Optimal. Leaf size=139 \[ \frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (b x^{n/2}-a\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}}}{x}-\frac {d \left (b x^{n/2}-a\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (1-\frac {b^2 x^n}{a^2}\right )^{-1/n} \, _2F_1\left (-\frac {1}{n},-\frac {1}{n};-\frac {1-n}{n};\frac {b^2 x^n}{a^2}\right )}{b^2 x} \]
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Rubi [A] time = 0.11, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {519, 452, 365, 364} \[ \frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (b x^{n/2}-a\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}}}{x}-\frac {d \left (b x^{n/2}-a\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (1-\frac {b^2 x^n}{a^2}\right )^{-1/n} \, _2F_1\left (-\frac {1}{n},-\frac {1}{n};-\frac {1-n}{n};\frac {b^2 x^n}{a^2}\right )}{b^2 x} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 452
Rule 519
Rubi steps
\begin {align*} \int \frac {\left (-a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (c+d x^n\right )}{x^2} \, dx &=\left (\left (-a+b x^{n/2}\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (-a^2+b^2 x^n\right )^{-1/n}\right ) \int \frac {\left (-a^2+b^2 x^n\right )^{-1+\frac {1}{n}} \left (c+d x^n\right )}{x^2} \, dx\\ &=\frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (-a+b x^{n/2}\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}}}{x}+\frac {\left (d \left (-a+b x^{n/2}\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (-a^2+b^2 x^n\right )^{-1/n}\right ) \int \frac {\left (-a^2+b^2 x^n\right )^{\frac {1}{n}}}{x^2} \, dx}{b^2}\\ &=\frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (-a+b x^{n/2}\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}}}{x}+\frac {\left (d \left (-a+b x^{n/2}\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (1-\frac {b^2 x^n}{a^2}\right )^{-1/n}\right ) \int \frac {\left (1-\frac {b^2 x^n}{a^2}\right )^{\frac {1}{n}}}{x^2} \, dx}{b^2}\\ &=\frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (-a+b x^{n/2}\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}}}{x}-\frac {d \left (-a+b x^{n/2}\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (1-\frac {b^2 x^n}{a^2}\right )^{-1/n} \, _2F_1\left (-\frac {1}{n},-\frac {1}{n};-\frac {1-n}{n};\frac {b^2 x^n}{a^2}\right )}{b^2 x}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 124, normalized size = 0.89 \[ \frac {\left (b x^{n/2}-a\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (1-\frac {b^2 x^n}{a^2}\right )^{-1/n} \left (c (n-1) \left (1-\frac {b^2 x^n}{a^2}\right )^{\frac {1}{n}}-d x^n \, _2F_1\left (\frac {n-1}{n},\frac {n-1}{n};2-\frac {1}{n};\frac {b^2 x^n}{a^2}\right )\right )}{a^2 (n-1) x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d x^{n} + c}{{\left (b x^{\frac {1}{2} \, n} + a\right )}^{\frac {n - 1}{n}} {\left (b x^{\frac {1}{2} \, n} - a\right )}^{\frac {n - 1}{n}} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (d x^{n} + c\right )} {\left (b x^{\frac {1}{2} \, n} + a\right )}^{\frac {1}{n} - 1} {\left (b x^{\frac {1}{2} \, n} - a\right )}^{\frac {1}{n} - 1}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{n}+c \right ) \left (b \,x^{\frac {n}{2}}-a \right )^{\frac {1}{n}-1} \left (b \,x^{\frac {n}{2}}+a \right )^{\frac {1}{n}-1}}{x^{2}}\, dx \]
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 \[ \int \frac {{\left (d x^{n} + c\right )} {\left (b x^{\frac {1}{2} \, n} + a\right )}^{\frac {1}{n} - 1} {\left (b x^{\frac {1}{2} \, n} - a\right )}^{\frac {1}{n} - 1}}{x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,x^{n/2}\right )}^{\frac {1}{n}-1}\,{\left (b\,x^{n/2}-a\right )}^{\frac {1}{n}-1}\,\left (c+d\,x^n\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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