3.968 \(\int \frac{x^2 \sqrt{2+b x^2}}{\sqrt{3+d x^2}} \, dx\)

Optimal. Leaf size=241 \[ -\frac{\sqrt{2} \sqrt{b x^2+2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{d^{3/2} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}+\frac{2 \sqrt{2} (3 b-d) \sqrt{b x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{3 b d^{3/2} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}+\frac{x \sqrt{b x^2+2} \sqrt{d x^2+3}}{3 d}-\frac{2 x (3 b-d) \sqrt{b x^2+2}}{3 b d \sqrt{d x^2+3}} \]

[Out]

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

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Rubi [A]  time = 0.445376, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{\sqrt{2} \sqrt{b x^2+2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{d^{3/2} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}+\frac{2 \sqrt{2} (3 b-d) \sqrt{b x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{3 b d^{3/2} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}+\frac{x \sqrt{b x^2+2} \sqrt{d x^2+3}}{3 d}-\frac{2 x (3 b-d) \sqrt{b x^2+2}}{3 b d \sqrt{d x^2+3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 51.6652, size = 218, normalized size = 0.9 \[ \frac{x \sqrt{b x^{2} + 2} \sqrt{d x^{2} + 3}}{3 d} - \frac{\sqrt{3} \sqrt{b x^{2} + 2} F\left (\operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{d} x}{3} \right )}\middle | - \frac{3 b}{2 d} + 1\right )}{d^{\frac{3}{2}} \sqrt{\frac{3 b x^{2} + 6}{2 d x^{2} + 6}} \sqrt{d x^{2} + 3}} - \frac{2 x \left (3 b - d\right ) \sqrt{b x^{2} + 2}}{3 b d \sqrt{d x^{2} + 3}} + \frac{2 \sqrt{3} \left (3 b - d\right ) \sqrt{b x^{2} + 2} E\left (\operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{d} x}{3} \right )}\middle | - \frac{3 b}{2 d} + 1\right )}{3 b d^{\frac{3}{2}} \sqrt{\frac{3 b x^{2} + 6}{2 d x^{2} + 6}} \sqrt{d x^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.190148, size = 127, normalized size = 0.53 \[ \frac{\sqrt{b} d x \sqrt{b x^2+2} \sqrt{d x^2+3}-2 i \sqrt{3} (3 b-2 d) F\left (i \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2}}\right )|\frac{2 d}{3 b}\right )+2 i \sqrt{3} (3 b-d) E\left (i \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2}}\right )|\frac{2 d}{3 b}\right )}{3 \sqrt{b} d^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*d*x*Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2] + (2*I)*Sqrt[3]*(3*b - d)*EllipticE
[I*ArcSinh[(Sqrt[b]*x)/Sqrt[2]], (2*d)/(3*b)] - (2*I)*Sqrt[3]*(3*b - 2*d)*Ellipt
icF[I*ArcSinh[(Sqrt[b]*x)/Sqrt[2]], (2*d)/(3*b)])/(3*Sqrt[b]*d^2)

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Maple [A]  time = 0.025, size = 306, normalized size = 1.3 \[{\frac{1}{ \left ( 3\,bd{x}^{4}+9\,b{x}^{2}+6\,d{x}^{2}+18 \right ) db}\sqrt{b{x}^{2}+2}\sqrt{d{x}^{2}+3} \left ({x}^{5}{b}^{2}d\sqrt{-d}+3\,{x}^{3}{b}^{2}\sqrt{-d}+2\,{x}^{3}bd\sqrt{-d}+3\,{\it EllipticF} \left ( 1/3\,x\sqrt{3}\sqrt{-d},1/2\,\sqrt{3}\sqrt{2}\sqrt{{\frac{b}{d}}} \right ) \sqrt{2}b\sqrt{b{x}^{2}+2}\sqrt{d{x}^{2}+3}-2\,{\it EllipticF} \left ( 1/3\,x\sqrt{3}\sqrt{-d},1/2\,\sqrt{3}\sqrt{2}\sqrt{{\frac{b}{d}}} \right ) \sqrt{2}d\sqrt{b{x}^{2}+2}\sqrt{d{x}^{2}+3}-6\,{\it EllipticE} \left ( 1/3\,x\sqrt{3}\sqrt{-d},1/2\,\sqrt{3}\sqrt{2}\sqrt{{\frac{b}{d}}} \right ) \sqrt{2}b\sqrt{b{x}^{2}+2}\sqrt{d{x}^{2}+3}+2\,{\it EllipticE} \left ( 1/3\,x\sqrt{3}\sqrt{-d},1/2\,\sqrt{3}\sqrt{2}\sqrt{{\frac{b}{d}}} \right ) \sqrt{2}d\sqrt{b{x}^{2}+2}\sqrt{d{x}^{2}+3}+6\,xb\sqrt{-d} \right ){\frac{1}{\sqrt{-d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b*x^2+2)^(1/2)*(d*x^2+3)^(1/2)*(x^5*b^2*d*(-d)^(1/2)+3*x^3*b^2*(-d)^(1/2)+2
*x^3*b*d*(-d)^(1/2)+3*EllipticF(1/3*x*3^(1/2)*(-d)^(1/2),1/2*3^(1/2)*2^(1/2)*(b/
d)^(1/2))*2^(1/2)*b*(b*x^2+2)^(1/2)*(d*x^2+3)^(1/2)-2*EllipticF(1/3*x*3^(1/2)*(-
d)^(1/2),1/2*3^(1/2)*2^(1/2)*(b/d)^(1/2))*2^(1/2)*d*(b*x^2+2)^(1/2)*(d*x^2+3)^(1
/2)-6*EllipticE(1/3*x*3^(1/2)*(-d)^(1/2),1/2*3^(1/2)*2^(1/2)*(b/d)^(1/2))*2^(1/2
)*b*(b*x^2+2)^(1/2)*(d*x^2+3)^(1/2)+2*EllipticE(1/3*x*3^(1/2)*(-d)^(1/2),1/2*3^(
1/2)*2^(1/2)*(b/d)^(1/2))*2^(1/2)*d*(b*x^2+2)^(1/2)*(d*x^2+3)^(1/2)+6*x*b*(-d)^(
1/2))/(b*d*x^4+3*b*x^2+2*d*x^2+6)/d/(-d)^(1/2)/b

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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