Optimal. Leaf size=119 \[ \frac {\sqrt {a+b x^n+c x^{2 n}}}{n}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+b x^n+c x^{2 n}}}\right )}{n}+\frac {b \tanh ^{-1}\left (\frac {b+2 c x^n}{2 \sqrt {c} \sqrt {a+b x^n+c x^{2 n}}}\right )}{2 \sqrt {c} n} \]
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Rubi [A] time = 0.10, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1357, 734, 843, 621, 206, 724} \[ \frac {\sqrt {a+b x^n+c x^{2 n}}}{n}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+b x^n+c x^{2 n}}}\right )}{n}+\frac {b \tanh ^{-1}\left (\frac {b+2 c x^n}{2 \sqrt {c} \sqrt {a+b x^n+c x^{2 n}}}\right )}{2 \sqrt {c} n} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 734
Rule 843
Rule 1357
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\sqrt {a+b x^n+c x^{2 n}}}{n}-\frac {\operatorname {Subst}\left (\int \frac {-2 a-b x}{x \sqrt {a+b x+c x^2}} \, dx,x,x^n\right )}{2 n}\\ &=\frac {\sqrt {a+b x^n+c x^{2 n}}}{n}+\frac {a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^n\right )}{n}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^n\right )}{2 n}\\ &=\frac {\sqrt {a+b x^n+c x^{2 n}}}{n}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^n}{\sqrt {a+b x^n+c x^{2 n}}}\right )}{n}+\frac {b \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^n}{\sqrt {a+b x^n+c x^{2 n}}}\right )}{n}\\ &=\frac {\sqrt {a+b x^n+c x^{2 n}}}{n}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+b x^n+c x^{2 n}}}\right )}{n}+\frac {b \tanh ^{-1}\left (\frac {b+2 c x^n}{2 \sqrt {c} \sqrt {a+b x^n+c x^{2 n}}}\right )}{2 \sqrt {c} n}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 110, normalized size = 0.92 \[ \frac {\sqrt {a+x^n \left (b+c x^n\right )}-\sqrt {a} \tanh ^{-1}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+x^n \left (b+c x^n\right )}}\right )+\frac {b \tanh ^{-1}\left (\frac {b+2 c x^n}{2 \sqrt {c} \sqrt {a+x^n \left (b+c x^n\right )}}\right )}{2 \sqrt {c}}}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.12, size = 658, normalized size = 5.53 \[ \left [\frac {b \sqrt {c} \log \left (-8 \, c^{2} x^{2 \, n} - 8 \, b c x^{n} - b^{2} - 4 \, a c - 4 \, {\left (2 \, c^{\frac {3}{2}} x^{n} + b \sqrt {c}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}\right ) + 2 \, \sqrt {a} c \log \left (-\frac {8 \, a b x^{n} + 8 \, a^{2} + {\left (b^{2} + 4 \, a c\right )} x^{2 \, n} - 4 \, {\left (\sqrt {a} b x^{n} + 2 \, a^{\frac {3}{2}}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{x^{2 \, n}}\right ) + 4 \, \sqrt {c x^{2 \, n} + b x^{n} + a} c}{4 \, c n}, -\frac {b \sqrt {-c} \arctan \left (\frac {{\left (2 \, \sqrt {-c} c x^{n} + b \sqrt {-c}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{2 \, {\left (c^{2} x^{2 \, n} + b c x^{n} + a c\right )}}\right ) - \sqrt {a} c \log \left (-\frac {8 \, a b x^{n} + 8 \, a^{2} + {\left (b^{2} + 4 \, a c\right )} x^{2 \, n} - 4 \, {\left (\sqrt {a} b x^{n} + 2 \, a^{\frac {3}{2}}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{x^{2 \, n}}\right ) - 2 \, \sqrt {c x^{2 \, n} + b x^{n} + a} c}{2 \, c n}, \frac {4 \, \sqrt {-a} c \arctan \left (\frac {{\left (\sqrt {-a} b x^{n} + 2 \, \sqrt {-a} a\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{2 \, {\left (a c x^{2 \, n} + a b x^{n} + a^{2}\right )}}\right ) + b \sqrt {c} \log \left (-8 \, c^{2} x^{2 \, n} - 8 \, b c x^{n} - b^{2} - 4 \, a c - 4 \, {\left (2 \, c^{\frac {3}{2}} x^{n} + b \sqrt {c}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}\right ) + 4 \, \sqrt {c x^{2 \, n} + b x^{n} + a} c}{4 \, c n}, \frac {2 \, \sqrt {-a} c \arctan \left (\frac {{\left (\sqrt {-a} b x^{n} + 2 \, \sqrt {-a} a\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{2 \, {\left (a c x^{2 \, n} + a b x^{n} + a^{2}\right )}}\right ) - b \sqrt {-c} \arctan \left (\frac {{\left (2 \, \sqrt {-c} c x^{n} + b \sqrt {-c}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{2 \, {\left (c^{2} x^{2 \, n} + b c x^{n} + a c\right )}}\right ) + 2 \, \sqrt {c x^{2 \, n} + b x^{n} + a} c}{2 \, c n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x^{2 \, n} + b x^{n} + a}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 125, normalized size = 1.05 \[ -\frac {\sqrt {a}\, \ln \left (\left (b \,{\mathrm e}^{n \ln \relax (x )}+2 a +2 \sqrt {b \,{\mathrm e}^{n \ln \relax (x )}+c \,{\mathrm e}^{2 n \ln \relax (x )}+a}\, \sqrt {a}\right ) {\mathrm e}^{-n \ln \relax (x )}\right )}{n}+\frac {b \ln \left (\frac {c \,{\mathrm e}^{n \ln \relax (x )}+\frac {b}{2}}{\sqrt {c}}+\sqrt {b \,{\mathrm e}^{n \ln \relax (x )}+c \,{\mathrm e}^{2 n \ln \relax (x )}+a}\right )}{2 \sqrt {c}\, n}+\frac {\sqrt {b \,{\mathrm e}^{n \ln \relax (x )}+c \,{\mathrm e}^{2 n \ln \relax (x )}+a}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x^{2 \, n} + b x^{n} + a}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a+b\,x^n+c\,x^{2\,n}}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + b x^{n} + c x^{2 n}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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