Optimal. Leaf size=98 \[ \frac {b \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,2;p+2;-\frac {d \left (b x^2+a\right )}{b c-a d},\frac {b x^2+a}{a}\right )}{2 a^2 (p+1)} \]
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Rubi [A] time = 0.09, antiderivative size = 98, 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 = {446, 137, 136} \[ \frac {b \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,2;p+2;-\frac {d \left (b x^2+a\right )}{b c-a d},\frac {b x^2+a}{a}\right )}{2 a^2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 136
Rule 137
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )^q}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^p (c+d x)^q}{x^2} \, 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 ) \operatorname {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^2} \, dx,x,x^2\right )\\ &=\frac {b \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,2;2+p;-\frac {d \left (a+b x^2\right )}{b c-a d},\frac {a+b x^2}{a}\right )}{2 a^2 (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 100, normalized size = 1.02 \[ \frac {\left (\frac {a}{b x^2}+1\right )^{-p} \left (a+b x^2\right )^p \left (\frac {c}{d x^2}+1\right )^{-q} \left (c+d x^2\right )^q F_1\left (-p-q+1;-p,-q;-p-q+2;-\frac {a}{b x^2},-\frac {c}{d x^2}\right )}{2 x^2 (p+q-1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q}}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{2}+a \right )^{p} \left (d \,x^{2}+c \right )^{q}}{x^{3}}\, dx \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q}}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^q}{x^3} \,d x \]
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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