Optimal. Leaf size=139 \[ \frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (-a+b x^{n/2}\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}}}{x}-\frac {d \left (-a+b x^{n/2}\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (1-\frac {b^2 x^n}{a^2}\right )^{-1/n} \, _2F_1\left (-\frac {1}{n},-\frac {1}{n};-\frac {1-n}{n};\frac {b^2 x^n}{a^2}\right )}{b^2 x} \]
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Rubi [A]
time = 0.08, antiderivative size = 167, normalized size of antiderivative = 1.20, number of steps
used = 4, number of rules used = 4, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {533, 463, 372,
371} \begin {gather*} \frac {a^2 d \left (b x^{n/2}-a\right )^{\frac {1}{n}-1} \left (a+b x^{n/2}\right )^{\frac {1}{n}-1} \left (1-\frac {b^2 x^n}{a^2}\right )^{-\frac {1-n}{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}-\frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (b x^{n/2}-a\right )^{\frac {1}{n}-1} \left (a+b x^{n/2}\right )^{\frac {1}{n}-1} \left (a^2-b^2 x^n\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rule 463
Rule 533
Rubi steps
\begin {align*} \int \frac {\left (-a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (c+d x^n\right )}{x^2} \, dx &=\left (\left (-a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (-a^2+b^2 x^n\right )^{-\frac {1-n}{n}}\right ) \int \frac {\left (-a^2+b^2 x^n\right )^{\frac {1-n}{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 )^{-1+\frac {1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a^2-b^2 x^n\right )}{x}+\frac {\left (d \left (-a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (-a^2+b^2 x^n\right )^{-\frac {1-n}{n}}\right ) \int \frac {\left (-a^2+b^2 x^n\right )^{1+\frac {1-n}{n}}}{x^2} \, dx}{b^2}\\ &=-\frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (-a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a^2-b^2 x^n\right )}{x}-\frac {\left (a^2 d \left (-a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (1-\frac {b^2 x^n}{a^2}\right )^{-\frac {1-n}{n}}\right ) \int \frac {\left (1-\frac {b^2 x^n}{a^2}\right )^{1+\frac {1-n}{n}}}{x^2} \, dx}{b^2}\\ &=-\frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (-a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a^2-b^2 x^n\right )}{x}+\frac {a^2 d \left (-a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (1-\frac {b^2 x^n}{a^2}\right )^{-\frac {1-n}{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 = 0.07, size = 124, normalized size = 0.89 \begin {gather*} \frac {\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} \left (c (-1+n) \left (1-\frac {b^2 x^n}{a^2}\right )^{\frac {1}{n}}-d x^n \, _2F_1\left (\frac {-1+n}{n},\frac {-1+n}{n};2-\frac {1}{n};\frac {b^2 x^n}{a^2}\right )\right )}{a^2 (-1+n) x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\left (-a +b \,x^{\frac {n}{2}}\right )^{\frac {1-n}{n}} \left (a +b \,x^{\frac {n}{2}}\right )^{\frac {1-n}{n}} \left (c +d \,x^{n}\right )}{x^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {c+d\,x^n}{x^2\,{\left (a+b\,x^{n/2}\right )}^{\frac {n-1}{n}}\,{\left (b\,x^{n/2}-a\right )}^{\frac {n-1}{n}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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