3.15 \(\int \frac{(A+B x^2) (d+e x^2)^q}{a+c x^4} \, dx\)

Optimal. Leaf size=169 \[ \frac{x \left (A-\frac{\sqrt{-a} B}{\sqrt{c}}\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{\sqrt{c} x^2}{\sqrt{-a}},-\frac{e x^2}{d}\right )}{2 a}+\frac{x \left (\frac{\sqrt{-a} B}{\sqrt{c}}+A\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};\frac{\sqrt{c} x^2}{\sqrt{-a}},-\frac{e x^2}{d}\right )}{2 a} \]

[Out]

((A - (Sqrt[-a]*B)/Sqrt[c])*x*(d + e*x^2)^q*AppellF1[1/2, 1, -q, 3/2, -((Sqrt[c]*x^2)/Sqrt[-a]), -((e*x^2)/d)]
)/(2*a*(1 + (e*x^2)/d)^q) + ((A + (Sqrt[-a]*B)/Sqrt[c])*x*(d + e*x^2)^q*AppellF1[1/2, 1, -q, 3/2, (Sqrt[c]*x^2
)/Sqrt[-a], -((e*x^2)/d)])/(2*a*(1 + (e*x^2)/d)^q)

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Rubi [A]  time = 0.218909, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1693, 430, 429} \[ \frac{x \left (A-\frac{\sqrt{-a} B}{\sqrt{c}}\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{\sqrt{c} x^2}{\sqrt{-a}},-\frac{e x^2}{d}\right )}{2 a}+\frac{x \left (\frac{\sqrt{-a} B}{\sqrt{c}}+A\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};\frac{\sqrt{c} x^2}{\sqrt{-a}},-\frac{e x^2}{d}\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A - (Sqrt[-a]*B)/Sqrt[c])*x*(d + e*x^2)^q*AppellF1[1/2, 1, -q, 3/2, -((Sqrt[c]*x^2)/Sqrt[-a]), -((e*x^2)/d)]
)/(2*a*(1 + (e*x^2)/d)^q) + ((A + (Sqrt[-a]*B)/Sqrt[c])*x*(d + e*x^2)^q*AppellF1[1/2, 1, -q, 3/2, (Sqrt[c]*x^2
)/Sqrt[-a], -((e*x^2)/d)])/(2*a*(1 + (e*x^2)/d)^q)

Rule 1693

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e*
x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[c*d^2 + a*e^2, 0] && Intege
rQ[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 (A+B x^2\right ) \left (d+e x^2\right )^q}{a+c x^4} \, dx &=\int \left (-\frac{\left (\sqrt{-a} B+A \sqrt{c}\right ) \left (d+e x^2\right )^q}{2 \sqrt{-a} \sqrt{c} \left (\sqrt{-a}-\sqrt{c} x^2\right )}+\frac{\left (\sqrt{-a} B-A \sqrt{c}\right ) \left (d+e x^2\right )^q}{2 \sqrt{-a} \sqrt{c} \left (\sqrt{-a}+\sqrt{c} x^2\right )}\right ) \, dx\\ &=-\left (\frac{1}{2} \left (\frac{A}{\sqrt{-a}}+\frac{B}{\sqrt{c}}\right ) \int \frac{\left (d+e x^2\right )^q}{\sqrt{-a}-\sqrt{c} x^2} \, dx\right )+\frac{1}{2} \left (\frac{a A}{(-a)^{3/2}}+\frac{B}{\sqrt{c}}\right ) \int \frac{\left (d+e x^2\right )^q}{\sqrt{-a}+\sqrt{c} x^2} \, dx\\ &=-\left (\frac{1}{2} \left (\left (\frac{A}{\sqrt{-a}}+\frac{B}{\sqrt{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}{\sqrt{-a}-\sqrt{c} x^2} \, dx\right )+\frac{1}{2} \left (\left (\frac{a A}{(-a)^{3/2}}+\frac{B}{\sqrt{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}{\sqrt{-a}+\sqrt{c} x^2} \, dx\\ &=\frac{\left (A-\frac{\sqrt{-a} B}{\sqrt{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{\sqrt{c} x^2}{\sqrt{-a}},-\frac{e x^2}{d}\right )}{2 a}-\frac{\left (\frac{A}{\sqrt{-a}}+\frac{B}{\sqrt{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{\sqrt{c} x^2}{\sqrt{-a}},-\frac{e x^2}{d}\right )}{2 \sqrt{-a}}\\ \end{align*}

Mathematica [F]  time = 0.395585, size = 0, normalized size = 0. \[ \int \frac{\left (A+B x^2\right ) \left (d+e x^2\right )^q}{a+c x^4} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( B{x}^{2}+A \right ) \left ( e{x}^{2}+d \right ) ^{q}}{c{x}^{4}+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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