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

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

[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 = 0.746629, 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 \[ \frac{x \sqrt{a-b x^2} (d e-c f)}{c \sqrt{c+d x^2} (a d+b c)}+\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} (d e-c f) E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{c d \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1} (a d+b c)}+\frac{\sqrt{a} f \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{a-b x^2} \sqrt{c+d x^2}} \]

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])

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Rubi in Sympy [A]  time = 113.146, size = 209, normalized size = 0.85 \[ - \frac{\sqrt{a} \sqrt{b} \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{c + d x^{2}} \left (c f - d e\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c}\right )}{c d \sqrt{1 + \frac{d x^{2}}{c}} \sqrt{a - b x^{2}} \left (a d + b c\right )} + \frac{\sqrt{a} f \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{1 + \frac{d x^{2}}{c}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{a - b x^{2}} \sqrt{c + d x^{2}}} - \frac{x \sqrt{a - b x^{2}} \left (c f - d e\right )}{c \sqrt{c + d x^{2}} \left (a d + b c\right )} \]

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

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Mathematica [C]  time = 1.23432, size = 220, normalized size = 0.89 \[ \frac{d x \sqrt{-\frac{b}{a}} \left (a-b x^2\right ) (d e-c f)+i b c \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (c f-d e) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )-i c f \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )}{c d \sqrt{-\frac{b}{a}} \sqrt{a-b x^2} \sqrt{c+d x^2} (a d+b c)} \]

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[Sq
rt[-(b/a)]*x], -((a*d)/(b*c))])/(Sqrt[-(b/a)]*c*d*(b*c + a*d)*Sqrt[a - b*x^2]*Sq
rt[c + d*x^2])

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Maple [A]  time = 0.073, size = 359, normalized size = 1.5 \[{\frac{1}{cd \left ( ad+bc \right ) \left ( bd{x}^{4}-ad{x}^{2}+c{x}^{2}b-ac \right ) } \left ( -{x}^{3}bcdf\sqrt{{\frac{b}{a}}}+{x}^{3}b{d}^{2}e\sqrt{{\frac{b}{a}}}-{\it EllipticF} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) acdf\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-{\it EllipticF} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) b{c}^{2}f\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+{\it EllipticE} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) b{c}^{2}f\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-{\it EllipticE} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) bcde\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+xacdf\sqrt{{\frac{b}{a}}}-xa{d}^{2}e\sqrt{{\frac{b}{a}}} \right ) \sqrt{-b{x}^{2}+a}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{{\frac{b}{a}}}}}} \]

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)+Elliptic
E(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*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)/(b/a)^(1/2)/c/d/(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. \[{\rm integral}\left (\frac{f x^{2} + e}{\sqrt{-b x^{2} + a}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}, x\right ) \]

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)^(3/2)), 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)