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

Optimal. Leaf size=283 \[ \frac{x \sqrt{c+d x^2} (3 a d f-2 b c f+b d e)}{3 d^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (3 a d f-2 b c f+b d e) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{e^{3/2} \sqrt{c+d x^2} (b c-3 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{b x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 d} \]

[Out]

((b*d*e - 2*b*c*f + 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*d^2*Sqrt[e + f*x^2]) + (b*x*S
qrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*d) - (Sqrt[e]*(b*d*e - 2*b*c*f + 3*a*d*f)*Sqr
t[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*d^2*Sqr
t[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - ((b*c - 3*a*d)*e^(
3/2)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3
*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 = 0.611093, antiderivative size = 283, 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 \[ \frac{x \sqrt{c+d x^2} (3 a d f-2 b c f+b d e)}{3 d^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (3 a d f-2 b c f+b d e) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{e^{3/2} \sqrt{c+d x^2} (b c-3 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{b x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

((b*d*e - 2*b*c*f + 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*d^2*Sqrt[e + f*x^2]) + (b*x*S
qrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*d) - (Sqrt[e]*(b*d*e - 2*b*c*f + 3*a*d*f)*Sqr
t[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*d^2*Sqr
t[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - ((b*c - 3*a*d)*e^(
3/2)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3
*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 = 69.5829, size = 252, normalized size = 0.89 \[ \frac{b x \sqrt{c + d x^{2}} \sqrt{e + f x^{2}}}{3 d} + \frac{\sqrt{c} \sqrt{e + f x^{2}} \left (3 a d - b c\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{3 d^{\frac{3}{2}} \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} - \frac{\sqrt{e} \sqrt{c + d x^{2}} \left (3 a d f - 2 b c f + b d e\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{3 d^{2} \sqrt{f} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}}} + \frac{x \sqrt{c + d x^{2}} \left (3 a d f - 2 b c f + b d e\right )}{3 d^{2} \sqrt{e + f x^{2}}} \]

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)**(1/2),x)

[Out]

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

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Mathematica [C]  time = 0.637933, size = 212, normalized size = 0.75 \[ \frac{i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (-3 a d f+2 b c f-b d e) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+b f x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right )-i b e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{3 d f \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.029, size = 394, normalized size = 1.4 \[{\frac{1}{ \left ( 3\,df{x}^{4}+3\,cf{x}^{2}+3\,de{x}^{2}+3\,ce \right ) df}\sqrt{f{x}^{2}+e}\sqrt{d{x}^{2}+c} \left ( \sqrt{-{\frac{d}{c}}}{x}^{5}bd{f}^{2}+\sqrt{-{\frac{d}{c}}}{x}^{3}bc{f}^{2}+\sqrt{-{\frac{d}{c}}}{x}^{3}bdef+\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bcef-\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bd{e}^{2}+3\,\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) adef-2\,\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bcef+\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bd{e}^{2}+\sqrt{-{\frac{d}{c}}}xbcef \right ){\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(f*x^2+e)^(1/2)*(d*x^2+c)^(1/2)*((-d/c)^(1/2)*x^5*b*d*f^2+(-d/c)^(1/2)*x^3*b
*c*f^2+(-d/c)^(1/2)*x^3*b*d*e*f+((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Elliptic
F(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c*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*d*e^2+3*((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*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*c*e*f+((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+(-d/c)^(1/2)*x*b*c*e*f)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)/d/(-d/c)^(1/2)/f

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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