Integrand size = 22, antiderivative size = 119 \[ \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x} \, dx=\frac {\sqrt {a+b x^n+c x^{2 n}}}{n}-\frac {\sqrt {a} \text {arctanh}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+b x^n+c x^{2 n}}}\right )}{n}+\frac {b \text {arctanh}\left (\frac {b+2 c x^n}{2 \sqrt {c} \sqrt {a+b x^n+c x^{2 n}}}\right )}{2 \sqrt {c} n} \]
-arctanh(1/2*(2*a+b*x^n)/a^(1/2)/(a+b*x^n+c*x^(2*n))^(1/2))*a^(1/2)/n+1/2* b*arctanh(1/2*(b+2*c*x^n)/c^(1/2)/(a+b*x^n+c*x^(2*n))^(1/2))/n/c^(1/2)+(a+ b*x^n+c*x^(2*n))^(1/2)/n
Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x} \, dx=\frac {2 \sqrt {a+x^n \left (b+c x^n\right )}+4 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {c} x^n-\sqrt {a+x^n \left (b+c x^n\right )}}{\sqrt {a}}\right )-\frac {b \log \left (n \left (b+2 c x^n-2 \sqrt {c} \sqrt {a+x^n \left (b+c x^n\right )}\right )\right )}{\sqrt {c}}}{2 n} \]
(2*Sqrt[a + x^n*(b + c*x^n)] + 4*Sqrt[a]*ArcTanh[(Sqrt[c]*x^n - Sqrt[a + x ^n*(b + c*x^n)])/Sqrt[a]] - (b*Log[n*(b + 2*c*x^n - 2*Sqrt[c]*Sqrt[a + x^n *(b + c*x^n)])])/Sqrt[c])/(2*n)
Time = 0.29 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1693, 1162, 25, 1269, 1092, 219, 1154, 219}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x} \, dx\) |
| \(\Big \downarrow \) 1693 |
| \(\displaystyle \frac {\int x^{-n} \sqrt {b x^n+c x^{2 n}+a}dx^n}{n}\) |
| \(\Big \downarrow \) 1162 |
| \(\displaystyle \frac {\sqrt {a+b x^n+c x^{2 n}}-\frac {1}{2} \int -\frac {x^{-n} \left (b x^n+2 a\right )}{\sqrt {b x^n+c x^{2 n}+a}}dx^n}{n}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {x^{-n} \left (b x^n+2 a\right )}{\sqrt {b x^n+c x^{2 n}+a}}dx^n+\sqrt {a+b x^n+c x^{2 n}}}{n}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {1}{2} \left (b \int \frac {1}{\sqrt {b x^n+c x^{2 n}+a}}dx^n+2 a \int \frac {x^{-n}}{\sqrt {b x^n+c x^{2 n}+a}}dx^n\right )+\sqrt {a+b x^n+c x^{2 n}}}{n}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\frac {1}{2} \left (2 b \int \frac {1}{4 c-x^{2 n}}d\frac {2 c x^n+b}{\sqrt {b x^n+c x^{2 n}+a}}+2 a \int \frac {x^{-n}}{\sqrt {b x^n+c x^{2 n}+a}}dx^n\right )+\sqrt {a+b x^n+c x^{2 n}}}{n}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{2} \left (2 a \int \frac {x^{-n}}{\sqrt {b x^n+c x^{2 n}+a}}dx^n+\frac {b \text {arctanh}\left (\frac {b+2 c x^n}{2 \sqrt {c} \sqrt {a+b x^n+c x^{2 n}}}\right )}{\sqrt {c}}\right )+\sqrt {a+b x^n+c x^{2 n}}}{n}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {b \text {arctanh}\left (\frac {b+2 c x^n}{2 \sqrt {c} \sqrt {a+b x^n+c x^{2 n}}}\right )}{\sqrt {c}}-4 a \int \frac {1}{4 a-x^{2 n}}d\frac {b x^n+2 a}{\sqrt {b x^n+c x^{2 n}+a}}\right )+\sqrt {a+b x^n+c x^{2 n}}}{n}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {b \text {arctanh}\left (\frac {b+2 c x^n}{2 \sqrt {c} \sqrt {a+b x^n+c x^{2 n}}}\right )}{\sqrt {c}}-2 \sqrt {a} \text {arctanh}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+b x^n+c x^{2 n}}}\right )\right )+\sqrt {a+b x^n+c x^{2 n}}}{n}\) |
(Sqrt[a + b*x^n + c*x^(2*n)] + (-2*Sqrt[a]*ArcTanh[(2*a + b*x^n)/(2*Sqrt[a ]*Sqrt[a + b*x^n + c*x^(2*n)])] + (b*ArcTanh[(b + 2*c*x^n)/(2*Sqrt[c]*Sqrt [a + b*x^n + c*x^(2*n)])])/Sqrt[c])/2)/n
3.6.72.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x ] - Simp[p/(e*(m + 2*p + 1)) Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x ] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ [Simplify[(m + 1)/n]]
Time = 0.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.05
| method | result | size |
| risch | \(\frac {\sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}+c \,{\mathrm e}^{2 n \ln \left (x \right )}}}{n}+\frac {b \ln \left (\frac {\frac {b}{2}+c \,{\mathrm e}^{n \ln \left (x \right )}}{\sqrt {c}}+\sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}+c \,{\mathrm e}^{2 n \ln \left (x \right )}}\right )}{2 n \sqrt {c}}-\frac {\sqrt {a}\, \ln \left (\left (2 a +b \,{\mathrm e}^{n \ln \left (x \right )}+2 \sqrt {a}\, \sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}+c \,{\mathrm e}^{2 n \ln \left (x \right )}}\right ) {\mathrm e}^{-n \ln \left (x \right )}\right )}{n}\) | \(125\) |
1/n*(a+b*exp(n*ln(x))+c*exp(n*ln(x))^2)^(1/2)+1/2/n*b*ln((1/2*b+c*exp(n*ln (x)))/c^(1/2)+(a+b*exp(n*ln(x))+c*exp(n*ln(x))^2)^(1/2))/c^(1/2)-1/n*a^(1/ 2)*ln((2*a+b*exp(n*ln(x))+2*a^(1/2)*(a+b*exp(n*ln(x))+c*exp(n*ln(x))^2)^(1 /2))/exp(n*ln(x)))
Time = 0.29 (sec) , antiderivative size = 658, normalized size of antiderivative = 5.53 \[ \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x} \, dx=\left [\frac {b \sqrt {c} \log \left (-8 \, c^{2} x^{2 \, n} - 8 \, b c x^{n} - b^{2} - 4 \, a c - 4 \, {\left (2 \, c^{\frac {3}{2}} x^{n} + b \sqrt {c}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}\right ) + 2 \, \sqrt {a} c \log \left (-\frac {8 \, a b x^{n} + 8 \, a^{2} + {\left (b^{2} + 4 \, a c\right )} x^{2 \, n} - 4 \, {\left (\sqrt {a} b x^{n} + 2 \, a^{\frac {3}{2}}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{x^{2 \, n}}\right ) + 4 \, \sqrt {c x^{2 \, n} + b x^{n} + a} c}{4 \, c n}, -\frac {b \sqrt {-c} \arctan \left (\frac {{\left (2 \, \sqrt {-c} c x^{n} + b \sqrt {-c}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{2 \, {\left (c^{2} x^{2 \, n} + b c x^{n} + a c\right )}}\right ) - \sqrt {a} c \log \left (-\frac {8 \, a b x^{n} + 8 \, a^{2} + {\left (b^{2} + 4 \, a c\right )} x^{2 \, n} - 4 \, {\left (\sqrt {a} b x^{n} + 2 \, a^{\frac {3}{2}}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{x^{2 \, n}}\right ) - 2 \, \sqrt {c x^{2 \, n} + b x^{n} + a} c}{2 \, c n}, \frac {4 \, \sqrt {-a} c \arctan \left (\frac {{\left (\sqrt {-a} b x^{n} + 2 \, \sqrt {-a} a\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{2 \, {\left (a c x^{2 \, n} + a b x^{n} + a^{2}\right )}}\right ) + b \sqrt {c} \log \left (-8 \, c^{2} x^{2 \, n} - 8 \, b c x^{n} - b^{2} - 4 \, a c - 4 \, {\left (2 \, c^{\frac {3}{2}} x^{n} + b \sqrt {c}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}\right ) + 4 \, \sqrt {c x^{2 \, n} + b x^{n} + a} c}{4 \, c n}, \frac {2 \, \sqrt {-a} c \arctan \left (\frac {{\left (\sqrt {-a} b x^{n} + 2 \, \sqrt {-a} a\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{2 \, {\left (a c x^{2 \, n} + a b x^{n} + a^{2}\right )}}\right ) - b \sqrt {-c} \arctan \left (\frac {{\left (2 \, \sqrt {-c} c x^{n} + b \sqrt {-c}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{2 \, {\left (c^{2} x^{2 \, n} + b c x^{n} + a c\right )}}\right ) + 2 \, \sqrt {c x^{2 \, n} + b x^{n} + a} c}{2 \, c n}\right ] \]
[1/4*(b*sqrt(c)*log(-8*c^2*x^(2*n) - 8*b*c*x^n - b^2 - 4*a*c - 4*(2*c^(3/2 )*x^n + b*sqrt(c))*sqrt(c*x^(2*n) + b*x^n + a)) + 2*sqrt(a)*c*log(-(8*a*b* x^n + 8*a^2 + (b^2 + 4*a*c)*x^(2*n) - 4*(sqrt(a)*b*x^n + 2*a^(3/2))*sqrt(c *x^(2*n) + b*x^n + a))/x^(2*n)) + 4*sqrt(c*x^(2*n) + b*x^n + a)*c)/(c*n), -1/2*(b*sqrt(-c)*arctan(1/2*(2*sqrt(-c)*c*x^n + b*sqrt(-c))*sqrt(c*x^(2*n) + b*x^n + a)/(c^2*x^(2*n) + b*c*x^n + a*c)) - sqrt(a)*c*log(-(8*a*b*x^n + 8*a^2 + (b^2 + 4*a*c)*x^(2*n) - 4*(sqrt(a)*b*x^n + 2*a^(3/2))*sqrt(c*x^(2 *n) + b*x^n + a))/x^(2*n)) - 2*sqrt(c*x^(2*n) + b*x^n + a)*c)/(c*n), 1/4*( 4*sqrt(-a)*c*arctan(1/2*(sqrt(-a)*b*x^n + 2*sqrt(-a)*a)*sqrt(c*x^(2*n) + b *x^n + a)/(a*c*x^(2*n) + a*b*x^n + a^2)) + b*sqrt(c)*log(-8*c^2*x^(2*n) - 8*b*c*x^n - b^2 - 4*a*c - 4*(2*c^(3/2)*x^n + b*sqrt(c))*sqrt(c*x^(2*n) + b *x^n + a)) + 4*sqrt(c*x^(2*n) + b*x^n + a)*c)/(c*n), 1/2*(2*sqrt(-a)*c*arc tan(1/2*(sqrt(-a)*b*x^n + 2*sqrt(-a)*a)*sqrt(c*x^(2*n) + b*x^n + a)/(a*c*x ^(2*n) + a*b*x^n + a^2)) - b*sqrt(-c)*arctan(1/2*(2*sqrt(-c)*c*x^n + b*sqr t(-c))*sqrt(c*x^(2*n) + b*x^n + a)/(c^2*x^(2*n) + b*c*x^n + a*c)) + 2*sqrt (c*x^(2*n) + b*x^n + a)*c)/(c*n)]
\[ \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x} \, dx=\int \frac {\sqrt {a + b x^{n} + c x^{2 n}}}{x}\, dx \]
Timed out. \[ \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x} \, dx=\int { \frac {\sqrt {c x^{2 \, n} + b x^{n} + a}}{x} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x} \, dx=\int \frac {\sqrt {a+b\,x^n+c\,x^{2\,n}}}{x} \,d x \]