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

Optimal. Leaf size=262 \[ -\frac{\sqrt{2} \sqrt{d x^2+2} (b-a f) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{f^{3/2} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{\sqrt{2} \sqrt{d x^2+2} (-3 a d f+6 b d-2 b f) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{3 d f^{3/2} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}-\frac{x \sqrt{d x^2+2} (-3 a d f+6 b d-2 b f)}{3 d f \sqrt{f x^2+3}}+\frac{b x \sqrt{d x^2+2} \sqrt{f x^2+3}}{3 f} \]

[Out]

-((6*b*d - 2*b*f - 3*a*d*f)*x*Sqrt[2 + d*x^2])/(3*d*f*Sqrt[3 + f*x^2]) + (b*x*Sq
rt[2 + d*x^2]*Sqrt[3 + f*x^2])/(3*f) + (Sqrt[2]*(6*b*d - 2*b*f - 3*a*d*f)*Sqrt[2
+ d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(3*d*f^(3/2)*
Sqrt[(2 + d*x^2)/(3 + f*x^2)]*Sqrt[3 + f*x^2]) - (Sqrt[2]*(b - a*f)*Sqrt[2 + d*x
^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(f^(3/2)*Sqrt[(2 +
d*x^2)/(3 + f*x^2)]*Sqrt[3 + f*x^2])

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Rubi [A]  time = 0.547819, antiderivative size = 262, 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 \[ -\frac{\sqrt{2} \sqrt{d x^2+2} (b-a f) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{f^{3/2} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{\sqrt{2} \sqrt{d x^2+2} (-3 a d f+6 b d-2 b f) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{3 d f^{3/2} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}-\frac{x \sqrt{d x^2+2} (-3 a d f+6 b d-2 b f)}{3 d f \sqrt{f x^2+3}}+\frac{b x \sqrt{d x^2+2} \sqrt{f x^2+3}}{3 f} \]

Antiderivative was successfully verified.

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

[Out]

-((6*b*d - 2*b*f - 3*a*d*f)*x*Sqrt[2 + d*x^2])/(3*d*f*Sqrt[3 + f*x^2]) + (b*x*Sq
rt[2 + d*x^2]*Sqrt[3 + f*x^2])/(3*f) + (Sqrt[2]*(6*b*d - 2*b*f - 3*a*d*f)*Sqrt[2
+ d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(3*d*f^(3/2)*
Sqrt[(2 + d*x^2)/(3 + f*x^2)]*Sqrt[3 + f*x^2]) - (Sqrt[2]*(b - a*f)*Sqrt[2 + d*x
^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(f^(3/2)*Sqrt[(2 +
d*x^2)/(3 + f*x^2)]*Sqrt[3 + f*x^2])

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Rubi in Sympy [A]  time = 59.8932, size = 245, normalized size = 0.94 \[ \frac{b x \sqrt{d x^{2} + 2} \sqrt{f x^{2} + 3}}{3 f} - \frac{\sqrt{3} \left (- a f + b\right ) \sqrt{d x^{2} + 2} F\left (\operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{f} x}{3} \right )}\middle | - \frac{3 d}{2 f} + 1\right )}{f^{\frac{3}{2}} \sqrt{\frac{3 d x^{2} + 6}{2 f x^{2} + 6}} \sqrt{f x^{2} + 3}} - \frac{x \sqrt{d x^{2} + 2} \left (- 3 a d f + 6 b d - 2 b f\right )}{3 d f \sqrt{f x^{2} + 3}} + \frac{\sqrt{3} \sqrt{d x^{2} + 2} \left (- 3 a d f + 6 b d - 2 b f\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{f} x}{3} \right )}\middle | - \frac{3 d}{2 f} + 1\right )}{3 d f^{\frac{3}{2}} \sqrt{\frac{3 d x^{2} + 6}{2 f x^{2} + 6}} \sqrt{f x^{2} + 3}} \]

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*sqrt(d*x**2 + 2)*sqrt(f*x**2 + 3)/(3*f) - sqrt(3)*(-a*f + b)*sqrt(d*x**2 + 2
)*elliptic_f(atan(sqrt(3)*sqrt(f)*x/3), -3*d/(2*f) + 1)/(f**(3/2)*sqrt((3*d*x**2
+ 6)/(2*f*x**2 + 6))*sqrt(f*x**2 + 3)) - x*sqrt(d*x**2 + 2)*(-3*a*d*f + 6*b*d -
2*b*f)/(3*d*f*sqrt(f*x**2 + 3)) + sqrt(3)*sqrt(d*x**2 + 2)*(-3*a*d*f + 6*b*d -
2*b*f)*elliptic_e(atan(sqrt(3)*sqrt(f)*x/3), -3*d/(2*f) + 1)/(3*d*f**(3/2)*sqrt(
(3*d*x**2 + 6)/(2*f*x**2 + 6))*sqrt(f*x**2 + 3))

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Mathematica [C]  time = 0.282959, size = 142, normalized size = 0.54 \[ \frac{i \sqrt{3} (3 d-2 f) (a f-2 b) F\left (i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )+i \sqrt{3} (-3 a d f+6 b d-2 b f) E\left (i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )+b \sqrt{d} f x \sqrt{d x^2+2} \sqrt{f x^2+3}}{3 \sqrt{d} f^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.028, size = 367, normalized size = 1.4 \[{\frac{1}{ \left ( 3\,df{x}^{4}+9\,d{x}^{2}+6\,f{x}^{2}+18 \right ) fd}\sqrt{d{x}^{2}+2}\sqrt{f{x}^{2}+3} \left ({x}^{5}b{d}^{2}f\sqrt{-f}+3\,\sqrt{2}{\it EllipticE} \left ( 1/3\,x\sqrt{3}\sqrt{-f},1/2\,\sqrt{3}\sqrt{2}\sqrt{{\frac{d}{f}}} \right ) adf\sqrt{d{x}^{2}+2}\sqrt{f{x}^{2}+3}+3\,{x}^{3}b{d}^{2}\sqrt{-f}+2\,{x}^{3}bdf\sqrt{-f}-6\,\sqrt{2}{\it EllipticE} \left ( 1/3\,x\sqrt{3}\sqrt{-f},1/2\,\sqrt{3}\sqrt{2}\sqrt{{\frac{d}{f}}} \right ) bd\sqrt{d{x}^{2}+2}\sqrt{f{x}^{2}+3}+2\,\sqrt{2}{\it EllipticE} \left ( 1/3\,x\sqrt{3}\sqrt{-f},1/2\,\sqrt{3}\sqrt{2}\sqrt{{\frac{d}{f}}} \right ) bf\sqrt{d{x}^{2}+2}\sqrt{f{x}^{2}+3}+3\,\sqrt{2}{\it EllipticF} \left ( 1/3\,x\sqrt{3}\sqrt{-f},1/2\,\sqrt{3}\sqrt{2}\sqrt{{\frac{d}{f}}} \right ) bd\sqrt{d{x}^{2}+2}\sqrt{f{x}^{2}+3}-2\,\sqrt{2}{\it EllipticF} \left ( 1/3\,x\sqrt{3}\sqrt{-f},1/2\,\sqrt{3}\sqrt{2}\sqrt{{\frac{d}{f}}} \right ) bf\sqrt{d{x}^{2}+2}\sqrt{f{x}^{2}+3}+6\,xbd\sqrt{-f} \right ){\frac{1}{\sqrt{-f}}}} \]

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]

1/3*(d*x^2+2)^(1/2)*(f*x^2+3)^(1/2)*(x^5*b*d^2*f*(-f)^(1/2)+3*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*(d*x^2+2)^(1/2
)*(f*x^2+3)^(1/2)+3*x^3*b*d^2*(-f)^(1/2)+2*x^3*b*d*f*(-f)^(1/2)-6*2^(1/2)*Ellipt
icE(1/3*x*3^(1/2)*(-f)^(1/2),1/2*3^(1/2)*2^(1/2)*(1/f*d)^(1/2))*b*d*(d*x^2+2)^(1
/2)*(f*x^2+3)^(1/2)+2*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*(d*x^2+2)^(1/2)*(f*x^2+3)^(1/2)+3*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*(d*x^2+2)^(1/2)*(f*x
^2+3)^(1/2)-2*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*(d*x^2+2)^(1/2)*(f*x^2+3)^(1/2)+6*x*b*d*(-f)^(1/2))/(d*f*x^4+3*d
*x^2+2*f*x^2+6)/f/(-f)^(1/2)/d

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

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)

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

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)

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

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)

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

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)