3.1144 \(\int \frac{\left (a+b x^2\right )^p \left (c+d x^2\right )^q}{x^4} \, dx\)

Optimal. Leaf size=84 \[ -\frac{\left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (-\frac{3}{2};-p,-q;-\frac{1}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.208077, antiderivative size = 84, 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 \[ -\frac{\left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (-\frac{3}{2};-p,-q;-\frac{1}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^p*(c + d*x^2)^q)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 28.9303, size = 70, normalized size = 0.83 \[ - \frac{\left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (1 + \frac{d x^{2}}{c}\right )^{- q} \left (a + b x^{2}\right )^{p} \left (c + d x^{2}\right )^{q} \operatorname{appellf_{1}}{\left (- \frac{3}{2},- p,- q,- \frac{1}{2},- \frac{b x^{2}}{a},- \frac{d x^{2}}{c} \right )}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**p*(d*x**2+c)**q/x**4,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(-3/2, -p, -q, -1/2, -b*x**2/a, -d*x**2/c)/(3*x**3)

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Mathematica [B]  time = 0.457249, size = 173, normalized size = 2.06 \[ \frac{a c \left (a+b x^2\right )^p \left (c+d x^2\right )^q F_1\left (-\frac{3}{2};-p,-q;-\frac{1}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{6 x^5 \left (b c p F_1\left (-\frac{1}{2};1-p,-q;\frac{1}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+a d q F_1\left (-\frac{1}{2};-p,1-q;\frac{1}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-3 a c x^3 F_1\left (-\frac{3}{2};-p,-q;-\frac{1}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x^2)^p*(c + d*x^2)^q)/x^4,x]

[Out]

(a*c*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[-3/2, -p, -q, -1/2, -((b*x^2)/a), -((d
*x^2)/c)])/(-3*a*c*x^3*AppellF1[-3/2, -p, -q, -1/2, -((b*x^2)/a), -((d*x^2)/c)]
+ 6*x^5*(b*c*p*AppellF1[-1/2, 1 - p, -q, 1/2, -((b*x^2)/a), -((d*x^2)/c)] + a*d*
q*AppellF1[-1/2, -p, 1 - q, 1/2, -((b*x^2)/a), -((d*x^2)/c)]))

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Maple [F]  time = 0.089, size = 0, normalized size = 0. \[ \int{\frac{ \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q}}{{x}^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^p*(d*x^2+c)^q/x^4,x)

[Out]

int((b*x^2+a)^p*(d*x^2+c)^q/x^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q/x^4,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^p*(d*x^2 + c)^q/x^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q}}{x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q/x^4,x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^p*(d*x^2 + c)^q/x^4, 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**4,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q/x^4,x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^p*(d*x^2 + c)^q/x^4, x)