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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]