Optimal. Leaf size=45 \[ -\frac{2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]
[Out]
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Rubi [A] time = 0.0450085, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(a + b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 7.20804, size = 42, normalized size = 0.93 \[ \frac{4 a e - 2 b d + x \left (2 b e - 4 c d\right )}{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/(c*x**2+b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.057332, size = 43, normalized size = 0.96 \[ \frac{4 a e-2 b d+2 b e x-4 c d x}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(a + b*x + c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.007, size = 45, normalized size = 1. \[ -2\,{\frac{bex-2\,cdx+2\,ae-bd}{\sqrt{c{x}^{2}+bx+a} \left ( 4\,ac-{b}^{2} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((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((e*x + d)/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.273531, size = 100, normalized size = 2.22 \[ -\frac{2 \, \sqrt{c x^{2} + b x + a}{\left (b d - 2 \, a e +{\left (2 \, c d - b e\right )} x\right )}}{a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^2 + b*x + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d + e x}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/(c*x**2+b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.216246, size = 77, normalized size = 1.71 \[ -\frac{2 \,{\left (\frac{{\left (2 \, c d - b e\right )} x}{b^{2} - 4 \, a c} + \frac{b d - 2 \, a e}{b^{2} - 4 \, a c}\right )}}{\sqrt{c x^{2} + b x + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^2 + b*x + a)^(3/2),x, algorithm="giac")
[Out]