3.51 \(\int \frac{e+f x^2}{\sqrt{a-b x^2} (c-d x^2)^{3/2}} \, dx\)

Optimal. Leaf size=242 \[ \frac{e \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{a-b x^2} \sqrt{c-d x^2}}-\frac{x \sqrt{a-b x^2} (c f+d e)}{c \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}} (c f+d e) E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{1-\frac{b x^2}{a}} \sqrt{c-d x^2} (b c-a d)} \]

[Out]

-(((d*e + c*f)*x*Sqrt[a - b*x^2])/(c*(b*c - a*d)*Sqrt[c - d*x^2])) + ((d*e + c*f)*Sqrt[a - b*x^2]*Sqrt[1 - (d*
x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*d)])/(Sqrt[c]*Sqrt[d]*(b*c - a*d)*Sqrt[1 - (b*x^2)/a]*
Sqrt[c - d*x^2]) + (e*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*
d)])/(Sqrt[c]*Sqrt[d]*Sqrt[a - b*x^2]*Sqrt[c - d*x^2])

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Rubi [A]  time = 0.237454, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {527, 524, 427, 426, 424, 421, 419} \[ -\frac{x \sqrt{a-b x^2} (c f+d e)}{c \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}} (c f+d e) E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{1-\frac{b x^2}{a}} \sqrt{c-d x^2} (b c-a d)}+\frac{e \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{a-b x^2} \sqrt{c-d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(e + f*x^2)/(Sqrt[a - b*x^2]*(c - d*x^2)^(3/2)),x]

[Out]

-(((d*e + c*f)*x*Sqrt[a - b*x^2])/(c*(b*c - a*d)*Sqrt[c - d*x^2])) + ((d*e + c*f)*Sqrt[a - b*x^2]*Sqrt[1 - (d*
x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*d)])/(Sqrt[c]*Sqrt[d]*(b*c - a*d)*Sqrt[1 - (b*x^2)/a]*
Sqrt[c - d*x^2]) + (e*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*
d)])/(Sqrt[c]*Sqrt[d]*Sqrt[a - b*x^2]*Sqrt[c - d*x^2])

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 524

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-(b/a), -(d/c)]))))))

Rule 427

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*x^2]
, Int[Sqrt[a + b*x^2]/Sqrt[1 + (d*x^2)/c], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]

Rule 426

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b*x^2)/a]
, Int[Sqrt[1 + (b*x^2)/a]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 0]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 421

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d*x^2)/c]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{e+f x^2}{\sqrt{a-b x^2} \left (c-d x^2\right )^{3/2}} \, dx &=-\frac{(d e+c f) x \sqrt{a-b x^2}}{c (b c-a d) \sqrt{c-d x^2}}-\frac{\int \frac{-c (b e+a f)+b (d e+c f) x^2}{\sqrt{a-b x^2} \sqrt{c-d x^2}} \, dx}{c (b c-a d)}\\ &=-\frac{(d e+c f) x \sqrt{a-b x^2}}{c (b c-a d) \sqrt{c-d x^2}}+\frac{e \int \frac{1}{\sqrt{a-b x^2} \sqrt{c-d x^2}} \, dx}{c}+\frac{(d e+c f) \int \frac{\sqrt{a-b x^2}}{\sqrt{c-d x^2}} \, dx}{c (b c-a d)}\\ &=-\frac{(d e+c f) x \sqrt{a-b x^2}}{c (b c-a d) \sqrt{c-d x^2}}+\frac{\left (e \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{1}{\sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}}} \, dx}{c \sqrt{c-d x^2}}+\frac{\left ((d e+c f) \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{\sqrt{a-b x^2}}{\sqrt{1-\frac{d x^2}{c}}} \, dx}{c (b c-a d) \sqrt{c-d x^2}}\\ &=-\frac{(d e+c f) x \sqrt{a-b x^2}}{c (b c-a d) \sqrt{c-d x^2}}+\frac{\left ((d e+c f) \sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{\sqrt{1-\frac{b x^2}{a}}}{\sqrt{1-\frac{d x^2}{c}}} \, dx}{c (b c-a d) \sqrt{1-\frac{b x^2}{a}} \sqrt{c-d x^2}}+\frac{\left (e \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{1}{\sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}}} \, dx}{c \sqrt{a-b x^2} \sqrt{c-d x^2}}\\ &=-\frac{(d e+c f) x \sqrt{a-b x^2}}{c (b c-a d) \sqrt{c-d x^2}}+\frac{(d e+c f) \sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} (b c-a d) \sqrt{1-\frac{b x^2}{a}} \sqrt{c-d x^2}}+\frac{e \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{a-b x^2} \sqrt{c-d x^2}}\\ \end{align*}

Mathematica [C]  time = 0.465846, size = 221, normalized size = 0.91 \[ \frac{i c f \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} (a d-b c) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{-\frac{b}{a}}\right ),\frac{a d}{b c}\right )+d x \sqrt{-\frac{b}{a}} \left (a-b x^2\right ) (c f+d e)+i b c \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} (c f+d e) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{c d \sqrt{-\frac{b}{a}} \sqrt{a-b x^2} \sqrt{c-d x^2} (a d-b c)} \]

Antiderivative was successfully verified.

[In]

Integrate[(e + f*x^2)/(Sqrt[a - b*x^2]*(c - d*x^2)^(3/2)),x]

[Out]

(Sqrt[-(b/a)]*d*(d*e + c*f)*x*(a - b*x^2) + I*b*c*(d*e + c*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*Elliptic
E[I*ArcSinh[Sqrt[-(b/a)]*x], (a*d)/(b*c)] + I*c*(-(b*c) + a*d)*f*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*Ellip
ticF[I*ArcSinh[Sqrt[-(b/a)]*x], (a*d)/(b*c)])/(Sqrt[-(b/a)]*c*d*(-(b*c) + a*d)*Sqrt[a - b*x^2]*Sqrt[c - d*x^2]
)

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Maple [A]  time = 0.048, size = 354, normalized size = 1.5 \begin{align*}{\frac{1}{c \left ( ad-bc \right ) \left ( bd{x}^{4}-ad{x}^{2}-bc{x}^{2}+ac \right ) } \left ( -{x}^{3}bcf\sqrt{{\frac{d}{c}}}-{x}^{3}bde\sqrt{{\frac{d}{c}}}+{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) ade\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{-{\frac{b{x}^{2}-a}{a}}}-{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) bce\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{-{\frac{b{x}^{2}-a}{a}}}-{\it EllipticE} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) acf\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{-{\frac{b{x}^{2}-a}{a}}}-{\it EllipticE} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) ade\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{-{\frac{b{x}^{2}-a}{a}}}+xacf\sqrt{{\frac{d}{c}}}+xade\sqrt{{\frac{d}{c}}} \right ) \sqrt{-b{x}^{2}+a}\sqrt{-d{x}^{2}+c}{\frac{1}{\sqrt{{\frac{d}{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^2+e)/(-d*x^2+c)^(3/2)/(-b*x^2+a)^(1/2),x)

[Out]

(-x^3*b*c*f*(d/c)^(1/2)-x^3*b*d*e*(d/c)^(1/2)+EllipticF(x*(d/c)^(1/2),(b*c/a/d)^(1/2))*a*d*e*(-(d*x^2-c)/c)^(1
/2)*(-(b*x^2-a)/a)^(1/2)-EllipticF(x*(d/c)^(1/2),(b*c/a/d)^(1/2))*b*c*e*(-(d*x^2-c)/c)^(1/2)*(-(b*x^2-a)/a)^(1
/2)-EllipticE(x*(d/c)^(1/2),(b*c/a/d)^(1/2))*a*c*f*(-(d*x^2-c)/c)^(1/2)*(-(b*x^2-a)/a)^(1/2)-EllipticE(x*(d/c)
^(1/2),(b*c/a/d)^(1/2))*a*d*e*(-(d*x^2-c)/c)^(1/2)*(-(b*x^2-a)/a)^(1/2)+x*a*c*f*(d/c)^(1/2)+x*a*d*e*(d/c)^(1/2
))*(-b*x^2+a)^(1/2)*(-d*x^2+c)^(1/2)/c/(d/c)^(1/2)/(a*d-b*c)/(b*d*x^4-a*d*x^2-b*c*x^2+a*c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f x^{2} + e}{\sqrt{-b x^{2} + a}{\left (-d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e)/(-d*x^2+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

integrate((f*x^2 + e)/(sqrt(-b*x^2 + a)*(-d*x^2 + c)^(3/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e)/(-d*x^2+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e + f x^{2}}{\sqrt{a - b x^{2}} \left (c - d x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**2+e)/(-d*x**2+c)**(3/2)/(-b*x**2+a)**(1/2),x)

[Out]

Integral((e + f*x**2)/(sqrt(a - b*x**2)*(c - d*x**2)**(3/2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f x^{2} + e}{\sqrt{-b x^{2} + a}{\left (-d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e)/(-d*x^2+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

integrate((f*x^2 + e)/(sqrt(-b*x^2 + a)*(-d*x^2 + c)^(3/2)), x)