3.825 \(\int \frac{\left (a+\frac{b}{x^2}\right )^p \left (c+\frac{d}{x^2}\right )^q}{x^3} \, dx\)

Optimal. Leaf size=85 \[ -\frac{\left (a+\frac{b}{x^2}\right )^{p+1} \left (c+\frac{d}{x^2}\right )^q \left (\frac{b \left (c+\frac{d}{x^2}\right )}{b c-a d}\right )^{-q} \, _2F_1\left (p+1,-q;p+2;-\frac{d \left (a+\frac{b}{x^2}\right )}{b c-a d}\right )}{2 b (p+1)} \]

[Out]

-((a + b/x^2)^(1 + p)*(c + d/x^2)^q*Hypergeometric2F1[1 + p, -q, 2 + p, -((d*(a
+ b/x^2))/(b*c - a*d))])/(2*b*(1 + p)*((b*(c + d/x^2))/(b*c - a*d))^q)

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Rubi [A]  time = 0.169153, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{\left (a+\frac{b}{x^2}\right )^{p+1} \left (c+\frac{d}{x^2}\right )^q \left (\frac{b \left (c+\frac{d}{x^2}\right )}{b c-a d}\right )^{-q} \, _2F_1\left (p+1,-q;p+2;-\frac{d \left (a+\frac{b}{x^2}\right )}{b c-a d}\right )}{2 b (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b/x^2)^p*(c + d/x^2)^q)/x^3,x]

[Out]

-((a + b/x^2)^(1 + p)*(c + d/x^2)^q*Hypergeometric2F1[1 + p, -q, 2 + p, -((d*(a
+ b/x^2))/(b*c - a*d))])/(2*b*(1 + p)*((b*(c + d/x^2))/(b*c - a*d))^q)

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Rubi in Sympy [A]  time = 24.1578, size = 66, normalized size = 0.78 \[ - \frac{\left (\frac{b \left (- c - \frac{d}{x^{2}}\right )}{a d - b c}\right )^{- q} \left (a + \frac{b}{x^{2}}\right )^{p + 1} \left (c + \frac{d}{x^{2}}\right )^{q}{{}_{2}F_{1}\left (\begin{matrix} - q, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{d \left (a + \frac{b}{x^{2}}\right )}{a d - b c}} \right )}}{2 b \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**p*(c+d/x**2)**q/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.524801, size = 109, normalized size = 1.28 \[ -\frac{\left (c x^2+d\right ) \left (a+\frac{b}{x^2}\right )^p \left (c+\frac{d}{x^2}\right )^q \left (\frac{d \left (a x^2+b\right )}{b \left (c x^2+d\right )}\right )^{-p} \, _2F_1\left (-p,-p-q-1;-p-q;\frac{(b c-a d) x^2}{b \left (c x^2+d\right )}\right )}{2 d x^2 (p+q+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b/x^2)^p*(c + d/x^2)^q)/x^3,x]

[Out]

-((a + b/x^2)^p*(c + d/x^2)^q*(d + c*x^2)*Hypergeometric2F1[-p, -1 - p - q, -p -
q, ((b*c - a*d)*x^2)/(b*(d + c*x^2))])/(2*d*(1 + p + q)*x^2*((d*(b + a*x^2))/(b
*(d + c*x^2)))^p)

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Maple [F]  time = 0.09, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \left ( c+{\frac{d}{{x}^{2}}} \right ) ^{q}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^p*(c+d/x^2)^q/x^3,x)

[Out]

int((a+b/x^2)^p*(c+d/x^2)^q/x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\right )}^{q}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^p*(c + d/x^2)^q/x^3,x, algorithm="maxima")

[Out]

integrate((a + b/x^2)^p*(c + d/x^2)^q/x^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (\frac{a x^{2} + b}{x^{2}}\right )^{p} \left (\frac{c x^{2} + d}{x^{2}}\right )^{q}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^p*(c + d/x^2)^q/x^3,x, algorithm="fricas")

[Out]

integral(((a*x^2 + b)/x^2)^p*((c*x^2 + d)/x^2)^q/x^3, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**p*(c+d/x**2)**q/x**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\right )}^{q}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^p*(c + d/x^2)^q/x^3,x, algorithm="giac")

[Out]

integrate((a + b/x^2)^p*(c + d/x^2)^q/x^3, x)