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

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

[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 = 0.755732, 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 \[ -\frac{x \sqrt{a-b x^2} (c f+d e)}{c \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}} (c f+d e) E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{1-\frac{b x^2}{a}} \sqrt{c-d x^2} (b c-a d)}+\frac{e \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{a-b x^2} \sqrt{c-d x^2}} \]

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])

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Rubi in Sympy [A]  time = 147.743, size = 202, normalized size = 0.83 \[ \frac{\sqrt{a} e \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{1 - \frac{d x^{2}}{c}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | \frac{a d}{b c}\right )}{\sqrt{b} c \sqrt{a - b x^{2}} \sqrt{c - d x^{2}}} + \frac{x \sqrt{a - b x^{2}} \left (c f + d e\right )}{c \sqrt{c - d x^{2}} \left (a d - b c\right )} - \frac{\sqrt{1 - \frac{d x^{2}}{c}} \sqrt{a - b x^{2}} \left (c f + d e\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | \frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{c - d x^{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**2+e)/(-d*x**2+c)**(3/2)/(-b*x**2+a)**(1/2),x)

[Out]

sqrt(a)*e*sqrt(1 - b*x**2/a)*sqrt(1 - d*x**2/c)*elliptic_f(asin(sqrt(b)*x/sqrt(a
)), a*d/(b*c))/(sqrt(b)*c*sqrt(a - b*x**2)*sqrt(c - d*x**2)) + x*sqrt(a - b*x**2
)*(c*f + d*e)/(c*sqrt(c - d*x**2)*(a*d - b*c)) - sqrt(1 - d*x**2/c)*sqrt(a - b*x
**2)*(c*f + d*e)*elliptic_e(asin(sqrt(d)*x/sqrt(c)), b*c/(a*d))/(sqrt(c)*sqrt(d)
*sqrt(1 - b*x**2/a)*sqrt(c - d*x**2)*(a*d - b*c))

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Mathematica [C]  time = 0.73136, size = 221, normalized size = 0.91 \[ \frac{d x \sqrt{-\frac{b}{a}} \left (a-b x^2\right ) (c f+d e)+i b c \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} (c f+d e) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+i c f \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{c d \sqrt{-\frac{b}{a}} \sqrt{a-b x^2} \sqrt{c-d x^2} (a d-b c)} \]

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]*EllipticE[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]*EllipticF[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 = 0.063, size = 354, normalized size = 1.5 \[{\frac{1}{c \left ( ad-bc \right ) \left ( bd{x}^{4}-ad{x}^{2}-c{x}^{2}b+ac \right ) } \left ( -{x}^{3}bcf\sqrt{{\frac{d}{c}}}-{x}^{3}bde\sqrt{{\frac{d}{c}}}+{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) ade\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{-{\frac{b{x}^{2}-a}{a}}}-{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) bce\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{-{\frac{b{x}^{2}-a}{a}}}-{\it EllipticE} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) acf\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{-{\frac{b{x}^{2}-a}{a}}}-{\it EllipticE} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) ade\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{-{\frac{b{x}^{2}-a}{a}}}+xacf\sqrt{{\frac{d}{c}}}+xade\sqrt{{\frac{d}{c}}} \right ) \sqrt{-b{x}^{2}+a}\sqrt{-d{x}^{2}+c}{\frac{1}{\sqrt{{\frac{d}{c}}}}}} \]

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)-Ellipti
cE(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)/(d/
c)^(1/2)/c/(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. \[ \int \frac{f x^{2} + e}{\sqrt{-b x^{2} + a}{\left (-d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^2 + e)/(sqrt(-b*x^2 + a)*(-d*x^2 + c)^(3/2)),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. \[{\rm integral}\left (-\frac{f x^{2} + e}{\sqrt{-b x^{2} + a}{\left (d x^{2} - c\right )} \sqrt{-d x^{2} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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