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

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

[Out]

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

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Rubi [A]  time = 0.328624, 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 \[ \frac{2 (e x)^{7/2} \left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{d}{c x^2}+1\right )^{-q} F_1\left (-\frac{7}{4};-p,-q;-\frac{3}{4};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{7 e} \]

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

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Rubi in Sympy [A]  time = 49.8652, size = 75, normalized size = 0.82 \[ \frac{2 \left (e x\right )^{\frac{7}{2}} \left (1 + \frac{b}{a x^{2}}\right )^{- p} \left (1 + \frac{d}{c x^{2}}\right )^{- q} \left (a + \frac{b}{x^{2}}\right )^{p} \left (c + \frac{d}{x^{2}}\right )^{q} \operatorname{appellf_{1}}{\left (- \frac{7}{4},- p,- q,- \frac{3}{4},- \frac{b}{a x^{2}},- \frac{d}{c x^{2}} \right )}}{7 e} \]

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

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Mathematica [B]  time = 0.851592, size = 260, normalized size = 2.86 \[ \frac{2 b d x (e x)^{5/2} (4 p+4 q-11) \left (a+\frac{b}{x^2}\right )^p \left (c+\frac{d}{x^2}\right )^q F_1\left (-p-q+\frac{7}{4};-p,-q;-p-q+\frac{11}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )}{(4 p+4 q-7) \left (4 x^2 \left (a d p F_1\left (-p-q+\frac{11}{4};1-p,-q;-p-q+\frac{15}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )+b c q F_1\left (-p-q+\frac{11}{4};-p,1-q;-p-q+\frac{15}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )\right )+b d (-4 p-4 q+11) F_1\left (-p-q+\frac{7}{4};-p,-q;-p-q+\frac{11}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )\right )} \]

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*(-11 + 4*p + 4*q)*(a + b/x^2)^p*(c + d/x^2)^q*x*(e*x)^(5/2)*AppellF1[7/4
- p - q, -p, -q, 11/4 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((-7 + 4*p + 4*q)*(b
*d*(11 - 4*p - 4*q)*AppellF1[7/4 - p - q, -p, -q, 11/4 - p - q, -((a*x^2)/b), -(
(c*x^2)/d)] + 4*x^2*(a*d*p*AppellF1[11/4 - p - q, 1 - p, -q, 15/4 - p - q, -((a*
x^2)/b), -((c*x^2)/d)] + b*c*q*AppellF1[11/4 - p - q, -p, 1 - q, 15/4 - p - q, -
((a*x^2)/b), -((c*x^2)/d)])))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \left ( c+{\frac{d}{{x}^{2}}} \right ) ^{q} \left ( ex \right ) ^{{\frac{5}{2}}}\, dx \]

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. \[ \int \left (e x\right )^{\frac{5}{2}}{\left (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\right )}^{q}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(e*x)*e^2*x^2*((a*x^2 + b)/x^2)^p*((c*x^2 + d)/x^2)^q, 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. \[ \int \left (e x\right )^{\frac{5}{2}}{\left (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\right )}^{q}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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