{2/3} b}\\ &=-\frac {x^2}{6 b \left (a+b x^6\right )}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{2/3} b^{4/3}}-\frac {\operatorname {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{36 a^{2/3} b^{4/3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{12 \sqrt [3]{a} b}\\ &=-\frac {x^2}{6 b \left (a+b x^6\right )}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{2/3} b^{4/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{36 a^{2/3} b^{4/3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}\right )}{6 a^{2/3} b^{4/3}}\\ &=-\frac {x^2}{6 b \left (a+b x^6\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{6 \sqrt {3} a^{2/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{2/3} b^{4/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{36 a^{2/3} b^{4/3}}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 197, normalized size = 1.39 \[ \frac {\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{a^{2/3}}-\frac {\log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{a^{2/3}}-\frac {\log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{a^{2/3}}-\frac {2 \sqrt {3} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{a^{2/3}}-\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt {3}\right )}{a^{2/3}}-\frac {6 \sqrt [3]{b} x^2}{a+b x^6}}{36 b^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x^6)^2,x]

[Out]

((-6*b^(1/3)*x^2)/(a + b*x^6) - (2*Sqrt[3]*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)])/a^(2/3) - (2*Sqrt[3]*ArcTa
n[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)])/a^(2/3) + (2*Log[a^(1/3) + b^(1/3)*x^2])/a^(2/3) - Log[a^(1/3) - Sqrt[3]*a
^(1/6)*b^(1/6)*x + b^(1/3)*x^2]/a^(2/3) - Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2]/a^(2/3))/(36*
b^(4/3))

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fricas [A]  time = 0.91, size = 409, normalized size = 2.88 \[ \left [-\frac {6 \, a^{2} b x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{6} + a^{2} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{6} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x^{2} - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{4} + \left (a^{2} b\right )^{\frac {2}{3}} x^{2} - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{6} + a}\right ) + {\left (b x^{6} + a\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{4} - \left (a^{2} b\right )^{\frac {2}{3}} x^{2} + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) - 2 \, {\left (b x^{6} + a\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{36 \, {\left (a^{2} b^{3} x^{6} + a^{3} b^{2}\right )}}, -\frac {6 \, a^{2} b x^{2} - 6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{6} + a^{2} b\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x^{2} - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) + {\left (b x^{6} + a\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{4} - \left (a^{2} b\right )^{\frac {2}{3}} x^{2} + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) - 2 \, {\left (b x^{6} + a\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{36 \, {\left (a^{2} b^{3} x^{6} + a^{3} b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^6+a)^2,x, algorithm="fricas")

[Out]

[-1/36*(6*a^2*b*x^2 - 3*sqrt(1/3)*(a*b^2*x^6 + a^2*b)*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x^6 - 3*(a^2*b)^(1/3)*
a*x^2 - a^2 + 3*sqrt(1/3)*(2*a*b*x^4 + (a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b))/(b*x^6 + a
)) + (b*x^6 + a)*(a^2*b)^(2/3)*log(a*b*x^4 - (a^2*b)^(2/3)*x^2 + (a^2*b)^(1/3)*a) - 2*(b*x^6 + a)*(a^2*b)^(2/3
)*log(a*b*x^2 + (a^2*b)^(2/3)))/(a^2*b^3*x^6 + a^3*b^2), -1/36*(6*a^2*b*x^2 - 6*sqrt(1/3)*(a*b^2*x^6 + a^2*b)*
sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) + (b
*x^6 + a)*(a^2*b)^(2/3)*log(a*b*x^4 - (a^2*b)^(2/3)*x^2 + (a^2*b)^(1/3)*a) - 2*(b*x^6 + a)*(a^2*b)^(2/3)*log(a
*b*x^2 + (a^2*b)^(2/3)))/(a^2*b^3*x^6 + a^3*b^2)]

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giac [A]  time = 0.20, size = 138, normalized size = 0.97 \[ -\frac {x^{2}}{6 \, {\left (b x^{6} + a\right )} b} - \frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{2} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{18 \, a b} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{18 \, a b^{2}} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{4} + x^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{36 \, a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^6+a)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 114, normalized size = 0.80 \[ -\frac {x^{2}}{6 \left (b \,x^{6}+a \right ) b}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{18 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}+\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}-\frac {\ln \left (x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{2}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{36 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(b*x^6+a)^2,x)

[Out]

-1/6*x^2/b/(b*x^6+a)+1/18/b^2/(a/b)^(2/3)*ln(x^2+(a/b)^(1/3))-1/36/b^2/(a/b)^(2/3)*ln(x^4-(a/b)^(1/3)*x^2+(a/b
)^(2/3))+1/18/b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x^2-1))

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maxima [A]  time = 2.36, size = 122, normalized size = 0.86 \[ -\frac {x^{2}}{6 \, {\left (b^{2} x^{6} + a b\right )}} + \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{18 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{4} - x^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{36 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{2} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{18 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^6+a)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.18, size = 118, normalized size = 0.83 \[ \frac {\ln \left (a^{1/3}+b^{1/3}\,x^2\right )}{18\,a^{2/3}\,b^{4/3}}-\frac {x^2}{6\,b\,\left (b\,x^6+a\right )}+\frac {\ln \left (\frac {2\,b\,x^2}{9}+\frac {a^{1/3}\,b^{2/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{9}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{36\,a^{2/3}\,b^{4/3}}-\frac {\ln \left (\frac {2\,b\,x^2}{9}-\frac {a^{1/3}\,b^{2/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{9}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{36\,a^{2/3}\,b^{4/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a + b*x^6)^2,x)

[Out]

log(a^(1/3) + b^(1/3)*x^2)/(18*a^(2/3)*b^(4/3)) - x^2/(6*b*(a + b*x^6)) + (log((2*b*x^2)/9 + (a^(1/3)*b^(2/3)*
(3^(1/2)*1i - 1))/9)*(3^(1/2)*1i - 1))/(36*a^(2/3)*b^(4/3)) - (log((2*b*x^2)/9 - (a^(1/3)*b^(2/3)*(3^(1/2)*1i
+ 1))/9)*(3^(1/2)*1i + 1))/(36*a^(2/3)*b^(4/3))

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sympy [A]  time = 0.78, size = 42, normalized size = 0.30 \[ - \frac {x^{2}}{6 a b + 6 b^{2} x^{6}} + \operatorname {RootSum} {\left (5832 t^{3} a^{2} b^{4} - 1, \left (t \mapsto t \log {\left (18 t a b + x^{2} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(b*x**6+a)**2,x)

[Out]

-x**2/(6*a*b + 6*b**2*x**6) + RootSum(5832*_t**3*a**2*b**4 - 1, Lambda(_t, _t*log(18*_t*a*b + x**2)))

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