Integrand size = 13, antiderivative size = 133 \[ \int \frac {x^7}{a+b x^6} \, dx=\frac {x^2}{2 b}+\frac {\sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} b^{4/3}}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 b^{4/3}}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 b^{4/3}} \]
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)
Time = 0.06 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.40 \[ \int \frac {x^7}{a+b x^6} \, dx=\frac {6 \sqrt [3]{b} x^2+2 \sqrt {3} \sqrt [3]{a} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt {3} \sqrt [3]{a} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )+\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{12 b^{4/3}} \]
(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)*L og[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))
Time = 0.32 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.02, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {807, 843, 750, 16, 1142, 25, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^7}{a+b x^6} \, dx\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {1}{2} \int \frac {x^6}{b x^6+a}dx^2\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2}{b}-\frac {a \int \frac {1}{b x^6+a}dx^2}{b}\right )\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2}{b}-\frac {a \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x^2}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x^2+\sqrt [3]{a}}dx^2}{3 a^{2/3}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2}{b}-\frac {a \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x^2}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2}{b}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x^2\right )}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2}{b}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x^2\right )}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2}{b}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2}{b}-\frac {a \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2+\frac {3 \int \frac {1}{-x^4-3}d\left (1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2}{b}-\frac {a \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2}{b}-\frac {a \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )\) |
(x^2/b - (a*(Log[a^(1/3) + b^(1/3)*x^2]/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3]* ArcTan[(1 - (2*b^(1/3)*x^2)/a^(1/3))/Sqrt[3]])/b^(1/3)) - Log[a^(2/3) - a^ (1/3)*b^(1/3)*x^2 + b^(2/3)*x^4]/(2*b^(1/3)))/(3*a^(2/3))))/b)/2
3.14.19.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[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]
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] - Simp[ 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]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.51 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.27
| method | result | size |
| risch | \(\frac {x^{2}}{2 b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\textit {\_R} \ln \left (x^{2}-\textit {\_R} \right )}{6 b}\) | \(36\) |
| default | \(\frac {x^{2}}{2 b}-\frac {\left (\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{2}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) a}{2 b}\) | \(112\) |
Time = 0.31 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.86 \[ \int \frac {x^7}{a+b x^6} \, dx=\frac {6 \, x^{2} + 2 \, \sqrt {3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x^{2} \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{4} + x^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) + 2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, b} \]
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)*log(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
Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.20 \[ \int \frac {x^7}{a+b x^6} \, dx=\operatorname {RootSum} {\left (216 t^{3} b^{4} + a, \left ( t \mapsto t \log {\left (- 6 t b + x^{2} \right )} \right )\right )} + \frac {x^{2}}{2 b} \]
Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.86 \[ \int \frac {x^7}{a+b x^6} \, dx=\frac {x^{2}}{2 \, b} - \frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{6 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {a \log \left (x^{4} - x^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {a \log \left (x^{2} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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))
Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.90 \[ \int \frac {x^7}{a+b x^6} \, dx=\frac {x^{2}}{2 \, b} + \frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{2} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{6 \, b} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{6 \, b^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{4} + x^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 \, b^{2}} \]
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
Time = 5.63 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.98 \[ \int \frac {x^7}{a+b x^6} \, dx=\frac {x^2}{2\,b}+\frac {{\left (-a\right )}^{1/3}\,\ln \left ({\left (-a\right )}^{10/3}+a^3\,b^{1/3}\,x^2\right )}{6\,b^{4/3}}+\frac {{\left (-a\right )}^{1/3}\,\ln \left (6\,a^3\,b\,x^2+6\,{\left (-a\right )}^{10/3}\,b^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{6\,b^{4/3}}-\frac {{\left (-a\right )}^{1/3}\,\ln \left (6\,a^3\,b\,x^2-6\,{\left (-a\right )}^{10/3}\,b^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{6\,b^{4/3}} \]
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))