Optimal. Leaf size=82 \[ -\frac {\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q} F_1\left (-\frac {1}{2};-p,-q;\frac {1}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{x} \]
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Rubi [A]
time = 0.05, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {525, 524}
\begin {gather*} -\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} F_1\left (-\frac {1}{2};-p,-q;\frac {1}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )^q}{x^2} \, dx &=\left (\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {b x^2}{a}\right )^p \left (c+d x^2\right )^q}{x^2} \, dx\\ &=\left (\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q}\right ) \int \frac {\left (1+\frac {b x^2}{a}\right )^p \left (1+\frac {d x^2}{c}\right )^q}{x^2} \, dx\\ &=-\frac {\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q} F_1\left (-\frac {1}{2};-p,-q;\frac {1}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{x}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 84, normalized size = 1.02 \begin {gather*} -\frac {\left (a+b x^2\right )^p \left (\frac {a+b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (\frac {c+d x^2}{c}\right )^{-q} F_1\left (-\frac {1}{2};-p,-q;\frac {1}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{2}+a \right )^{p} \left (d \,x^{2}+c \right )^{q}}{x^{2}}\, dx\]
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 {{\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^q}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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