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

Optimal. Leaf size=258 \[ -\frac{\sqrt{c+d x^2} (2 b e-a f) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{\sqrt{e} f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{c+d x^2} (b e-a f)}{e f \sqrt{e+f x^2}}+\frac{x \sqrt{c+d x^2} (2 b e-a f)}{e f \sqrt{e+f x^2}}+\frac{b \sqrt{e} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

[Out]

-(((b*e - a*f)*x*Sqrt[c + d*x^2])/(e*f*Sqrt[e + f*x^2])) + ((2*b*e - a*f)*x*Sqrt
[c + d*x^2])/(e*f*Sqrt[e + f*x^2]) - ((2*b*e - a*f)*Sqrt[c + d*x^2]*EllipticE[Ar
cTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(Sqrt[e]*f^(3/2)*Sqrt[(e*(c + d*x^2
))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b*Sqrt[e]*Sqrt[c + d*x^2]*EllipticF[ArcT
an[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e +
f*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 0.55995, antiderivative size = 258, 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 \[ -\frac{\sqrt{c+d x^2} (2 b e-a f) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{\sqrt{e} f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{c+d x^2} (b e-a f)}{e f \sqrt{e+f x^2}}+\frac{x \sqrt{c+d x^2} (2 b e-a f)}{e f \sqrt{e+f x^2}}+\frac{b \sqrt{e} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

-(((b*e - a*f)*x*Sqrt[c + d*x^2])/(e*f*Sqrt[e + f*x^2])) + ((2*b*e - a*f)*x*Sqrt
[c + d*x^2])/(e*f*Sqrt[e + f*x^2]) - ((2*b*e - a*f)*Sqrt[c + d*x^2]*EllipticE[Ar
cTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(Sqrt[e]*f^(3/2)*Sqrt[(e*(c + d*x^2
))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b*Sqrt[e]*Sqrt[c + d*x^2]*EllipticF[ArcT
an[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(f^(3/2)*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 = 73.7741, size = 230, normalized size = 0.89 \[ \frac{b \sqrt{e} \sqrt{c + d x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{f^{\frac{3}{2}} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}}} + \frac{\sqrt{c} \sqrt{d} \sqrt{e + f x^{2}} \left (a f - 2 b e\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{e f^{2} \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} - \frac{d x \sqrt{e + f x^{2}} \left (a f - 2 b e\right )}{e f^{2} \sqrt{c + d x^{2}}} + \frac{x \sqrt{c + d x^{2}} \left (a f - b e\right )}{e f \sqrt{e + f x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.593644, size = 208, normalized size = 0.81 \[ \frac{f x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) (a f-b e)-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (a d f+b c f-2 b d e) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-i d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (2 b e-a f) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{e f^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.035, size = 393, normalized size = 1.5 \[{\frac{1}{ \left ( df{x}^{4}+cf{x}^{2}+de{x}^{2}+ce \right ){f}^{2}e}\sqrt{d{x}^{2}+c}\sqrt{f{x}^{2}+e} \left ({x}^{3}ad{f}^{2}\sqrt{-{\frac{d}{c}}}-\sqrt{-{\frac{d}{c}}}{x}^{3}bdef+{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) \sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}adef+\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bcef-2\,\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bd{e}^{2}-\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) adef+2\,\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bd{e}^{2}+xac{f}^{2}\sqrt{-{\frac{d}{c}}}-\sqrt{-{\frac{d}{c}}}xbcef \right ){\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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