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

Optimal. Leaf size=91 \[ \frac{2 (e x)^{5/2} \left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{d}{c x^2}+1\right )^{-q} F_1\left (-\frac{5}{4};-p,-q;-\frac{1}{4};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{5 e} \]

[Out]

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

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Rubi [A]  time = 0.335237, 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 \[ \frac{2 (e x)^{5/2} \left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{d}{c x^2}+1\right )^{-q} F_1\left (-\frac{5}{4};-p,-q;-\frac{1}{4};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{5 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 49.8174, size = 75, normalized size = 0.82 \[ \frac{2 \left (e x\right )^{\frac{5}{2}} \left (1 + \frac{b}{a x^{2}}\right )^{- p} \left (1 + \frac{d}{c x^{2}}\right )^{- q} \left (a + \frac{b}{x^{2}}\right )^{p} \left (c + \frac{d}{x^{2}}\right )^{q} \operatorname{appellf_{1}}{\left (- \frac{5}{4},- p,- q,- \frac{1}{4},- \frac{b}{a x^{2}},- \frac{d}{c x^{2}} \right )}}{5 e} \]

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)**(3/2),x)

[Out]

2*(e*x)**(5/2)*(1 + b/(a*x**2))**(-p)*(1 + d/(c*x**2))**(-q)*(a + b/x**2)**p*(c
+ d/x**2)**q*appellf1(-5/4, -p, -q, -1/4, -b/(a*x**2), -d/(c*x**2))/(5*e)

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Mathematica [B]  time = 0.866296, size = 260, normalized size = 2.86 \[ \frac{2 b d x (e x)^{3/2} (4 p+4 q-9) \left (a+\frac{b}{x^2}\right )^p \left (c+\frac{d}{x^2}\right )^q F_1\left (-p-q+\frac{5}{4};-p,-q;-p-q+\frac{9}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )}{(4 p+4 q-5) \left (4 x^2 \left (a d p F_1\left (-p-q+\frac{9}{4};1-p,-q;-p-q+\frac{13}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )+b c q F_1\left (-p-q+\frac{9}{4};-p,1-q;-p-q+\frac{13}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )\right )+b d (-4 p-4 q+9) F_1\left (-p-q+\frac{5}{4};-p,-q;-p-q+\frac{9}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*b*d*(-9 + 4*p + 4*q)*(a + b/x^2)^p*(c + d/x^2)^q*x*(e*x)^(3/2)*AppellF1[5/4 -
p - q, -p, -q, 9/4 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((-5 + 4*p + 4*q)*(b*d
*(9 - 4*p - 4*q)*AppellF1[5/4 - p - q, -p, -q, 9/4 - p - q, -((a*x^2)/b), -((c*x
^2)/d)] + 4*x^2*(a*d*p*AppellF1[9/4 - p - q, 1 - p, -q, 13/4 - p - q, -((a*x^2)/
b), -((c*x^2)/d)] + b*c*q*AppellF1[9/4 - p - q, -p, 1 - q, 13/4 - p - q, -((a*x^
2)/b), -((c*x^2)/d)])))

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \left ( c+{\frac{d}{{x}^{2}}} \right ) ^{q} \left ( ex \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(e*x)*e*x*((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)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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