3.1319 \(\int \frac {x^7}{a+b x^6} \, dx\)

Optimal. Leaf size=133 \[ \frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 b^{4/3}}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 b^{4/3}}+\frac {\sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} b^{4/3}}+\frac {x^2}{2 b} \]

[Out]

1/2*x^2/b-1/6*a^(1/3)*ln(a^(1/3)+b^(1/3)*x^2)/b^(4/3)+1/12*a^(1/3)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)
/b^(4/3)+1/6*a^(1/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x^2)/a^(1/3)*3^(1/2))/b^(4/3)*3^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 133, 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, 321, 200, 31, 634, 617, 204, 628} \[ \frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 b^{4/3}}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 b^{4/3}}+\frac {\sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} b^{4/3}}+\frac {x^2}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x^6),x]

[Out]

x^2/(2*b) + (a^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))])/(2*Sqrt[3]*b^(4/3)) - (a^(1/3)*Log[a
^(1/3) + b^(1/3)*x^2])/(6*b^(4/3)) + (a^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4])/(12*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 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 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}{a+b x^6} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{a+b x^3} \, dx,x,x^2\right )\\ &=\frac {x^2}{2 b}-\frac {a \operatorname {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,x^2\right )}{2 b}\\ &=\frac {x^2}{2 b}-\frac {\sqrt [3]{a} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,x^2\right )}{6 b}-\frac {\sqrt [3]{a} \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 )}{6 b}\\ &=\frac {x^2}{2 b}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 b^{4/3}}+\frac {\sqrt [3]{a} \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 )}{12 b^{4/3}}-\frac {a^{2/3} \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 )}{4 b}\\ &=\frac {x^2}{2 b}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 b^{4/3}}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 b^{4/3}}-\frac {\sqrt [3]{a} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}\right )}{2 b^{4/3}}\\ &=\frac {x^2}{2 b}+\frac {\sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} b^{4/3}}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 b^{4/3}}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 b^{4/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 186, normalized size = 1.40 \[ \frac {-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\sqrt [3]{a} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\sqrt [3]{a} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+2 \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt {3}\right )+6 \sqrt [3]{b} x^2}{12 b^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x^6),x]

[Out]

(6*b^(1/3)*x^2 + 2*Sqrt[3]*a^(1/3)*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] + 2*Sqrt[3]*a^(1/3)*ArcTan[Sqrt[3]
+ (2*b^(1/6)*x)/a^(1/6)] - 2*a^(1/3)*Log[a^(1/3) + b^(1/3)*x^2] + a^(1/3)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6
)*x + b^(1/3)*x^2] + a^(1/3)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(12*b^(4/3))

________________________________________________________________________________________

fricas [A]  time = 0.88, size = 114, normalized size = 0.86 \[ \frac {6 \, x^{2} + 2 \, \sqrt {3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x^{2} \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{4} + x^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) + 2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(6*x^2 + 2*sqrt(3)*(-a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x^2*(-a/b)^(2/3) - sqrt(3)*a)/a) - (-a/b)^(1/3)*l
og(x^4 + x^2*(-a/b)^(1/3) + (-a/b)^(2/3)) + 2*(-a/b)^(1/3)*log(x^2 - (-a/b)^(1/3)))/b

________________________________________________________________________________________

giac [A]  time = 0.16, size = 120, normalized size = 0.90 \[ \frac {x^{2}}{2 \, b} + \frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{2} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{6 \, b} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{6 \, b^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{4} + x^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/b + 1/6*(-a/b)^(1/3)*log(abs(x^2 - (-a/b)^(1/3)))/b - 1/6*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*(2
*x^2 + (-a/b)^(1/3))/(-a/b)^(1/3))/b^2 - 1/12*(-a*b^2)^(1/3)*log(x^4 + x^2*(-a/b)^(1/3) + (-a/b)^(2/3))/b^2

________________________________________________________________________________________

maple [A]  time = 0.00, size = 108, normalized size = 0.81 \[ \frac {x^{2}}{2 b}-\frac {\sqrt {3}\, a \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}-\frac {a \ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}+\frac {a \ln \left (x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{2}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/b*x^2-1/6*a/b^2/(a/b)^(2/3)*ln(x^2+(a/b)^(1/3))+1/12*a/b^2/(a/b)^(2/3)*ln(x^4-(a/b)^(1/3)*x^2+(a/b)^(2/3))
-1/6*a/b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x^2-1))

________________________________________________________________________________________

maxima [A]  time = 2.47, size = 115, normalized size = 0.86 \[ \frac {x^{2}}{2 \, b} - \frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{6 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {a \log \left (x^{4} - x^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {a \log \left (x^{2} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/b - 1/6*sqrt(3)*a*arctan(1/3*sqrt(3)*(2*x^2 - (a/b)^(1/3))/(a/b)^(1/3))/(b^2*(a/b)^(2/3)) + 1/12*a*log
(x^4 - x^2*(a/b)^(1/3) + (a/b)^(2/3))/(b^2*(a/b)^(2/3)) - 1/6*a*log(x^2 + (a/b)^(1/3))/(b^2*(a/b)^(2/3))

________________________________________________________________________________________

mupad [B]  time = 1.25, size = 130, normalized size = 0.98 \[ \frac {x^2}{2\,b}+\frac {{\left (-a\right )}^{1/3}\,\ln \left ({\left (-a\right )}^{10/3}+a^3\,b^{1/3}\,x^2\right )}{6\,b^{4/3}}+\frac {{\left (-a\right )}^{1/3}\,\ln \left (6\,a^3\,b\,x^2+6\,{\left (-a\right )}^{10/3}\,b^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{6\,b^{4/3}}-\frac {{\left (-a\right )}^{1/3}\,\ln \left (6\,a^3\,b\,x^2-6\,{\left (-a\right )}^{10/3}\,b^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{6\,b^{4/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a + b*x^6),x)

[Out]

x^2/(2*b) + ((-a)^(1/3)*log((-a)^(10/3) + a^3*b^(1/3)*x^2))/(6*b^(4/3)) + ((-a)^(1/3)*log(6*a^3*b*x^2 + 6*(-a)
^(10/3)*b^(2/3)*((3^(1/2)*1i)/2 - 1/2))*((3^(1/2)*1i)/2 - 1/2))/(6*b^(4/3)) - ((-a)^(1/3)*log(6*a^3*b*x^2 - 6*
(-a)^(10/3)*b^(2/3)*((3^(1/2)*1i)/2 + 1/2))*((3^(1/2)*1i)/2 + 1/2))/(6*b^(4/3))

________________________________________________________________________________________

sympy [A]  time = 0.43, size = 27, normalized size = 0.20 \[ \operatorname {RootSum} {\left (216 t^{3} b^{4} + a, \left (t \mapsto t \log {\left (- 6 t b + x^{2} \right )} \right )\right )} + \frac {x^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(216*_t**3*b**4 + a, Lambda(_t, _t*log(-6*_t*b + x**2))) + x**2/(2*b)

________________________________________________________________________________________