3.832 \(\int \frac{\left (a+\frac{b}{x^2}\right )^p \left (c+\frac{d}{x^2}\right )^q}{(e x)^{5/2}} \, dx\)

Optimal. Leaf size=91 \[ -\frac{2 \left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{d}{c x^2}+1\right )^{-q} F_1\left (\frac{3}{4};-p,-q;\frac{7}{4};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{3 e (e x)^{3/2}} \]

[Out]

(-2*(a + b/x^2)^p*(c + d/x^2)^q*AppellF1[3/4, -p, -q, 7/4, -(b/(a*x^2)), -(d/(c*
x^2))])/(3*e*(1 + b/(a*x^2))^p*(1 + d/(c*x^2))^q*(e*x)^(3/2))

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Rubi [A]  time = 0.344689, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{2 \left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{d}{c x^2}+1\right )^{-q} F_1\left (\frac{3}{4};-p,-q;\frac{7}{4};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{3 e (e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b/x^2)^p*(c + d/x^2)^q)/(e*x)^(5/2),x]

[Out]

(-2*(a + b/x^2)^p*(c + d/x^2)^q*AppellF1[3/4, -p, -q, 7/4, -(b/(a*x^2)), -(d/(c*
x^2))])/(3*e*(1 + b/(a*x^2))^p*(1 + d/(c*x^2))^q*(e*x)^(3/2))

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Rubi in Sympy [A]  time = 50.274, size = 73, normalized size = 0.8 \[ - \frac{2 \left (1 + \frac{b}{a x^{2}}\right )^{- p} \left (1 + \frac{d}{c x^{2}}\right )^{- q} \left (a + \frac{b}{x^{2}}\right )^{p} \left (c + \frac{d}{x^{2}}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{3}{4},- p,- q,\frac{7}{4},- \frac{b}{a x^{2}},- \frac{d}{c x^{2}} \right )}}{3 e \left (e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**p*(c+d/x**2)**q/(e*x)**(5/2),x)

[Out]

-2*(1 + b/(a*x**2))**(-p)*(1 + d/(c*x**2))**(-q)*(a + b/x**2)**p*(c + d/x**2)**q
*appellf1(3/4, -p, -q, 7/4, -b/(a*x**2), -d/(c*x**2))/(3*e*(e*x)**(3/2))

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Mathematica [B]  time = 0.824265, size = 260, normalized size = 2.86 \[ \frac{2 b d x (4 p+4 q-1) \left (a+\frac{b}{x^2}\right )^p \left (c+\frac{d}{x^2}\right )^q F_1\left (-p-q-\frac{3}{4};-p,-q;-p-q+\frac{1}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )}{(e x)^{5/2} (4 p+4 q+3) \left (4 x^2 \left (a d p F_1\left (-p-q+\frac{1}{4};1-p,-q;-p-q+\frac{5}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )+b c q F_1\left (-p-q+\frac{1}{4};-p,1-q;-p-q+\frac{5}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )\right )+b d (-4 p-4 q+1) F_1\left (-p-q-\frac{3}{4};-p,-q;-p-q+\frac{1}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b/x^2)^p*(c + d/x^2)^q)/(e*x)^(5/2),x]

[Out]

(2*b*d*(-1 + 4*p + 4*q)*(a + b/x^2)^p*(c + d/x^2)^q*x*AppellF1[-3/4 - p - q, -p,
 -q, 1/4 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((3 + 4*p + 4*q)*(e*x)^(5/2)*(b*d
*(1 - 4*p - 4*q)*AppellF1[-3/4 - p - q, -p, -q, 1/4 - p - q, -((a*x^2)/b), -((c*
x^2)/d)] + 4*x^2*(a*d*p*AppellF1[1/4 - p - q, 1 - p, -q, 5/4 - p - q, -((a*x^2)/
b), -((c*x^2)/d)] + b*c*q*AppellF1[1/4 - p - q, -p, 1 - q, 5/4 - p - q, -((a*x^2
)/b), -((c*x^2)/d)])))

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{1 \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \left ( c+{\frac{d}{{x}^{2}}} \right ) ^{q} \left ( ex \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^p*(c+d/x^2)^q/(e*x)^(5/2),x)

[Out]

int((a+b/x^2)^p*(c+d/x^2)^q/(e*x)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\right )}^{q}}{\left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^p*(c + d/x^2)^q/(e*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((a + b/x^2)^p*(c + d/x^2)^q/(e*x)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^p*(c + d/x^2)^q/(e*x)^(5/2),x, algorithm="fricas")

[Out]

integral(((a*x^2 + b)/x^2)^p*((c*x^2 + d)/x^2)^q/(sqrt(e*x)*e^2*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((a+b/x**2)**p*(c+d/x**2)**q/(e*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\right )}^{q}}{\left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^p*(c + d/x^2)^q/(e*x)^(5/2),x, algorithm="giac")

[Out]

integrate((a + b/x^2)^p*(c + d/x^2)^q/(e*x)^(5/2), x)