3.170 \(\int \left (a+\frac{b}{x}\right )^p \left (c+\frac{d}{x}\right )^q \, dx\)

Optimal. Leaf size=96 \[ -\frac{b \left (a+\frac{b}{x}\right )^{p+1} \left (c+\frac{d}{x}\right )^q \left (\frac{b \left (c+\frac{d}{x}\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,2;p+2;-\frac{d \left (a+\frac{b}{x}\right )}{b c-a d},\frac{a+\frac{b}{x}}{a}\right )}{a^2 (p+1)} \]

[Out]

-((b*(a + b/x)^(1 + p)*(c + d/x)^q*AppellF1[1 + p, -q, 2, 2 + p, -((d*(a + b/x))
/(b*c - a*d)), (a + b/x)/a])/(a^2*(1 + p)*((b*(c + d/x))/(b*c - a*d))^q))

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Rubi [A]  time = 0.180777, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{b \left (a+\frac{b}{x}\right )^{p+1} \left (c+\frac{d}{x}\right )^q \left (\frac{b \left (c+\frac{d}{x}\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,2;p+2;-\frac{d \left (a+\frac{b}{x}\right )}{b c-a d},\frac{a+\frac{b}{x}}{a}\right )}{a^2 (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^p*(c + d/x)^q,x]

[Out]

-((b*(a + b/x)^(1 + p)*(c + d/x)^q*AppellF1[1 + p, -q, 2, 2 + p, -((d*(a + b/x))
/(b*c - a*d)), (a + b/x)/a])/(a^2*(1 + p)*((b*(c + d/x))/(b*c - a*d))^q))

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Rubi in Sympy [A]  time = 23.2342, size = 66, normalized size = 0.69 \[ - \frac{b \left (\frac{b \left (- c - \frac{d}{x}\right )}{a d - b c}\right )^{- q} \left (a + \frac{b}{x}\right )^{p + 1} \left (c + \frac{d}{x}\right )^{q} \operatorname{appellf_{1}}{\left (p + 1,2,- q,p + 2,\frac{a + \frac{b}{x}}{a},\frac{d \left (a + \frac{b}{x}\right )}{a d - b c} \right )}}{a^{2} \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**p*(c+d/x)**q,x)

[Out]

-b*(b*(-c - d/x)/(a*d - b*c))**(-q)*(a + b/x)**(p + 1)*(c + d/x)**q*appellf1(p +
1, 2, -q, p + 2, (a + b/x)/a, d*(a + b/x)/(a*d - b*c))/(a**2*(p + 1))

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Mathematica [B]  time = 0.493105, 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.

[In]  Integrate[(a + b/x)^p*(c + d/x)^q,x]

[Out]

(b*d*(-2 + p + q)*(a + b/x)^p*(c + d/x)^q*x*AppellF1[1 - p - q, -p, -q, 2 - p -
q, -((a*x)/b), -((c*x)/d)])/((-1 + p + q)*(-(b*d*(-2 + p + q)*AppellF1[1 - p - q
, -p, -q, 2 - p - q, -((a*x)/b), -((c*x)/d)]) + x*(a*d*p*AppellF1[2 - p - q, 1 -
p, -q, 3 - p - q, -((a*x)/b), -((c*x)/d)] + b*c*q*AppellF1[2 - p - q, -p, 1 - q
, 3 - p - q, -((a*x)/b), -((c*x)/d)])))

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Maple [F]  time = 0.142, size = 0, normalized size = 0. \[ \int \left ( a+{\frac{b}{x}} \right ) ^{p} \left ( c+{\frac{d}{x}} \right ) ^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^p*(c+d/x)^q,x)

[Out]

int((a+b/x)^p*(c+d/x)^q,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a + \frac{b}{x}\right )}^{p}{\left (c + \frac{d}{x}\right )}^{q}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^p*(c + d/x)^q,x, algorithm="maxima")

[Out]

integrate((a + b/x)^p*(c + d/x)^q, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\left (\frac{a x + b}{x}\right )^{p} \left (\frac{c x + d}{x}\right )^{q}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^p*(c + d/x)^q,x, algorithm="fricas")

[Out]

integral(((a*x + b)/x)^p*((c*x + d)/x)^q, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a + \frac{b}{x}\right )^{p} \left (c + \frac{d}{x}\right )^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**p*(c+d/x)**q,x)

[Out]

Integral((a + b/x)**p*(c + d/x)**q, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a + \frac{b}{x}\right )}^{p}{\left (c + \frac{d}{x}\right )}^{q}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^p*(c + d/x)^q,x, algorithm="giac")

[Out]

integrate((a + b/x)^p*(c + d/x)^q, x)