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

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 = 0.11591, 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, Rules used = {497, 511, 510} \[ \frac{x (e x)^m \left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{d}{c x^2}+1\right )^{-q} F_1\left (\frac{1}{2} (-m-1);-p,-q;\frac{1-m}{2};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{m+1} \]

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)

Rule 497

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Dist[(e*x)^m*
(x^(-1))^m, Subst[Int[((a + b/x^n)^p*(c + d/x^n)^q)/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, m, p,
q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0] &&  !RationalQ[m]

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x^2}\right )^p \left (c+\frac{d}{x^2}\right )^q (e x)^m \, dx &=-\left (\left (\left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int x^{-2-m} \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx,x,\frac{1}{x}\right )\right )\\ &=-\left (\left (\left (a+\frac{b}{x^2}\right )^p \left (1+\frac{b}{a x^2}\right )^{-p} \left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int x^{-2-m} \left (1+\frac{b x^2}{a}\right )^p \left (c+d x^2\right )^q \, dx,x,\frac{1}{x}\right )\right )\\ &=-\left (\left (\left (a+\frac{b}{x^2}\right )^p \left (1+\frac{b}{a x^2}\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (1+\frac{d}{c x^2}\right )^{-q} \left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int x^{-2-m} \left (1+\frac{b x^2}{a}\right )^p \left (1+\frac{d x^2}{c}\right )^q \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{\left (a+\frac{b}{x^2}\right )^p \left (1+\frac{b}{a x^2}\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (1+\frac{d}{c x^2}\right )^{-q} x (e x)^m F_1\left (\frac{1}{2} (-1-m);-p,-q;\frac{1-m}{2};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{1+m}\\ \end{align*}

Mathematica [A]  time = 0.128659, size = 115, normalized size = 1.1 \[ \frac{x (e x)^m \left (a+\frac{b}{x^2}\right )^p \left (\frac{a x^2}{b}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{c x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2} (m-2 p-2 q+1);-p,-q;\frac{1}{2} (m-2 p-2 q+3);-\frac{a x^2}{b},-\frac{c x^2}{d}\right )}{m-2 p-2 q+1} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(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 - 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)*(1 + (a*x^2)/b)^p*(1 + (c*x^2)/d)^q)

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Maple [F]  time = 0.227, size = 0, normalized size = 0. \begin{align*} \int \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \left ( c+{\frac{d}{{x}^{2}}} \right ) ^{q} \left ( ex \right ) ^{m}\, dx \end{align*}

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. \begin{align*} \int \left (e x\right )^{m}{\left (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\right )}^{q}\,{d x} \end{align*}

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, 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. \begin{align*}{\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 ) \end{align*}

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, 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. \begin{align*} \text{Timed out} \end{align*}

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

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, algorithm="giac")

[Out]

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