Optimal. Leaf size=96 \[ -\frac {b \left (a+\frac {b}{x}\right )^{p+1} \left (c+\frac {d}{x}\right )^q \left (\frac {b \left (c+\frac {d}{x}\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,2;p+2;-\frac {d \left (a+\frac {b}{x}\right )}{b c-a d},\frac {a+\frac {b}{x}}{a}\right )}{a^2 (p+1)} \]
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Rubi [A] time = 0.06, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {375, 137, 136} \[ -\frac {b \left (a+\frac {b}{x}\right )^{p+1} \left (c+\frac {d}{x}\right )^q \left (\frac {b \left (c+\frac {d}{x}\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,2;p+2;-\frac {d \left (a+\frac {b}{x}\right )}{b c-a d},\frac {a+\frac {b}{x}}{a}\right )}{a^2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 136
Rule 137
Rule 375
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^p \left (c+\frac {d}{x}\right )^q \, dx &=-\operatorname {Subst}\left (\int \frac {(a+b x)^p (c+d x)^q}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=-\left (\left (\left (c+\frac {d}{x}\right )^q \left (\frac {b \left (c+\frac {d}{x}\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,\frac {1}{x}\right )\right )\\ &=-\frac {b \left (a+\frac {b}{x}\right )^{1+p} \left (c+\frac {d}{x}\right )^q \left (\frac {b \left (c+\frac {d}{x}\right )}{b c-a d}\right )^{-q} F_1\left (1+p;-q,2;2+p;-\frac {d \left (a+\frac {b}{x}\right )}{b c-a d},\frac {a+\frac {b}{x}}{a}\right )}{a^2 (1+p)}\\ \end {align*}
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Mathematica [B] time = 0.37, size = 206, normalized size = 2.15 \[ \frac {b d x (p+q-2) \left (a+\frac {b}{x}\right )^p \left (c+\frac {d}{x}\right )^q F_1\left (-p-q+1;-p,-q;-p-q+2;-\frac {a x}{b},-\frac {c x}{d}\right )}{(p+q-1) \left (x \left (a d p F_1\left (-p-q+2;1-p,-q;-p-q+3;-\frac {a x}{b},-\frac {c x}{d}\right )+b c q F_1\left (-p-q+2;-p,1-q;-p-q+3;-\frac {a x}{b},-\frac {c x}{d}\right )\right )-b d (p+q-2) F_1\left (-p-q+1;-p,-q;-p-q+2;-\frac {a x}{b},-\frac {c x}{d}\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\frac {a x + b}{x}\right )^{p} \left (\frac {c x + d}{x}\right )^{q}, x\right ) \]
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 \[ \int {\left (a + \frac {b}{x}\right )}^{p} {\left (c + \frac {d}{x}\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \left (a +\frac {b}{x}\right )^{p} \left (c +\frac {d}{x}\right )^{q}\, dx \]
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 \[ \int {\left (a + \frac {b}{x}\right )}^{p} {\left (c + \frac {d}{x}\right )}^{q}\,{d x} \]
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 \[ \int {\left (a+\frac {b}{x}\right )}^p\,{\left (c+\frac {d}{x}\right )}^q \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + \frac {b}{x}\right )^{p} \left (c + \frac {d}{x}\right )^{q}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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