Optimal. Leaf size=100 \[ -\frac {b^2 \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^q \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} F_1\left (1+p;-q,3;2+p;-\frac {d \left (a+b x^2\right )}{b c-a d},\frac {a+b x^2}{a}\right )}{2 a^3 (1+p)} \]
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Rubi [A]
time = 0.07, antiderivative size = 100, 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 = {457, 142, 141}
\begin {gather*} -\frac {b^2 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,3;p+2;-\frac {d \left (b x^2+a\right )}{b c-a d},\frac {b x^2+a}{a}\right )}{2 a^3 (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 141
Rule 142
Rule 457
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )^q}{x^5} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^p (c+d x)^q}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \left (\left (c+d x^2\right )^q \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q}\right ) \text {Subst}\left (\int \frac {(a+b x)^p \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^q}{x^3} \, dx,x,x^2\right )\\ &=-\frac {b^2 \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^q \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} F_1\left (1+p;-q,3;2+p;-\frac {d \left (a+b x^2\right )}{b c-a d},\frac {a+b x^2}{a}\right )}{2 a^3 (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 100, normalized size = 1.00 \begin {gather*} \frac {\left (1+\frac {a}{b x^2}\right )^{-p} \left (1+\frac {c}{d x^2}\right )^{-q} \left (a+b x^2\right )^p \left (c+d x^2\right )^q F_1\left (2-p-q;-p,-q;3-p-q;-\frac {a}{b x^2},-\frac {c}{d x^2}\right )}{2 (-2+p+q) x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{2}+a \right )^{p} \left (d \,x^{2}+c \right )^{q}}{x^{5}}\, dx\]
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^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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