3.411 \(\int \frac{(d+e x^2)^q}{x^2 (a+b x^2+c x^4)} \, dx\)

Optimal. Leaf size=264 \[ -\frac{c x \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{a \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c x \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{a \left (\sqrt{b^2-4 a c}+b\right )}-\frac{\left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (-\frac{1}{2},-q;\frac{1}{2};-\frac{e x^2}{d}\right )}{a x} \]

[Out]

-((c*(1 + b/Sqrt[b^2 - 4*a*c])*x*(d + e*x^2)^q*AppellF1[1/2, 1, -q, 3/2, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]), -
((e*x^2)/d)])/(a*(b - Sqrt[b^2 - 4*a*c])*(1 + (e*x^2)/d)^q)) - (c*(1 - b/Sqrt[b^2 - 4*a*c])*x*(d + e*x^2)^q*Ap
pellF1[1/2, 1, -q, 3/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), -((e*x^2)/d)])/(a*(b + Sqrt[b^2 - 4*a*c])*(1 + (e*
x^2)/d)^q) - ((d + e*x^2)^q*Hypergeometric2F1[-1/2, -q, 1/2, -((e*x^2)/d)])/(a*x*(1 + (e*x^2)/d)^q)

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Rubi [A]  time = 0.552902, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1303, 365, 364, 1692, 430, 429} \[ -\frac{c x \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{a \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c x \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{a \left (\sqrt{b^2-4 a c}+b\right )}-\frac{\left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (-\frac{1}{2},-q;\frac{1}{2};-\frac{e x^2}{d}\right )}{a x} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^q/(x^2*(a + b*x^2 + c*x^4)),x]

[Out]

-((c*(1 + b/Sqrt[b^2 - 4*a*c])*x*(d + e*x^2)^q*AppellF1[1/2, 1, -q, 3/2, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]), -
((e*x^2)/d)])/(a*(b - Sqrt[b^2 - 4*a*c])*(1 + (e*x^2)/d)^q)) - (c*(1 - b/Sqrt[b^2 - 4*a*c])*x*(d + e*x^2)^q*Ap
pellF1[1/2, 1, -q, 3/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), -((e*x^2)/d)])/(a*(b + Sqrt[b^2 - 4*a*c])*(1 + (e*
x^2)/d)^q) - ((d + e*x^2)^q*Hypergeometric2F1[-1/2, -q, 1/2, -((e*x^2)/d)])/(a*x*(1 + (e*x^2)/d)^q)

Rule 1303

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x^2)^q, (f*x)^m/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b^2
- 4*a*c, 0] &&  !IntegerQ[q] && IntegerQ[m]

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 1692

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rule 430

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

Rule 429

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

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^q}{x^2 \left (a+b x^2+c x^4\right )} \, dx &=\int \left (\frac{\left (d+e x^2\right )^q}{a x^2}+\frac{\left (-b-c x^2\right ) \left (d+e x^2\right )^q}{a \left (a+b x^2+c x^4\right )}\right ) \, dx\\ &=\frac{\int \frac{\left (d+e x^2\right )^q}{x^2} \, dx}{a}+\frac{\int \frac{\left (-b-c x^2\right ) \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx}{a}\\ &=\frac{\int \left (\frac{\left (-c-\frac{b c}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q}{b-\sqrt{b^2-4 a c}+2 c x^2}+\frac{\left (-c+\frac{b c}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q}{b+\sqrt{b^2-4 a c}+2 c x^2}\right ) \, dx}{a}+\frac{\left (\left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q}\right ) \int \frac{\left (1+\frac{e x^2}{d}\right )^q}{x^2} \, dx}{a}\\ &=-\frac{\left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} \, _2F_1\left (-\frac{1}{2},-q;\frac{1}{2};-\frac{e x^2}{d}\right )}{a x}-\frac{\left (c \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{\left (d+e x^2\right )^q}{b+\sqrt{b^2-4 a c}+2 c x^2} \, dx}{a}-\frac{\left (c \left (1+\frac{b}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{\left (d+e x^2\right )^q}{b-\sqrt{b^2-4 a c}+2 c x^2} \, dx}{a}\\ &=-\frac{\left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} \, _2F_1\left (-\frac{1}{2},-q;\frac{1}{2};-\frac{e x^2}{d}\right )}{a x}-\frac{\left (c \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q}\right ) \int \frac{\left (1+\frac{e x^2}{d}\right )^q}{b+\sqrt{b^2-4 a c}+2 c x^2} \, dx}{a}-\frac{\left (c \left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q}\right ) \int \frac{\left (1+\frac{e x^2}{d}\right )^q}{b-\sqrt{b^2-4 a c}+2 c x^2} \, dx}{a}\\ &=-\frac{c \left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) x \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{a \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) x \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{a \left (b+\sqrt{b^2-4 a c}\right )}-\frac{\left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} \, _2F_1\left (-\frac{1}{2},-q;\frac{1}{2};-\frac{e x^2}{d}\right )}{a x}\\ \end{align*}

Mathematica [F]  time = 0.221173, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x^2\right )^q}{x^2 \left (a+b x^2+c x^4\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(d + e*x^2)^q/(x^2*(a + b*x^2 + c*x^4)),x]

[Out]

Integrate[(d + e*x^2)^q/(x^2*(a + b*x^2 + c*x^4)), x]

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( e{x}^{2}+d \right ) ^{q}}{{x}^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^q/x^2/(c*x^4+b*x^2+a),x)

[Out]

int((e*x^2+d)^q/x^2/(c*x^4+b*x^2+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{q}}{{\left (c x^{4} + b x^{2} + a\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^q/x^2/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)^q/((c*x^4 + b*x^2 + a)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e x^{2} + d\right )}^{q}}{c x^{6} + b x^{4} + a x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^q/x^2/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

integral((e*x^2 + d)^q/(c*x^6 + b*x^4 + a*x^2), 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((e*x**2+d)**q/x**2/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{q}}{{\left (c x^{4} + b x^{2} + a\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^q/x^2/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^q/((c*x^4 + b*x^2 + a)*x^2), x)