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

Optimal. Leaf size=209 \[ \frac{\sqrt{c} \sqrt{a+b x^2} (b e-a f) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{a+b x^2} (d e-c f) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

[Out]

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

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Rubi [A]  time = 0.306223, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\sqrt{c} \sqrt{a+b x^2} (b e-a f) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{a+b x^2} (d e-c f) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

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

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Rubi in Sympy [A]  time = 35.6353, size = 173, normalized size = 0.83 \[ \frac{\sqrt{a} \sqrt{c + d x^{2}} \left (a f - b e\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{\sqrt{b} c \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}} \left (a d - b c\right )} - \frac{\sqrt{a + b x^{2}} \left (c f - d e\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (a d - b c\right )} \]

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

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Mathematica [C]  time = 0.64809, size = 212, normalized size = 1.01 \[ \frac{d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) (d e-c f)-i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (c f-d e) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )-i c f \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{c d \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2} (a d-b c)} \]

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[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 = 0.042, size = 349, normalized size = 1.7 \[{\frac{1}{cd \left ( ad-bc \right ) \left ( bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac \right ) } \left ( -{x}^{3}bcdf\sqrt{-{\frac{b}{a}}}+{x}^{3}b{d}^{2}e\sqrt{-{\frac{b}{a}}}+{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) acdf\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) b{c}^{2}f\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) b{c}^{2}f\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) bcde\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-xacdf\sqrt{-{\frac{b}{a}}}+xa{d}^{2}e\sqrt{-{\frac{b}{a}}} \right ) \sqrt{d{x}^{2}+c}\sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]

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))*(d*x^2+c)^(1/2)*(b*x^
2+a)^(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)