Optimal. Leaf size=85 \[ -\frac {\left (a+\frac {b}{x^2}\right )^{1+p} \left (c+\frac {d}{x^2}\right )^q \left (\frac {b \left (c+\frac {d}{x^2}\right )}{b c-a d}\right )^{-q} \, _2F_1\left (1+p,-q;2+p;-\frac {d \left (a+\frac {b}{x^2}\right )}{b c-a d}\right )}{2 b (1+p)} \]
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Rubi [A]
time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {455, 72, 71}
\begin {gather*} -\frac {\left (a+\frac {b}{x^2}\right )^{p+1} \left (c+\frac {d}{x^2}\right )^q \left (\frac {b \left (c+\frac {d}{x^2}\right )}{b c-a d}\right )^{-q} \, _2F_1\left (p+1,-q;p+2;-\frac {d \left (a+\frac {b}{x^2}\right )}{b c-a d}\right )}{2 b (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 455
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^3} \, dx &=-\left (\frac {1}{2} \text {Subst}\left (\int (a+b x)^p (c+d x)^q \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \left (\left (c+\frac {d}{x^2}\right )^q \left (\frac {b \left (c+\frac {d}{x^2}\right )}{b c-a d}\right )^{-q}\right ) \text {Subst}\left (\int (a+b x)^p \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^q \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {\left (a+\frac {b}{x^2}\right )^{1+p} \left (c+\frac {d}{x^2}\right )^q \left (\frac {b \left (c+\frac {d}{x^2}\right )}{b c-a d}\right )^{-q} \, _2F_1\left (1+p,-q;2+p;-\frac {d \left (a+\frac {b}{x^2}\right )}{b c-a d}\right )}{2 b (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 110, normalized size = 1.29 \begin {gather*} -\frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \left (1+\frac {a x^2}{b}\right )^{-p} \left (d+c x^2\right ) \left (1+\frac {c x^2}{d}\right )^p \, _2F_1\left (-p,-1-p-q;-p-q;\frac {(b c-a d) x^2}{b \left (d+c x^2\right )}\right )}{2 d (1+p+q) x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (\frac {b}{x^{2}}+a \right )^{p} \left (c +\frac {d}{x^{2}}\right )^{q}}{x^{3}}\, dx\]
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 {{\left (a+\frac {b}{x^2}\right )}^p\,{\left (c+\frac {d}{x^2}\right )}^q}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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