Optimal. Leaf size=330 \[ -\frac{2 f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )}+\frac{6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )}{\left (2 d e-b f^2\right )^4}-\frac{2 e f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac{6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{\left (2 d e-b f^2\right )^4}-\frac{a e f^2-b d f^2+d^2 e}{\left (2 d e-b f^2\right )^2 \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^2} \]
[Out]
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Rubi [A] time = 0.70135, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{2 f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )}+\frac{6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )}{\left (2 d e-b f^2\right )^4}-\frac{2 e f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac{6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{\left (2 d e-b f^2\right )^4}-\frac{a e f^2-b d f^2+d^2 e}{\left (2 d e-b f^2\right )^2 \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(-3),x]
[Out]
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Rubi in Sympy [A] time = 117.511, size = 306, normalized size = 0.93 \[ \frac{6 e f^{2} \left (4 a e^{2} - b^{2} f^{2}\right ) \log{\left (d + f \left (\frac{e x}{f} + \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right ) \right )}}{\left (b f^{2} - 2 d e\right )^{4}} - \frac{6 e f^{2} \left (4 a e^{2} - b^{2} f^{2}\right ) \log{\left (b f + e \left (\frac{2 e x}{f} + 2 \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right ) \right )}}{\left (b f^{2} - 2 d e\right )^{4}} + \frac{2 e f \left (4 a e^{2} - b^{2} f^{2}\right )}{\left (b f + e \left (\frac{2 e x}{f} + 2 \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right )\right ) \left (b f^{2} - 2 d e\right )^{3}} + \frac{2 f^{2} \left (4 a e^{2} - b^{2} f^{2}\right )}{\left (d + f \left (\frac{e x}{f} + \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right )\right ) \left (b f^{2} - 2 d e\right )^{3}} - \frac{a e f^{2} - b d f^{2} + d^{2} e}{\left (d + f \left (\frac{e x}{f} + \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right )\right )^{2} \left (b f^{2} - 2 d e\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(d+e*x+f*(a+b*x+e**2*x**2/f**2)**(1/2))**3,x)
[Out]
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Mathematica [B] time = 1.54139, size = 665, normalized size = 2.02 \[ -\frac{3 \left (4 a^2 e^3 f^4+a e f^2 \left (-b^2 f^4-4 b d e f^2+4 d^2 e^2\right )+b^2 d f^4 \left (b f^2-d e\right )\right )}{\left (b f^2-2 d e\right )^4 \left (-f^2 (a+b x)+d^2+2 d e x\right )}-\frac{2 f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )} \left (-2 e^2 \left (3 a^2 f^4-a d f^2 (5 d+9 e x)+d^2 e x (3 d+4 e x)\right )+b^2 \left (a f^6-e f^4 x (d+2 e x)\right )+b e f^2 \left (-a d f^2-9 a e f^2 x-3 d^3+d^2 e x+8 d e^2 x^2\right )+b^3 f^6 x\right )}{\left (b f^2-2 d e\right )^3 \left (-f^2 (a+b x)+d^2+2 d e x\right )^2}+\frac{3 \left (4 a e^3 f^2-b^2 e f^4\right ) \log \left (-f^2 (a+b x)+d^2+2 d e x\right )}{\left (b f^2-2 d e\right )^4}-\frac{3 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (-f^2 (a+b x)+d^2+2 d e x\right )}{\left (b f^2-2 d e\right )^4}+\frac{3 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (b f^2 \left (2 d f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+a f^2+d^2-2 d e x\right )+2 d^2 e \left (e x-f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}\right )-2 a e f^2 \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+2 d+e x\right )+b^2 f^4 x\right )}{\left (b f^2-2 d e\right )^4}-\frac{3 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (2 e \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+e x\right )+b f^2\right )}{\left (b f^2-2 d e\right )^4}-\frac{2 \left (a e f^2-b d f^2+d^2 e\right )^3}{\left (b f^2-2 d e\right )^4 \left (-f^2 (a+b x)+d^2+2 d e x\right )^2}+\frac{4 e^3 x}{\left (2 d e-b f^2\right )^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(-3),x]
[Out]
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Maple [B] time = 0.205, size = 295147, normalized size = 894.4 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(d+e*x+f*(a+b*x+e^2*x^2/f^2)^(1/2))^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (e x + \sqrt{b x + \frac{e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + sqrt(b*x + e^2*x^2/f^2 + a)*f + d)^(-3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 10.5942, size = 2638, normalized size = 7.99 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + sqrt(b*x + e^2*x^2/f^2 + a)*f + d)^(-3),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(d+e*x+f*(a+b*x+e**2*x**2/f**2)**(1/2))**3,x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + sqrt(b*x + e^2*x^2/f^2 + a)*f + d)^(-3),x, algorithm="giac")
[Out]