Optimal. Leaf size=187 \[ \frac{e (e f-d g) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{2 \left (c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]
[Out]
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Rubi [A] time = 0.326243, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{e (e f-d g) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{2 \left (c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(f + g*x)/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 55.5514, size = 180, normalized size = 0.96 \[ \frac{e \left (d g - e f\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{\left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}}} + \frac{2 \left (- a g \left (b e - 2 c d\right ) - c x \left (2 a e g - b d g - b e f + 2 c d f\right ) + f \left (- 2 a c e + b^{2} e - b c d\right )\right )}{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \left (a e^{2} - b d e + c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)/(e*x+d)/(c*x**2+b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.546335, size = 217, normalized size = 1.16 \[ \frac{2 b (a e g+c d (f-g x)-c e f x)+4 c (-a d g+a e (f+g x)+c d f x)-2 b^2 e f}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \left (e (b d-a e)-c d^2\right )}+\frac{e (e f-d g) \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{e (d g-e f) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(f + g*x)/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]
[Out]
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Maple [B] time = 0.014, size = 1261, normalized size = 6.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x+f)/(e*x+d)/(c*x^2+b*x+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x + f)/((c*x^2 + b*x + a)^(3/2)*(e*x + d)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 2.01024, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x + f)/((c*x^2 + b*x + a)^(3/2)*(e*x + d)),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((g*x+f)/(e*x+d)/(c*x**2+b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.282408, size = 767, normalized size = 4.1 \[ -\frac{2 \,{\left (\frac{{\left (2 \, c^{3} d^{3} f - b c^{2} d^{3} g - 3 \, b c^{2} d^{2} f e + b^{2} c d^{2} g e + 2 \, a c^{2} d^{2} g e + b^{2} c d f e^{2} + 2 \, a c^{2} d f e^{2} - 3 \, a b c d g e^{2} - a b c f e^{3} + 2 \, a^{2} c g e^{3}\right )} x}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}} + \frac{b c^{2} d^{3} f - 2 \, a c^{2} d^{3} g - 2 \, b^{2} c d^{2} f e + 2 \, a c^{2} d^{2} f e + 3 \, a b c d^{2} g e + b^{3} d f e^{2} - a b c d f e^{2} - a b^{2} d g e^{2} - 2 \, a^{2} c d g e^{2} - a b^{2} f e^{3} + 2 \, a^{2} c f e^{3} + a^{2} b g e^{3}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}}\right )}}{\sqrt{c x^{2} + b x + a}} - \frac{2 \,{\left (d g e - f e^{2}\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e - a e^{2}}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-c d^{2} + b d e - a e^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x + f)/((c*x^2 + b*x + a)^(3/2)*(e*x + d)),x, algorithm="giac")
[Out]