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

Optimal. Leaf size=105 \[ \frac{(e x)^{m+1} \left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{d}{c x^2}+1\right )^{-q} F_1\left (\frac{1}{2} (-m-1);-p,-q;\frac{1-m}{2};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{e (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.317676, antiderivative size = 101, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{x (e x)^m \left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{d}{c x^2}+1\right )^{-q} F_1\left (\frac{1}{2} (-m-1);-p,-q;\frac{1-m}{2};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{m+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.7598, size = 90, normalized size = 0.86 \[ \frac{\left (e x\right )^{m} \left (1 + \frac{b}{a x^{2}}\right )^{- p} \left (1 + \frac{d}{c x^{2}}\right )^{- q} \left (a + \frac{b}{x^{2}}\right )^{p} \left (c + \frac{d}{x^{2}}\right )^{q} \left (\frac{1}{x}\right )^{m} \left (\frac{1}{x}\right )^{- m - 1} \operatorname{appellf_{1}}{\left (- \frac{m}{2} - \frac{1}{2},- p,- q,- \frac{m}{2} + \frac{1}{2},- \frac{b}{a x^{2}},- \frac{d}{c x^{2}} \right )}}{m + 1} \]

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)**m,x)

[Out]

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

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Mathematica [B]  time = 0.779692, size = 284, normalized size = 2.7 \[ \frac{b d x (e x)^m (m-2 p-2 q+3) \left (a+\frac{b}{x^2}\right )^p \left (c+\frac{d}{x^2}\right )^q F_1\left (\frac{1}{2} (m-2 p-2 q+1);-p,-q;\frac{1}{2} (m-2 p-2 q+3);-\frac{a x^2}{b},-\frac{c x^2}{d}\right )}{(m-2 p-2 q+1) \left (2 x^2 \left (a d p F_1\left (\frac{1}{2} (m-2 p-2 q+3);1-p,-q;\frac{1}{2} (m-2 p-2 q+5);-\frac{a x^2}{b},-\frac{c x^2}{d}\right )+b c q F_1\left (\frac{1}{2} (m-2 p-2 q+3);-p,1-q;\frac{1}{2} (m-2 p-2 q+5);-\frac{a x^2}{b},-\frac{c x^2}{d}\right )\right )+b d (m-2 p-2 q+3) F_1\left (\frac{1}{2} (m-2 p-2 q+1);-p,-q;\frac{1}{2} (m-2 p-2 q+3);-\frac{a x^2}{b},-\frac{c x^2}{d}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.26, size = 0, normalized size = 0. \[ \int \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \left ( c+{\frac{d}{{x}^{2}}} \right ) ^{q} \left ( ex \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x)^m*((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)**m,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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