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

Optimal. Leaf size=271 \[ \frac{f x \sqrt{c+d x^2} (2 b c-a d)}{c d^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (2 b c-a d) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c d^2 \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{c d \sqrt{c+d x^2}}+\frac{b e^{3/2} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

[Out]

((2*b*c - a*d)*f*x*Sqrt[c + d*x^2])/(c*d^2*Sqrt[e + f*x^2]) - ((b*c - a*d)*x*Sqr
t[e + f*x^2])/(c*d*Sqrt[c + d*x^2]) - ((2*b*c - a*d)*Sqrt[e]*Sqrt[f]*Sqrt[c + d*
x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*d^2*Sqrt[(e*(c
+ d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b*e^(3/2)*Sqrt[c + d*x^2]*Ellipti
cF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*d*Sqrt[f]*Sqrt[(e*(c + d*x^
2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 0.58857, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{f x \sqrt{c+d x^2} (2 b c-a d)}{c d^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (2 b c-a d) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c d^2 \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{c d \sqrt{c+d x^2}}+\frac{b e^{3/2} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

((2*b*c - a*d)*f*x*Sqrt[c + d*x^2])/(c*d^2*Sqrt[e + f*x^2]) - ((b*c - a*d)*x*Sqr
t[e + f*x^2])/(c*d*Sqrt[c + d*x^2]) - ((2*b*c - a*d)*Sqrt[e]*Sqrt[f]*Sqrt[c + d*
x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*d^2*Sqrt[(e*(c
+ d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b*e^(3/2)*Sqrt[c + d*x^2]*Ellipti
cF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*d*Sqrt[f]*Sqrt[(e*(c + d*x^
2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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Rubi in Sympy [A]  time = 70.3996, size = 224, normalized size = 0.83 \[ \frac{b e^{\frac{3}{2}} \sqrt{c + d x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{c d \sqrt{f} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}}} - \frac{x \sqrt{e + f x^{2}} \left (a d - 2 b c\right )}{c d \sqrt{c + d x^{2}}} + \frac{x \sqrt{e + f x^{2}} \left (a d - b c\right )}{c d \sqrt{c + d x^{2}}} + \frac{\sqrt{e + f x^{2}} \left (a d - 2 b c\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{\sqrt{c} d^{\frac{3}{2}} \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*e**(3/2)*sqrt(c + d*x**2)*elliptic_f(atan(sqrt(f)*x/sqrt(e)), 1 - d*e/(c*f))/(
c*d*sqrt(f)*sqrt(e*(c + d*x**2)/(c*(e + f*x**2)))*sqrt(e + f*x**2)) - x*sqrt(e +
f*x**2)*(a*d - 2*b*c)/(c*d*sqrt(c + d*x**2)) + x*sqrt(e + f*x**2)*(a*d - b*c)/(
c*d*sqrt(c + d*x**2)) + sqrt(e + f*x**2)*(a*d - 2*b*c)*elliptic_e(atan(sqrt(d)*x
/sqrt(c)), -c*f/(d*e) + 1)/(sqrt(c)*d**(3/2)*sqrt(c*(e + f*x**2)/(e*(c + d*x**2)
))*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.561719, size = 192, normalized size = 0.71 \[ \frac{-(b c-a d) \left (x \sqrt{\frac{d}{c}} \left (e+f x^2\right )-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (2 b c-a d) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{c^2 \left (\frac{d}{c}\right )^{3/2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*(2*b*c - a*d)*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSin
h[Sqrt[d/c]*x], (c*f)/(d*e)] - (b*c - a*d)*(Sqrt[d/c]*x*(e + f*x^2) - I*e*Sqrt[1
+ (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]
))/(c^2*(d/c)^(3/2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [A]  time = 0.058, size = 328, normalized size = 1.2 \[{\frac{1}{d \left ( df{x}^{4}+cf{x}^{2}+de{x}^{2}+ce \right ) c}\sqrt{f{x}^{2}+e}\sqrt{d{x}^{2}+c} \left ({x}^{3}adf\sqrt{-{\frac{d}{c}}}-{x}^{3}bcf\sqrt{-{\frac{d}{c}}}+{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) ade\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}-{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bce\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}-{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) ade\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}+2\,{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bce\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}+xade\sqrt{-{\frac{d}{c}}}-xbce\sqrt{-{\frac{d}{c}}} \right ){\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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