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

Optimal. Leaf size=247 \[ \frac{\sqrt{a} f \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{a-b x^2} \sqrt{c+d x^2}}+\frac{x \sqrt{a-b x^2} (d e-c f)}{c \sqrt{c+d x^2} (a d+b c)}+\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} (d e-c f) E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{c d \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1} (a d+b c)} \]

[Out]

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

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Rubi [A]  time = 0.239852, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {527, 524, 427, 426, 424, 421, 419} \[ \frac{x \sqrt{a-b x^2} (d e-c f)}{c \sqrt{c+d x^2} (a d+b c)}+\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} (d e-c f) E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{c d \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1} (a d+b c)}+\frac{\sqrt{a} f \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{a-b x^2} \sqrt{c+d x^2}} \]

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]) + (Sqrt[a]*Sqrt[b]*(d*e - c*f)*Sqrt[1 - (b*x^2
)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(c*d*(b*c + a*d)*Sqrt[a - b*x^2]*
Sqrt[1 + (d*x^2)/c]) + (Sqrt[a]*f*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]
], -((a*d)/(b*c))])/(Sqrt[b]*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{f \int \frac{1}{\sqrt{a-b x^2} \sqrt{c+d x^2}} \, dx}{d}+\frac{(b (d e-c f)) \int \frac{\sqrt{c+d x^2}}{\sqrt{a-b x^2}} \, dx}{c d (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 (b (d e-c f) \sqrt{1-\frac{b x^2}{a}}\right ) \int \frac{\sqrt{c+d x^2}}{\sqrt{1-\frac{b x^2}{a}}} \, dx}{c d (b c+a d) \sqrt{a-b x^2}}+\frac{\left (f \sqrt{1+\frac{d x^2}{c}}\right ) \int \frac{1}{\sqrt{a-b x^2} \sqrt{1+\frac{d x^2}{c}}} \, dx}{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 (b (d e-c f) \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2}\right ) \int \frac{\sqrt{1+\frac{d x^2}{c}}}{\sqrt{1-\frac{b x^2}{a}}} \, dx}{c d (b c+a d) \sqrt{a-b x^2} \sqrt{1+\frac{d x^2}{c}}}+\frac{\left (f \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}{d \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{\sqrt{a} \sqrt{b} (d e-c f) \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{c d (b c+a d) \sqrt{a-b x^2} \sqrt{1+\frac{d x^2}{c}}}+\frac{\sqrt{a} f \sqrt{1-\frac{b x^2}{a}} \sqrt{1+\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{a-b x^2} \sqrt{c+d x^2}}\\ \end{align*}

Mathematica [C]  time = 0.69438, size = 220, normalized size = 0.89 \[ \frac{-i c f \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{-\frac{b}{a}}\right ),-\frac{a d}{b c}\right )+d x \sqrt{-\frac{b}{a}} \left (a-b x^2\right ) (d e-c f)+i b c \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (c f-d e) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )}{c d \sqrt{-\frac{b}{a}} \sqrt{a-b x^2} \sqrt{c+d x^2} (a d+b c)} \]

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]*Ellip
ticE[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]*El
lipticF[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 = 359, normalized size = 1.5 \begin{align*}{\frac{1}{cd \left ( ad+bc \right ) \left ( bd{x}^{4}-ad{x}^{2}+bc{x}^{2}-ac \right ) } \left ( -{x}^{3}bcdf\sqrt{{\frac{b}{a}}}+{x}^{3}b{d}^{2}e\sqrt{{\frac{b}{a}}}-{\it EllipticF} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) acdf\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-{\it EllipticF} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) b{c}^{2}f\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+{\it EllipticE} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) b{c}^{2}f\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-{\it EllipticE} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) bcde\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+xacdf\sqrt{{\frac{b}{a}}}-xa{d}^{2}e\sqrt{{\frac{b}{a}}} \right ) \sqrt{-b{x}^{2}+a}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{{\frac{b}{a}}}}}} \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*d*f*(b/a)^(1/2)+x^3*b*d^2*e*(b/a)^(1/2)-EllipticF(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*a*c*d*f*(-(b*x^2-a
)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-EllipticF(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*b*c^2*f*(-(b*x^2-a)/a)^(1/2)*((d*x^2+
c)/c)^(1/2)+EllipticE(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*b*c^2*f*(-(b*x^2-a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-Ellipti
cE(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*b*c*d*e*(-(b*x^2-a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+x*a*c*d*f*(b/a)^(1/2)-x*a*
d^2*e*(b/a)^(1/2))*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/c/d/(b/a)^(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)