Optimal. Leaf size=28 \[ -\frac{a e-c d x}{a c \sqrt{a+c x^2}} \]
[Out]
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Rubi [A] time = 0.0318201, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{a e-c d x}{a c \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(a + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 4.07493, size = 22, normalized size = 0.79 \[ - \frac{a e - c d x}{a c \sqrt{a + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/(c*x**2+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0284599, size = 27, normalized size = 0.96 \[ \frac{c d x-a e}{a c \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(a + c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.005, size = 27, normalized size = 1. \[ -{\frac{-cdx+ae}{ac}{\frac{1}{\sqrt{c{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/(c*x^2+a)^(3/2),x)
[Out]
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Maxima [A] time = 0.716031, size = 42, normalized size = 1.5 \[ \frac{d x}{\sqrt{c x^{2} + a} a} - \frac{e}{\sqrt{c x^{2} + a} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^2 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221072, size = 47, normalized size = 1.68 \[ \frac{{\left (c d x - a e\right )} \sqrt{c x^{2} + a}}{a c^{2} x^{2} + a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^2 + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.71059, size = 46, normalized size = 1.64 \[ e \left (\begin{cases} - \frac{1}{c \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{2}}{2 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + \frac{d x}{a^{\frac{3}{2}} \sqrt{1 + \frac{c x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/(c*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.215611, size = 32, normalized size = 1.14 \[ \frac{\frac{d x}{a} - \frac{e}{c}}{\sqrt{c x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^2 + a)^(3/2),x, algorithm="giac")
[Out]