3.1143 \(\int \frac{\left (a+b x^2\right )^p \left (c+d x^2\right )^q}{x^2} \, dx\)

Optimal. Leaf size=82 \[ -\frac{\left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (-\frac{1}{2};-p,-q;\frac{1}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x} \]

[Out]

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

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Rubi [A]  time = 0.202976, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{\left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (-\frac{1}{2};-p,-q;\frac{1}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.3653, size = 65, normalized size = 0.79 \[ - \frac{\left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (1 + \frac{d x^{2}}{c}\right )^{- q} \left (a + b x^{2}\right )^{p} \left (c + d x^{2}\right )^{q} \operatorname{appellf_{1}}{\left (- \frac{1}{2},- p,- q,\frac{1}{2},- \frac{b x^{2}}{a},- \frac{d x^{2}}{c} \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(1 + b*x**2/a)**(-p)*(1 + d*x**2/c)**(-q)*(a + b*x**2)**p*(c + d*x**2)**q*appel
lf1(-1/2, -p, -q, 1/2, -b*x**2/a, -d*x**2/c)/x

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Mathematica [B]  time = 0.355485, size = 171, normalized size = 2.09 \[ -\frac{a c \left (a+b x^2\right )^p \left (c+d x^2\right )^q F_1\left (-\frac{1}{2};-p,-q;\frac{1}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{a c x F_1\left (-\frac{1}{2};-p,-q;\frac{1}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+2 x^3 \left (b c p F_1\left (\frac{1}{2};1-p,-q;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+a d q F_1\left (\frac{1}{2};-p,1-q;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((a*c*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[-1/2, -p, -q, 1/2, -((b*x^2)/a), -((
d*x^2)/c)])/(a*c*x*AppellF1[-1/2, -p, -q, 1/2, -((b*x^2)/a), -((d*x^2)/c)] + 2*x
^3*(b*c*p*AppellF1[1/2, 1 - p, -q, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + a*d*q*Appe
llF1[1/2, -p, 1 - q, 3/2, -((b*x^2)/a), -((d*x^2)/c)])))

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \[ \int{\frac{ \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q}}{{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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