3.54 \(\int \left (a+b x^2\right ) \sqrt{2+d x^2} \sqrt{3+f x^2} \, dx\)

Optimal. Leaf size=356 \[ \frac{x \sqrt{d x^2+2} \left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right )}{15 d^2 f \sqrt{f x^2+3}}-\frac{\sqrt{2} \sqrt{d x^2+2} \left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{15 d^2 f^{3/2} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}-\frac{\sqrt{2} \sqrt{d x^2+2} (-10 a d f+3 b d+2 b f) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{5 d f^{3/2} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{x \sqrt{d x^2+2} \sqrt{f x^2+3} (5 a d f+3 b d-4 b f)}{15 d f}+\frac{b x \left (d x^2+2\right )^{3/2} \sqrt{f x^2+3}}{5 d} \]

[Out]

((5*a*d*f*(3*d + 2*f) - 2*b*(9*d^2 - 6*d*f + 4*f^2))*x*Sqrt[2 + d*x^2])/(15*d^2*
f*Sqrt[3 + f*x^2]) + ((3*b*d - 4*b*f + 5*a*d*f)*x*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2
])/(15*d*f) + (b*x*(2 + d*x^2)^(3/2)*Sqrt[3 + f*x^2])/(5*d) - (Sqrt[2]*(5*a*d*f*
(3*d + 2*f) - 2*b*(9*d^2 - 6*d*f + 4*f^2))*Sqrt[2 + d*x^2]*EllipticE[ArcTan[(Sqr
t[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(15*d^2*f^(3/2)*Sqrt[(2 + d*x^2)/(3 + f*x^2)
]*Sqrt[3 + f*x^2]) - (Sqrt[2]*(3*b*d + 2*b*f - 10*a*d*f)*Sqrt[2 + d*x^2]*Ellipti
cF[ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(5*d*f^(3/2)*Sqrt[(2 + d*x^2)/
(3 + f*x^2)]*Sqrt[3 + f*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.925818, antiderivative size = 356, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{x \sqrt{d x^2+2} \left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right )}{15 d^2 f \sqrt{f x^2+3}}-\frac{\sqrt{2} \sqrt{d x^2+2} \left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{15 d^2 f^{3/2} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}-\frac{\sqrt{2} \sqrt{d x^2+2} (-10 a d f+3 b d+2 b f) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{5 d f^{3/2} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{x \sqrt{d x^2+2} \sqrt{f x^2+3} (5 a d f+3 b d-4 b f)}{15 d f}+\frac{b x \left (d x^2+2\right )^{3/2} \sqrt{f x^2+3}}{5 d} \]

Antiderivative was successfully verified.

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

[Out]

((5*a*d*f*(3*d + 2*f) - 2*b*(9*d^2 - 6*d*f + 4*f^2))*x*Sqrt[2 + d*x^2])/(15*d^2*
f*Sqrt[3 + f*x^2]) + ((3*b*d - 4*b*f + 5*a*d*f)*x*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2
])/(15*d*f) + (b*x*(2 + d*x^2)^(3/2)*Sqrt[3 + f*x^2])/(5*d) - (Sqrt[2]*(5*a*d*f*
(3*d + 2*f) - 2*b*(9*d^2 - 6*d*f + 4*f^2))*Sqrt[2 + d*x^2]*EllipticE[ArcTan[(Sqr
t[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(15*d^2*f^(3/2)*Sqrt[(2 + d*x^2)/(3 + f*x^2)
]*Sqrt[3 + f*x^2]) - (Sqrt[2]*(3*b*d + 2*b*f - 10*a*d*f)*Sqrt[2 + d*x^2]*Ellipti
cF[ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(5*d*f^(3/2)*Sqrt[(2 + d*x^2)/
(3 + f*x^2)]*Sqrt[3 + f*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 101.835, size = 350, normalized size = 0.98 \[ \frac{b x \left (d x^{2} + 2\right )^{\frac{3}{2}} \sqrt{f x^{2} + 3}}{5 d} + \frac{x \sqrt{d x^{2} + 2} \sqrt{f x^{2} + 3} \left (5 a d f + 3 b d - 4 b f\right )}{15 d f} - \frac{x \sqrt{d x^{2} + 2} \left (- 15 a d^{2} f - 10 a d f^{2} + 18 b d^{2} - 12 b d f + 8 b f^{2}\right )}{15 d^{2} f \sqrt{f x^{2} + 3}} + \frac{\sqrt{3} \sqrt{d x^{2} + 2} \left (- 15 a d^{2} f - 10 a d f^{2} + 18 b d^{2} - 12 b d f + 8 b f^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{f} x}{3} \right )}\middle | - \frac{3 d}{2 f} + 1\right )}{15 d^{2} f^{\frac{3}{2}} \sqrt{\frac{3 d x^{2} + 6}{2 f x^{2} + 6}} \sqrt{f x^{2} + 3}} - \frac{2 \sqrt{2} \sqrt{f x^{2} + 3} \left (- 10 a d f + 3 b d + 2 b f\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{d} x}{2} \right )}\middle | 1 - \frac{2 f}{3 d}\right )}{15 d^{\frac{3}{2}} f \sqrt{\frac{2 f x^{2} + 6}{3 d x^{2} + 6}} \sqrt{d x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x*(d*x**2 + 2)**(3/2)*sqrt(f*x**2 + 3)/(5*d) + x*sqrt(d*x**2 + 2)*sqrt(f*x**2
+ 3)*(5*a*d*f + 3*b*d - 4*b*f)/(15*d*f) - x*sqrt(d*x**2 + 2)*(-15*a*d**2*f - 10*
a*d*f**2 + 18*b*d**2 - 12*b*d*f + 8*b*f**2)/(15*d**2*f*sqrt(f*x**2 + 3)) + sqrt(
3)*sqrt(d*x**2 + 2)*(-15*a*d**2*f - 10*a*d*f**2 + 18*b*d**2 - 12*b*d*f + 8*b*f**
2)*elliptic_e(atan(sqrt(3)*sqrt(f)*x/3), -3*d/(2*f) + 1)/(15*d**2*f**(3/2)*sqrt(
(3*d*x**2 + 6)/(2*f*x**2 + 6))*sqrt(f*x**2 + 3)) - 2*sqrt(2)*sqrt(f*x**2 + 3)*(-
10*a*d*f + 3*b*d + 2*b*f)*elliptic_f(atan(sqrt(2)*sqrt(d)*x/2), 1 - 2*f/(3*d))/(
15*d**(3/2)*f*sqrt((2*f*x**2 + 6)/(3*d*x**2 + 6))*sqrt(d*x**2 + 2))

_______________________________________________________________________________________

Mathematica [C]  time = 0.533412, size = 186, normalized size = 0.52 \[ \frac{i \sqrt{3} \left (2 b \left (9 d^2-6 d f+4 f^2\right )-5 a d f (3 d+2 f)\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )+\sqrt{d} f x \sqrt{d x^2+2} \sqrt{f x^2+3} \left (5 a d f+3 b d \left (f x^2+1\right )+2 b f\right )+i \sqrt{3} (3 d-2 f) (5 a d f-6 b d+2 b f) F\left (i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )}{15 d^{3/2} f^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d]*f*x*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2]*(2*b*f + 5*a*d*f + 3*b*d*(1 + f*x^2
)) + I*Sqrt[3]*(-5*a*d*f*(3*d + 2*f) + 2*b*(9*d^2 - 6*d*f + 4*f^2))*EllipticE[I*
ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)] + I*Sqrt[3]*(3*d - 2*f)*(-6*b*d + 2*b
*f + 5*a*d*f)*EllipticF[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)])/(15*d^(3/2
)*f^2)

_______________________________________________________________________________________

Maple [B]  time = 0.033, size = 775, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(d*x^2+2)^(1/2)*(f*x^2+3)^(1/2)*(3*x^7*b*d^3*f^2*(-f)^(1/2)+5*x^5*a*d^3*f^2
*(-f)^(1/2)+12*x^5*b*d^3*f*(-f)^(1/2)+8*x^5*b*d^2*f^2*(-f)^(1/2)+15*x^3*a*d^3*f*
(-f)^(1/2)+10*x^3*a*d^2*f^2*(-f)^(1/2)+15*2^(1/2)*EllipticF(1/3*x*3^(1/2)*(-f)^(
1/2),1/2*3^(1/2)*2^(1/2)*(1/f*d)^(1/2))*a*d^2*f*(f*x^2+3)^(1/2)*(d*x^2+2)^(1/2)-
10*2^(1/2)*EllipticF(1/3*x*3^(1/2)*(-f)^(1/2),1/2*3^(1/2)*2^(1/2)*(1/f*d)^(1/2))
*a*d*f^2*(f*x^2+3)^(1/2)*(d*x^2+2)^(1/2)+15*2^(1/2)*EllipticE(1/3*x*3^(1/2)*(-f)
^(1/2),1/2*3^(1/2)*2^(1/2)*(1/f*d)^(1/2))*a*d^2*f*(f*x^2+3)^(1/2)*(d*x^2+2)^(1/2
)+10*2^(1/2)*EllipticE(1/3*x*3^(1/2)*(-f)^(1/2),1/2*3^(1/2)*2^(1/2)*(1/f*d)^(1/2
))*a*d*f^2*(f*x^2+3)^(1/2)*(d*x^2+2)^(1/2)+9*x^3*b*d^3*(-f)^(1/2)+30*x^3*b*d^2*f
*(-f)^(1/2)+4*x^3*b*d*f^2*(-f)^(1/2)+9*2^(1/2)*EllipticF(1/3*x*3^(1/2)*(-f)^(1/2
),1/2*3^(1/2)*2^(1/2)*(1/f*d)^(1/2))*b*d^2*(f*x^2+3)^(1/2)*(d*x^2+2)^(1/2)-18*2^
(1/2)*EllipticF(1/3*x*3^(1/2)*(-f)^(1/2),1/2*3^(1/2)*2^(1/2)*(1/f*d)^(1/2))*b*d*
f*(f*x^2+3)^(1/2)*(d*x^2+2)^(1/2)+8*2^(1/2)*EllipticF(1/3*x*3^(1/2)*(-f)^(1/2),1
/2*3^(1/2)*2^(1/2)*(1/f*d)^(1/2))*b*f^2*(f*x^2+3)^(1/2)*(d*x^2+2)^(1/2)-18*2^(1/
2)*EllipticE(1/3*x*3^(1/2)*(-f)^(1/2),1/2*3^(1/2)*2^(1/2)*(1/f*d)^(1/2))*b*d^2*(
f*x^2+3)^(1/2)*(d*x^2+2)^(1/2)+12*2^(1/2)*EllipticE(1/3*x*3^(1/2)*(-f)^(1/2),1/2
*3^(1/2)*2^(1/2)*(1/f*d)^(1/2))*b*d*f*(f*x^2+3)^(1/2)*(d*x^2+2)^(1/2)-8*2^(1/2)*
EllipticE(1/3*x*3^(1/2)*(-f)^(1/2),1/2*3^(1/2)*2^(1/2)*(1/f*d)^(1/2))*b*f^2*(f*x
^2+3)^(1/2)*(d*x^2+2)^(1/2)+30*x*a*d^2*f*(-f)^(1/2)+18*x*b*d^2*(-f)^(1/2)+12*x*b
*d*f*(-f)^(1/2))/(d*f*x^4+3*d*x^2+2*f*x^2+6)/f/d^2/(-f)^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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